Section monotone.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Proof with auto with typeclass_instances.
intros x y Hxy.
destruct (total_order_T x 0) as [[Hx|Hx]|Hx].
3: now apply round_le_pos.
(* x < 0 *)
unfold round.
destruct (Rlt_or_le y 0) as [Hy|Hy].
(* . y < 0 *)
rewrite <- (Ropp_involutive x), <- (Ropp_involutive y).
rewrite (scaled_mantissa_opp (-x)), (scaled_mantissa_opp (-y)).
rewrite (cexp_opp (-x)), (cexp_opp (-y)).
apply Ropp_le_cancel.
rewrite <- 2!F2R_Zopp.
apply (round_le_pos (Zrnd_opp rnd) (-y) (-x)).
rewrite <- Ropp_0.
now apply Ropp_lt_contravar.
now apply Ropp_le_contravar.
(* . 0 <= y *)
apply Rle_trans with 0%R.
apply F2R_le_0. simpl.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
simpl.
rewrite <- (Rmult_0_l (bpow (- fexp (mag beta x)))).
apply Rmult_le_compat_r.
apply bpow_ge_0.
now apply Rlt_le.
apply F2R_ge_0. simpl.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
apply Rmult_le_pos.
exact Hy.
apply bpow_ge_0.
(* x = 0 *)
rewrite Hx.
rewrite round_0...
apply F2R_ge_0.
simpl.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
apply Rmult_le_pos.
now rewrite <- Hx.
apply bpow_ge_0.
Qed.
End monotone.

Theorem round_abs_abs :
forall P : R -> R -> Prop,
( forall rnd (Hr : Valid_rnd rnd) x, (0 <= x)%R -> P x (round rnd x) ) ->
forall rnd {Hr : Valid_rnd rnd} x, P (Rabs x) (Rabs (round rnd x)).