Theorem: FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux
Proof state before:
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:
any_goals
  simp only [fastAdmitsAux]
Proof state after:
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