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] | intro a a2 | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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_biUnion, Finset.mem_union] | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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 Exists.intro a | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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 psi_ih V V' Ο | case a.h.e'_2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVarFr... | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β 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 x a1 | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi 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] | apply h1 | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | case a.h.e'_2.a.h1.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi 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] | right | case a.h.e'_2.a.h1.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_2.a.h1.a.h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pr... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h1.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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 a.h.e'_2.a.h1.a.h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pr... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h1.a.h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x ph... |
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 a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi 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] | apply h2 | case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (predVa... | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi 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 [Finset.mem_biUnion, Finset.mem_union] at a1 | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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 Exists.elim a1 | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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 a a2 | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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_biUnion, Finset.mem_union] | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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 Exists.intro a | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_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 (pred... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi ... |
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
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 [predVarSet] at 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 (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 [subAux] | 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 [I'] | 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 [Interpretation.usingPred] | 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 [Holds] | 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] | first | apply forall_congr' | apply exists_congr | 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] | intro d | 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... | 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:
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 ih | 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... | case h.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 (predVarFree... | 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 forall_congr' | 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] | apply exists_congr | 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] | intro v a1 | case h.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 (predVarFree... | case h.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 (predVarFree... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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 ... |
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.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 (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.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 ... |
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 c3 =>
rfl
case _ c2 c3 =>
contradiction
case _ c2 c3 =>
obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β c3] at 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),... | 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 c3 =>
rfl | 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 c3 =>
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] | case _ c2 c3 =>
obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β c3] at s1
simp only [Finset.mem_union] at s1
simp only [isFreeIn_iff_mem_freeVarSet] at a1
obtain s2 := Finset.mem_image_of_mem (Fu... | 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 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] | rfl | 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] | obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Ο x x) (freeVarSet 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 c3] at 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] | simp only [Finset.mem_union] at 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] | 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 x x) 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 [if_neg 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] | 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 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] | 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] | 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] | by_cases c2 : 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 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] | simp only [if_pos c2] | 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... | 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 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 =... |
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 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 =... |
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 c2] | case neg
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... | case neg
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 neg
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 neg
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... | 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 neg
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 _ c3 =>
obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi
simp only [s1] at c1
simp only [β c3] at c1
simp only [Finset.mem_union] at c1
simp only [isFreeIn_iff_mem_freeVarSet] at a1
obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο (Ο 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 : 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 _ 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
... |
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