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/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
simp only [fastAdmitsAux] at h1
|
case def_
v u : VarName
T : Finset VarName
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)
|
case def_
v u : VarName
T : Finset VarName
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S ∪ T
⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
simp only [fastAdmitsAux]
|
case def_
v u : VarName
T : Finset VarName
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S ∪ T
⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)
|
case def_
v u : VarName
T : Finset VarName
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S ∪ T
⊢ v ∈ a✝ → u ∉ S
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S ∪ T
⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
simp at h1
|
v u : VarName
T : Finset VarName
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → u ∉ S ∪ T
⊢ v ∈ xs → u ∉ S
|
v u : VarName
T : Finset VarName
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → u ∉ S ∧ u ∉ T
⊢ v ∈ xs → u ∉ S
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → u ∉ S ∪ T
⊢ v ∈ xs → u ∉ S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
tauto
|
v u : VarName
T : Finset VarName
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → u ∉ S ∧ u ∉ T
⊢ v ∈ xs → u ∉ S
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → u ∉ S ∧ u ∉ T
⊢ v ∈ xs → u ∉ S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
simp at h1
|
v u : VarName
T : Finset VarName
x y : VarName
S : Finset VarName
h1 : v = x ∨ v = y → u ∉ S ∪ T
⊢ v = x ∨ v = y → u ∉ S
|
v u : VarName
T : Finset VarName
x y : VarName
S : Finset VarName
h1 : v = x ∨ v = y → u ∉ S ∧ u ∉ T
⊢ v = x ∨ v = y → u ∉ S
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
x y : VarName
S : Finset VarName
h1 : v = x ∨ v = y → u ∉ S ∪ T
⊢ v = x ∨ v = y → u ∉ S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
tauto
|
v u : VarName
T : Finset VarName
x y : VarName
S : Finset VarName
h1 : v = x ∨ v = y → u ∉ S ∧ u ∉ T
⊢ v = x ∨ v = y → u ∉ S
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
x y : VarName
S : Finset VarName
h1 : v = x ∨ v = y → u ∉ S ∧ u ∉ T
⊢ v = x ∨ v = y → u ∉ S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
tauto
|
v u : VarName
T : Finset VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) phi
⊢ fastAdmitsAux v u S phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) phi
⊢ fastAdmitsAux v u S phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
tauto
|
v u : VarName
T : Finset VarName
phi psi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
psi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) psi → fastAdmitsAux v u S psi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) phi ∧ fastAdmitsAux v u (S ∪ T) psi
⊢ fastAdmitsAux v u S phi ∧ fastAdmitsAux v u S psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
phi psi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
psi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) psi → fastAdmitsAux v u S psi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) phi ∧ fastAdmitsAux v u (S ∪ T) psi
⊢ fastAdmitsAux v u S phi ∧ fastAdmitsAux v u S psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
simp only [Finset.union_right_comm S T {x}] at h1
|
v u : VarName
T : Finset VarName
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
|
v u : VarName
T : Finset VarName
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders
|
[544, 1]
|
[575, 10]
|
tauto
|
v u : VarName
T : Finset VarName
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
induction F generalizing binders
|
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
⊢ u ∉ binders
case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
h2 : isFreeIn v true_
⊢ u ∉ binders
case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
h2 : isFreeIn v false_
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
all_goals
simp only [fastAdmitsAux] at h1
simp only [isFreeIn] at h2
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
⊢ u ∉ binders
case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
h2 : isFreeIn v true_
⊢ u ∉ binders
case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
h2 : isFreeIn v false_
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
⊢ u ∉ binders
case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
h2 : isFreeIn v true_
⊢ u ∉ binders
case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
h2 : isFreeIn v false_
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
all_goals
tauto
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
simp only [fastAdmitsAux] at h1
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
simp only [isFreeIn] at h2
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
cases h2
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h2 : ¬v = x ∧ isFreeIn v phi
⊢ u ∉ binders
|
case intro
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
left✝ : ¬v = x
right✝ : isFreeIn v phi
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h2 : ¬v = x ∧ isFreeIn v phi
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
cases h1
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ u ∉ binders
|
case inl
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h✝ : v = x
⊢ u ∉ binders
case inr
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h✝ : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
case inl h1 =>
contradiction
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : v = x
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : v = x
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
contradiction
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : v = x
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : v = x
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
apply phi_ih
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ u ∉ binders
|
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u binders phi
case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ isFreeIn v phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
exact fastAdmitsAux_del_binders phi v u binders {x} h1
|
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u binders phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u binders phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
exact h2_right
|
case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ isFreeIn v phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ isFreeIn v phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
|
[579, 1]
|
[603, 10]
|
tauto
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_mem_binders
|
[606, 1]
|
[615, 51]
|
contrapose! h2
|
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : u ∈ binders
⊢ ¬isFreeIn v F
|
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : u ∈ binders
⊢ ¬isFreeIn v F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_mem_binders
|
[606, 1]
|
[615, 51]
|
exact fastAdmitsAux_isFreeIn F v u binders h1 h2
|
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
induction F generalizing binders
|
F : Formula
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders F
⊢ toIsBoundAux binders F = toIsBoundAux binders (fastReplaceFree v u F)
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ toIsBoundAux binders (pred_const_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_const_ a✝¹ a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ toIsBoundAux binders (pred_var_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_var_ a✝¹ a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
⊢ toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (eq_ a✝¹ a✝))
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders true_
