simp only [I']
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
⊢ (I' D I V'' E τ).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 (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))
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
⊢ (Interpretation.usingPred D I fun X ds =>
          if h : (τ X ds.length).isSome = true then
            if ds.length = ((τ X ds.length).get ⋯).1.length then
              Holds D I (Function.updateListITE V'' ((τ X ds.length).get ⋯).1 ds) E ((τ X ds.length).get ⋯).2
            else I.pred_var_ X ds
          else I.pred_var_ X ds).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 (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))