intro H0. rewrite H0. left. rewrite <- Zmod_plus_distr_l. rewrite Zplus_comm with (n := a). rewrite <- Zmod_plus_distr_l.
rewrite Zplus_comm with (m := a). ring. omega. omega. intro H0. rewrite H0. right. rewrite <- Zmod_plus_distr_l.
rewrite Zplus_comm with (n := a). rewrite <- Zmod_plus_distr_l. rewrite Zplus_comm with (m := a). ring. omega. omega. omega.
omega. Qed.

Lemma div_m_1 : forall m : Z, 0 < m -> 2 ^ Zlog_sup m / m = 1.
intros m H. elim (Zle_lt_or_eq 1 m). intro H0. replace (2 ^ Zlog_sup m) with (1 * m + (2 ^ Zlog_sup m - m)).