⊢ toIsBoundAux binders true_ = toIsBoundAux binders (fastReplaceFree v u true_)
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders false_
⊢ toIsBoundAux binders false_ = toIsBoundAux binders (fastReplaceFree v u false_)
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝.not_
⊢ toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝.not_)
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
⊢ toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.imp_ a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
⊢ toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.and_ a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
⊢ toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.or_ a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
⊢ toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.iff_ a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
⊢ toIsBoundAux binders (forall_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (forall_ a✝¹ a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
⊢ toIsBoundAux binders (exists_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (exists_ a✝¹ a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders F
⊢ toIsBoundAux binders F = toIsBoundAux binders (fastReplaceFree v u F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
all_goals
simp only [fastAdmitsAux] at h2
simp only [fastReplaceFree]
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ toIsBoundAux binders (pred_const_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_const_ a✝¹ a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ toIsBoundAux binders (pred_var_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_var_ a✝¹ a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
⊢ toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (eq_ a✝¹ a✝))
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders true_
⊢ toIsBoundAux binders true_ = toIsBoundAux binders (fastReplaceFree v u true_)
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders false_
⊢ toIsBoundAux binders false_ = toIsBoundAux binders (fastReplaceFree v u false_)
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝.not_
⊢ toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝.not_)
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
⊢ toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.imp_ a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
⊢ toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.and_ a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
⊢ toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.or_ a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
⊢ toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.iff_ a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
⊢ toIsBoundAux binders (forall_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (forall_ a✝¹ a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
⊢ toIsBoundAux binders (exists_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (exists_ a✝¹ a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (pred_const_ a✝¹ a✝) =
toIsBoundAux binders (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (pred_var_ a✝¹ a✝) =
toIsBoundAux binders (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ v = a✝ → u ∉ binders
⊢ toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝))
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝).not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).imp_ (fastReplaceFree v u a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).and_ (fastReplaceFree v u a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).or_ (fastReplaceFree v u a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).iff_ (fastReplaceFree v u a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ toIsBoundAux binders (forall_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ toIsBoundAux binders (exists_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ toIsBoundAux binders (pred_const_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_const_ a✝¹ a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ toIsBoundAux binders (pred_var_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_var_ a✝¹ a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
⊢ toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (eq_ a✝¹ a✝))
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders true_
⊢ toIsBoundAux binders true_ = toIsBoundAux binders (fastReplaceFree v u true_)
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders false_
⊢ toIsBoundAux binders false_ = toIsBoundAux binders (fastReplaceFree v u false_)
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝.not_
⊢ toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝.not_)
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
⊢ toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.imp_ a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
⊢ toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.and_ a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
⊢ toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.or_ a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
⊢ toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.iff_ a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
⊢ toIsBoundAux binders (forall_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (forall_ a✝¹ a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
⊢ toIsBoundAux binders (exists_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (exists_ a✝¹ a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
any_goals
simp only [toIsBoundAux]
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (pred_const_ a✝¹ a✝) =
toIsBoundAux binders (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (pred_var_ a✝¹ a✝) =
toIsBoundAux binders (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ v = a✝ → u ∉ binders
⊢ toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝))
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝).not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).imp_ (fastReplaceFree v u a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).and_ (fastReplaceFree v u a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).or_ (fastReplaceFree v u a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).iff_ (fastReplaceFree v u a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ toIsBoundAux binders (forall_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ toIsBoundAux binders (exists_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ BoolFormula.pred_const_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.pred_const_ a✝¹
(List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ BoolFormula.pred_var_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.pred_var_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ v = a✝ → u ∉ binders
⊢ BoolFormula.eq_ (decide (a✝¹ ∈ binders)) (decide (a✝ ∈ binders)) =
BoolFormula.eq_ (decide ((if v = a✝¹ then u else a✝¹) ∈ binders)) (decide ((if v = a✝ then u else a✝) ∈ binders))
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝
⊢ (toIsBoundAux binders a✝).not_ = (toIsBoundAux binders (fastReplaceFree v u a✝)).not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ (toIsBoundAux binders a✝¹).imp_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).imp_ (toIsBoundAux binders (fastReplaceFree v u a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ (toIsBoundAux binders a✝¹).and_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).and_ (toIsBoundAux binders (fastReplaceFree v u a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ (toIsBoundAux binders a✝¹).or_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).or_ (toIsBoundAux binders (fastReplaceFree v u a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ (toIsBoundAux binders a✝¹).iff_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).iff_ (toIsBoundAux binders (fastReplaceFree v u a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {a✝¹}) a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {a✝¹}) a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (pred_const_ a✝¹ a✝) =
