Section monotone_abs.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Proof with auto with typeclass_instances.
intros x y.
apply round_abs_abs...
clear rnd valid_rnd y.
intros rnd' Hrnd y Hy.
apply round_ge_generic...
Qed.
Proof with auto with typeclass_instances.
intros x y.
apply round_abs_abs...
clear rnd valid_rnd x.
intros rnd' Hrnd x Hx.
apply round_le_generic...
Qed.
End monotone_abs.
Proof with auto with typeclass_instances.
intros x.
apply sym_eq.
unfold Rabs at 2.
destruct (Rcase_abs x) as [Hx|Hx].
rewrite round_ZR_opp.
apply Rabs_left1.
rewrite <- (round_0 Ztrunc).
apply round_le...
now apply Rlt_le.
apply Rabs_pos_eq.
rewrite <- (round_0 Ztrunc).
apply round_le...
now apply Rge_le.
Qed.
Proof with auto with typeclass_instances.
intros x.
apply sym_eq.
unfold Rabs at 2.
destruct (Rcase_abs x) as [Hx|Hx].
rewrite round_AW_opp.
apply Rabs_left1.
rewrite <- (round_0 Zaway).
apply round_le...
now apply Rlt_le.
apply Rabs_pos_eq.
rewrite <- (round_0 Zaway).
apply round_le...
now apply Rge_le.
Qed.
Proof with auto with typeclass_instances.
intros rnd Zrnd x.
destruct (total_order_T x 0) as [[Hx|Hx]|Hx].
rewrite <- (Ropp_involutive x).
destruct (round_DN_or_UP rnd (- - x)) as [Hr|Hr] ; rewrite Hr.
rewrite round_DN_opp.
apply generic_format_opp.
apply generic_format_round_pos...
now apply Ropp_0_gt_lt_contravar.
rewrite round_UP_opp.
apply generic_format_opp.
apply generic_format_round_pos...
now apply Ropp_0_gt_lt_contravar.
rewrite Hx.
rewrite round_0...
apply generic_format_0.
now apply generic_format_round_pos.
Qed.

Theorem round_DN_pt :
forall x,
Rnd_DN_pt generic_format x (round Zfloor x).