url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [Function.updateITE]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = if v = x then d else V v
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [if_neg s1]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = if v = x then d else V v
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = V v
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = if v = x then d else V v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
exact h2 v a1
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = V v
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = V v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [subAux]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (subAux c Ο Ο (def_ X xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (subAux c Ο Ο (def_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
induction E generalizing V V' Ο
|
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs))
|
case nil
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
case cons
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' tailβ Ο) V' tailβ (def_ X xs) β Holds D I V tailβ (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (headβ :: tailβ) Ο) V' (headβ :: tailβ) (def_ X xs) β
Holds D I V (headβ :: tailβ) (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
case nil =>
simp only [Holds]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [isFreeIn] at h1
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [Holds]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
have s1 : (List.map V' xs) = (List.map (V β Ο) xs)
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [List.map_eq_map_iff]
|
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
intro x a1
|
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
exact h1 x a1
|
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [s1]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
clear s1
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
split_ifs
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
|
case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : X = E_hd.name β§ xs.length = E_hd.args.length
hβ : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : X = E_hd.name β§ xs.length = E_hd.args.length
hβ : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
hβ : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
hβ : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
case _ c1 c2 =>
simp only [List.length_map] at c2
contradiction
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
case _ c1 c2 =>
simp only [List.length_map] at c2
contradiction
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
have s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
case s2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
apply Holds_coincide_Var
|
case s2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
intro x a1
|
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) x =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
apply Function.updateListITE_map_mem_ext
|
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) x =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
case s2.h1.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y
case s2.h1.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ E_hd.args.length = xs.length
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ x β E_hd.args
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) x =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [β s2]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
clear s2
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
apply Holds_coincide_PredVar
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = I.pred_const_
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length E_hd.q β ((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp
|
case s2.h1.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
tauto
|
case s2.h1.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ E_hd.args.length = xs.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ E_hd.args.length = xs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [isFreeIn_iff_mem_freeVarSet] at a1
|
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ x β E_hd.args
|
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ x β E_hd.args
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [β List.mem_toFinset]
|
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args
|
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args.toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
apply Finset.mem_of_subset E_hd.h1 a1
|
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args.toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [I']
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = I.pred_const_
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) (E_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
I.pred_const_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = I.pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [Interpretation.usingPred]
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) (E_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
I.pred_const_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) (E_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
I.pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
intro P ds a1
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length E_hd.q β ((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds)
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length E_hd.q
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length E_hd.q β ((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [predVarOccursIn_iff_mem_predVarSet] at a1
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length E_hd.q
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β E_hd.q.predVarSet
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length E_hd.q
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [E_hd.h2] at a1
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β E_hd.q.predVarSet
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β β
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β E_hd.q.predVarSet
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp at a1
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β β
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β β
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [List.length_map] at c2
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
contradiction
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [List.length_map] at c2
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
contradiction
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
obtain s2 := E_ih V V' Ο
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [isFreeIn] at s2
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β x β xs, V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
specialize s2 h1 h2
