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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply Holds_coincide_Var | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ u phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ u phi) | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
⊢ ∀ (v_1 : VarName), isFreeIn v_1 (exists_ u phi) → Function.updateITE V u a v_1 = Function.updateITE V v a v_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | intro x a1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
⊢ ∀ (v_1 : VarName), isFreeIn v_1 (exists_ u phi) → Function.updateITE V u a v_1 = Function.updateITE V v a v_1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1 : isFreeIn x (exists_ u phi)
⊢ Function.updateITE V u a x = Function.updateITE V v a x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at a1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1 : isFreeIn x (exists_ u phi)
⊢ Function.updateITE V u a x = Function.updateITE V v a x | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1 : ¬x = u ∧ isFreeIn x phi
⊢ Function.updateITE V u a x = Function.updateITE V v a x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases a1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1 : ¬x = u ∧ isFreeIn x phi
⊢ Function.updateITE V u a x = Function.updateITE V v a x | case h1.intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
left✝ : ¬x = u
right✝ : isFreeIn x phi
⊢ Function.updateITE V u a x = Function.updateITE V v a x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case h1.intro a1_left a1_right =>
simp only [Function.updateITE]
simp only [if_neg a1_left]
split_ifs
case pos c2 =>
subst c2
tauto
case neg c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
⊢ Function.updateITE V u a x = Function.updateITE V v a x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
⊢ Function.updateITE V u a x = Function.updateITE V v a x | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
⊢ (if x = u then a else V x) = if x = v then a else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [if_neg a1_left] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
⊢ (if x = u then a else V x) = if x = v then a else V x | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
⊢ V x = if x = v then a else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
⊢ V x = if x = v then a else V x | case pos
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
h✝ : x = v
⊢ V x = a
case neg
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
h✝ : ¬x = v
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case pos c2 =>
subst c2
tauto | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
c2 : x = v
⊢ V x = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case neg c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
c2 : ¬x = v
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | subst c2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
c2 : x = v
⊢ V x = a | D : Type
I : Interpretation D
u : VarName
a : D
head✝ : Definition
tail✝ : List Definition
phi : Formula
V : VarAssignment D
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn x F →
¬isBoundIn x F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V x a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u x F))
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn x phi →
¬isBoundIn x phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u x phi))
h2_right : ¬isBoundIn x phi
h1 : x ≠ u → ¬isFreeIn x phi
h2_left : x ≠ u
⊢ V x = a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | tauto | D : Type
I : Interpretation D
u : VarName
a : D
head✝ : Definition
tail✝ : List Definition
phi : Formula
V : VarAssignment D
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn x F →
¬isBoundIn x F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V x a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u x F))
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn x phi →
¬isBoundIn x phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u x phi))
h2_right : ¬isBoundIn x phi
h1 : x ≠ u → ¬isFreeIn x phi
h2_left : x ≠ u
⊢ V x = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
x : VarName
a1_left : ¬x = u
a1_right : isFreeIn x phi
c2 : ¬x = v
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ x (Sub.Var.One.Rec.fastReplaceFree u v phi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi) ↔
∃ d,
Holds D I (Function.updateITE (Function.updateITE V v a) x d) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi) ↔
∃ d,
Holds D I (Function.updateITE (Function.updateITE V v a) x d) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ ∀ (a_1 : D),
Holds D I (Function.updateITE (Function.updateITE V u a) x a_1) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V v a) x a_1) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | intro d | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ ∀ (a_1 : D),
Holds D I (Function.updateITE (Function.updateITE V u a) x a_1) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V v a) x a_1) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V v a) x d) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE_comm V v x d a h2_left] | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V v a) x d) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V x d) v a) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE_comm V u x d a c1] | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V x d) v a) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V x d) u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V x d) v a) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply phi_ih | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V x d) u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V x d) v a) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h.h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ ¬isFreeIn v phi
