Proof with auto with typeclass_instances.
intros x.
destruct (generic_format_EM beta fexp x) as [Fx|Fx].
rewrite !round_generic...
apply pred_le_id.
case (Req_dec (round beta fexp Zceil x) 0); intros Zx.
rewrite Zx; unfold pred; rewrite Ropp_0.
unfold succ; rewrite Rle_bool_true;[idtac|now right].
rewrite Rplus_0_l; unfold ulp; rewrite Req_bool_true; trivial.
case negligible_exp_spec'.
intros (H1,H2).
contradict Zx; apply round_neq_0_negligible_exp...
intros L; apply Fx; rewrite L; apply generic_format_0.
intros (n,(H1,Hn)); rewrite H1.
case (Rle_or_lt (- bpow (fexp n)) (round beta fexp Zfloor x)); trivial; intros K.
absurd (round beta fexp Zceil x <= - bpow (fexp n))%R.
apply Rlt_not_le.
rewrite Zx, <- Ropp_0.
apply Ropp_lt_contravar, bpow_gt_0.
apply round_UP_le_gt_DN; try assumption.
apply generic_format_opp, generic_format_bpow.
now apply valid_exp.
assert (let u := round beta fexp Zceil x in pred u < u)%R as Hup.
now apply pred_lt_id.
apply le_round_DN_lt_UP...
apply generic_format_pred...
now apply round_UP_pt.
Qed.

(* Specifications: *)
Theorem pred_UP_eq_DN :
forall x,  ~ F x ->
(pred (round beta fexp Zceil x) = round beta fexp Zfloor x)%R.
Theorem pred_UP_eq_DN :
forall x,  ~ F x ->
(pred (round beta fexp Zceil x) = round beta fexp Zfloor x)%R.