url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | intro v a1 | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : ∀ v ∈ vs, Var.lc_at j v
⊢ ∀ x ∈ vs, V x = ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) x | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : ∀ v ∈ vs, Var.lc_at j v
v : Var
a1 : v ∈ vs
⊢ V v = ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | specialize h2 v a1 | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : ∀ v ∈ vs, Var.lc_at j v
v : Var
a1 : v ∈ vs
⊢ V v = ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) v | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
v : Var
a1 : v ∈ vs
h2 : Var.lc_at j v
⊢ V v = ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
v : Var
a1 : v ∈ vs
h2 : Var.lc_at j v
⊢ V v = ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) v | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
v : Var
a1 : v ∈ vs
h2 : Var.lc_at j v
⊢ V v = V (Var.close j (free_ z) (Var.openList j [free_ z] v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | cases v | case a.h.e'_4
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
v : Var
a1 : v ∈ vs
h2 : Var.lc_at j v
⊢ V v = V (Var.close j (free_ z) (Var.openList j [free_ z] v)) | case a.h.e'_4.free_
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
a✝ : String
a1 : free_ a✝ ∈ vs
h2 : Var.lc_at j (free_ a✝)
⊢ V (free_ a✝) = V (Var.close j (free_ z) (Var.openList j [free_ z] (free_ a✝)))
case a.h.e'_4.bound_
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
a✝ : ℕ
a1 : bound_ a✝ ∈ vs
h2 : Var.lc_at j (bound_ a✝)
⊢ V (bound_ a✝) = V (Var.close j (free_ z) (Var.openList j [free_ z] (bound_ a✝))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | case _ x =>
simp only [Var.openList]
simp only [Var.close]
have s1 : ¬ free_ x = free_ z
intro contra
apply h1
simp only [← contra]
exact a1
simp only [if_neg s1] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ V (free_ x) = V (Var.close j (free_ z) (Var.openList j [free_ z] (free_ x))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | case _ i =>
simp only [Var.lc_at] at h2
simp only [Var.openList]
simp
split_ifs
simp only [Var.close]
simp only [if_pos h2] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : Var.lc_at j (bound_ i)
⊢ V (bound_ i) = V (Var.close j (free_ z) (Var.openList j [free_ z] (bound_ i))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Var.openList] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ V (free_ x) = V (Var.close j (free_ z) (Var.openList j [free_ z] (free_ x))) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ V (free_ x) = V (Var.close j (free_ z) (free_ x)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Var.close] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ V (free_ x) = V (Var.close j (free_ z) (free_ x)) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | have s1 : ¬ free_ x = free_ z | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ ¬free_ x = free_ z
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | intro contra | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
⊢ ¬free_ x = free_ z
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
contra : free_ x = free_ z
⊢ False
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | apply h1 | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
contra : free_ x = free_ z
⊢ False
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
contra : free_ x = free_ z
⊢ free_ z ∈ vs
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [← contra] | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
contra : free_ x = free_ z
⊢ free_ z ∈ vs
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
contra : free_ x = free_ z
⊢ free_ x ∈ vs
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | exact a1 | case s1
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
contra : free_ x = free_ z
⊢ free_ x ∈ vs
D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [if_neg s1] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
x : String
a1 : free_ x ∈ vs
h2 : Var.lc_at j (free_ x)
s1 : ¬free_ x = free_ z
⊢ V (free_ x) = V (if free_ x = free_ z then bound_ j else free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Var.lc_at] at h2 | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : Var.lc_at j (bound_ i)
⊢ V (bound_ i) = V (Var.close j (free_ z) (Var.openList j [free_ z] (bound_ i))) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) = V (Var.close j (free_ z) (Var.openList j [free_ z] (bound_ i))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Var.openList] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) = V (Var.close j (free_ z) (Var.openList j [free_ z] (bound_ i))) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) =
