rewrite two_power_nat_correct. rewrite IHi. replace (Z_of_nat (S i)) with (1 + Z_of_nat i).
rewrite Zpower_exp. replace (2 ^ 1) with 2. trivial. trivial. omega. omega. rewrite inj_S.
omega. Qed.

Lemma Zlt_pow_lt :
forall i j : Z, 0 <= i -> i < j -> 2 ^ i < 2 ^ j.