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

Theorem pred_UP_le_DN :
forall x, (pred (round beta fexp Zceil x) <= round beta fexp Zfloor x)%R.