case h.h2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ ¬isBoundIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply forall_congr' | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi) ↔
∀ (d : D),
Holds D I (Function.updateITE (Function.updateITE V v a) x d) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ ∀ (a_1 : D),
Holds D I (Function.updateITE (Function.updateITE V u a) x a_1) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V v a) x a_1) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply exists_congr | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V u a) x d) (head✝ :: tail✝) phi) ↔
∃ d,
Holds D I (Function.updateITE (Function.updateITE V v a) x d) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) | case h
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
⊢ ∀ (a_1 : D),
Holds D I (Function.updateITE (Function.updateITE V u a) x a_1) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V v a) x a_1) (head✝ :: tail✝)
(Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact h1 h2_left | case h.h1
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ ¬isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact h2_right | case h.h2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : ¬u = x
d : D
⊢ ¬isBoundIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) [] (def_ X xs) ↔
Holds D I (Function.updateITE V v a) [] (Sub.Var.One.Rec.fastReplaceFree u v (def_ X xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) [] (def_ X xs) ↔
Holds D I (Function.updateITE V v a) [] (def_ X (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) [] (def_ X xs) ↔
Holds D I (Function.updateITE V v a) [] (def_ X (List.map (fun x => if u = x then v else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) (hd :: tl) (def_ X xs) ↔
Holds D I (Function.updateITE V v a) (hd :: tl) (Sub.Var.One.Rec.fastReplaceFree u v (def_ X xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) (hd :: tl) (def_ X xs) ↔
Holds D I (Function.updateITE V v a) (hd :: tl) (def_ X (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) (hd :: tl) (def_ X xs) ↔
Holds D I (Function.updateITE V v a) (hd :: tl) (def_ X (List.map (fun x => if u = x then v else x) xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I (Function.updateListITE (Function.updateITE V u a) hd.args (List.map (Function.updateITE V u a) xs)) tl
hd.q
else Holds D I (Function.updateITE V u a) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V v a) hd.args
(List.map (Function.updateITE V v a) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q
else Holds D I (Function.updateITE V v a) tl (def_ X (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | unfold Function.updateITE | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I (Function.updateListITE (Function.updateITE V u a) hd.args (List.map (Function.updateITE V u a) xs)) tl
hd.q
else Holds D I (Function.updateITE V u a) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V v a) hd.args
(List.map (Function.updateITE V v a) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q
else Holds D I (Function.updateITE V v a) tl (def_ X (List.map (fun x => if u = x then v else x) xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I
(Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs))
tl hd.q
else Holds D I (fun c => if c = u then a else V c) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length then
Holds D I
(Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q
else Holds D I (fun c => if c = v then a else V c) tl (def_ X (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I
(Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs))
tl hd.q
else Holds D I (fun c => if c = u then a else V c) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length then
Holds D I
(Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q
else Holds D I (fun c => if c = v then a else V c) tl (def_ X (List.map (fun x => if u = x then v else x) xs)) | case a.h₁.a
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ X = hd.name ∧ xs.length = hd.args.length ↔
X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
case a.h₂.a
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs))
tl hd.q ↔
Holds D I
(Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q
case a.h₃.a
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length)
⊢ Holds D I (fun c => if c = u then a else V c) tl (def_ X xs) ↔
Holds D I (fun c => if c = v then a else V c) tl (def_ X (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ =>
simp | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ X = hd.name ∧ xs.length = hd.args.length ↔
X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ X = hd.name ∧ xs.length = hd.args.length ↔
X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs))
tl hd.q ↔
Holds D I
(Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs))
tl hd.q ↔
Holds D I
(Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply Holds_coincide_Var | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs))
tl hd.q ↔
Holds D I
(Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
tl hd.q | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ ∀ (v_1 : VarName),
isFreeIn v_1 hd.q →
Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs) v_1 =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)) v_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | intro v' a1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ ∀ (v_1 : VarName),
isFreeIn v_1 hd.q →
Function.updateListITE (fun c => if c = u then a else V c) hd.args
