simp only [subAux]
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 V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 :
  ∀
    x ∈
      if h : (τ X xs.length).isSome = true then
        ((τ X xs.length).get ⋯).2.freeVarSet \ ((τ X xs.length).get ⋯).1.toFinset
      else ∅,
    V'' x = V x
⊢ Holds D (I' D I V'' E τ) V' E (pred_var_ X xs) ↔ Holds D I V E (subAux c τ σ (pred_var_ X xs))
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 V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 :
  ∀
    x ∈
      if h : (τ X xs.length).isSome = true then
        ((τ X xs.length).get ⋯).2.freeVarSet \ ((τ X xs.length).get ⋯).1.toFinset
      else ∅,
    V'' x = V x
⊢ Holds D (I' D I V'' E τ) V' E (pred_var_ X 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 (List.map σ xs)) c
            ((τ X xs.length).get ⋯).2
        else pred_var_ X (List.map σ xs)
      else pred_var_ X (List.map σ xs))