Proof with auto with typeclass_instances.
intros rnd Hrnd x.
destruct (round_ZR_or_AW rnd x) as [Hr|Hr] ; rewrite Hr ; clear Hr rnd Hrnd.
left.
now apply ln_beta_round_ZR.
intros Zr.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, round_0...
destruct (ln_beta beta x) as (ex, Ex) ; simpl.
specialize (Ex Zx).
rewrite <- ln_beta_abs.
rewrite <- round_AW_abs.
destruct (Zle_or_lt ex (fexp ex)) as [He|He].
right.
rewrite Zmax_r with (1 := He).
rewrite round_AW_pos with (1 := Rabs_pos _).
now apply round_UP_small_pos.
destruct (round_bounded_large_pos Zaway _ ex He Ex) as (H1,[H2|H2]).
left.
apply ln_beta_unique.
rewrite <- round_AW_abs, Rabs_Rabsolu.
now split.
right.
now rewrite Zmax_l with (1 := Zlt_le_weak _ _ He).
Qed.

Theorem scaled_mantissa_DN :
forall x,
(0 < round Zfloor x)%R ->
scaled_mantissa (round Zfloor x) = Z2R (Zfloor (scaled_mantissa x)).