(List.map (fun c => if c = u then a else V c) xs) v_1 =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)) v_1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)) v' | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map ((fun c => if c = v then a else V c) ∘ fun x => if u = x then v else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | have s1 : (List.map ((fun (a_1 : VarName) => if a_1 = v then a else V a_1) ∘ fun (x : VarName) => if u = x then v else x) xs) = (List.map (fun (a_1 : VarName) => if a_1 = u then a else V a_1) xs) | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map ((fun c => if c = v then a else V c) ∘ fun x => if u = x then v else x) xs) v' | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map ((fun c => if c = v then a else V c) ∘ fun x => if u = x then v else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | {
simp only [List.map_eq_map_iff]
intro x a2
simp only [eq_comm]
simp
split_ifs
case _ =>
rfl
case _ =>
contradiction
case _ c1 c2 =>
subst c2
contradiction
case _ =>
rfl
} | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map ((fun c => if c = v then a else V c) ∘ fun x => if u = x then v else x) xs) v' | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map ((fun c => if c = v then a else V c) ∘ fun x => if u = x then v else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [s1] | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args
(List.map ((fun c => if c = v then a else V c) ∘ fun x => if u = x then v else x) xs) v' | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ Function.updateListITE (fun c => if c = u then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' =
Function.updateListITE (fun c => if c = v then a else V c) hd.args (List.map (fun c => if c = u then a else V c) xs)
v' | case h1.h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ v' ∈ hd.args
case h1.h2
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [List.map_eq_map_iff] | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ xs, ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) x = if x = u then a else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | intro x a2 | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ xs, ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) x = if x = u then a else V x | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) x = if x = u then a else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [eq_comm] | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) x = if x = u then a else V x | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if u = x then a else V x) = ((fun a_1 => if v = a_1 then a else V a_1) ∘ fun x => if u = x then v else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if u = x then a else V x) = ((fun a_1 => if v = a_1 then a else V a_1) ∘ fun x => if u = x then v else x) x | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if u = x then a else V x) = if v = if u = x then v else x then a else V (if u = x then v else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs | case s1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if u = x then a else V x) = if v = if u = x then v else x then a else V (if u = x then v else x) | case pos
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : u = x
h✝ : v = v
⊢ a = a
case neg
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : u = x
h✝ : ¬v = v
⊢ a = V v
case pos
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : ¬u = x
h✝ : v = x
⊢ V x = a
case neg
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : ¬u = x
h✝ : ¬v = x
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : u = x
h✝ : v = v
⊢ a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ =>
contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : u = x
h✝ : ¬v = v
⊢ a = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
subst c2
contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1✝ : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c1 : ¬u = x
c2 : v = x
⊢ V x = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : ¬u = x
h✝ : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : u = x
h✝ : v = v
⊢ a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : u = x
h✝ : ¬v = v
⊢ a = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | subst c2 | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1✝ : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c1 : ¬u = x
c2 : v = x
⊢ V x = a | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1✝ : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
a2 : v ∈ xs
c1 : ¬u = v
⊢ V v = a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1✝ : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
a2 : v ∈ xs
c1 : ¬u = v
⊢ V v = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝¹ : ¬u = x
h✝ : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case h1.h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ v' ∈ hd.args | case h1.h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [← List.mem_toFinset] | case h1.h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args | case h1.h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply Finset.mem_of_subset hd.h1 a1 | case h1.h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp at c1 | case h1.h2
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length | case h1.h2
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases c1 | case h1.h2
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length | case h1.h2.intro
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
left✝ : X = hd.name
right✝ : xs.length = hd.args.length
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case intro c1_left c1_right =>
simp
simp only [eq_comm]
exact c1_right | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1_left : X = hd.name
c1_right : xs.length = hd.args.length
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1_left : X = hd.name