V
(Var.close j (free_ z)
(if i < j then bound_ i
else if x : i - j < [free_ z].length then [free_ z][i - j] else bound_ (i - j - [free_ z].length + j))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) =
V
(Var.close j (free_ z)
(if i < j then bound_ i
else if x : i - j < [free_ z].length then [free_ z][i - j] else bound_ (i - j - [free_ z].length + j))) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) =
V (Var.close j (free_ z) (if i < j then bound_ i else if i - j = 0 then free_ z else bound_ (i - j - 1 + j))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | split_ifs | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) =
V (Var.close j (free_ z) (if i < j then bound_ i else if i - j = 0 then free_ z else bound_ (i - j - 1 + j))) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) = V (Var.close j (free_ z) (bound_ i)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Var.close] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) = V (Var.close j (free_ z) (bound_ i)) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [if_pos h2] | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
i : ℕ
a1 : bound_ i ∈ vs
h2 : i < j
⊢ V (bound_ i) = V (if i < j then bound_ i else bound_ (i + 1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [occursIn] at h1 | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi.not_
h2 : Formula.lc_at j phi.not_
⊢ Holds D I V phi.not_ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi.not_) | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi.not_
⊢ Holds D I V phi.not_ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Formula.lc_at] at h2 | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi.not_
⊢ Holds D I V phi.not_ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi.not_) | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi
⊢ Holds D I V phi.not_ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Holds] | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi
⊢ Holds D I V phi.not_ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi.not_) | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi
⊢ ¬Holds D I V phi ↔ ¬Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | congr! 1 | D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi
⊢ ¬Holds D I V phi ↔ ¬Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) | case a.h.e'_1.a
D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi
⊢ Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | exact phi_ih V j h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
z : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j phi
⊢ Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [occursIn] at h1 | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) (phi.imp_ psi)
h2 : Formula.lc_at j (phi.imp_ psi)
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1 : ¬(occursIn (free_ z) phi ∨ occursIn (free_ z) psi)
h2 : Formula.lc_at j (phi.imp_ psi)
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | push_neg at h1 | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1 : ¬(occursIn (free_ z) phi ∨ occursIn (free_ z) psi)
h2 : Formula.lc_at j (phi.imp_ psi)
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h2 : Formula.lc_at j (phi.imp_ psi)
h1 : ¬occursIn (free_ z) phi ∧ ¬occursIn (free_ z) psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Formula.lc_at] at h2 | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h2 : Formula.lc_at j (phi.imp_ psi)
h1 : ¬occursIn (free_ z) phi ∧ ¬occursIn (free_ z) psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h2 : Formula.lc_at j phi ∧ Formula.lc_at j psi
h1 : ¬occursIn (free_ z) phi ∧ ¬occursIn (free_ z) psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | cases h1 | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h2 : Formula.lc_at j phi ∧ Formula.lc_at j psi
h1 : ¬occursIn (free_ z) phi ∧ ¬occursIn (free_ z) psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) | case intro
D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h2 : Formula.lc_at j phi ∧ Formula.lc_at j psi
left✝ : ¬occursIn (free_ z) phi
right✝ : ¬occursIn (free_ z) psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | cases h2 | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h2 : Formula.lc_at j phi ∧ Formula.lc_at j psi
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) | case intro