toIsBoundAux binders (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (pred_var_ a✝¹ a✝) =
toIsBoundAux binders (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ v = a✝ → u ∉ binders
⊢ toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝))
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝).not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).imp_ (fastReplaceFree v u a✝))
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).and_ (fastReplaceFree v u a✝))
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).or_ (fastReplaceFree v u a✝))
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders a✝¹ → toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹)
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).iff_ (fastReplaceFree v u a✝))
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ toIsBoundAux binders (forall_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → fastAdmitsAux v u binders a✝ → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ toIsBoundAux binders (exists_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case eq_ x y =>
simp
constructor
case left | right =>
split_ifs
case pos c1 =>
subst c1
tauto
case neg c1 =>
rfl
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case not_ phi phi_ih =>
tauto
|
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi
⊢ (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi
⊢ (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp
tauto
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ (toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ (toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp only [fastAdmitsAux] at h2
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp only [fastReplaceFree]
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp only [toIsBoundAux]
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ List.map (fun v => decide (v ∈ binders)) xs =
List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp only [List.map_eq_map_iff]
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ List.map (fun v => decide (v ∈ binders)) xs =
List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ List.map (fun v => decide (v ∈ binders)) xs =
List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
intro x a1
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
by_cases c1 : v = x
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
|
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
case neg
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
subst c1
|
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
|
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ (if v = v then u else v) ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ (if v = v then u else v) ∈ binders
|
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ u ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ (if v = v then u else v) ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
tauto
|
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ u ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ u ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp only [if_neg c1]
|
case neg
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
constructor
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
|
case left
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
case right
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case left | right =>
split_ifs
case pos c1 =>
subst c1
tauto
case neg c1 =>
rfl
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
split_ifs
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
|
case pos
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
h✝ : v = y
⊢ y ∈ binders ↔ u ∈ binders
case neg
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
h✝ : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case pos c1 =>
subst c1
tauto
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : v = y
⊢ y ∈ binders ↔ u ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : v = y
⊢ y ∈ binders ↔ u ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case neg c1 =>
rfl
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
subst c1
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : v = y
⊢ y ∈ binders ↔ u ∈ binders
|
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = v → u ∉ binders
⊢ v ∈ binders ↔ u ∈ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : v = y
⊢ y ∈ binders ↔ u ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
tauto
|
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = v → u ∉ binders
⊢ v ∈ binders ↔ u ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = v → u ∉ binders
⊢ v ∈ binders ↔ u ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
rfl
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
tauto
|
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi
⊢ (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi
⊢ (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ (toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ (toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
tauto
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
split_ifs
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v u phi))
|
case pos
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h✝ : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
case neg
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h✝ : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v u phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
case pos c1 =>
rfl
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
rfl
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp only [toIsBoundAux]
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
apply phi_ih
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
|
case h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
case h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
simp
|
case h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
|
case h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
tauto
|
case h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
|
[619, 1]
|
[674, 14]
|
tauto
|
case h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
induction F generalizing binders
|
F : Formula
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders F = toIsBoundAux binders (fastReplaceFree v u F)
⊢ fastAdmitsAux v u binders F
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (pred_const_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_const_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (pred_var_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_var_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (eq_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders true_ = toIsBoundAux binders (fastReplaceFree v u true_)
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders false_ = toIsBoundAux binders (fastReplaceFree v u false_)
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝.not_)
