Proof with auto with typeclass_instances.
intros x.
destruct (round_ZR_or_AW rnd x) as [H|H] ; rewrite H ; clear H ; intros Zr.
rewrite ln_beta_round_ZR with (1 := Zr).
apply Zle_refl.
apply ln_beta_le_abs.
contradict Zr.
rewrite Zr.
apply round_0...
rewrite <- round_AW_abs.
rewrite round_AW_pos.
apply round_UP_pt.
apply Rabs_pos.
Qed.

Theorem canonic_exp_round_ge :
forall x,
round rnd x <> 0%R ->
(canonic_exp x <= canonic_exp (round rnd x))%Z.