D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
left✝ : Formula.lc_at j phi
right✝ : Formula.lc_at j psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Holds] | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (phi.imp_ psi)) | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
⊢ Holds D I V phi → Holds D I V psi ↔
Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) →
Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | congr! 1 | D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
⊢ Holds D I V phi → Holds D I V psi ↔
Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) →
Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi) | case a.h.a
D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
⊢ Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi)
case a.h'.a
D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
a✝ : Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi)
⊢ Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | exact phi_ih V j h1_left h2_left | case a.h.a
D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
⊢ Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | exact psi_ih V j h1_right h2_right | case a.h'.a
D : Type
I : Interpretation D
z : String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) psi →
Formula.lc_at j psi → (Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi))
V : VarAssignment D
j : ℕ
h1_left : ¬occursIn (free_ z) phi
h1_right : ¬occursIn (free_ z) psi
h2_left : Formula.lc_at j phi
h2_right : Formula.lc_at j psi
a✝ : Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi)
⊢ Holds D I V psi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [occursIn] at h1 | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) (forall_ x phi)
h2 : Formula.lc_at j (forall_ x phi)
⊢ Holds D I V (forall_ x phi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (forall_ x phi)) | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j (forall_ x phi)
⊢ Holds D I V (forall_ x phi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (forall_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Formula.lc_at] at h2 | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at j (forall_ x phi)
⊢ Holds D I V (forall_ x phi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (forall_ x phi)) | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ Holds D I V (forall_ x phi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (forall_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Formula.openList] | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ Holds D I V (forall_ x phi) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (forall_ x phi)) | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ Holds D I V (forall_ x phi) ↔
Holds D I (V ∘ Var.close j (free_ z)) (forall_ x (Formula.openList (j + 1) [free_ z] phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [Holds] | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ Holds D I V (forall_ x phi) ↔
Holds D I (V ∘ Var.close j (free_ z)) (forall_ x (Formula.openList (j + 1) [free_ z] phi)) | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔
∀ (d : D), Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | apply forall_congr' | D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔
∀ (d : D), Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ ∀ (a : D),
Holds D I (shift D V a) phi ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) a) (Formula.openList (j + 1) [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | intro d | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
⊢ ∀ (a : D),
Holds D I (shift D V a) phi ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) a) (Formula.openList (j + 1) [free_ z] phi) | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
⊢ Holds D I (shift D V d) phi ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | specialize phi_ih (shift D V d) (j + 1) h1 h2 | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) phi →
Formula.lc_at j phi → (Holds D I V phi ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] phi))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
⊢ Holds D I (shift D V d) phi ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
phi_ih :
Holds D I (shift D V d) phi ↔
Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi)
⊢ Holds D I (shift D V d) phi ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [phi_ih] | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
phi_ih :
Holds D I (shift D V d) phi ↔
Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi)