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.imp_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.and_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.or_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.iff_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (forall_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (forall_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (exists_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (exists_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders F = toIsBoundAux binders (fastReplaceFree v u F)
⊢ fastAdmitsAux v u binders F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
all_goals
simp only [fastReplaceFree] at h2
simp only [fastAdmitsAux]
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (pred_const_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_const_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (pred_var_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_var_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (eq_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders true_ = toIsBoundAux binders (fastReplaceFree v u true_)
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders false_ = toIsBoundAux binders (fastReplaceFree v u false_)
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝.not_)
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.imp_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.and_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.or_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.iff_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (forall_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (forall_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (exists_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (exists_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (pred_const_ a✝¹ a✝) =
toIsBoundAux binders (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (pred_var_ a✝¹ a✝) =
toIsBoundAux binders (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝))
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝).not_
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).imp_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).and_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).or_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).iff_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (forall_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (exists_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (pred_const_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_const_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (pred_var_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (pred_var_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (eq_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders true_ = toIsBoundAux binders (fastReplaceFree v u true_)
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders false_ = toIsBoundAux binders (fastReplaceFree v u false_)
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝.not_)
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.imp_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.and_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.or_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders (fastReplaceFree v u (a✝¹.iff_ a✝))
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (forall_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (forall_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (exists_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (exists_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
any_goals
simp only [toIsBoundAux] at h2
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (pred_const_ a✝¹ a✝) =
toIsBoundAux binders (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (pred_var_ a✝¹ a✝) =
toIsBoundAux binders (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝))
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝).not_
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).imp_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).and_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).or_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).iff_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (forall_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (exists_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.pred_const_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.pred_const_ a✝¹
(List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.pred_var_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.pred_var_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.eq_ (decide (a✝¹ ∈ binders)) (decide (a✝ ∈ binders)) =
BoolFormula.eq_ (decide ((if v = a✝¹ then u else a✝¹) ∈ binders)) (decide ((if v = a✝ then u else a✝) ∈ binders))
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : (toIsBoundAux binders a✝).not_ = (toIsBoundAux binders (fastReplaceFree v u a✝)).not_
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders a✝¹).imp_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).imp_ (toIsBoundAux binders (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders a✝¹).and_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).and_ (toIsBoundAux binders (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders a✝¹).or_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).or_ (toIsBoundAux binders (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders a✝¹).iff_ (toIsBoundAux binders a✝) =
(toIsBoundAux binders (fastReplaceFree v u a✝¹)).iff_ (toIsBoundAux binders (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {a✝¹}) a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {a✝¹}) a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (pred_const_ a✝¹ a✝) =
toIsBoundAux binders (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (pred_var_ a✝¹ a✝) =
toIsBoundAux binders (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (eq_ a✝¹ a✝) = toIsBoundAux binders (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝))
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders a✝.not_ = toIsBoundAux binders (fastReplaceFree v u a✝).not_
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.imp_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).imp_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.and_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).and_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.or_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).or_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders a✝¹ = toIsBoundAux binders (fastReplaceFree v u a✝¹) → fastAdmitsAux v u binders a✝¹
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (a✝¹.iff_ a✝) = toIsBoundAux binders ((fastReplaceFree v u a✝¹).iff_ (fastReplaceFree v u a✝))
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (forall_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (binders : Finset VarName),
v ∉ binders → toIsBoundAux binders a✝ = toIsBoundAux binders (fastReplaceFree v u a✝) → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (exists_ a✝¹ a✝) =
toIsBoundAux binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v u a✝))
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp at h2
simp only [List.map_eq_map_iff] at h2
intro a1
specialize h2 v a1
simp at h2
tauto
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
⊢ v ∈ xs → u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
⊢ v ∈ xs → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case eq_ x y =>
simp at h2
cases h2
case intro h2_left h2_right =>
intros a1
cases a1
case inl a1 =>
subst a1
simp at h2_left
tauto
case inr a1 =>
subst a1
simp at h2_right
tauto
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
⊢ v = x ∨ v = y → u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
⊢ v = x ∨ v = y → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case not_ phi phi_ih =>
simp at h2
exact phi_ih binders h1 h2
