Section FTZ_round.
(** Rounding with FTZ *)
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Definition Zrnd_FTZ x :=
if Rle_bool 1 (Rabs x) then rnd x else Z0.
Global Instance valid_rnd_FTZ : Valid_rnd Zrnd_FTZ.
Proof with auto with typeclass_instances.
split.
(* *)
intros x y Hxy.
unfold Zrnd_FTZ.
case Rle_bool_spec ; intros Hx ;
case Rle_bool_spec ; intros Hy.
4: easy.
(* 1 <= |x| *)
now apply Zrnd_le.
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
apply Rle_trans with (-1)%R. 2: now apply IZR_le.
destruct (Rabs_ge_inv _ _ Hx) as [Hx1|Hx1].
exact Hx1.
elim Rle_not_lt with (1 := Hx1).
apply Rle_lt_trans with (2 := Hy).
apply Rle_trans with (1 := Hxy).
apply RRle_abs.
(* |x| < 1 *)
rewrite <- (Zrnd_IZR rnd 0).
apply Zrnd_le...
apply Rle_trans with 1%R.
now apply IZR_le.
destruct (Rabs_ge_inv _ _ Hy) as [Hy1|Hy1].
elim Rle_not_lt with (1 := Hy1).
apply Rlt_le_trans with (2 := Hxy).
apply (Rabs_def2 _ _ Hx).
exact Hy1.
(* *)
intros n.
unfold Zrnd_FTZ.
rewrite Zrnd_IZR...
case Rle_bool_spec.
easy.
rewrite <- abs_IZR.
intros H.
generalize (lt_IZR _ 1 H).
clear.
now case n ; trivial ; simpl ; intros [p|p|].
Qed.

Theorem round_FTZ_small :
forall x : R,
(Rabs x < bpow (emin + prec - 1))%R ->
round beta FTZ_exp Zrnd_FTZ x = 0%R.