url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
all_goals
simp only [isBoundIn] at h1
simp only [fastAdmitsAux]
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝.not_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.imp_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.and_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.or_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.iff_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (forall_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (exists_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝.not_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.imp_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.and_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.or_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.iff_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (forall_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (exists_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
all_goals
tauto
|
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
simp only [isBoundIn] at h1
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
simp only [fastAdmitsAux]
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
push_neg at h1
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
cases h1
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
case intro
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
left✝ : u ≠ x
right✝ : ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
right
|
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
|
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
apply phi_ih (binders ∪ {x}) h1_right
|
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
|
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
simp
|
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x}
|
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
tauto
|
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux
|
[362, 1]
|
[385, 10]
|
tauto
|
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits
|
[388, 1]
|
[396, 7]
|
simp only [fastAdmits]
|
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmits v u F
|
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmitsAux v u ∅ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmits v u F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits
|
[388, 1]
|
[396, 7]
|
apply not_isBoundIn_imp_fastAdmitsAux F v u ∅ h1
|
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmitsAux v u ∅ F
|
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ fastAdmitsAux v u ∅ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits
|
[388, 1]
|
[396, 7]
|
simp
|
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
h1 : ¬isBoundIn u F
⊢ u ∉ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
induction F generalizing binders
|
F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t F)
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
all_goals
simp only [occursIn] at h1
simp only [fastReplaceFree]
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬False
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders true_
case false_
v t : VarName
binders : Finset VarName
h1 : ¬False
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝).not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
any_goals
simp only [fastAdmitsAux]
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬False
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders true_
case false_
v t : VarName
binders : Finset VarName
h1 : ¬False
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝).not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
h2 : v ∉ binders
⊢ ((t = if v = a✝¹ then t else a✝¹) ∨ t = if v = a✝ then t else a✝) → v ∉ binders
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬False
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders true_
case false_
v t : VarName
binders : Finset VarName
h1 : ¬False
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders false_
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝).not_
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
push_neg at h1
cases h1
case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastAdmitsAux]
subst c1
right
apply not_isFreeIn_imp_fastAdmitsAux
intro contra
apply h1_right
apply isFreeIn_imp_occursIn
exact contra
case neg c1 =>
simp only [fastAdmitsAux]
right
apply phi_ih (binders ∪ {x}) h1_right
simp
tauto
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
all_goals
tauto
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
h2 : v ∉ binders
⊢ ((t = if v = a✝¹ then t else a✝¹) ∨ t = if v = a✝ then t else a✝) → v ∉ binders
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
h2 : v ∉ binders
⊢ ((t = if v = a✝¹ then t else a✝¹) ∨ t = if v = a✝ then t else a✝) → v ∉ binders
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
simp only [occursIn] at h1
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
simp only [fastReplaceFree]
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
simp only [fastAdmitsAux]
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
push_neg at h1
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h1 : ¬(t = x ∨ occursIn t phi)
h2 : v ∉ binders
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
cases h1
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
case intro
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
left✝ : t ≠ x
right✝ : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1 : t ≠ x ∧ ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
case intro h1_left h1_right =>
split_ifs
case pos c1 =>
simp only [fastAdmitsAux]
subst c1
right
apply not_isFreeIn_imp_fastAdmitsAux
intro contra
apply h1_right
apply isFreeIn_imp_occursIn
exact contra
case neg c1 =>
simp only [fastAdmitsAux]
right
apply phi_ih (binders ∪ {x}) h1_right
simp
tauto
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
split_ifs
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
|
case pos
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi)
case neg
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
h✝ : ¬v = x
⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
case pos c1 =>
simp only [fastAdmitsAux]
subst c1
right
apply not_isFreeIn_imp_fastAdmitsAux
intro contra
apply h1_right
apply isFreeIn_imp_occursIn
exact contra
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
case neg c1 =>
simp only [fastAdmitsAux]
right
apply phi_ih (binders ∪ {x}) h1_right
simp
tauto
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
simp only [fastAdmitsAux]
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi)
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ fastAdmitsAux t v binders (exists_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
subst c1
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi
|
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
right
|
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi
|
case h
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ fastAdmitsAux t v (binders ∪ {v}) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
apply not_isFreeIn_imp_fastAdmitsAux
|
case h
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ fastAdmitsAux t v (binders ∪ {v}) phi
|
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ ¬isFreeIn t phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ fastAdmitsAux t v (binders ∪ {v}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
intro contra
|
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ ¬isFreeIn t phi
|
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ False
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
⊢ ¬isFreeIn t phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
apply h1_right
|
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ False
