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stringclasses 147
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stringlengths 1
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stringlengths 6
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stringlengths 1
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ x (fastReplaceFree v t phi)) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | intro d | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (fastReplaceFree v t phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | cases h1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | case h.inl
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h✝ : v = x
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi)
case h.inr
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h✝ : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case inl h1 =>
contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : v = x
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case inr h1 =>
simp only [Function.updateITE_comm V v x d (V' t) c1]
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
simp only [Function.updateITE]
simp
push_neg
intros v' a1 a2
simp only [if_neg a2]
exact h2 v' a1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply forall_congr' | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : v = x
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Function.updateITE_comm V v x d (V' t) c1] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ Holds D I (Function.updateITE (Function.updateITE V x d) v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ Holds D I (Function.updateITE (Function.updateITE V x d) v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders ∪ {x}, V' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Function.updateITE] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders ∪ {x}, V' v = Function.updateITE V x d v | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders ∪ {x}, V' v = if v = x then d else V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders ∪ {x}, V' v = if v = x then d else V v | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders, ¬v = x → V' v = if v = x then d else V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | push_neg | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders, ¬v = x → V' v = if v = x then d else V v | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders, v ≠ x → V' v = if v = x then d else V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | intros v' a1 a2 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
⊢ ∀ v ∉ binders, v ≠ x → V' v = if v = x then d else V v | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
v' : VarName
a1 : v' ∉ binders
a2 : v' ≠ x
⊢ V' v' = if v' = x then d else V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [if_neg a2] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
v' : VarName
a1 : v' ∉ binders
a2 : v' ≠ x
⊢ V' v' = if v' = x then d else V v' | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
v' : VarName
a1 : v' ∉ binders
a2 : v' ≠ x
⊢ V' v' = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact h2 v' a1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : fastAdmitsAux v t (binders ∪ {x}) phi
v' : VarName
a1 : v' ∉ binders
a2 : v' ≠ x
⊢ V' v' = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | unfold Function.updateITE | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V v (V' t)) hd.args (List.map (Function.updateITE V v (V' t)) xs))
tl hd.q
else Holds D I (Function.updateITE V v (V' t)) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs))) tl hd.q
else Holds D I V tl (def_ X (List.map (fun x => if v = x then t else x) xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I
(Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs))
tl hd.q
else Holds D I (fun c => if c = v then V' t else V c) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs))) tl hd.q
else Holds D I V tl (def_ X (List.map (fun x => if v = x then t else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D I
(Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs))
tl hd.q
else Holds D I (fun c => if c = v then V' t else V c) tl (def_ X xs)) ↔
if X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs))) tl hd.q
else Holds D I V tl (def_ X (List.map (fun x => if v = x then t else x) xs)) | case a.h₁.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ X = hd.name ∧ xs.length = hd.args.length ↔
X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
case a.h₂.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs))) tl hd.q
case a.h₃.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ Holds D I (fun c => if c = v then V' t else V c) tl (def_ X xs) ↔
Holds D I V tl (def_ X (List.map (fun x => if v = x then t else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ =>
simp | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ X = hd.name ∧ xs.length = hd.args.length ↔
X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ X = hd.name ∧ xs.length = hd.args.length ↔
X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply Holds_coincide_Var | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs))) tl hd.q | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
⊢ ∀ (v_1 : VarName),
isFreeIn v_1 hd.q →
Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs) v_1 =
Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs)) v_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | intro v' a1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
⊢ ∀ (v_1 : VarName),
isFreeIn v_1 hd.q →
Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs) v_1 =
Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs)) v_1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs)) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map V (List.map (fun x => if v = x then t else x) xs)) v' | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [eq_comm] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if c = v then V' t else V c) hd.args