⊢ Holds D I (shift D V d) phi ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
phi_ih :
Holds D I (shift D V d) phi ↔
Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi)
⊢ Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi) ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | congr! 1 | case h
D : Type
I : Interpretation D
z x : String
phi : Formula
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
phi_ih :
Holds D I (shift D V d) phi ↔
Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi)
⊢ Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi) ↔
Holds D I (shift D (V ∘ Var.close j (free_ z)) d) (Formula.openList (j + 1) [free_ z] phi) | case h.a.h.e'_3
D : Type
I : Interpretation D
z x : String
phi : Formula
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
phi_ih :
Holds D I (shift D V d) phi ↔
Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi)
⊢ shift D V d ∘ Var.close (j + 1) (free_ z) = shift D (V ∘ Var.close j (free_ z)) d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | apply extracted_1 | case h.a.h.e'_3
D : Type
I : Interpretation D
z x : String
phi : Formula
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) phi
h2 : Formula.lc_at (j + 1) phi
d : D
phi_ih :
Holds D I (shift D V d) phi ↔
Holds D I (shift D V d ∘ Var.close (j + 1) (free_ z)) (Formula.openList (j + 1) [free_ z] phi)
⊢ shift D V d ∘ Var.close (j + 1) (free_ z) = shift D (V ∘ Var.close j (free_ z)) d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | induction F generalizing j | F : Formula
j : ℕ
zs : List String
h1 : lc_at j F
⊢ openList j (List.map free_ zs) F = F | case pred_
zs : List String
a✝¹ : String
a✝ : List Var
j : ℕ
h1 : lc_at j (pred_ a✝¹ a✝)
⊢ openList j (List.map free_ zs) (pred_ a✝¹ a✝) = pred_ a✝¹ a✝
case not_
zs : List String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), lc_at j a✝ → openList j (List.map free_ zs) a✝ = a✝
j : ℕ
h1 : lc_at j a✝.not_
⊢ openList j (List.map free_ zs) a✝.not_ = a✝.not_
case imp_
zs : List String
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (j : ℕ), lc_at j a✝¹ → openList j (List.map free_ zs) a✝¹ = a✝¹
a_ih✝ : ∀ (j : ℕ), lc_at j a✝ → openList j (List.map free_ zs) a✝ = a✝
j : ℕ
h1 : lc_at j (a✝¹.imp_ a✝)
⊢ openList j (List.map free_ zs) (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case forall_
zs : List String
a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), lc_at j a✝ → openList j (List.map free_ zs) a✝ = a✝
j : ℕ
h1 : lc_at j (forall_ a✝¹ a✝)
⊢ openList j (List.map free_ zs) (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case pred_ X vs =>
simp only [Formula.lc_at] at h1
simp only [Formula.openList]
simp
apply List.map_mem_id
intro v a1
specialize h1 v a1
cases v
case _ x =>
simp only [Var.openList]
case _ i =>
cases i
case zero =>
simp only [Var.lc_at] at h1
simp only [Var.openList]
simp only [if_pos h1]
case succ i =>
simp only [Var.lc_at] at h1
simp only [Var.openList]
simp only [if_pos h1] | zs : List String
X : String
vs : List Var
j : ℕ
h1 : lc_at j (pred_ X vs)
⊢ openList j (List.map free_ zs) (pred_ X vs) = pred_ X vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case not_ phi phi_ih =>
simp only [Formula.lc_at] at h1
simp only [Formula.openList]
congr!
exact phi_ih j h1 | zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi.not_
⊢ openList j (List.map free_ zs) phi.not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case forall_ x phi phi_ih =>
simp only [Formula.lc_at] at h1
simp only [Formula.openList]
simp
exact phi_ih (j + 1) h1 | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j (forall_ x phi)
⊢ openList j (List.map free_ zs) (forall_ x phi) = forall_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.lc_at] at h1 | zs : List String
X : String
vs : List Var
j : ℕ
h1 : lc_at j (pred_ X vs)
⊢ openList j (List.map free_ zs) (pred_ X vs) = pred_ X vs | zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ openList j (List.map free_ zs) (pred_ X vs) = pred_ X vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.openList] | zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ openList j (List.map free_ zs) (pred_ X vs) = pred_ X vs | zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ pred_ X (List.map (Var.openList j (List.map free_ zs)) vs) = pred_ X vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp | zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ pred_ X (List.map (Var.openList j (List.map free_ zs)) vs) = pred_ X vs | zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ List.map (Var.openList j (List.map free_ zs)) vs = vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | apply List.map_mem_id | zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ List.map (Var.openList j (List.map free_ zs)) vs = vs | case h1
zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ ∀ x ∈ vs, Var.openList j (List.map free_ zs) x = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | intro v a1 | case h1
zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
⊢ ∀ x ∈ vs, Var.openList j (List.map free_ zs) x = x | case h1
zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
v : Var
a1 : v ∈ vs
⊢ Var.openList j (List.map free_ zs) v = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | specialize h1 v a1 | case h1
zs : List String
X : String
vs : List Var
j : ℕ
h1 : ∀ v ∈ vs, Var.lc_at j v
v : Var
a1 : v ∈ vs
⊢ Var.openList j (List.map free_ zs) v = v | case h1
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
h1 : Var.lc_at j v
⊢ Var.openList j (List.map free_ zs) v = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | cases v | case h1
zs : List String
X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
h1 : Var.lc_at j v
⊢ Var.openList j (List.map free_ zs) v = v | case h1.free_
zs : List String
X : String
vs : List Var
j : ℕ
a✝ : String
a1 : free_ a✝ ∈ vs
h1 : Var.lc_at j (free_ a✝)
⊢ Var.openList j (List.map free_ zs) (free_ a✝) = free_ a✝
case h1.bound_
zs : List String
X : String
vs : List Var
j a✝ : ℕ
a1 : bound_ a✝ ∈ vs
h1 : Var.lc_at j (bound_ a✝)
⊢ Var.openList j (List.map free_ zs) (bound_ a✝) = bound_ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case _ x =>
simp only [Var.openList] | zs : List String
X : String
vs : List Var
j : ℕ
x : String
a1 : free_ x ∈ vs
h1 : Var.lc_at j (free_ x)
⊢ Var.openList j (List.map free_ zs) (free_ x) = free_ x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case _ i =>
cases i
case zero =>
simp only [Var.lc_at] at h1
simp only [Var.openList]
simp only [if_pos h1]
case succ i =>
simp only [Var.lc_at] at h1
simp only [Var.openList]
simp only [if_pos h1] | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ i ∈ vs
h1 : Var.lc_at j (bound_ i)
⊢ Var.openList j (List.map free_ zs) (bound_ i) = bound_ i | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Var.openList] | zs : List String
X : String
vs : List Var
j : ℕ
x : String
a1 : free_ x ∈ vs
h1 : Var.lc_at j (free_ x)
⊢ Var.openList j (List.map free_ zs) (free_ x) = free_ x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | cases i | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ i ∈ vs
h1 : Var.lc_at j (bound_ i)
⊢ Var.openList j (List.map free_ zs) (bound_ i) = bound_ i | case zero
zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : Var.lc_at j (bound_ 0)
⊢ Var.openList j (List.map free_ zs) (bound_ 0) = bound_ 0
case succ
zs : List String
X : String
vs : List Var
j n✝ : ℕ
a1 : bound_ (n✝ + 1) ∈ vs
h1 : Var.lc_at j (bound_ (n✝ + 1))
⊢ Var.openList j (List.map free_ zs) (bound_ (n✝ + 1)) = bound_ (n✝ + 1) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case zero =>
simp only [Var.lc_at] at h1
simp only [Var.openList]
simp only [if_pos h1] | zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : Var.lc_at j (bound_ 0)
⊢ Var.openList j (List.map free_ zs) (bound_ 0) = bound_ 0 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | case succ i =>
simp only [Var.lc_at] at h1
simp only [Var.openList]
simp only [if_pos h1] | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ (i + 1) ∈ vs
h1 : Var.lc_at j (bound_ (i + 1))
⊢ Var.openList j (List.map free_ zs) (bound_ (i + 1)) = bound_ (i + 1) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Var.lc_at] at h1 | zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : Var.lc_at j (bound_ 0)
⊢ Var.openList j (List.map free_ zs) (bound_ 0) = bound_ 0 | zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : 0 < j
⊢ Var.openList j (List.map free_ zs) (bound_ 0) = bound_ 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Var.openList] | zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : 0 < j
⊢ Var.openList j (List.map free_ zs) (bound_ 0) = bound_ 0 | zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : 0 < j
⊢ (if 0 < j then bound_ 0
else
if x : 0 - j < (List.map free_ zs).length then (List.map free_ zs)[0 - j]
else bound_ (0 - j - (List.map free_ zs).length + j)) =
bound_ 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [if_pos h1] | zs : List String
X : String
vs : List Var
j : ℕ
a1 : bound_ 0 ∈ vs
h1 : 0 < j
⊢ (if 0 < j then bound_ 0
else
if x : 0 - j < (List.map free_ zs).length then (List.map free_ zs)[0 - j]
