intros i j Hi Hij. replace j with (i + (j - i)). rewrite Zpower_exp.
apply Zle_lt_trans with (2 ^ i * 2 ^ 0). simpl. omega. apply Zmult_lt_compat_l. apply lt_0_Zpow.
assumption. replace (j - i) with (j - i - 1 + 1).  rewrite Zpower_exp. replace (2 ^ 1) with 2.
replace (2 ^ 0) with 1. cut (0 < 2 ^ (j - i - 1)). omega. apply lt_0_Zpow. omega. trivial. trivial.
omega. omega.  ring. omega. omega. ring. Qed.

(* Specifications: *)
Lemma Zle_pow_le :
forall i j : Z, 0 <= i -> i <= j -> 2 ^ i <= 2 ^ j.