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β x β xs, V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β x β xs, V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [β s2]
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
apply Holds_coincide_PredVar
|
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs)
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = (I' D I V'' E_tl Ο).pred_const_
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (def_ X xs) β
((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [I']
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = (I' D I V'' E_tl Ο).pred_const_
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) (E_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
(Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E_tl ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = (I' D I V'' E_tl Ο).pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [Interpretation.usingPred]
|
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) (E_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
(Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E_tl ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) (E_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
(Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E_tl ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
intro P ds a1
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (def_ X xs) β
((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds)
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (def_ X xs) β
((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
|
[123, 1]
|
[434, 44]
|
simp only [predVarOccursIn] at a1
|
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem
|
[437, 1]
|
[449, 9]
|
apply substitution_theorem_aux
|
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ Holds D (I' D I V E Ο) V E F β Holds D I V E (sub c Ο F)
|
case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β (x : VarName), isFreeIn x F β V x = V (id x)
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β x β F.predVarSet.biUnion (predVarFreeVarSet Ο), V x = V x
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ Holds D (I' D I V E Ο) V E F β Holds D I V E (sub c Ο F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem
|
[437, 1]
|
[449, 9]
|
simp
|
case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β (x : VarName), isFreeIn x F β V x = V (id x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β (x : VarName), isFreeIn x F β V x = V (id x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem
|
[437, 1]
|
[449, 9]
|
simp
|
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β x β F.predVarSet.biUnion (predVarFreeVarSet Ο), V x = V x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β x β F.predVarSet.biUnion (predVarFreeVarSet Ο), V x = V x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_is_valid
|
[452, 1]
|
[464, 11]
|
simp only [IsValid] at h1
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : F.IsValid
β’ (sub c Ο F).IsValid
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub c Ο F).IsValid
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : F.IsValid
β’ (sub c Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_is_valid
|
[452, 1]
|
[464, 11]
|
simp only [IsValid]
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub c Ο F).IsValid
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο F)
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub c Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_is_valid
|
[452, 1]
|
[464, 11]
|
intro D I V E
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο F)
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (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
β’ Holds D I V E (sub c Ο F)
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_is_valid
|
[452, 1]
|
[464, 11]
|
simp only [β substitution_theorem]
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (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
β’ Holds D I V E (sub c Ο F)
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (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
β’ Holds D (I' D I V E Ο) V E F
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (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
β’ Holds D I V E (sub c Ο F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean
|
FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_is_valid
|
[452, 1]
|
[464, 11]
|
apply h1
|
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (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
β’ Holds D (I' D I V E Ο) V E F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (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
β’ Holds D (I' D I V E Ο) V E F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
induction h1 generalizing V
|
D : Type
I J : Interpretation D
V : VarAssignment D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h1 : IsSub P zs H A B
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
β’ Holds D I V E B β Holds D J V E A
|
case pred_const_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
Xβ : PredName
xsβ : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_const_ Xβ xsβ) β Holds D J V E (pred_const_ Xβ xsβ)
case pred_not_occurs_in
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
Xβ : PredName
xsβ : List VarName
aβ : Β¬(Xβ = P β§ xsβ.length = zs.length)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_var_ Xβ xsβ) β Holds D J V E (pred_var_ Xβ xsβ)
case pred_occurs_in
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
Xβ : PredName
tsβ : List VarName
aβΒΉ : Xβ = P β§ tsβ.length = zs.length
aβ : Var.All.Rec.admits (Function.updateListITE id zs tsβ) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs tsβ) H) β Holds D J V E (pred_var_ P tsβ)
case eq_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
xβ yβ : VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (eq_ xβ yβ) β Holds D J V E (eq_ xβ yβ)
case true_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E true_ β Holds D J V E true_
case false_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E false_ β Holds D J V E false_
case not_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
phiβ phi'β : Formula
aβ : IsSub P zs H phiβ phi'β
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E phi'β.not_ β Holds D J V E phiβ.not_
case imp_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSub P zs H phiβ phi'β
aβ : IsSub P zs H psiβ psi'β
a_ihβΒΉ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E psi'β β Holds D J V E psiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (phi'β.imp_ psi'β) β Holds D J V E (phiβ.imp_ psiβ)
case and_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSub P zs H phiβ phi'β
aβ : IsSub P zs H psiβ psi'β
a_ihβΒΉ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E psi'β β Holds D J V E psiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (phi'β.and_ psi'β) β Holds D J V E (phiβ.and_ psiβ)
case or_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSub P zs H phiβ phi'β
aβ : IsSub P zs H psiβ psi'β
