Definition var_env'_dash (ve : var_env') (n : nat) :=
  if leb N n then ve (n - N) else false.Definition var_env''_dash (ve : var_env'') :=
  var_env'_to_env'' N (2 * N) (var_env'_dash (var_env''_to_env' ve)).Fixpoint be_dash (be : bool_expr) : bool_expr :=
  match be with
  | Zero => Zero
  | One => One
  | Var x => Var (N_of_nat (N + nat_of_N x))
  | Neg be' => Neg (be_dash be')
  | Or be1 be2 => Or (be_dash be1) (be_dash be2)
  | ANd be1 be2 => ANd (be_dash be1) (be_dash be2)
  | Impl be1 be2 => Impl (be_dash be1) (be_dash be2)
  | Iff be1 be2 => Iff (be_dash be1) (be_dash be2)
  end.Fixpoint renamef (f : ad -> ad) (be : bool_expr) {struct be} : bool_expr :=
  match be with
  | Zero => Zero
  | One => One
  | Var x => Var (f x)
  | Neg be' => Neg (renamef f be')
  | Or be1 be2 => Or (renamef f be1) (renamef f be2)
  | ANd be1 be2 => ANd (renamef f be1) (renamef f be2)
  | Impl be1 be2 => Impl (renamef f be1) (renamef f be2)
  | Iff be1 be2 => Iff (renamef f be1) (renamef f be2)
  end.Definition renfnat (n m : nat) := if leb n m then m else m + N.Definition renfnad (n : nat) (x : ad) := N_of_nat (renfnat n (nat_of_N x)).

(* Specifications: *)
Lemma dash_renf : forall be : bool_expr, be_ok (var_lu 0 N) be -> be_dash be = renamef (renfnad N) be.
Lemma dash_be_ok : forall be : bool_expr, be_ok (var_lu 0 N) be -> be_ok (var_lu N (2 * N)) (be_dash be).
Lemma eval_dash_lemma1 : forall (be : bool_expr) (ve : var_env'), eval_be' be ve = eval_be' (be_dash be) (var_env'_dash ve).