c1_right : xs.length = hd.args.length
⊢ hd.args.length = (List.map (fun c => if c = u then a else V c) xs).length | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1_left : X = hd.name
c1_right : xs.length = hd.args.length
⊢ hd.args.length = xs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [eq_comm] | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1_left : X = hd.name
c1_right : xs.length = hd.args.length
⊢ hd.args.length = xs.length | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1_left : X = hd.name
c1_right : xs.length = hd.args.length
⊢ xs.length = hd.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact c1_right | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
v' : VarName
a1 : isFreeIn v' hd.q
s1 :
List.map ((fun a_1 => if a_1 = v then a else V a_1) ∘ fun x => if u = x then v else x) xs =
List.map (fun a_1 => if a_1 = u then a else V a_1) xs
c1_left : X = hd.name
c1_right : xs.length = hd.args.length
⊢ xs.length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | apply ih | D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length)
⊢ Holds D I (fun c => if c = u then a else V c) tl (def_ X xs) ↔
Holds D I (fun c => if c = v then a else V c) tl (def_ X (List.map (fun x => if u = x then v else x) xs)) | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length)
⊢ ¬isFreeIn v (def_ X xs)
case h2
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length)
⊢ ¬isBoundIn v (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact h1 | case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length)
⊢ ¬isFreeIn v (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact h2 | case h2
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tl F ↔
Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length)
⊢ ¬isBoundIn v (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | induction h1 generalizing V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F F' : Formula
h1 : AlphaEqv F F'
⊢ Holds D I V E F ↔ Holds D I V E F' | case rename_forall_
D : Type
I : Interpretation D
E : Env
F F' phi✝ : Formula
x✝ y✝ : VarName
a✝¹ : ¬isFreeIn y✝ phi✝
a✝ : ¬isBoundIn y✝ phi✝
V : VarAssignment D
⊢ Holds D I V E (forall_ x✝ phi✝) ↔ Holds D I V E (forall_ y✝ (Sub.Var.One.Rec.fastReplaceFree x✝ y✝ phi✝))
case rename_exists_
D : Type
I : Interpretation D
E : Env
F F' phi✝ : Formula
x✝ y✝ : VarName
a✝¹ : ¬isFreeIn y✝ phi✝
a✝ : ¬isBoundIn y✝ phi✝
V : VarAssignment D
⊢ Holds D I V E (exists_ x✝ phi✝) ↔ Holds D I V E (exists_ y✝ (Sub.Var.One.Rec.fastReplaceFree x✝ y✝ phi✝))
case compat_not_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ : Formula
a✝ : AlphaEqv phi✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I V E phi✝.not_ ↔ Holds D I V E phi'✝.not_
case compat_imp_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ psi✝ psi'✝ : Formula
a✝¹ : AlphaEqv phi✝ phi'✝
a✝ : AlphaEqv psi✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I V E (phi✝.imp_ psi✝) ↔ Holds D I V E (phi'✝.imp_ psi'✝)
case compat_and_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ psi✝ psi'✝ : Formula
a✝¹ : AlphaEqv phi✝ phi'✝
a✝ : AlphaEqv psi✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I V E (phi✝.and_ psi✝) ↔ Holds D I V E (phi'✝.and_ psi'✝)
case compat_or_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ psi✝ psi'✝ : Formula
a✝¹ : AlphaEqv phi✝ phi'✝
a✝ : AlphaEqv psi✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I V E (phi✝.or_ psi✝) ↔ Holds D I V E (phi'✝.or_ psi'✝)
case compat_iff_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ psi✝ psi'✝ : Formula
a✝¹ : AlphaEqv phi✝ phi'✝
a✝ : AlphaEqv psi✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I V E (phi✝.iff_ psi✝) ↔ Holds D I V E (phi'✝.iff_ psi'✝)
case compat_forall_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ : Formula
x✝ : VarName
a✝ : AlphaEqv phi✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I V E (forall_ x✝ phi✝) ↔ Holds D I V E (forall_ x✝ phi'✝)
case compat_exists_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ : Formula
x✝ : VarName
a✝ : AlphaEqv phi✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I V E (exists_ x✝ phi✝) ↔ Holds D I V E (exists_ x✝ phi'✝)
case refl_
D : Type
I : Interpretation D
E : Env
F F' phi✝ : Formula
V : VarAssignment D
⊢ Holds D I V E phi✝ ↔ Holds D I V E phi✝
case symm_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ : Formula
a✝ : AlphaEqv phi✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I V E phi'✝ ↔ Holds D I V E phi✝
case trans_
D : Type
I : Interpretation D
E : Env
F F' phi✝ phi'✝ phi''✝ : Formula
a✝¹ : AlphaEqv phi✝ phi'✝
a✝ : AlphaEqv phi'✝ phi''✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I V E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I V E phi'✝ ↔ Holds D I V E phi''✝
V : VarAssignment D
⊢ Holds D I V E phi✝ ↔ Holds D I V E phi''✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | case rename_forall_ h1_phi h1_x h1_y h1_1 h1_2 | rename_exists_ h1_phi h1_x h1_y h1_1 h1_2 =>
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
exact replace_empty_Holds D I V E h1_x h1_y h1_phi d h1_1 h1_2 | D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ Holds D I V E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_y (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | case compat_not_ h1_phi h1_phi' _ h1_ih =>
simp only [Holds]
congr! 1
exact h1_ih V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | case compat_forall_ h1_phi h1_psi h1_x _ h1_ih | compat_exists_ h1_phi h1_psi h1_x _ h1_ih =>
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
exact h1_ih (Function.updateITE V h1_x d) | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ Holds D I V E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | case refl_ h1 =>
