Proof with auto with typeclass_instances.
intros choice x d u H.
apply Rle_antisym.
destruct (generic_format_EM beta fexp x) as [Fx|Fx].
rewrite round_generic...
apply round_DN_pt; trivial; now right.
apply round_N_le_midp.
apply round_DN_pt...
apply Rlt_le_trans with (1:=H).
right; apply f_equal2; trivial; apply f_equal.
now apply sym_eq, succ_DN_eq_UP.
apply round_ge_generic; try apply round_DN_pt...
Qed.

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