|
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ occursIn t phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ False
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
apply isFreeIn_imp_occursIn
|
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ occursIn t phi
|
case h.h1.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ isFreeIn t phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ occursIn t phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
exact contra
|
case h.h1.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ isFreeIn t phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1.h1
v t : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_right : ¬occursIn t phi
h1_left : t ≠ v
contra : isFreeIn t phi
⊢ isFreeIn t phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
simp only [fastAdmitsAux]
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
right
|
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
|
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
apply phi_ih (binders ∪ {x}) h1_right
|
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
|
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
simp
|
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
|
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
tauto
|
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
v t x : VarName
phi : Formula
phi_ih :
∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi)
binders : Finset VarName
h2 : v ∉ binders
h1_left : t ≠ x
h1_right : ¬occursIn t phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
|
[400, 1]
|
[437, 10]
|
tauto
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
h2 : v ∉ binders
⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits
|
[440, 1]
|
[448, 7]
|
simp only [fastAdmits]
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (fastReplaceFree v t F)
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (fastReplaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits
|
[440, 1]
|
[448, 7]
|
apply fastReplaceFree_aux_fastAdmitsAux F v t ∅ h1
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F)
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ v ∉ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits
|
[440, 1]
|
[448, 7]
|
simp
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ v ∉ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ v ∉ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
induction F generalizing binders
|
F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders F)
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
all_goals
simp only [occursIn] at h1
simp only [replaceFreeAux]
simp only [fastAdmitsAux]
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
|
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
⊢ ((t = if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) ∨ t = if v = a✝ ∧ a✝ ∉ binders then t else a✝) → v ∉ binders
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))
case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝))
case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝))
case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders true_)
case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders false_)
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝.not_)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝))
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝))
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝))
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝))
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝))
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝))
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case eq_ x y =>
push_neg at h1
intros a1
split_ifs at a1
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
case _ c1 c2 =>
cases c2
case intro c2_left c2_right =>
subst c2_left
exact c2_right
case _ c1 c2 =>
tauto
|
v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
all_goals
tauto
|
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
simp only [occursIn] at h1
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
simp only [replaceFreeAux]
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
simp only [fastAdmitsAux]
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))
|
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
simp
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs → v ∉ binders
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
intro x a1 a2
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
by_cases c1 : v = x ∧ x ∉ binders
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
⊢ v ∉ binders
|
case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
cases c1
|
case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders
|
case pos.intro
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x ∧ x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case _ c1_left c1_right =>
subst c1_left
exact c1_right
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
subst c1_left
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
exact c1_right
|
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
a1 : v ∈ xs
a2 : (v = v → v ∈ binders) → v = t
c1_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
push_neg at c1
|
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders
|
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x → x ∈ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : ¬(v = x ∧ x ∉ binders)
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
specialize a2 c1
|
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x → x ∈ binders
⊢ v ∉ binders
|
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
a2 : (v = x → x ∈ binders) → x = t
c1 : v = x → x ∈ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
subst a2
|
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders
|
case neg
v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v t : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : t ∉ xs
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
a2 : x = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
contradiction
|
case neg
v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
v : VarName
X : DefName
xs : List VarName
binders : Finset VarName
x : VarName
a1 : x ∈ xs
c1 : v = x → x ∈ binders
h1 : x ∉ xs
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
push_neg at h1
|
v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : ¬(t = x ∨ t = y)
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
intros a1
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
split_ifs at a1
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y
⊢ v ∉ binders
|
case pos
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
h✝¹ : v = x ∧ x ∉ binders
h✝ : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
case neg
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
h✝¹ : v = x ∧ x ∉ binders
h✝ : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
case pos
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
h✝¹ : ¬(v = x ∧ x ∉ binders)
h✝ : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
case neg
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
h✝¹ : ¬(v = x ∧ x ∉ binders)
h✝ : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case _ c1 c2 =>
cases c1
case intro c1_left c1_right =>
subst c1_left
exact c1_right
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case _ c1 c2 =>
cases c2
case intro c2_left c2_right =>
subst c2_left
exact c2_right
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case _ c1 c2 =>
tauto
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
cases c1
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
|
case intro
