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stringclasses 147
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stringlengths 7
101
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stringlengths 1
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| start
stringlengths 6
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stringlengths 6
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stringlengths 1
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stringlengths 3
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| state_after
stringlengths 6
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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 | 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/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | induction F generalizing binders | F : Formula
v t : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders F = F | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v t : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders true_ = true_
case false_
v t : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders false_ = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders a✝.not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | any_goals
simp only [replaceFreeAux] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
v t : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders true_ = true_
case false_
v t : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders false_ = false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders a✝.not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = def_ a✝¹ a✝ | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ pred_const_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = pred_const_ a✝¹ a✝
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ pred_var_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = pred_var_ a✝¹ a✝
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ eq_ (if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) (if v = a✝ ∧ a✝ ∉ binders then t else a✝) = eq_ a✝¹ a✝
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders a✝).not_ = a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders a✝¹).imp_ (replaceFreeAux v t binders a✝) = a✝¹.imp_ a✝
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders a✝¹).and_ (replaceFreeAux v t binders a✝) = a✝¹.and_ a✝
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders a✝¹).or_ (replaceFreeAux v t binders a✝) = a✝¹.or_ a✝
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders a✝¹).iff_ (replaceFreeAux v t binders a✝) = a✝¹.iff_ a✝
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ forall_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) = forall_ a✝¹ a✝
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ exists_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) = exists_ a✝¹ a✝
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp
simp only [List.map_eq_self_iff]
simp
intro x _ a2 a3
subst a2
contradiction | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | case eq_ x y =>
simp
constructor
case left | right =>
intro a1 a2
subst a1
contradiction | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | case not_ phi phi_ih =>
tauto | v t : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders phi).not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto | v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) = phi.iff_ psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp
apply phi_ih
simp
left
exact h1 | v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp only [replaceFreeAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = def_ a✝¹ a✝ | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) = def_ X xs | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs = xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp only [List.map_eq_self_iff] | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs = xs | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ ∀ x ∈ xs, (if v = x ∧ x ∉ binders then t else x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ ∀ x ∈ xs, (if v = x ∧ x ∉ binders then t else x) = x | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ ∀ x ∈ xs, v = x → x ∉ binders → t = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | intro x _ a2 a3 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ ∀ x ∈ xs, v = x → x ∉ binders → t = x | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
x : VarName
a✝ : x ∈ xs
a2 : v = x
a3 : x ∉ binders
⊢ t = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | subst a2 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
x : VarName
a✝ : x ∈ xs
a2 : v = x
a3 : x ∉ binders
⊢ t = x | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
a✝ : v ∈ xs
a3 : v ∉ binders
⊢ t = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | contradiction | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∈ binders
a✝ : v ∈ xs
a3 : v ∉ binders
⊢ t = v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) = eq_ x y | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ (v = x → x ∉ binders → t = x) ∧ (v = y → y ∉ binders → t = y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | constructor | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ (v = x → x ∉ binders → t = x) ∧ (v = y → y ∉ binders → t = y) | case left
v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v = x → x ∉ binders → t = x
case right
v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v = y → y ∉ binders → t = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | case left | right =>
intro a1 a2
subst a1
contradiction | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v = y → y ∉ binders → t = y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | intro a1 a2 | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v = y → y ∉ binders → t = y | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
a1 : v = y
a2 : y ∉ binders
⊢ t = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | subst a1 | v t x y : VarName
binders : Finset VarName
h1 : v ∈ binders
a1 : v = y
a2 : y ∉ binders
⊢ t = y | v t x : VarName
binders : Finset VarName
h1 : v ∈ binders
a2 : v ∉ binders
⊢ t = v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | contradiction | v t x : VarName
binders : Finset VarName
h1 : v ∈ binders
a2 : v ∉ binders
⊢ t = v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | tauto | v t : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders phi).not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp | v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi
binders : Finset VarName
h1 : v ∈ binders
⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) = phi.iff_ psi | v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders phi = phi ∧ replaceFreeAux v t binders psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | tauto | v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t binders phi = phi ∧ replaceFreeAux v t binders psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp | v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi | v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | apply phi_ih | v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi | case h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∪ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | simp | case h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∪ {x} | case h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∨ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | left | case h1
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∨ v = x | case h1.h
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders | [122, 1] | [160, 13] | exact h1 | case h1.h
v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | induction F generalizing binders | F : Formula
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders F = fastReplaceFree v t F | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = fastReplaceFree v t (pred_const_ a✝¹ a✝)
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = fastReplaceFree v t (pred_var_ a✝¹ a✝)
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = fastReplaceFree v t (eq_ a✝¹ a✝)
case true_
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders true_ = fastReplaceFree v t true_
case false_
