Proof with auto with typeclass_instances.
intros x y Fy Hlt.
apply round_DN_pt...
apply Rnot_lt_le.
contradict Hlt.
apply RIneq.Rle_not_lt.
apply round_UP_pt...
now apply Rlt_le.
Qed.

(* Specifications: *)
(* was lt_DN_le_UP *)
Theorem round_UP_le_gt_DN :
forall x y, F y ->
(round beta fexp Zfloor x < y -> round beta fexp Zceil x <= y)%R.
(* was lt_DN_le_UP *)
Theorem round_UP_le_gt_DN :
forall x y, F y ->
(round beta fexp Zfloor x < y -> round beta fexp Zceil x <= y)%R.