intro Hq. replace (Zlog_sup q) with 0. omega. generalize Hq. elim q. trivial. intros r Hr.
absurd (Zpos r <= 0). apply Zlt_not_le. unfold Zlt. trivial. assumption. intros r Hr. trivial.
apply Zlog_sup_correct1. generalize Hp. elim p. trivial. intros r Hr. absurd (Zpos r <= 0).
apply Zlt_not_le. unfold Zlt. trivial. assumption. intros r Hr. trivial. intro H0.
elim (Zle_or_lt (Zlog_sup p) (Zlog_sup q)); intro H1. assumption. cut (Zlog_sup q <= Zlog_sup p - 1).
intro H2. cut (2 ^ Zlog_sup q <= 2 ^ (Zlog_sup p - 1)). intro H3. absurd (p <= q). apply Zlt_not_le.
apply Zle_lt_trans with (2 ^ Zlog_sup q). elim (Zlog_sup_correct2 q). omega. omega.
elim (Zlog_sup_correct2 p). omega. assumption. assumption. apply Zle_pow_le. apply Zlog_sup_correct1.
omega. assumption. omega. Qed.

Lemma Z_pow_plus : forall x : Z, 0 <= x -> 2 ^ x + 2 ^ x = 2 ^ (x + 1).
intros. rewrite Zplus_diag_eq_mult_2. rewrite Zpower_exp. trivial. omega. omega. Qed.