Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self (FOL/NV/Sub/Var/One/Rec/Admits.lean)

State before:
F : Formula
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders F
Tactic:
induction F generalizing binders
State after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝)

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_var_ a✝¹ a✝)

case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (eq_ a✝¹ a✝)

case true_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders true_

case false_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders false_

case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)

State before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝)

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_var_ a✝¹ a✝)

case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (eq_ a✝¹ a✝)

case true_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders true_

case false_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders false_

case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)
Tactic:
all_goals
  simp only [fastAdmitsAux]
State after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → v ∉ binders

case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ fastAdmitsAux v v (binders ∪ {a✝¹}) a✝

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ fastAdmitsAux v v (binders ∪ {a✝¹}) a✝

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

State before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → v ∉ binders

case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
Tactic:
all_goals
  tauto
State after:
no goals

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)
Tactic:
simp only [fastAdmitsAux]
State after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

State before:
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
by_cases c1 : v = x
State after:
case pos
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi

case neg
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi

State before:
case pos
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
left
State after:
case pos.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x

State before:
case pos.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x
Tactic:
exact c1
State after:
no goals

State before:
case neg
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
right
State after:
case neg.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ fastAdmitsAux v v (binders ∪ {x}) phi

State before:
case neg.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
apply phi_ih
State after:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}

State before:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
Tactic:
simp
State after:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x

State before:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
Tactic:
tauto
State after:
no goals

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
Tactic:
tauto
State after:
no goals