apply Zle_trans with (2 ^ Zlog_sup m / m). apply Zdiv_le. assumption. cut (a mod 2 ^ Zlog_sup m < 2 ^ Zlog_sup m). omega.
apply Zmod_lt_z_m. apply lt_0_Zpow. apply Zlog_sup_correct1. assumption. rewrite div_m_1. omega. assumption. apply Zle_0_div.
apply Zmod_le_0_z. apply lt_0_Zpow. apply Zlog_sup_correct1. assumption. assumption. Qed.

(* <KEEP TOGETHER - approx_0_m2 *)
Theorem approx_0_m2 : forall (y b : Zpls) (a : Z_ y) (x : Z_ (a * b + a)),
x / y - 2 <= x / a * (a * b / y) / b <= x / y.