Notation pow2 := (Nat.pow 2).Fixpoint Npow2 (n : nat) : N :=
  match n with
    | O => 1
    | S n' => 2 * Npow2 n'
  end%N.

Lemma pow2_S_z: forall n, Z.of_nat (pow2 (S n)) = (2 * Z.of_nat (pow2 n))%Z.