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders false_ = fastReplaceFree v t false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders a✝.not_ = fastReplaceFree v t a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = fastReplaceFree v t (a✝¹.imp_ a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = fastReplaceFree v t (a✝¹.and_ a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = fastReplaceFree v t (a✝¹.or_ a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = fastReplaceFree v t (a✝¹.iff_ a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = fastReplaceFree v t (forall_ a✝¹ a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = fastReplaceFree v t (exists_ a✝¹ a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = fastReplaceFree v t (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | any_goals
simp only [replaceFreeAux]
simp only [fastReplaceFree] | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = fastReplaceFree v t (pred_const_ a✝¹ a✝)
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = fastReplaceFree v t (pred_var_ a✝¹ a✝)
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = fastReplaceFree v t (eq_ a✝¹ a✝)
case true_
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders true_ = fastReplaceFree v t true_
case false_
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders false_ = fastReplaceFree v t false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders a✝.not_ = fastReplaceFree v t a✝.not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = fastReplaceFree v t (a✝¹.imp_ a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = fastReplaceFree v t (a✝¹.and_ a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = fastReplaceFree v t (a✝¹.or_ a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = fastReplaceFree v t (a✝¹.iff_ a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = fastReplaceFree v t (forall_ a✝¹ a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = fastReplaceFree v t (exists_ a✝¹ a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = fastReplaceFree v t (def_ a✝¹ a✝) | case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ pred_const_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) =
pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ pred_var_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) =
pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ eq_ (if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) (if v = a✝ ∧ a✝ ∉ binders then t else a✝) =
eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝).not_ = (fastReplaceFree v t a✝).not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).imp_ (replaceFreeAux v t binders a✝) =
(fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).and_ (replaceFreeAux v t binders a✝) =
(fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).or_ (replaceFreeAux v t binders a✝) =
(fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).iff_ (replaceFreeAux v t binders a✝) =
(fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ forall_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) =
if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ exists_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) =
if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) =
def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr!
case _ x =>
constructor
case mp =>
tauto
case mpr =>
intro a1
subst a1
tauto | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) =
def_ X (List.map (fun x => if v = x then t else x) xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case eq_ x y =>
congr!
case _ | _ =>
constructor
case mp =>
tauto
case mpr =>
intro a1
subst a1
tauto | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) =
eq_ (if v = x then t else x) (if v = y then t else y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case not_ phi phi_ih =>
tauto | v t : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders phi).not_ = (fastReplaceFree v t phi).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto | v t : VarName
phi psi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) =
(fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
split_ifs
case pos c1 =>
congr! 1
apply replaceFreeAux_mem_binders
simp
right
exact c1
case neg c1 =>
congr! 1
apply phi_ih
simp
tauto | v t x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi
binders : Finset VarName
h1 : v ∉ binders
⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) =
if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | simp only [replaceFreeAux] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = fastReplaceFree v t (def_ a✝¹ a✝) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = fastReplaceFree v t (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | simp only [fastReplaceFree] | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = fastReplaceFree v t (def_ a✝¹ a✝) | case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) =
def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | congr! | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) =
def_ X (List.map (fun x => if v = x then t else x) xs) | case h.e'_2.h.e'_3.h.h₁.a
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x✝ : VarName
⊢ v = x✝ ∧ x✝ ∉ binders ↔ v = x✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case _ x =>
constructor
case mp =>
tauto
case mpr =>
intro a1
subst a1
tauto | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x ∧ x ∉ binders ↔ v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | constructor | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x ∧ x ∉ binders ↔ v = x | case mp
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x ∧ x ∉ binders → v = x
case mpr
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x → v = x ∧ x ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case mp =>
tauto | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x ∧ x ∉ binders → v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case mpr =>
intro a1
subst a1
tauto | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x → v = x ∧ x ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | tauto | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x ∧ x ∉ binders → v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | intro a1 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
⊢ v = x → v = x ∧ x ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
a1 : v = x
⊢ v = x ∧ x ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | subst a1 | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
x : VarName
a1 : v = x
⊢ v = x ∧ x ∉ binders | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = v ∧ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | tauto | v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = v ∧ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | congr! | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) =
eq_ (if v = x then t else x) (if v = y then t else y) | case h.e'_1.h₁.a
v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = x ∧ x ∉ binders ↔ v = x
case h.e'_2.h₁.a
v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y ∧ y ∉ binders ↔ v = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case _ | _ =>
constructor
case mp =>
tauto
case mpr =>
intro a1
subst a1
tauto | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y ∧ y ∉ binders ↔ v = y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | constructor | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y ∧ y ∉ binders ↔ v = y | case mp
v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y ∧ y ∉ binders → v = y
case mpr
v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y → v = y ∧ y ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case mp =>
tauto | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y ∧ y ∉ binders → v = y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | case mpr =>
intro a1
subst a1
tauto | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y → v = y ∧ y ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | tauto | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y ∧ y ∉ binders → v = y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | intro a1 | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = y → v = y ∧ y ∉ binders | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v = y
⊢ v = y ∧ y ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | subst a1 | v t x y : VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v = y
⊢ v = y ∧ y ∉ binders | v t x : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = v ∧ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree | [163, 1] | [216, 12] | tauto | v t x : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = v ∧ v ∉ binders | no goals |
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