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_add_binders | [515, 1] | [541, 10] | right | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | case h
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {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_add_binders | [515, 1] | [541, 10] | simp only [Finset.union_right_comm_assoc] | case h
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi | case h
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ fastAdmitsAux v u (S ∪ {x} ∪ T) phi | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ fastAdmitsAux v u (S ∪ (T ∪ {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_add_binders | [515, 1] | [541, 10] | exact phi_ih | case h
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ fastAdmitsAux v u (S ∪ {x} ∪ T) phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi
⊢ fastAdmitsAux v u (S ∪ {x} ∪ T) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | simp only [Finset.mem_union] | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders | [515, 1] | [541, 10] | tauto | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
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] | induction F generalizing S | F : Formula
v u : VarName
S T : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) F
⊢ fastAdmitsAux v u S F | case pred_const_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
S T : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) F
⊢ fastAdmitsAux v u S F
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] | all_goals
simp only [fastAdmitsAux] at h1
simp only [fastAdmitsAux] | case pred_const_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u... | case pred_const_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S ∪ T
⊢ v ∈ a✝ → u ∉ S
case pred_var_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S ∪ T
⊢ v ∈ a✝ → u ∉ S
case eq_
v u : VarName
T : Finset ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
a✝¹ : PredNam... |
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] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
simp at h1
tauto | v u : VarName
T : Finset VarName
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → u ∉ S ∪ 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 ∪ 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] | case eq_ x y =>
simp at h1
tauto | 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 | 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 ∪ 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] | case not_ phi phi_ih =>
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] | 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 =>
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
⊢ fas... | 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 : fastAd... |
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] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [Finset.union_right_comm S T {x}] at h1
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 ∪ T ∪ {x}) 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 ∪ T ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
T... |
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
⊢ fas... | 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 : fastAd... |
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
T... |
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
T... |
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... | 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... | 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 ... | 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
bind... |
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 ... | 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 :... |
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), fastAdmit... | 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 ∉ bind... |
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
... | 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
TACT... |
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
⊢ fastAdmi... |
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... |
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✝ : Li... | 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✝ : Li... | 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✝ : L... | 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_ ... |
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✝ : L... | 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 t... | 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 els... |
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) ... |
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_ = (toIsBou... | 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... |
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... | 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... |
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 (fu... |
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 ∉... | 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) ... |
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_ = (toIsBou... | 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... |
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... | 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... | 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... |
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... | 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... |
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_ (dec... | 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
⊢ Bool... | 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 ∨ fas... |
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.f... | 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 ∨ fas... |
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.f... | 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 ∨ fas... |
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.... | 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.... | 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 ∨ fas... |
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.... | 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... | 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 ∨ fas... |
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... | 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 ∉ ... | 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 ∨ fas... |
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 ∉ ... | 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 ∉ ... | 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 =... |
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 ∉ ... | 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 =... |
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
⊢ fast... | 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 =... |
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✝ : Li... | 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✝ : Li... | 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✝ :... | 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_... |
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✝ :... | 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✝))
⊢ ... | 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✝ → ... |
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 ... |
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 ... |
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... | 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... |
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 (fas... | 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... |
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... |
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 ... |
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 ... |
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 the... | 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:
|
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