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case intro c1_left c1_right =>
subst c1_left
exact c1_right
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
subst c1_left
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
|
v t y : VarName
binders : Finset VarName
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
exact c1_right
|
v t y : VarName
binders : Finset VarName
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t y : VarName
binders : Finset VarName
c2 : v = y ∧ y ∉ binders
a1 : t = t ∨ t = t
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
cases c1
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
|
case intro
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
left✝ : v = x
right✝ : x ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : v = x ∧ x ∉ binders
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case intro c1_left c1_right =>
subst c1_left
exact c1_right
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
subst c1_left
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
|
v t y : VarName
binders : Finset VarName
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
c1_left : v = x
c1_right : x ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
exact c1_right
|
v t y : VarName
binders : Finset VarName
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t y : VarName
binders : Finset VarName
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = t ∨ t = y
h1 : t ≠ v ∧ t ≠ y
c1_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
cases c2
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
|
case intro
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
left✝ : v = y
right✝ : y ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : v = y ∧ y ∉ binders
a1 : t = x ∨ t = t
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
case intro c2_left c2_right =>
subst c2_left
exact c2_right
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
subst c2_left
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders
|
v t x : VarName
binders : Finset VarName
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
h1 : t ≠ x ∧ t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
c2_left : v = y
c2_right : y ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
exact c2_right
|
v t x : VarName
binders : Finset VarName
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
h1 : t ≠ x ∧ t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x : VarName
binders : Finset VarName
c1 : ¬(v = x ∧ x ∉ binders)
a1 : t = x ∨ t = t
h1 : t ≠ x ∧ t ≠ v
c2_right : v ∉ binders
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
tauto
|
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t x y : VarName
binders : Finset VarName
h1 : t ≠ x ∧ t ≠ y
c1 : ¬(v = x ∧ x ∉ binders)
c2 : ¬(v = y ∧ y ∉ binders)
a1 : t = x ∨ t = y
⊢ v ∉ binders
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux
|
[452, 1]
|
[500, 10]
|
tauto
|
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits
|
[503, 1]
|
[511, 48]
|
simp only [replaceFree]
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFree v t F)
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFreeAux v t ∅ F)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFree v t F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits
|
[503, 1]
|
[511, 48]
|
simp only [fastAdmits]
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFreeAux v t ∅ F)
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmits t v (replaceFreeAux v t ∅ F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits
|
[503, 1]
|
[511, 48]
|
exact replaceFreeAux_fastAdmitsAux F v t ∅ h1
|
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
induction F generalizing S
|
F : Formula
v u : VarName
S T : Finset VarName
h1 : fastAdmitsAux v u S F
h2 : u ∉ T
⊢ fastAdmitsAux v u (S ∪ T) F
|
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : fastAdmitsAux v u S true_
⊢ fastAdmitsAux v u (S ∪ T) true_
case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : fastAdmitsAux v u S false_
⊢ fastAdmitsAux v u (S ∪ T) false_
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝.not_
⊢ fastAdmitsAux v u (S ∪ T) a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
S T : Finset VarName
h1 : fastAdmitsAux v u S F
h2 : u ∉ T
⊢ fastAdmitsAux v u (S ∪ T) F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
all_goals
simp only [fastAdmitsAux] at h1
simp only [fastAdmitsAux]
|
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : fastAdmitsAux v u S true_
⊢ fastAdmitsAux v u (S ∪ T) true_
case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : fastAdmitsAux v u S false_
⊢ fastAdmitsAux v u (S ∪ T) false_
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝.not_
⊢ fastAdmitsAux v u (S ∪ T) a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
|
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S
⊢ v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝
⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝
⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : fastAdmitsAux v u S true_
⊢ fastAdmitsAux v u (S ∪ T) true_
case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : fastAdmitsAux v u S false_
⊢ fastAdmitsAux v u (S ∪ T) false_
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝.not_
⊢ fastAdmitsAux v u (S ∪ T) a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp
cases h1
case inl h1 =>
tauto
case inr h1 =>
specialize phi_ih (S ∪ {x}) h1
right
simp only [Finset.union_right_comm_assoc]
exact phi_ih
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
any_goals
simp only [Finset.mem_union]
|
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S
⊢ v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
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 pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S
⊢ v = a✝¹ ∨ v = a✝ → ¬(u ∈ S ∨ u ∈ T)
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S
⊢ v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → u ∉ S ∪ T
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
all_goals
tauto
|
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S
⊢ v = a✝¹ ∨ v = a✝ → ¬(u ∈ S ∨ u ∈ T)
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ S
⊢ v = a✝¹ ∨ v = a✝ → ¬(u ∈ S ∨ u ∈ T)
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝
⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
simp only [fastAdmitsAux] at h1
|
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
|
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
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : fastAdmitsAux v u S (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
simp only [fastAdmitsAux]
|
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
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ → u ∉ S
⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
simp
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
cases h1
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
case inl
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h✝ : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
case inr
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h✝ : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
case inl h1 =>
tauto
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
case inr h1 =>
specialize phi_ih (S ∪ {x}) h1
right
simp only [Finset.union_right_comm_assoc]
exact phi_ih
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
tauto
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Rec/Admits.lean
|
FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders
|
[515, 1]
|
[541, 10]
|
specialize phi_ih (S ∪ {x}) h1
|
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : fastAdmitsAux v u (S ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
TACTIC:
|
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