else bound_ (0 - j - (List.map free_ zs).length + j)) =
bound_ 0 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Var.lc_at] at h1 | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ (i + 1) ∈ vs
h1 : Var.lc_at j (bound_ (i + 1))
⊢ Var.openList j (List.map free_ zs) (bound_ (i + 1)) = bound_ (i + 1) | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ (i + 1) ∈ vs
h1 : i + 1 < j
⊢ Var.openList j (List.map free_ zs) (bound_ (i + 1)) = bound_ (i + 1) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Var.openList] | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ (i + 1) ∈ vs
h1 : i + 1 < j
⊢ Var.openList j (List.map free_ zs) (bound_ (i + 1)) = bound_ (i + 1) | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ (i + 1) ∈ vs
h1 : i + 1 < j
⊢ (if i + 1 < j then bound_ (i + 1)
else
if x : i + 1 - j < (List.map free_ zs).length then (List.map free_ zs)[i + 1 - j]
else bound_ (i + 1 - j - (List.map free_ zs).length + j)) =
bound_ (i + 1) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [if_pos h1] | zs : List String
X : String
vs : List Var
j i : ℕ
a1 : bound_ (i + 1) ∈ vs
h1 : i + 1 < j
⊢ (if i + 1 < j then bound_ (i + 1)
else
if x : i + 1 - j < (List.map free_ zs).length then (List.map free_ zs)[i + 1 - j]
else bound_ (i + 1 - j - (List.map free_ zs).length + j)) =
bound_ (i + 1) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.lc_at] at h1 | zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi.not_
⊢ openList j (List.map free_ zs) phi.not_ = phi.not_ | zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi
⊢ openList j (List.map free_ zs) phi.not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.openList] | zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi
⊢ openList j (List.map free_ zs) phi.not_ = phi.not_ | zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi
⊢ (openList j (List.map free_ zs) phi).not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | congr! | zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi
⊢ (openList j (List.map free_ zs) phi).not_ = phi.not_ | case h.e'_1
zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi
⊢ openList j (List.map free_ zs) phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | exact phi_ih j h1 | case h.e'_1
zs : List String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j phi
⊢ openList j (List.map free_ zs) phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.lc_at] at h1 | zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1 : lc_at j (phi.imp_ psi)
⊢ openList j (List.map free_ zs) (phi.imp_ psi) = phi.imp_ psi | zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1 : lc_at j phi ∧ lc_at j psi
⊢ openList j (List.map free_ zs) (phi.imp_ psi) = phi.imp_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.openList] | zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1 : lc_at j phi ∧ lc_at j psi
⊢ openList j (List.map free_ zs) (phi.imp_ psi) = phi.imp_ psi | zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1 : lc_at j phi ∧ lc_at j psi
⊢ (openList j (List.map free_ zs) phi).imp_ (openList j (List.map free_ zs) psi) = phi.imp_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | cases h1 | zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1 : lc_at j phi ∧ lc_at j psi
⊢ (openList j (List.map free_ zs) phi).imp_ (openList j (List.map free_ zs) psi) = phi.imp_ psi | case intro
zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
left✝ : lc_at j phi
right✝ : lc_at j psi
⊢ (openList j (List.map free_ zs) phi).imp_ (openList j (List.map free_ zs) psi) = phi.imp_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | congr! | zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1_left : lc_at j phi
h1_right : lc_at j psi
⊢ (openList j (List.map free_ zs) phi).imp_ (openList j (List.map free_ zs) psi) = phi.imp_ psi | case h.e'_1
zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1_left : lc_at j phi
h1_right : lc_at j psi
⊢ openList j (List.map free_ zs) phi = phi
case h.e'_2
zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1_left : lc_at j phi
h1_right : lc_at j psi
⊢ openList j (List.map free_ zs) psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | exact phi_ih j h1_left | case h.e'_1
zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1_left : lc_at j phi
h1_right : lc_at j psi
⊢ openList j (List.map free_ zs) phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | exact psi_ih j h1_right | case h.e'_2
zs : List String
phi psi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
