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] | case _ c3 =>
apply h1
tauto | 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] | obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi | 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 [β c3] at c1 | 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 [Finset.mem_union] at c1 | 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 [isFreeIn_iff_mem_freeVarSet] at 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),... | 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] | obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο (Ο v) (Ο 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),... | 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 [Function.updateITE] at s2 | 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 [ite_self] at s2 | 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] | exfalso | 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] | apply c1 | 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] | left | 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),... | case h
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 (predVarFreeVar... | 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 s2 | case h
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 (predVarFreeVar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
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... |
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 h1 | 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),... | case a
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 (predVarFreeVar... | 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] | tauto | case a
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 (predVarFreeVar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
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... |
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 v a1 | case h.h2
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 (predVarFree... | case h.h2
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 (predVarFree... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
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 ... |
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 | case h.h2
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 (predVarFree... | case pos
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 (predVarFreeV... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
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 ... |
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 =>
simp only [Function.updateITE]
split_ifs
case _ c2 =>
obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi
simp only [β s1] at c2
obtain s2 := fresh_not_mem x c ((freeVarSet (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi)) βͺ (Fins... | 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] | case _ c1 =>
simp only [Finset.mem_union] at c1
push_neg at c1
cases c1
case _ c1_left c1_right =>
have s1 : Β¬ v = x
intro contra
apply c1_right
subst contra
exact a1
simp only [Function.updateITE]
simp only [if_neg s1]
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 [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] | split_ifs | 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),... | case pos
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 (predVarFreeV... | 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] | case _ c2 =>
obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi
simp only [β s1] at c2
obtain s2 := fresh_not_mem x c ((freeVarSet (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β... | 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] | case _ c2 =>
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] | obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi | 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 [β s1] at c2 | 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] | obtain s2 := fresh_not_mem x c ((freeVarSet (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi)) βͺ (Finset.biUnion (predVarSet phi) (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),... | 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 [β c2] at s2 | 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 [Finset.mem_union] at s2 | 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] | push_neg at s2 | 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] | cases s2 | 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),... | case intro
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 (predVarFre... | 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] | case _ s2_left s2_right =>
contradiction | 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] | contradiction | 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] | 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 [Finset.mem_union] at c1 | 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] | push_neg at c1 | 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] | cases c1 | 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),... | case intro
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 (predVarFre... | 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] | case _ c1_left c1_right =>
have s1 : Β¬ v = x
intro contra
apply c1_right
subst contra
exact a1
simp only [Function.updateITE]
simp only [if_neg s1]
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] | have s1 : Β¬ v = x | 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),... | case 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 (predVarFreeVa... | 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] | intro contra | case 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 (predVarFreeVa... | case 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 (predVarFreeVa... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 = ... |
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 c1_right | case 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 (predVarFreeVa... | case 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 (predVarFreeVa... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 = ... |
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] | subst contra | case 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 (predVarFreeVa... | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V''... | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 = ... |
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 a1 | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V''... | 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:
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
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
... |
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/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | have : isInv (id : Ξ± β Ξ±) id := by
apply And.intro
. intro x
simp
. intro y
simp | Ξ± : Sort u_1
β’ invertible id | Ξ± : Sort u_1
this : isInv id id
β’ invertible id | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Sort u_1
β’ invertible id
TACTIC:
|
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