(List.map (fun c => if c = v then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (fun c => if v = c then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | have s1 :
(List.map (fun (x : VarName) =>
if v = x then V' t else V x) xs) =
(List.map (V ∘ fun (x : VarName) =>
if v = x then t else x) xs) | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (fun c => if v = c then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (fun c => if v = c then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | {
simp only [List.map_eq_map_iff]
intro x a2
simp
split_ifs
case _ c2 =>
apply h2
subst c2
exact h1 a2
case _ c2 =>
rfl
} | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (fun c => if v = c then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (fun c => if v = c then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [s1] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (fun c => if v = c then V' t else V c) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (V ∘ fun x => if v = x then t else x) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ Function.updateListITE (fun c => if v = c then V' t else V c) hd.args
(List.map (V ∘ fun x => if v = x then t else x) xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if v = x then t else x) xs) v' | case h1.h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ v' ∈ hd.args
case h1.h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ hd.args.length = (List.map (V ∘ fun x => if v = x then t else x) xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [List.map_eq_map_iff] | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ xs, (if v = x then V' t else V x) = (V ∘ fun x => if v = x then t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | intro x a2 | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ xs, (if v = x then V' t else V x) = (V ∘ fun x => if v = x then t else x) x | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if v = x then V' t else V x) = (V ∘ fun x => if v = x then t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if v = x then V' t else V x) = (V ∘ fun x => if v = x then t else x) x | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if v = x then V' t else V x) = V (if v = x then t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | split_ifs | case s1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
⊢ (if v = x then V' t else V x) = V (if v = x then t else x) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝ : v = x
⊢ V' t = V t
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
h✝ : ¬v = x
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c2 =>
apply h2
subst c2
exact h1 a2 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c2 : v = x
⊢ V' t = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c2 =>
rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c2 : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply h2 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c2 : v = x
⊢ V' t = V t | case a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c2 : v = x
⊢ t ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | subst c2 | case a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c2 : v = x
⊢ t ∉ binders | case a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
a2 : v ∈ xs
⊢ t ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact h1 a2 | case a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
a2 : v ∈ xs
⊢ t ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a2 : x ∈ xs
c2 : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case h1.h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ v' ∈ hd.args | case h1.h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [← List.mem_toFinset] | case h1.h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args | case h1.h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact Finset.mem_of_subset hd.h1 a1 | case h1.h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp at c1 | case h1.h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
⊢ hd.args.length = (List.map (V ∘ fun x => if v = x then t else x) xs).length | case h1.h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ hd.args.length = (List.map (V ∘ fun x => if v = x then t else x) xs).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | case h1.h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ hd.args.length = (List.map (V ∘ fun x => if v = x then t else x) xs).length | case h1.h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ hd.args.length = xs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | tauto | case h1.h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (fun x => if v = x then V' t else V x) xs = List.map (V ∘ fun x => if v = x then t else x) xs
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ hd.args.length = xs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply ih V binders | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ Holds D I (fun c => if c = v then V' t else V c) tl (def_ X xs) ↔
Holds D I V tl (def_ X (List.map (fun x => if v = x then t else x) xs)) | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ fastAdmitsAux v t binders (def_ X xs)
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ ∀ v ∉ binders, V' v = V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [fastAdmitsAux] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ fastAdmitsAux v t binders (def_ X xs) | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ v ∈ xs → t ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact h1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ v ∈ xs → t ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact h2 | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tl F ↔ Holds D I V tl (fastReplaceFree v t F))
