Proof with auto with typeclass_instances.
intros x y Fx Fy H.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
now apply succ_le_lt_aux.
unfold succ; rewrite Rle_bool_false; try assumption.
case (Rle_or_lt y 0); intros Hy.
rewrite <- (Ropp_involutive y).
apply Ropp_le_contravar.
apply le_pred_pos_lt.
now apply generic_format_opp.
now apply generic_format_opp.
split.
rewrite <- Ropp_0; now apply Ropp_le_contravar.
now apply Ropp_lt_contravar.
apply Rle_trans with (-0)%R.
apply Ropp_le_contravar.
apply pred_pos_ge_0.
rewrite <- Ropp_0; now apply Ropp_lt_contravar.
now apply generic_format_opp.
rewrite Ropp_0; now left.
Qed.

(** Error of a rounding, expressed in number of ulps *)
(** false for x=0 in the FLX format *)
(* was ulp_error *)
Theorem error_lt_ulp :
forall rnd { Zrnd : Valid_rnd rnd } x,
(x <> 0)%R ->
(Rabs (round beta fexp rnd x - x) < ulp x)%R.