psi_ih : ∀ (j : ℕ), lc_at j psi → openList j (List.map free_ zs) psi = psi
j : ℕ
h1_left : lc_at j phi
h1_right : lc_at j psi
⊢ openList j (List.map free_ zs) psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.lc_at] at h1 | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at j (forall_ x phi)
⊢ openList j (List.map free_ zs) (forall_ x phi) = forall_ x phi | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at (j + 1) phi
⊢ openList j (List.map free_ zs) (forall_ x phi) = forall_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp only [Formula.openList] | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at (j + 1) phi
⊢ openList j (List.map free_ zs) (forall_ x phi) = forall_ x phi | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at (j + 1) phi
⊢ forall_ x (openList (j + 1) (List.map free_ zs) phi) = forall_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | simp | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at (j + 1) phi
⊢ forall_ x (openList (j + 1) (List.map free_ zs) phi) = forall_ x phi | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at (j + 1) phi
⊢ openList (j + 1) (List.map free_ zs) phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.Formula.OpenListLC | [1207, 1] | [1256, 28] | exact phi_ih (j + 1) h1 | zs : List String
x : String
phi : Formula
phi_ih : ∀ (j : ℕ), lc_at j phi → openList j (List.map free_ zs) phi = phi
j : ℕ
h1 : lc_at (j + 1) phi
⊢ openList (j + 1) (List.map free_ zs) phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | induction F generalizing j zs | F : Formula
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) F) ↔ Formula.lc_at (j + zs.length) F | case pred_
a✝¹ : String
a✝ : List Var
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝)) ↔ Formula.lc_at (j + zs.length) (pred_ a✝¹ a✝)
case not_
a✝ : Formula
a_ih✝ :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) a✝) ↔ Formula.lc_at (j + zs.length) a✝
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) a✝.not_) ↔ Formula.lc_at (j + zs.length) a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) a✝¹) ↔ Formula.lc_at (j + zs.length) a✝¹
a_ih✝ :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) a✝) ↔ Formula.lc_at (j + zs.length) a✝
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝)) ↔ Formula.lc_at (j + zs.length) (a✝¹.imp_ a✝)
case forall_
a✝¹ : String
a✝ : Formula
a_ih✝ :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) a✝) ↔ Formula.lc_at (j + zs.length) a✝
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)) ↔
Formula.lc_at (j + zs.length) (forall_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case not_ phi phi_ih =>
simp only [Formula.openList]
simp only [Formula.lc_at]
apply phi_ih | phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi.not_) ↔ Formula.lc_at (j + zs.length) phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case forall_ _ phi phi_ih =>
simp only [Formula.openList]
simp only [Formula.lc_at]
simp only [phi_ih]
have s1 : j + 1 + List.length zs = j + List.length zs + 1
linarith;
simp only [s1] | a✝ : String
phi : Formula
phi_ih :
∀ (j : ℕ) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) ↔ Formula.lc_at (j + zs.length) phi
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (forall_ a✝ phi)) ↔
Formula.lc_at (j + zs.length) (forall_ a✝ phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | X : String
vs : List Var
j : ℕ
zs : List String
⊢ Formula.lc_at j (Formula.openList j (List.map free_ zs) (pred_ X vs)) ↔ Formula.lc_at (j + zs.length) (pred_ X vs) | X : String
vs : List Var
j : ℕ
zs : List String
⊢ Formula.lc_at j (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)) ↔
Formula.lc_at (j + zs.length) (pred_ X vs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | X : String
vs : List Var
j : ℕ
zs : List String
⊢ Formula.lc_at j (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)) ↔
Formula.lc_at (j + zs.length) (pred_ X vs) | X : String
vs : List Var
j : ℕ
zs : List String
⊢ (∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v) ↔ ∀ v ∈ vs, Var.lc_at (j + zs.length) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | constructor | X : String
vs : List Var
j : ℕ
zs : List String
⊢ (∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v) ↔ ∀ v ∈ vs, Var.lc_at (j + zs.length) v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
⊢ (∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v) → ∀ v ∈ vs, Var.lc_at (j + zs.length) v
case mpr
X : String
vs : List Var
j : ℕ
zs : List String
⊢ (∀ v ∈ vs, Var.lc_at (j + zs.length) v) → ∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | intro a1 v a2 | case mp
