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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 |
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 |
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 |
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) |
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 |
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 (S ∪ T) (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
a✝¹ a✝ : VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (eq_ a✝¹ a✝)
case true_
v u : VarName
T S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) true_
⊢ fastAdmitsAux v u S true_
case false_
v u : VarName
T S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) false_
⊢ fastAdmitsAux v u S false_
case not_
v u : VarName
T : Finset VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝.not_
⊢ fastAdmitsAux v u S a✝.not_
case imp_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (exists_ a✝¹ a✝)
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✝) |
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 (S ∪ T) (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
a✝¹ a✝ : VarName
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (eq_ a✝¹ a✝)
case true_
v u : VarName
T S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) true_
⊢ fastAdmitsAux v u S true_
case false_
v u : VarName
T S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) false_
⊢ fastAdmitsAux v u S false_
case not_
v u : VarName
T : Finset VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝.not_
⊢ fastAdmitsAux v u S a✝.not_
case imp_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u S (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u S (exists_ a✝¹ a✝)
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 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 VarName
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T
⊢ v = a✝¹ ∨ v = a✝ → u ∉ S
case not_
v u : VarName
T : Finset VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝
⊢ fastAdmitsAux v u S a✝
case imp_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
case and_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
case or_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
case iff_
v u : VarName
T : Finset VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
case forall_
v u : VarName
T : Finset VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝
⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝
case exists_
v u : VarName
T : Finset VarName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝
S : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝
⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {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 |
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 |
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 |
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 |
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
⊢ fastAdmitsAux v u S phi ∧ fastAdmitsAux v u S psi | no goals |
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 |
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✝) |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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✝)) |
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✝)) |
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✝)) |
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 |
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 |
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 |
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✝)) |
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✝)) |
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✝)) |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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) |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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) |
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 |
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)) |
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 |
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 |
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)) |
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) |
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 |
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 |
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 |
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 |
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✝) |
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 |
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 |
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 |
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 |
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 |
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 |
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✝) |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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