Proof with auto with typeclass_instances.
intros choice x d u H.
apply Rle_antisym.
apply round_le_generic; try apply round_UP_pt...
destruct (generic_format_EM beta fexp x) as [Fx|Fx].
rewrite round_generic...
apply round_UP_pt; trivial; now right.
apply round_N_ge_midp.
apply round_UP_pt...
apply Rle_lt_trans with (2:=H).
right; apply f_equal2; trivial; rewrite Rplus_comm; apply f_equal2; trivial.
now apply pred_UP_eq_DN.
Qed.

(* Specifications: *)
Lemma round_N_eq_UP_pt: forall choice x d u,
Rnd_DN_pt F x d -> Rnd_UP_pt F x u ->
((d+u)/2 < x)%R ->
round beta fexp (Znearest choice) x = u.
Lemma round_N_eq_UP_pt: forall choice x d u,
Rnd_DN_pt F x d -> Rnd_UP_pt F x u ->
((d+u)/2 < x)%R ->
round beta fexp (Znearest choice) x = u.