simp only [List.map_eq_map_iff]
case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
    admitsAux σ binders F →
      (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
        (∀ v ∈ binders, v = σ' v) →
          (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ List.map V xs = List.map (V' ∘ σ') xs
case a.h.e'_4
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula),
    admitsAux σ binders F →
      (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) →
        (∀ v ∈ binders, v = σ' v) →
          (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F))
X : PredName
xs : List VarName
V V' : VarAssignment D
σ σ' : VarName → VarName
binders : Finset VarName
h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders
h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)
h2' : ∀ v ∈ binders, v = σ' v
h3 : ∀ v ∉ binders, σ' v = σ v
⊢ ∀ x ∈ xs, V x = (V' ∘ σ') x