Theorem: FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree
Proof state before:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = fastReplaceFree v t (pred_const_ a✝¹ a✝)

case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = fastReplaceFree v t (pred_var_ a✝¹ a✝)

case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = fastReplaceFree v t (eq_ a✝¹ a✝)

case true_
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders true_ = fastReplaceFree v t true_

case false_
v t : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ replaceFreeAux v t binders false_ = fastReplaceFree v t false_

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

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

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

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

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

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

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

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

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

case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).imp_ (replaceFreeAux v t binders a✝) =
    (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), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).and_ (replaceFreeAux v t binders a✝) =
    (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), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).or_ (replaceFreeAux v t binders a✝) =
    (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), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ (replaceFreeAux v t binders a✝¹).iff_ (replaceFreeAux v t binders a✝) =
    (fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝)

case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ forall_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) =
    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), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ exists_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) =
    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 : v ∉ binders
⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) =
    def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)