Proof with auto with typeclass_instances.
intros choice x d u Hd Hu H.
assert (H0:(d = round beta fexp Zfloor x)%R).
apply Rnd_DN_pt_unicity with (1:=Hd).
apply round_DN_pt...
rewrite H0.
apply round_N_eq_DN.
rewrite <- H0.
rewrite Rnd_UP_pt_unicity with F x (round beta fexp Zceil x) u; try assumption.
apply round_UP_pt...
Qed.

(* Specifications: *)
Lemma round_N_eq_UP: forall choice x,
let d:=round beta fexp Zfloor x in
let u:=round beta fexp Zceil x in
((d+u)/2 < x)%R ->
round beta fexp (Znearest choice) x = u.
Lemma round_N_eq_UP: forall choice x,
let d:=round beta fexp Zfloor x in
let u:=round beta fexp Zceil x in
((d+u)/2 < x)%R ->
round beta fexp (Znearest choice) x = u.