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),... | 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),... | 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))... |
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),... | 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),... | 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))... |
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),... | 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))... |
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)... | 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)... | 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... |
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)... | 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... | 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... |
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 ... | 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... |
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 ... | 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... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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
... | 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 : Var... |
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
... | 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
... | 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 ... |
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
... | 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
... | 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 ... |
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
... | 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 ... | 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 ... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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
... | 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 : Var... |
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 ... | 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 : Var... |
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 ... | 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 : Var... |
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 ... | 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
... | 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 : Var... |
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
... | 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
... | 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 ... |
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
... | 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
... | 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),
... |
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
... | 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))... | 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),
... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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
... | 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 : Var... |
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))... | 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),
... |
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))... | 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),
... |
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))... | 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))... | 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),
... |
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))... | 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))... | 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),
... |
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))... | 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),
... |
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
... | 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
... | 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 ... |
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
... | 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 ... |
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
... | 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
... | 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 ... |
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
... | 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
... | 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 ... |
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
... | 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
... | 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 ... |
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
... | 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 ... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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 ... | 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 : Var... |
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 ... | 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
... | 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 : Var... |
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
... | 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
... | 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 ... |
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
... | 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 ... |
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
... | 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
... | 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 ... |
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
... | 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 ... |
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 : Fo... | 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)
... |
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_cons... | 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 ... | 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... |
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... | 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 : ... |
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 [Hol... | 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 ... | 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 : ... |
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 ... | 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... |
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... | 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 : Var... |
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 :
... | 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 ... |
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 ... | 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 :... | 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 : ... |
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... | 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... | 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 : ... |
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... | 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 : ... |
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.lengt... | 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... | 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 : ... |
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... | 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 ... | 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 : ... |
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 ... | 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)... |
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 ... | 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 ... | 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)... |
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 ... | 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 ... | 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)... |
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 ... | 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... | 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)... |
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... | 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... | 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 : ... |
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... | 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 : PredNa... | 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 :... |
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 : PredNa... | 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 : PredNa... | 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)
... |
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 : PredNa... | 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 : PredNa... | 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)
... |
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 : PredNa... | 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_va... | 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)
... |
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_va... | 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_va... | 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)... |
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_va... | 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_va... | 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)... |
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_va... | 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_va... | 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)... |
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_va... | 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... | 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)... |
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... | 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... | 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 : ... |
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... | 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 : ... |
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... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 ... | 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 : ... |
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 ... | 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 : ... |
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 ... | 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... |
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... | 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 : Var... |
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 :
... | 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 :
... | 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 ... |
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 :
... | 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... | 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 ... |
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... | 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_va... |
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