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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | tauto | case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | simp only [occursFreeIn] at h1 | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (forall_ x phi) → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi) | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | simp only [Holds] | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi) | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔ ∀ (d : D), Holds D I (shift D V' d) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | apply forall_congr' | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔ ∀ (d : D), Holds D I (shift D V' d) phi | case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ ∀ (a : D), Holds D I (shift D V a) phi ↔ Holds D I (shift D V' a) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | intro d | case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ ∀ (a : D), Holds D I (shift D V a) phi ↔ Holds D I (shift D V' a) phi | case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ Holds D I (shift D V d) phi ↔ Holds D I (shift D V' d) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | apply phi_ih | case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ Holds D I (shift D V d) phi ↔ Holds D I (shift D V' d) phi | case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ ∀ (v : Var), occursFreeIn v phi → shift D V d v = shift D V' d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | intro v a1 | case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ ∀ (v : Var), occursFreeIn v phi → shift D V d v = shift D V' d v | case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
v : Var
a1 : occursFreeIn v phi
⊢ shift D V d v = shift D V' d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | cases v | case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
v : Var
a1 : occursFreeIn v phi
⊢ shift D V d v = shift D V' d v | case h.h1.free_
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a✝ : String
a1 : occursFreeIn (free_ a✝) phi
⊢ shift D V d (free_ a✝) = shift D V' d (free_ a✝)
case h.h1.bound_
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a✝ : ℕ
a1 : occursFreeIn (bound_ a✝) phi
⊢ shift D V d (bound_ a✝) = shift D V' d (bound_ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | case free_ x =>
simp only [shift]
apply h1
exact a1 | D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ shift D V d (free_ x) = shift D V' d (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | case bound_ i =>
cases i
case zero =>
simp only [shift]
case succ i =>
simp only [shift]
apply h1
simp only [lift]
exact a1 | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) phi
⊢ shift D V d (bound_ i) = shift D V' d (bound_ i) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | simp only [shift] | D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ shift D V d (free_ x) = shift D V' d (free_ x) | D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ V (free_ x) = V' (free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | apply h1 | D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ V (free_ x) = V' (free_ x) | case a
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ occursFreeIn (lift (free_ x)) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | exact a1 | case a
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ occursFreeIn (lift (free_ x)) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | cases i | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) phi
⊢ shift D V d (bound_ i) = shift D V' d (bound_ i) | case zero
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0)
case succ
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
n✝ : ℕ
a1 : occursFreeIn (bound_ (n✝ + 1)) phi
⊢ shift D V d (bound_ (n✝ + 1)) = shift D V' d (bound_ (n✝ + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | case zero =>
simp only [shift] | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | case succ i =>
simp only [shift]
apply h1
simp only [lift]
exact a1 | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ shift D V d (bound_ (i + 1)) = shift D V' d (bound_ (i + 1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | simp only [shift] | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | simp only [shift] | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ shift D V d (bound_ (i + 1)) = shift D V' d (bound_ (i + 1)) | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ V (bound_ i) = V' (bound_ i) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | apply h1 | D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ V (bound_ i) = V' (bound_ i) | case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (lift (bound_ i)) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | simp only [lift] | case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (lift (bound_ i)) phi | case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (bound_ (i + 1)) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsCoincideVar | [959, 1] | [1015, 17] | exact a1 | case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (bound_ (i + 1)) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | apply forall_congr' | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (shift D V d) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F | case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D), Holds D I (shift D V a) F ↔ Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | intro d | case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D), Holds D I (shift D V a) F ↔ Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F | case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (shift D V d) F ↔ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | apply HoldsCoincideVar | case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (shift D V d) F ↔ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ ∀ (v : Var), occursFreeIn v F → shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | intro v a1 | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ ∀ (v : Var), occursFreeIn v F → shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = Function.updateITE V (free_ z) d (Var.openList 0 [free_ z] v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [Function.updateITE] | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = Function.updateITE V (free_ z) d (Var.openList 0 [free_ z] v) | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = if Var.openList 0 [free_ z] v = free_ z then d else V (Var.openList 0 [free_ z] v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | cases v | case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = if Var.openList 0 [free_ z] v = free_ z then d else V (Var.openList 0 [free_ z] v) | case h.h1.free_
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a✝ : String
a1 : occursFreeIn (free_ a✝) F
⊢ shift D V d (free_ a✝) =
if Var.openList 0 [free_ z] (free_ a✝) = free_ z then d else V (Var.openList 0 [free_ z] (free_ a✝))
case h.h1.bound_
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a✝ : ℕ
a1 : occursFreeIn (bound_ a✝) F
⊢ shift D V d (bound_ a✝) =
if Var.openList 0 [free_ z] (bound_ a✝) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | case _ x' =>
simp only [shift]
simp only [Var.openList]
split_ifs
case _ c1 =>
simp at c1
simp only [← c1] at h1
contradiction
case _ c1 =>
rfl | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ shift D V d (free_ x') =
if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x')) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | case _ i =>
cases i
case zero =>
simp only [shift]
simp only [Var.openList]
simp
case succ i =>
simp only [shift]
simp only [Var.openList]
simp | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) F
⊢ shift D V d (bound_ i) =
if Var.openList 0 [free_ z] (bound_ i) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ i)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [shift] | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ shift D V d (free_ x') =
if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x')) | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x')) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [Var.openList] | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x')) | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if free_ x' = free_ z then d else V (free_ x') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | split_ifs | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if free_ x' = free_ z then d else V (free_ x') | case pos
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
h✝ : free_ x' = free_ z
⊢ V (free_ x') = d
case neg
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
h✝ : ¬free_ x' = free_ z
⊢ V (free_ x') = V (free_ x') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | case _ c1 =>
simp at c1
simp only [← c1] at h1
contradiction | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : free_ x' = free_ z
⊢ V (free_ x') = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | case _ c1 =>
rfl | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : ¬free_ x' = free_ z
⊢ V (free_ x') = V (free_ x') | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp at c1 | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : free_ x' = free_ z
⊢ V (free_ x') = d | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : x' = z
⊢ V (free_ x') = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [← c1] at h1 | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : x' = z
⊢ V (free_ x') = d | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : x' = z
h1 : ¬occursFreeIn (free_ x') F
⊢ V (free_ x') = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | contradiction | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : x' = z
h1 : ¬occursFreeIn (free_ x') F
⊢ V (free_ x') = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | rfl | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : ¬free_ x' = free_ z
⊢ V (free_ x') = V (free_ x') | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | cases i | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) F
⊢ shift D V d (bound_ i) =
if Var.openList 0 [free_ z] (bound_ i) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ i)) | case zero
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ shift D V d (bound_ 0) =
if Var.openList 0 [free_ z] (bound_ 0) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ 0))
case succ
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
n✝ : ℕ
a1 : occursFreeIn (bound_ (n✝ + 1)) F
⊢ shift D V d (bound_ (n✝ + 1)) =
