Section omega_compat.
Ltac omega ::= lia.
induction n as [|n IHn]; simpl; intuition.
rewrite <- IHn; clear IHn.
case_eq (Npow2 n); intuition.
Qed.
End omega_compat.

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