split_ifs
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
⊢ (if h : (τ X xs.length).isSome = true then
      if xs.length = ((τ X xs.length).get ⋯).1.length then
        Holds D I (Function.updateListITE V' ((τ X xs.length).get ⋯).1 (List.map V xs)) E ((τ X xs.length).get ⋯).2
      else I.pred_var_ X (List.map V xs)
    else I.pred_var_ X (List.map V xs)) ↔
    Holds D I V E
      (if h : (τ X xs.length).isSome = true then
        if xs.length = ((τ X xs.length).get ⋯).1.length then
          Var.All.Rec.Fresh.sub (Function.updateListITE id ((τ X xs.length).get ⋯).1 xs) c ((τ X xs.length).get ⋯).2
        else pred_var_ X xs
      else pred_var_ X xs)
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
h✝¹ : (τ X xs.length).isSome = true
h✝ : xs.length = ((τ X xs.length).get ⋯).1.length
⊢ Holds D I (Function.updateListITE V' ((τ X xs.length).get ⋯).1 (List.map V xs)) E ((τ X xs.length).get ⋯).2 ↔
    Holds D I V E
      (Var.All.Rec.Fresh.sub (Function.updateListITE id ((τ X xs.length).get ⋯).1 xs) c ((τ X xs.length).get ⋯).2)

case neg
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
h✝¹ : (τ X xs.length).isSome = true
h✝ : ¬xs.length = ((τ X xs.length).get ⋯).1.length
⊢ I.pred_var_ X (List.map V xs) ↔ Holds D I V E (pred_var_ X xs)

case neg
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
h✝ : ¬(τ X xs.length).isSome = true
⊢ I.pred_var_ X (List.map V xs) ↔ Holds D I V E (pred_var_ X xs)