Section round_large.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
End round_large.
Proof with auto with typeclass_instances.
intros x Zr.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, round_0...
apply mag_unique.
destruct (mag beta x) as (ex, Ex) ; simpl.
specialize (Ex Zx).
rewrite <- round_ZR_abs.
split.
apply round_large_pos_ge_bpow...
rewrite round_ZR_abs.
now apply Rabs_pos_lt.
apply Ex.
apply Rle_lt_trans with (2 := proj2 Ex).
rewrite round_ZR_DN.
apply round_DN_pt.
apply Rabs_pos.
Qed.
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 mag_round_ZR.
intros Zr.
destruct (Req_dec x 0) as [Zx|Zx].
rewrite Zx, round_0...
destruct (mag beta x) as (ex, Ex) ; simpl.
specialize (Ex Zx).
rewrite <- mag_abs.
rewrite <- round_AW_abs.
destruct (Zle_or_lt ex (fexp ex)) as [He|He].
right.
rewrite Z.max_r with (1 := He).
rewrite round_AW_UP 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 mag_unique.
rewrite <- round_AW_abs, Rabs_Rabsolu.
now split.
right.
now rewrite Z.max_l with (1 := Zlt_le_weak _ _ He).
Qed.

Theorem mag_DN :
forall x,
(0 < round Zfloor x)%R ->
(mag beta (round Zfloor x) = mag beta x :> Z).