if Var.openList 0 [free_ z] (bound_ (n✝ + 1)) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ (n✝ + 1))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | case zero =>
simp only [shift]
simp only [Var.openList]
simp | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ shift D V d (bound_ 0) =
if Var.openList 0 [free_ z] (bound_ 0) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ 0)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | case succ i =>
simp only [shift]
simp only [Var.openList]
simp | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) F
⊢ shift D V d (bound_ (i + 1)) =
if Var.openList 0 [free_ z] (bound_ (i + 1)) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ (i + 1))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [shift] | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ shift D V d (bound_ 0) =
if Var.openList 0 [free_ z] (bound_ 0) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ 0)) | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ d = if Var.openList 0 [free_ z] (bound_ 0) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ 0)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [Var.openList] | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ d = if Var.openList 0 [free_ z] (bound_ 0) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ 0)) | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ d =
if
(if 0 < 0 then bound_ 0
else if x : 0 - 0 < [free_ z].length then [free_ z][0 - 0] else bound_ (0 - 0 - [free_ z].length + 0)) =
free_ z then
d
else
V
(if 0 < 0 then bound_ 0
else if x : 0 - 0 < [free_ z].length then [free_ z][0 - 0] else bound_ (0 - 0 - [free_ z].length + 0)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a1 : occursFreeIn (bound_ 0) F
⊢ d =
if
(if 0 < 0 then bound_ 0
else if x : 0 - 0 < [free_ z].length then [free_ z][0 - 0] else bound_ (0 - 0 - [free_ z].length + 0)) =
free_ z then
d
else
V
(if 0 < 0 then bound_ 0
else if x : 0 - 0 < [free_ z].length then [free_ z][0 - 0] else bound_ (0 - 0 - [free_ z].length + 0)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [shift] | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) F
⊢ shift D V d (bound_ (i + 1)) =
if Var.openList 0 [free_ z] (bound_ (i + 1)) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ (i + 1))) | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) F
⊢ V (bound_ i) =
if Var.openList 0 [free_ z] (bound_ (i + 1)) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ (i + 1))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp only [Var.openList] | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) F
⊢ V (bound_ i) =
if Var.openList 0 [free_ z] (bound_ (i + 1)) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ (i + 1))) | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) F
⊢ V (bound_ i) =
if
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < [free_ z].length then [free_ z][i + 1 - 0]
else bound_ (i + 1 - 0 - [free_ z].length + 0)) =
free_ z then
d
else
V
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < [free_ z].length then [free_ z][i + 1 - 0] else bound_ (i + 1 - 0 - [free_ z].length + 0)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsShift | [1018, 1] | [1053, 11] | simp | D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) F
⊢ V (bound_ i) =
if
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < [free_ z].length then [free_ z][i + 1 - 0]
else bound_ (i + 1 - 0 - [free_ z].length + 0)) =
free_ z then
d
else
V
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < [free_ z].length then [free_ z][i + 1 - 0] else bound_ (i + 1 - 0 - [free_ z].length + 0)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | simp only [Holds] | D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ Holds D I V (forall_ x F) ↔ ∀ (d : D), Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) | D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (shift D V d) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | simp only [HoldsShift D I V F z h1] | D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (shift D V d) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) | D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | apply forall_congr' | D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D),
Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) a) (Formula.openList 0 [free_ z] F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | intro d | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D),
Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) a) (Formula.openList 0 [free_ z] F) | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | obtain s1 := HoldsOpenList D I (Function.updateITE V (free_ z) d) 0 [z] F | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
s1 :
Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 (List.map free_ [z])) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 (List.map free_ [z]) F)
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | simp at s1 | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
s1 :
Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 (List.map free_ [z])) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 (List.map free_ [z]) F)
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
s1 :
Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F)
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsForall | [1056, 1] | [1072, 11] | exact s1 | case h