|
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
⊢ fastAdmitsAux v u binders phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
⊢ fastAdmitsAux v u binders phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp at h2
tauto
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp only [fastReplaceFree] at h2
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp only [fastAdmitsAux]
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp only [toIsBoundAux] at h2
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
⊢ v ∈ a✝ → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
⊢ v ∈ xs → u ∉ binders
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
List.map (fun v => decide (v ∈ binders)) xs =
List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
⊢ v ∈ xs → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
⊢ v ∈ xs → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp only [List.map_eq_map_iff] at h2
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
List.map (fun v => decide (v ∈ binders)) xs =
List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
⊢ v ∈ xs → u ∉ binders
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
⊢ v ∈ xs → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
List.map (fun v => decide (v ∈ binders)) xs =
List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
⊢ v ∈ xs → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
intro a1
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
⊢ v ∈ xs → u ∉ binders
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
a1 : v ∈ xs
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
⊢ v ∈ xs → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
specialize h2 v a1
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
a1 : v ∈ xs
⊢ u ∉ binders
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v ∈ xs
h2 : decide (v ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) v
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
a1 : v ∈ xs
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v ∈ xs
h2 : decide (v ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) v
⊢ u ∉ binders
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v ∈ xs
h2 : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v ∈ xs
h2 : decide (v ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) v
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
tauto
|
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v ∈ xs
h2 : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
a1 : v ∈ xs
h2 : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
⊢ v = x ∨ v = y → u ∉ binders
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
⊢ v = x ∨ v = y → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
⊢ v = x ∨ v = y → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
cases h2
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
⊢ v = x ∨ v = y → u ∉ binders
|
case intro
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
left✝ : x ∈ binders ↔ (if v = x then u else x) ∈ binders
right✝ : y ∈ binders ↔ (if v = y then u else y) ∈ binders
⊢ v = x ∨ v = y → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
⊢ v = x ∨ v = y → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case intro h2_left h2_right =>
intros a1
cases a1
case inl a1 =>
subst a1
simp at h2_left
tauto
case inr a1 =>
subst a1
simp at h2_right
tauto
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
⊢ v = x ∨ v = y → u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
⊢ v = x ∨ v = y → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
intros a1
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
⊢ v = x ∨ v = y → u ∉ binders
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x ∨ v = y
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
⊢ v = x ∨ v = y → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
cases a1
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x ∨ v = y
⊢ u ∉ binders
|
case inl
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h✝ : v = x
⊢ u ∉ binders
case inr
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h✝ : v = y
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x ∨ v = y
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case inl a1 =>
subst a1
simp at h2_left
tauto
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case inr a1 =>
subst a1
simp at h2_right
tauto
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = y
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = y
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
subst a1
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x
⊢ u ∉ binders
|
v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = x
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2_left
|
v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders
|
v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
tauto
|
v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
h2_left : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
subst a1
|
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = y
⊢ u ∉ binders
|
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : y ∈ binders ↔ (if v = y then u else y) ∈ binders
a1 : v = y
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2_right
|
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders
|
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ (if v = v then u else v) ∈ binders
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
tauto
|
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2_left : x ∈ binders ↔ (if v = x then u else x) ∈ binders
h2_right : v ∈ binders ↔ u ∈ binders
⊢ u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2
|
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
⊢ fastAdmitsAux v u binders phi
|
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
⊢ fastAdmitsAux v u binders phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
⊢ fastAdmitsAux v u binders phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
exact phi_ih binders h1 h2
|
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
⊢ fastAdmitsAux v u binders phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
⊢ fastAdmitsAux v u binders phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
simp at h2
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
(toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
(toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
tauto
|
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi psi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
psi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi) → fastAdmitsAux v u binders psi
binders : Finset VarName
h1 : v ∉ binders
h2 :
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
⊢ fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
split_ifs at h2
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v u phi))
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
case pos
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h✝ : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
case neg
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h✝ : ¬v = x
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 :
BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
toIsBoundAux binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v u phi))
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
case pos c1 =>
left
exact c1
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
|
[677, 1]
|
[740, 17]
|
left
|
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
case h
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName),
v ∉ binders →
toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
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