a_ihβΒΉ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E psi'β β Holds D J V E psiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (phi'β.or_ psi'β) β Holds D J V E (phiβ.or_ psiβ)
case iff_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSub P zs H phiβ phi'β
aβ : IsSub P zs H psiβ psi'β
a_ihβΒΉ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E psi'β β Holds D J V E psiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (phi'β.iff_ psi'β) β Holds D J V E (phiβ.iff_ psiβ)
case forall_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
xβ : VarName
phiβ phi'β : Formula
aβΒΉ : Β¬isFreeIn xβ H
aβ : IsSub P zs H phiβ phi'β
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (forall_ xβ phi'β) β Holds D J V E (forall_ xβ phiβ)
case exists_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
xβ : VarName
phiβ phi'β : Formula
aβΒΉ : Β¬isFreeIn xβ H
aβ : IsSub P zs H phiβ phi'β
a_ihβ :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E phi'β β Holds D J V E phiβ)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (exists_ xβ phi'β) β Holds D J V E (exists_ xβ phiβ)
case def_
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
Xβ : DefName
xsβ : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (def_ Xβ xsβ) β Holds D J V E (def_ Xβ xsβ)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
V : VarAssignment D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h1 : IsSub P zs H A B
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
β’ Holds D I V E B β Holds D J V E A
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case pred_const_ h1_X h1_ts =>
simp only [Holds]
simp only [h3_const]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_const_ h1_X h1_ts) β Holds D J V E (pred_const_ h1_X h1_ts)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_const_ h1_X h1_ts) β Holds D J V E (pred_const_ h1_X h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case pred_occurs_in h1_X h1_ts h1_1 h1_2 =>
obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id zs h1_ts) H h1_2
obtain s2 := Function.updateListITE_comp id V zs h1_ts
simp only [s2] at s1
simp at s1
specialize h2 h1_X (List.map V h1_ts)
simp only [s1] at h2
simp only [Holds]
apply h2
simp
exact h1_1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case eq_ h1_x h1_y =>
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x h1_y : VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (eq_ h1_x h1_y) β Holds D J V E (eq_ h1_x h1_y)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x h1_y : VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (eq_ h1_x h1_y) β Holds D J V E (eq_ h1_x h1_y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case true_ | false_ =>
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E false_ β Holds D J V E false_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E false_ β Holds D J V E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case not_ h1_phi h1_phi' _ h1_ih =>
simp only [Holds]
congr! 1
exact h1_ih V h2
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi'.not_ β Holds D J V E h1_phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi'.not_ β Holds D J V E h1_phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case
forall_ h1_x h1_phi h1_phi' h1_1 _ h1_ih
| exists_ h1_x h1_phi h1_phi' h1_1 _ h1_ih =>
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply h1_ih
intro Q ds a1
specialize h2 Q ds a1
have s1 :
Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H :=
by
apply Holds_coincide_Var
intro v a1
apply Function.updateListITE_updateIte
intro contra
subst contra
contradiction
simp only [h2] at s1
exact s1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (exists_ h1_x h1_phi') β Holds D J V E (exists_ h1_x h1_phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (exists_ h1_x h1_phi') β Holds D J V E (exists_ h1_x h1_phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_const_ h1_X h1_ts) β Holds D J V E (pred_const_ h1_X h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ I.pred_const_ h1_X (List.map V h1_ts) β J.pred_const_ h1_X (List.map V h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_const_ h1_X h1_ts) β Holds D J V E (pred_const_ h1_X h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [h3_const]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ I.pred_const_ h1_X (List.map V h1_ts) β J.pred_const_ h1_X (List.map V h1_ts)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ I.pred_const_ h1_X (List.map V h1_ts) β J.pred_const_ h1_X (List.map V h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp at h1_1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : Β¬(h1_X = P β§ h1_ts.length = zs.length)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_var_ h1_X h1_ts) β Holds D J V E (pred_var_ h1_X h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ Holds D I V E (pred_var_ h1_X h1_ts) β Holds D J V E (pred_var_ h1_X h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : Β¬(h1_X = P β§ h1_ts.length = zs.length)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (pred_var_ h1_X h1_ts) β Holds D J V E (pred_var_ h1_X h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply Holds_coincide_PredVar
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ Holds D I V E (pred_var_ h1_X h1_ts) β Holds D J V E (pred_var_ h1_X h1_ts)
|
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ I.pred_const_ = J.pred_const_
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β (I.pred_var_ P ds β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ Holds D I V E (pred_var_ h1_X h1_ts) β Holds D J V E (pred_var_ h1_X h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h3_const
|
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ I.pred_const_ = J.pred_const_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ I.pred_const_ = J.pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro X ds a1
|
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β (I.pred_var_ P ds β J.pred_var_ P ds)
|
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts)
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β (I.pred_var_ P ds β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [predVarOccursIn] at a1
|
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts)
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1 : X = h1_X β§ ds.length = h1_ts.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts)
β’ I.pred_var_ X ds β J.pred_var_ X ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
cases a1
|
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1 : X = h1_X β§ ds.length = h1_ts.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
case h2.intro
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
leftβ : X = h1_X
rightβ : ds.length = h1_ts.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1 : X = h1_X β§ ds.length = h1_ts.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
subst a1_left
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1_left : X = h1_X
a1_right : ds.length = h1_ts.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
h1_1 : h1_X = P β Β¬h1_ts.length = zs.length
X : PredName