D : Type
I : Interpretation D
V : VarAssignment D
x : String
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
s1 :
Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F)
⊢ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F ↔
Holds D I (Function.updateITE V (free_ z) d) (Formula.openList 0 [free_ z] F) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | funext v | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d ∘ Var.close (j + 1) (free_ z) = shift D (V ∘ Var.close j (free_ z)) d | case h
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
v : Var
⊢ (shift D V d ∘ Var.close (j + 1) (free_ z)) v = shift D (V ∘ Var.close j (free_ z)) d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp | case h
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
v : Var
⊢ (shift D V d ∘ Var.close (j + 1) (free_ z)) v = shift D (V ∘ Var.close j (free_ z)) d v | case h
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
v : Var
⊢ shift D V d (Var.close (j + 1) (free_ z) v) = shift D (V ∘ Var.close j (free_ z)) d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | cases v | case h
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
v : Var
⊢ shift D V d (Var.close (j + 1) (free_ z) v) = shift D (V ∘ Var.close j (free_ z)) d v | case h.free_
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
a✝ : String
⊢ shift D V d (Var.close (j + 1) (free_ z) (free_ a✝)) = shift D (V ∘ Var.close j (free_ z)) d (free_ a✝)
case h.bound_
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
a✝ : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ a✝)) = shift D (V ∘ Var.close j (free_ z)) d (bound_ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case _ x =>
conv =>
rhs
simp only [shift]
simp
simp only [Var.close]
simp
split_ifs
case _ c1 =>
simp only [shift]
simp
case _ c1 =>
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (Var.close (j + 1) (free_ z) (free_ x)) = shift D (V ∘ Var.close j (free_ z)) d (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case _ i =>
cases i
case zero =>
conv =>
rhs
simp only [shift]
simp only [Var.close]
simp
simp only [shift]
case succ i =>
conv =>
rhs
simp only [shift]
simp
simp only [Var.close]
simp only [Var.close]
split
case _ c1 =>
have s1 : i < j
linarith
simp only [if_pos s1]
simp only [shift]
case _ c1 =>
simp at c1
have s1 : ¬ i < j
linarith
simp only [if_neg s1]
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ i)) = shift D (V ∘ Var.close j (free_ z)) d (bound_ i) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | conv =>
rhs
simp only [shift]
simp | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (Var.close (j + 1) (free_ z) (free_ x)) = shift D (V ∘ Var.close j (free_ z)) d (free_ x) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (Var.close (j + 1) (free_ z) (free_ x)) = V (Var.close j (free_ z) (free_ x)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [Var.close] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (Var.close (j + 1) (free_ z) (free_ x)) = V (Var.close j (free_ z) (free_ x)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (if free_ x = free_ z then bound_ (j + 1) else 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.extracted_1 | [1075, 1] | [1126, 26] | simp | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (if free_ x = free_ z then bound_ (j + 1) else free_ x) =
V (if free_ x = free_ z then bound_ j else free_ x) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (if x = z then bound_ (j + 1) else free_ x) = V (if x = z then bound_ j else free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | split_ifs | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
⊢ shift D V d (if x = z then bound_ (j + 1) else free_ x) = V (if x = z then bound_ j else free_ x) | case pos
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
h✝ : x = z
⊢ shift D V d (bound_ (j + 1)) = V (bound_ j)
case neg
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
h✝ : ¬x = z
⊢ shift D V d (free_ x) = V (free_ x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case _ c1 =>
simp only [shift]
simp | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
c1 : x = z
⊢ shift D V d (bound_ (j + 1)) = V (bound_ j) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case _ c1 =>
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
c1 : ¬x = z
⊢ shift D V d (free_ x) = V (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
c1 : x = z
⊢ shift D V d (bound_ (j + 1)) = V (bound_ j) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
x : String
c1 : ¬x = z
⊢ shift D V d (free_ x) = V (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | cases i | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ i)) = shift D (V ∘ Var.close j (free_ z)) d (bound_ i) | case zero
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ 0)) = shift D (V ∘ Var.close j (free_ z)) d (bound_ 0)
case succ
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
n✝ : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ (n✝ + 1))) = shift D (V ∘ Var.close j (free_ z)) d (bound_ (n✝ + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case zero =>
conv =>
rhs
simp only [shift]
simp only [Var.close]
simp
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ 0)) = shift D (V ∘ Var.close j (free_ z)) d (bound_ 0) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case succ i =>
