Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_fastReplaceFree
Proof state before:
case pred_const_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (pred_const_ a✝¹ a✝))

case pred_var_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (pred_var_ a✝¹ a✝))

case eq_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (eq_ a✝¹ a✝))

case true_
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t true_)

case false_
v t : VarName
h1 : ¬v = t
⊢ ¬isFreeIn v (fastReplaceFree v t false_)

case not_
v t : VarName
h1 : ¬v = t
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t a✝.not_)

case imp_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.imp_ a✝))

case and_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.and_ a✝))

case or_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.or_ a✝))

case iff_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (a✝¹.iff_ a✝))

case forall_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (forall_ a✝¹ a✝))

case exists_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ a✝¹ a✝))

case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ ¬isFreeIn v (fastReplaceFree v t (def_ a✝¹ a✝))
Tactic:
any_goals
  simp only [fastReplaceFree]
  simp only [isFreeIn]
Proof state after:
case pred_const_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝

case pred_var_
v t : VarName
h1 : ¬v = t
a✝¹ : PredName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝

case eq_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : VarName
⊢ ¬((v = if v = a✝¹ then t else a✝¹) ∨ v = if v = a✝ then t else a✝)

case true_
v t : VarName
h1 : ¬v = t
⊢ ¬False

case false_
v t : VarName
h1 : ¬v = t
⊢ ¬False

case not_
v t : VarName
h1 : ¬v = t
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t a✝)

case imp_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))

case and_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))

case or_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))

case iff_
v t : VarName
h1 : ¬v = t
a✝¹ a✝ : Formula
a_ih✝¹ : ¬isFreeIn v (fastReplaceFree v t a✝¹)
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬(isFreeIn v (fastReplaceFree v t a✝¹) ∨ isFreeIn v (fastReplaceFree v t a✝))

case forall_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (forall_ a✝¹ a✝))

case exists_
v t : VarName
h1 : ¬v = t
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬isFreeIn v (fastReplaceFree v t a✝)
⊢ ¬isFreeIn v (fastReplaceFree v t (exists_ a✝¹ a✝))

case def_
v t : VarName
h1 : ¬v = t
a✝¹ : DefName
a✝ : List VarName
⊢ v ∉ List.map (fun x => if v = x then t else x) a✝