Proof with auto with typeclass_instances.
intros x Fx.
apply Rle_antisym.
now apply pred_UP_le_DN.
apply le_pred_lt; try apply generic_format_round...
pose proof round_DN_UP_lt _ _ _ Fx as HE.
now apply Rlt_trans with (1 := proj1 HE) (2 := proj2 HE).
Qed.

(* Specifications: *)
Theorem succ_DN_eq_UP :
forall x,  ~ F x ->
(succ (round beta fexp Zfloor x) = round beta fexp Zceil x)%R.
Theorem succ_DN_eq_UP :
forall x,  ~ F x ->
(succ (round beta fexp Zfloor x) = round beta fexp Zceil x)%R.