conv =>
rhs
simp only [shift]
simp
simp only [Var.close]
simp only [Var.close]
split
case _ c1 =>
have s1 : i < j
linarith
simp only [if_pos s1]
simp only [shift]
case _ c1 =>
simp at c1
have s1 : ¬ i < j
linarith
simp only [if_neg s1]
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ (i + 1))) = shift D (V ∘ Var.close j (free_ z)) d (bound_ (i + 1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | conv =>
rhs
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ 0)) = shift D (V ∘ Var.close j (free_ z)) d (bound_ 0) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ 0)) = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [Var.close] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ 0)) = d | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (if 0 < j + 1 then bound_ 0 else bound_ (0 + 1)) = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (if 0 < j + 1 then bound_ 0 else bound_ (0 + 1)) = d | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (bound_ 0) = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
⊢ shift D V d (bound_ 0) = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | conv =>
rhs
simp only [shift]
simp
simp only [Var.close] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ (i + 1))) = shift D (V ∘ Var.close j (free_ z)) d (bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ (i + 1))) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [Var.close] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (Var.close (j + 1) (free_ z) (bound_ (i + 1))) = V (if i < j then bound_ i else bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (if i + 1 < j + 1 then bound_ (i + 1) else bound_ (i + 1 + 1)) =
V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | split | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
⊢ shift D V d (if i + 1 < j + 1 then bound_ (i + 1) else bound_ (i + 1 + 1)) =
V (if i < j then bound_ i else bound_ (i + 1)) | case inl
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
h✝ : i + 1 < j + 1
⊢ shift D V d (bound_ (i + 1)) = V (if i < j then bound_ i else bound_ (i + 1))
case inr
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
h✝ : ¬i + 1 < j + 1
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | case _ c1 =>
have s1 : i < j
linarith
simp only [if_pos s1]
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
⊢ shift D V d (bound_ (i + 1)) = 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.extracted_1 | [1075, 1] | [1126, 26] | case _ c1 =>
simp at c1
have s1 : ¬ i < j
linarith
simp only [if_neg s1]
simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : ¬i + 1 < j + 1
⊢ shift D V d (bound_ (i + 1 + 1)) = 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.extracted_1 | [1075, 1] | [1126, 26] | have s1 : i < j | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
⊢ shift D V d (bound_ (i + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | case s1
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
⊢ i < j
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
s1 : i < j
⊢ shift D V d (bound_ (i + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | linarith | case s1
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
⊢ i < j
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
s1 : i < j
⊢ shift D V d (bound_ (i + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
s1 : i < j
⊢ shift D V d (bound_ (i + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [if_pos s1] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
s1 : i < j
⊢ shift D V d (bound_ (i + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
s1 : i < j
⊢ shift D V d (bound_ (i + 1)) = V (bound_ i) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : i + 1 < j + 1
s1 : i < j
⊢ shift D V d (bound_ (i + 1)) = V (bound_ i) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp at c1 | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : ¬i + 1 < j + 1
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | have s1 : ¬ i < j | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | case s1
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
⊢ ¬i < j
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
s1 : ¬i < j
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | linarith | case s1
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
⊢ ¬i < j
D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
s1 : ¬i < j
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
s1 : ¬i < j
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [if_neg s1] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
s1 : ¬i < j
⊢ shift D V d (bound_ (i + 1 + 1)) = V (if i < j then bound_ i else bound_ (i + 1)) | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
s1 : ¬i < j
⊢ shift D V d (bound_ (i + 1 + 1)) = V (bound_ (i + 1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.extracted_1 | [1075, 1] | [1126, 26] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
z : String
d : D
i : ℕ
c1 : j ≤ i
s1 : ¬i < j
⊢ shift D V d (bound_ (i + 1 + 1)) = V (bound_ (i + 1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | induction F generalizing V j | D : Type
I : Interpretation D
V : VarAssignment D
z : String
F : Formula
j : ℕ
h1 : ¬occursIn (free_ z) F
h2 : Formula.lc_at j F
⊢ Holds D I V F ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] F) | case pred_
