apply Zpos_eq_Z_of_nat_o_nat_of_P. intros p H. absurd (0 <= Zneg p). unfold Zle. tauto.
assumption. Qed.

Lemma Zlt_0_power_nat : forall i : nat, 0 < Zpower_nat 2 i.
simple induction i. replace (Zpower_nat 2 0) with 1. omega. trivial. intros n H.