simp only [List.map_eq_map_iff]
case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
    fastAdmitsAux v t binders F →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ List.map (Function.updateITE V v (V' t)) xs = List.map (V ∘ fun x => if v = x then t else x) xs
case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
    fastAdmitsAux v t binders F →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ∀ x ∈ xs, Function.updateITE V v (V' t) x = (V ∘ fun x => if v = x then t else x) x