D : Type
I : Interpretation D
z a✝¹ : String
a✝ : List Var
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) (pred_ a✝¹ a✝)
h2 : Formula.lc_at j (pred_ a✝¹ a✝)
⊢ Holds D I V (pred_ a✝¹ a✝) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ a✝¹ a✝))
case not_
D : Type
I : Interpretation D
z : String
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) a✝ →
Formula.lc_at j a✝ → (Holds D I V a✝ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] a✝))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) a✝.not_
h2 : Formula.lc_at j a✝.not_
⊢ Holds D I V a✝.not_ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] a✝.not_)
case imp_
D : Type
I : Interpretation D
z : String
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) a✝¹ →
Formula.lc_at j a✝¹ → (Holds D I V a✝¹ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) a✝ →
Formula.lc_at j a✝ → (Holds D I V a✝ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] a✝))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) (a✝¹.imp_ a✝)
h2 : Formula.lc_at j (a✝¹.imp_ a✝)
⊢ Holds D I V (a✝¹.imp_ a✝) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (a✝¹.imp_ a✝))
case forall_
D : Type
I : Interpretation D
z a✝¹ : String
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (j : ℕ),
¬occursIn (free_ z) a✝ →
Formula.lc_at j a✝ → (Holds D I V a✝ ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] a✝))
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) (forall_ a✝¹ a✝)
h2 : Formula.lc_at j (forall_ a✝¹ a✝)
⊢ Holds D I V (forall_ a✝¹ a✝) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (forall_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | case pred_ X vs =>
simp only [occursIn] at h1
simp only [Formula.lc_at] at h2
simp only [Formula.openList]
simp only [Holds]
congr! 1
simp
simp only [List.map_eq_map_iff]
intro v a1
specialize h2 v a1
simp
cases v
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]
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 : ¬occursIn (free_ z) (pred_ X vs)
h2 : Formula.lc_at j (pred_ X vs)
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ X vs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | case not_ phi phi_ih =>
simp only [occursIn] at h1
simp only [Formula.lc_at] at h2
simp only [Holds]
congr! 1
exact phi_ih V j h1 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.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_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | case forall_ x phi phi_ih =>
simp only [occursIn] at h1
simp only [Formula.lc_at] at h2
simp only [Formula.openList]
simp only [Holds]
apply forall_congr'
intro d
specialize phi_ih (shift D V d) (j + 1) h1 h2
simp only [phi_ih]
congr! 1
apply extracted_1 | 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)) | 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
vs : List Var
V : VarAssignment D
j : ℕ
h1 : ¬occursIn (free_ z) (pred_ X vs)
h2 : Formula.lc_at j (pred_ X vs)
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ X vs)) | D : Type
I : Interpretation D
z X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : Formula.lc_at j (pred_ X vs)
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ X vs)) |
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
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : Formula.lc_at j (pred_ X vs)
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ X vs)) | 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
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ X vs)) |
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
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : ∀ v ∈ vs, Var.lc_at j v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (Formula.openList j [free_ z] (pred_ X vs)) | 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
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (pred_ X (List.map (Var.openList j [free_ z]) vs)) |
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
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : ∀ v ∈ vs, Var.lc_at j v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I (V ∘ Var.close j (free_ z)) (pred_ X (List.map (Var.openList j [free_ z]) vs)) | 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
⊢ I.pred_ X (List.map V vs) ↔ I.pred_ X (List.map (V ∘ Var.close j (free_ z)) (List.map (Var.openList j [free_ z]) vs)) |
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 X : String
vs : List Var
V : VarAssignment D
j : ℕ
h1 : free_ z ∉ vs
h2 : ∀ v ∈ vs, Var.lc_at j v
⊢ I.pred_ X (List.map V vs) ↔ I.pred_ X (List.map (V ∘ Var.close j (free_ z)) (List.map (Var.openList j [free_ z]) vs)) | 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
⊢ List.map V vs = List.map (V ∘ Var.close j (free_ z)) (List.map (Var.openList j [free_ z]) vs) |
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
h2 : ∀ v ∈ vs, Var.lc_at j v
⊢ List.map V vs = List.map (V ∘ Var.close j (free_ z)) (List.map (Var.openList j [free_ z]) vs) | 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
⊢ List.map V vs = List.map ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsClose | [1129, 1] | [1204, 22] | simp only [List.map_eq_map_iff] | 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
⊢ List.map V vs = List.map ((V ∘ Var.close j (free_ z)) ∘ Var.openList j [free_ z]) vs | 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 |