X : String
vs : List Var
j : ℕ
zs : List String
⊢ (∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v) → ∀ v ∈ vs, Var.lc_at (j + zs.length) v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
a1 : ∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v
v : Var
a2 : v ∈ vs
⊢ Var.lc_at (j + zs.length) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | specialize a1 (Var.openList j (List.map free_ zs) v) | case mp
X : String
vs : List Var
j : ℕ
zs : List String
a1 : ∀ v ∈ List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v
v : Var
a2 : v ∈ vs
⊢ Var.lc_at (j + zs.length) v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 :
Var.openList j (List.map free_ zs) v ∈ List.map (Var.openList j (List.map free_ zs)) vs →
Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a1 | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 :
Var.openList j (List.map free_ zs) v ∈ List.map (Var.openList j (List.map free_ zs)) vs →
Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 :
∀ x ∈ vs,
Var.openList j (List.map free_ zs) x = Var.openList j (List.map free_ zs) v →
Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | specialize a1 v a2 | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 :
∀ x ∈ vs,
Var.openList j (List.map free_ zs) x = Var.openList j (List.map free_ zs) v →
Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 :
Var.openList j (List.map free_ zs) v = Var.openList j (List.map free_ zs) v →
Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a1 | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 :
Var.openList j (List.map free_ zs) v = Var.openList j (List.map free_ zs) v →
Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | cases v | case mp
X : String
vs : List Var
j : ℕ
zs : List String
v : Var
a2 : v ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) v)
⊢ Var.lc_at (j + zs.length) v | case mp.free_
X : String
vs : List Var
j : ℕ
zs : List String
a✝ : String
a2 : free_ a✝ ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (free_ a✝))
⊢ Var.lc_at (j + zs.length) (free_ a✝)
case mp.bound_
X : String
vs : List Var
j : ℕ
zs : List String
a✝ : ℕ
a2 : bound_ a✝ ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (bound_ a✝))
⊢ Var.lc_at (j + zs.length) (bound_ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ x =>
simp only [Var.lc_at] | X : String
vs : List Var
j : ℕ
zs : List String
x : String
a2 : free_ x ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (free_ x))
⊢ Var.lc_at (j + zs.length) (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ i =>
simp only [Var.lc_at]
simp only [Var.openList] at a1
split at a1
case _ c1 =>
linarith
case _ c1 =>
split at a1
case _ c2 =>
simp at c2
exact lt_add_of_tsub_lt_left c2
case _ c2 =>
simp only [Var.lc_at] at a1
simp at a1 | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (bound_ i))
⊢ Var.lc_at (j + zs.length) (bound_ i) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.lc_at] | X : String
vs : List Var
j : ℕ
zs : List String
x : String
a2 : free_ x ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (free_ x))
⊢ Var.lc_at (j + zs.length) (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.lc_at] | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (bound_ i))
⊢ Var.lc_at (j + zs.length) (bound_ i) | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (bound_ i))
⊢ i < j + zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.openList] at a1 | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
a1 : Var.lc_at j (Var.openList j (List.map free_ zs) (bound_ i))
⊢ i < j + zs.length | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
a1 :
Var.lc_at j
(if i < j then bound_ i
else
if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j))
⊢ i < j + zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | split at a1 | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
a1 :
Var.lc_at j
(if i < j then bound_ i
else
if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j))
⊢ i < j + zs.length | case inl
X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
h✝ : i < j
a1 : Var.lc_at j (bound_ i)
⊢ i < j + zs.length
case inr
X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
h✝ : ¬i < j
a1 :
Var.lc_at j
(if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j))
⊢ i < j + zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c1 =>
linarith | X : String
vs : List Var
j : ℕ
zs : List String
i : ℕ
a2 : bound_ i ∈ vs
c1 : i < j
a1 : Var.lc_at j (bound_ i)
⊢ i < j + zs.length | no goals |