Proof with auto with typeclass_instances.
intros z Hz1.
replace (F2R d) with (round beta fexp Zfloor z).
apply round_DN_pt...
case (Rnd_DN_UP_pt_split _ _ _ _ Hd Hu (round beta fexp Zfloor z)).
apply generic_format_round...
intros Y; apply Rle_antisym; trivial.
apply round_DN_pt...
apply Hd.
apply Hz1.
intros Y; absurd (z < z)%R.
auto with real.
apply Rlt_le_trans with (1:=proj2 Hz1), Rle_trans with (1:=Y).
apply round_DN_pt...
Qed.

(* Specifications: *)
Lemma UP_odd_d_aux: forall z, (F2R d< z <= F2R u)%R ->
Rnd_UP_pt (generic_format beta fexp) z (F2R u).
Lemma UP_odd_d_aux: forall z, (F2R d< z <= F2R u)%R ->
Rnd_UP_pt (generic_format beta fexp) z (F2R u).