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