ds : List D
a1_left : X = h1_X
a1_right : ds.length = h1_ts.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply h3_var
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ Β¬(X = P β§ ds.length = zs.length)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ I.pred_var_ X ds β J.pred_var_ X ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ Β¬(X = P β§ ds.length = zs.length)
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ X = P β Β¬ds.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ Β¬(X = P β§ ds.length = zs.length)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro a2
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ X = P β Β¬ds.length = zs.length
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
a2 : X = P
β’ Β¬ds.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
β’ X = P β Β¬ds.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
subst a2
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
a2 : X = P
β’ Β¬ds.length = zs.length
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : X = X β Β¬h1_ts.length = zs.length
β’ Β¬ds.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_ts : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h1_1 : X = P β Β¬h1_ts.length = zs.length
a2 : X = P
β’ Β¬ds.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp at h1_1
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : X = X β Β¬h1_ts.length = zs.length
β’ Β¬ds.length = zs.length
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
β’ Β¬ds.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : X = X β Β¬h1_ts.length = zs.length
β’ Β¬ds.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro contra
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
β’ Β¬ds.length = zs.length
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ False
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
β’ Β¬ds.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply h1_1
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ False
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ h1_ts.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ False
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
trans List.length ds
|
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ h1_ts.length = zs.length
|
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ h1_ts.length = ds.length
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ ds.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ h1_ts.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [eq_comm]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ h1_ts.length = ds.length
|
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ ds.length = h1_ts.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ h1_ts.length = ds.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact a1_right
|
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ ds.length = h1_ts.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ ds.length = h1_ts.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact contra
|
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ ds.length = zs.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h1_ts : List VarName
V : VarAssignment D
X : PredName
ds : List D
a1_right : ds.length = h1_ts.length
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = X β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h2 :
β (Q : PredName) (ds : List D),
Q = X β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ X ds)
h1_1 : Β¬h1_ts.length = zs.length
contra : ds.length = zs.length
β’ ds.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id zs h1_ts) H h1_2
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s1 :
Holds D I (V β Function.updateListITE id zs h1_ts) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
obtain s2 := Function.updateListITE_comp id V zs h1_ts
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s1 :
Holds D I (V β Function.updateListITE id zs h1_ts) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s1 :
Holds D I (V β Function.updateListITE id zs h1_ts) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s1 :
Holds D I (V β Function.updateListITE id zs h1_ts) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [s2] at s1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s1 :
Holds D I (V β Function.updateListITE id zs h1_ts) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE (V β id) zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s1 :
Holds D I (V β Function.updateListITE id zs h1_ts) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp at s1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE (V β id) zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE (V β id) zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
specialize h2 h1_X (List.map V h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [s1] at h2
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β J.pred_var_ P (List.map V h1_ts)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply h2
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β J.pred_var_ P (List.map V h1_ts)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ (List.map V h1_ts).length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β J.pred_var_ P (List.map V h1_ts)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ (List.map V h1_ts).length = zs.length
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ h1_ts.length = zs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ (List.map V h1_ts).length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h1_1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ h1_ts.length = zs.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ h1_ts.length = zs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x h1_y : VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (eq_ h1_x h1_y) β Holds D J V E (eq_ h1_x h1_y)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x h1_y : VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (eq_ h1_x h1_y) β Holds D J V E (eq_ h1_x h1_y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E false_ β Holds D J V E false_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E false_ β Holds D J V E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi'.not_ β Holds D J V E h1_phi.not_
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Β¬Holds D I V E h1_phi' β Β¬Holds D J V E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi'.not_ β Holds D J V E h1_phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
congr! 1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Β¬Holds D I V E h1_phi' β Β¬Holds D J V E h1_phi
|
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi' β Holds D J V E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Β¬Holds D I V E h1_phi' β Β¬Holds D J V E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h1_ih V h2
|
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi' β Holds D J V E h1_phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi' β Holds D J V E h1_phi
TACTIC:
|
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