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a✝ : ¬(X = hd.name ∧ (List.map (fun x => if v = x then t else x) xs).length = hd.args.length)
⊢ ∀ v ∉ binders, V' v = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem | [1139, 1] | [1153, 7] | simp only [fastAdmits] at h1 | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F : Formula
h1 : fastAdmits v t F
⊢ Holds D I (Function.updateITE V v (V t)) E F ↔ Holds D I V E (fastReplaceFree v t F) | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F : Formula
h1 : fastAdmitsAux v t ∅ F
⊢ Holds D I (Function.updateITE V v (V t)) E F ↔ Holds D I V E (fastReplaceFree v t F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem | [1139, 1] | [1153, 7] | apply substitution_theorem_aux D I V V E v t ∅ F h1 | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F : Formula
h1 : fastAdmitsAux v t ∅ F
⊢ Holds D I (Function.updateITE V v (V t)) E F ↔ Holds D I V E (fastReplaceFree v t F) | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F : Formula
h1 : fastAdmitsAux v t ∅ F
⊢ ∀ v ∉ ∅, V v = V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem | [1139, 1] | [1153, 7] | simp | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F : Formula
h1 : fastAdmitsAux v t ∅ F
⊢ ∀ v ∉ ∅, V v = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_is_valid | [1156, 1] | [1168, 11] | simp only [IsValid] at h2 | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : F.IsValid
⊢ (fastReplaceFree v t F).IsValid | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (fastReplaceFree v t F).IsValid |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_is_valid | [1156, 1] | [1168, 11] | simp only [IsValid] | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (fastReplaceFree v t F).IsValid | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (fastReplaceFree v t F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_is_valid | [1156, 1] | [1168, 11] | intro D I V E | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (fastReplaceFree v t F) | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (fastReplaceFree v t F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_is_valid | [1156, 1] | [1168, 11] | simp only [← substitution_theorem D I V E v t F h1] | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (fastReplaceFree v t F) | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I (Function.updateITE V v (V t)) E F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_is_valid | [1156, 1] | [1168, 11] | apply h2 | v t : VarName
F : Formula
h1 : fastAdmits v t F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I (Function.updateITE V v (V t)) E F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | cases v | v : Var
⊢ Var.lc_at 0 v ↔ v.isFree | case free_
a✝ : String
⊢ Var.lc_at 0 (free_ a✝) ↔ (free_ a✝).isFree
case bound_
a✝ : ℕ
⊢ Var.lc_at 0 (bound_ a✝) ↔ (bound_ a✝).isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | case free_ x =>
simp only [Var.lc_at]
simp only [isFree] | x : String
⊢ Var.lc_at 0 (free_ x) ↔ (free_ x).isFree | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | case bound_ i =>
simp only [Var.lc_at]
simp only [isFree]
simp | i : ℕ
⊢ Var.lc_at 0 (bound_ i) ↔ (bound_ i).isFree | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | simp only [Var.lc_at] | x : String
⊢ Var.lc_at 0 (free_ x) ↔ (free_ x).isFree | x : String
⊢ True ↔ (free_ x).isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | simp only [isFree] | x : String
⊢ True ↔ (free_ x).isFree | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | simp only [Var.lc_at] | i : ℕ
⊢ Var.lc_at 0 (bound_ i) ↔ (bound_ i).isFree | i : ℕ
⊢ i < 0 ↔ (bound_ i).isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | simp only [isFree] | i : ℕ
⊢ i < 0 ↔ (bound_ i).isFree | i : ℕ
⊢ i < 0 ↔ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_zero_iff_is_free | [171, 1] | [182, 9] | simp | i : ℕ
⊢ i < 0 ↔ False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | induction vs | vs : List Var
h1 : ∀ v ∈ vs, Var.lc_at 0 v
⊢ ∃ xs, vs = List.map free_ xs | case nil
h1 : ∀ v ∈ [], Var.lc_at 0 v
⊢ ∃ xs, [] = List.map free_ xs
case cons
head✝ : Var
tail✝ : List Var
tail_ih✝ : (∀ v ∈ tail✝, Var.lc_at 0 v) → ∃ xs, tail✝ = List.map free_ xs
h1 : ∀ v ∈ head✝ :: tail✝, Var.lc_at 0 v
⊢ ∃ xs, head✝ :: tail✝ = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | case nil =>
apply Exists.intro []
simp | h1 : ∀ v ∈ [], Var.lc_at 0 v
⊢ ∃ xs, [] = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | case cons hd tl ih =>
simp at h1
cases h1
case intro h1_left h1_right =>
specialize ih h1_right
apply Exists.elim ih
intro xs a1
cases hd
case free_ x =>
apply Exists.intro (x :: xs)
simp
exact a1
case bound_ i =>
simp only [Var.lc_at] at h1_left
simp at h1_left | hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v
⊢ ∃ xs, hd :: tl = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | apply Exists.intro [] | h1 : ∀ v ∈ [], Var.lc_at 0 v
⊢ ∃ xs, [] = List.map free_ xs | h1 : ∀ v ∈ [], Var.lc_at 0 v
⊢ [] = List.map free_ [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | simp | h1 : ∀ v ∈ [], Var.lc_at 0 v
⊢ [] = List.map free_ [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | simp at h1 | hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v
⊢ ∃ xs, hd :: tl = List.map free_ xs | hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a
⊢ ∃ xs, hd :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | cases h1 | hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a
⊢ ∃ xs, hd :: tl = List.map free_ xs | case intro
hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
left✝ : Var.lc_at 0 hd
right✝ : ∀ a ∈ tl, Var.lc_at 0 a
⊢ ∃ xs, hd :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | case intro h1_left h1_right =>
specialize ih h1_right
apply Exists.elim ih
intro xs a1
cases hd
case free_ x =>
apply Exists.intro (x :: xs)
simp
exact a1
case bound_ i =>
simp only [Var.lc_at] at h1_left
