url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
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.fastRe... | 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.Re... |
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.Re... | 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.Re... |
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.Re... | 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.Re... |
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.Re... | 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.... |
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.fastRe... | 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.fastRe... | 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.fastRe... |
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.fastRe... | 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.fastRe... |
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.fastRe... | 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.R... |
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.fastRe... | 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.fastRe... | 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.fastRe... | 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)... |
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)... | 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.fastRe... | 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.fastRe... | 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.fastRe... |
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.fastRe... | 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... |
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... | 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... |
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... | 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... |
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... | 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... |
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... | 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.... |
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.fastRe... | 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... |
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.fastRe... | 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... |
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.... | 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.... | 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))
... | 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))
... |
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))
... | 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))
... |
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))
... | 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))
... |
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))
... | 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.fastReplaceF... |
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))
... | 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))
... | 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))
... | 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))
... |
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))
... | 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 ... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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.fastReplaceFr... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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 ... |
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 ... | 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... |
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))
... | 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))
... | 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))
... | 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))
... | 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))
... | 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))
... | 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))
... | 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))
... |
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))
... | 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))
... | 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.fastReplaceFr... | 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.fastReplaceFr... |
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.fastReplaceFr... | 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.fastReplaceFr... |
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.fastReplaceFr... | 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.fastReplaceFr... | 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.fastReplaceFr... |
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.fastReplaceFr... | 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.fastRep... |
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))
... | 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))
... | 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))
... |
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))
... | 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))
... |
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))
... | 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))
... | 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 ... |
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 ... | 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 ... | 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... |
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 ... |
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 ... | 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 ... | 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 ... |
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 ... |
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 : 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 ... |
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 ... | 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 ... |
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 ... | 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 ... | 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 ... |
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 ... | 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_ps... |
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_ps... | 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 ... | 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 ... |
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 ... |
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
... | 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
... |
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
... | 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
... | 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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