apply Function.updateListITE_mem_eq_len V' V v zs (List.map V xs) c3
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V' x = V x
h1 :
  if h : (τ X xs.length).isSome = true then
    xs.length = ((τ X xs.length).get ⋯).1.length →
      binders ∩ (((τ X xs.length).get ⋯).2.freeVarSet \ ((τ X xs.length).get ⋯).1.toFinset) = ∅
  else True
c1 : (τ X xs.length).isSome = true
c2 : xs.length = ((τ X xs.length).get ⋯).1.length
opt : Option (List VarName × Formula) := τ X xs.length
val : List VarName × Formula := opt.get c1
zs : List VarName := val.1
H : Formula := val.2
s1 :
  Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs xs) c H) ↔
    Holds D I (Function.updateListITE V zs (List.map V xs)) E H
v : VarName
a1 : isFreeIn v ((τ X xs.length).get ⋯).2
c3 : v ∈ zs
⊢ Function.updateListITE V' ((τ X xs.length).get ⋯).1 (List.map V xs) v = Function.updateListITE V zs (List.map V xs) v
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V' x = V x
h1 :
  if h : (τ X xs.length).isSome = true then
    xs.length = ((τ X xs.length).get ⋯).1.length →
      binders ∩ (((τ X xs.length).get ⋯).2.freeVarSet \ ((τ X xs.length).get ⋯).1.toFinset) = ∅
  else True
c1 : (τ X xs.length).isSome = true
c2 : xs.length = ((τ X xs.length).get ⋯).1.length
opt : Option (List VarName × Formula) := τ X xs.length
val : List VarName × Formula := opt.get c1
zs : List VarName := val.1
H : Formula := val.2
s1 :
  Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs xs) c H) ↔
    Holds D I (Function.updateListITE V zs (List.map V xs)) E H
v : VarName
a1 : isFreeIn v ((τ X xs.length).get ⋯).2
c3 : v ∈ zs
⊢ zs.length = (List.map V xs).length