simp at h1_left | hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
⊢ ∃ xs, hd :: tl = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | specialize ih h1_right | hd : Var
tl : List Var
ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
⊢ ∃ xs, hd :: tl = List.map free_ xs | hd : Var
tl : List Var
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
⊢ ∃ xs, hd :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | apply Exists.elim ih | hd : Var
tl : List Var
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
⊢ ∃ xs, hd :: tl = List.map free_ xs | hd : Var
tl : List Var
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | intro xs a1 | hd : Var
tl : List Var
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs | hd : Var
tl : List Var
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
⊢ ∃ xs, hd :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | cases hd | hd : Var
tl : List Var
h1_left : Var.lc_at 0 hd
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
⊢ ∃ xs, hd :: tl = List.map free_ xs | case free_
tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
a✝ : String
h1_left : Var.lc_at 0 (free_ a✝)
⊢ ∃ xs, free_ a✝ :: tl = List.map free_ xs
case bound_
tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
a✝ : ℕ
h1_left : Var.lc_at 0 (bound_ a✝)
⊢ ∃ xs, bound_ a✝ :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | case free_ x =>
apply Exists.intro (x :: xs)
simp
exact a1 | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
x : String
h1_left : Var.lc_at 0 (free_ x)
⊢ ∃ xs, free_ x :: tl = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | case bound_ i =>
simp only [Var.lc_at] at h1_left
simp at h1_left | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
i : ℕ
h1_left : Var.lc_at 0 (bound_ i)
⊢ ∃ xs, bound_ i :: tl = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | apply Exists.intro (x :: xs) | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
x : String
h1_left : Var.lc_at 0 (free_ x)
⊢ ∃ xs, free_ x :: tl = List.map free_ xs | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
x : String
h1_left : Var.lc_at 0 (free_ x)
⊢ free_ x :: tl = List.map free_ (x :: xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | simp | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
x : String
h1_left : Var.lc_at 0 (free_ x)
⊢ free_ x :: tl = List.map free_ (x :: xs) | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
x : String
h1_left : Var.lc_at 0 (free_ x)
⊢ tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | exact a1 | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
x : String
h1_left : Var.lc_at 0 (free_ x)
⊢ tl = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | simp only [Var.lc_at] at h1_left | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
i : ℕ
h1_left : Var.lc_at 0 (bound_ i)
⊢ ∃ xs, bound_ i :: tl = List.map free_ xs | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
i : ℕ
h1_left : i < 0
⊢ ∃ xs, bound_ i :: tl = List.map free_ xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.free_var_list_to_string_list | [186, 1] | [210, 24] | simp at h1_left | tl : List Var
h1_right : ∀ a ∈ tl, Var.lc_at 0 a
ih : ∃ xs, tl = List.map free_ xs
xs : List String
a1 : tl = List.map free_ xs
i : ℕ
h1_left : i < 0
⊢ ∃ xs, bound_ i :: tl = List.map free_ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | cases v | j : ℕ
z : String
v : Var
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z} | case free_
j : ℕ
z a✝ : String
⊢ (Var.open j (free_ z) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ {free_ z}
case bound_
j : ℕ
z : String
a✝ : ℕ
⊢ (Var.open j (free_ z) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | case free_ x =>
simp only [Var.open]
simp only [Var.freeVarSet]
simp | j : ℕ
z x : String
⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | case bound_ i =>
simp only [Var.open]
split_ifs
case _ c1 =>
simp only [Var.freeVarSet]
simp
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp | j : ℕ
z : String
i : ℕ
⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp only [Var.open] | j : ℕ
z x : String
⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z} | j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp only [Var.freeVarSet] | j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z} | j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp | j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp only [Var.open] | j : ℕ
z : String
i : ℕ
⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | j : ℕ
z : String
i : ℕ
⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | split_ifs | j : ℕ
z : String
i : ℕ
⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | case pos
j : ℕ
z : String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
case pos
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : i = j
⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
case neg
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i = j
⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | case _ c1 =>
simp only [Var.freeVarSet]
simp | j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | case _ c1 c2 =>
simp only [Var.freeVarSet]
simp | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | case _ c1 c2 =>
simp only [Var.freeVarSet]
simp | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp only [Var.freeVarSet] | j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp | j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp only [Var.freeVarSet] | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ {free_ z} ⊆ ∅ ∪ {free_ z} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ {free_ z} ⊆ ∅ ∪ {free_ z} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.VarOpenFreeVarSet | [216, 1] | [238, 11] | simp only [Var.freeVarSet] | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} | j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅ ∪ {free_ z} |
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