apply Holds_coincide_Var
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
  ∀ (V : VarAssignment D) (F : Formula),
    ¬isFreeIn v F →
      ¬isBoundIn v F →
        (Holds D I (Function.updateITE V u a) tl F ↔
          Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ Holds D I
      (Function.updateListITE (fun c => if c = u then a else V c) hd.args
        (List.map (fun c => if c = u then a else V c) xs))
      tl hd.q ↔
    Holds D I
      (Function.updateListITE (fun c => if c = v then a else V c) hd.args
        (List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)))
      tl hd.q
case h1
D : Type
I : Interpretation D
u v : VarName
a : D
hd : Definition
tl : List Definition
ih :
  ∀ (V : VarAssignment D) (F : Formula),
    ¬isFreeIn v F →
      ¬isBoundIn v F →
        (Holds D I (Function.updateITE V u a) tl F ↔
          Holds D I (Function.updateITE V v a) tl (Sub.Var.One.Rec.fastReplaceFree u v F))
X : DefName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (def_ X xs)
c1 : X = hd.name ∧ (List.map (fun x => if u = x then v else x) xs).length = hd.args.length
⊢ ∀ (v_1 : VarName),
    isFreeIn v_1 hd.q →
      Function.updateListITE (fun c => if c = u then a else V c) hd.args
          (List.map (fun c => if c = u then a else V c) xs) v_1 =
        Function.updateListITE (fun c => if c = v then a else V c) hd.args
          (List.map (fun c => if c = v then a else V c) (List.map (fun x => if u = x then v else x) xs)) v_1