rfl | D : Type
I : Interpretation D
E : Env
F F' h1 : Formula
V : VarAssignment D
⊢ Holds D I V E h1 ↔ Holds D I V E h1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | case symm_ h1_phi h1_phi' _ h1_ih =>
symm
exact h1_ih V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ Holds D I V E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_y (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi)) | D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_y d) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_y d) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE V h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_y a) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | intro d | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE V h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_y a) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_y d) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact replace_empty_Holds D I V E h1_x h1_y h1_phi d h1_1 h1_2 | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_y d) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ (∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_y d) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE V h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_y a) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | apply exists_congr | D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_y d) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi : Formula
h1_x h1_y : VarName
h1_1 : ¬isFreeIn h1_y h1_phi
h1_2 : ¬isBoundIn h1_y h1_phi
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE V h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_y a) E (Sub.Var.One.Rec.fastReplaceFree h1_x h1_y h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I V E h1_phi ↔ ¬Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | congr! 1 | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I V E h1_phi ↔ ¬Holds D I V E h1_phi' | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E (h1_phi.iff_ h1_psi) ↔ Holds D I V E (h1_phi'.iff_ h1_psi') | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I V E h1_phi ↔ Holds D I V E h1_psi) ↔ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | congr! 1 | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I V E h1_phi ↔ Holds D I V E h1_psi) ↔ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_psi ↔ Holds D I V E h1_psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_1 V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_2 V | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_psi h1_psi' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_psi h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I V E h1_psi ↔ Holds D I V E h1_psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ Holds D I V E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_psi) | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔ ∃ d, Holds D I (Function.updateITE V h1_x d) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔ ∃ d, Holds D I (Function.updateITE V h1_x d) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | intro d | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih (Function.updateITE V h1_x d) | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) E h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | apply exists_congr | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi) ↔ ∃ d, Holds D I (Function.updateITE V h1_x d) E h1_psi | case h
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_psi : Formula
h1_x : VarName
a✝ : AlphaEqv h1_phi h1_psi
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_psi
V : VarAssignment D
⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi ↔ Holds D I (Function.updateITE V h1_x a) E h1_psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | rfl | D : Type
I : Interpretation D
E : Env
F F' h1 : Formula
V : VarAssignment D
⊢ Holds D I V E h1 ↔ Holds D I V E h1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | symm | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' : Formula
a✝ : AlphaEqv h1_phi h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | trans Holds D I V E h1_phi' | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi'' | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi'' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_1 V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.Holds_iff_alphaEqv_Holds | [348, 1] | [389, 22] | exact h1_ih_2 V | D : Type
I : Interpretation D
E : Env
F F' h1_phi h1_phi' h1_phi'' : Formula
a✝¹ : AlphaEqv h1_phi h1_phi'
a✝ : AlphaEqv h1_phi' h1_phi''
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I V E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I V E h1_phi' ↔ Holds D I V E h1_phi''
V : VarAssignment D
⊢ Holds D I V E h1_phi' ↔ Holds D I V E h1_phi'' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | induction xs | xs : List VarName
⊢ isAlphaEqvVarList [] xs xs | case nil
⊢ isAlphaEqvVarList [] [] []
case cons
head✝ : VarName
tail✝ : List VarName
tail_ih✝ : isAlphaEqvVarList [] tail✝ tail✝
⊢ isAlphaEqvVarList [] (head✝ :: tail✝) (head✝ :: tail✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | case nil =>
simp only [isAlphaEqvVarList] | ⊢ isAlphaEqvVarList [] [] [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | simp only [isAlphaEqvVarList] | ⊢ isAlphaEqvVarList [] [] [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqvVarListId | [441, 1] | [452, 15] | simp only [isAlphaEqvVarList] | hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVarList [] (hd :: tl) (hd :: tl) | hd : VarName
tl : List VarName
ih : isAlphaEqvVarList [] tl tl
⊢ isAlphaEqvVar [] hd hd ∧ isAlphaEqvVarList [] tl tl |
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