Proof with auto with typeclass_instances.
intros x Fx.
rewrite <- pred_UP_eq_DN; trivial.
rewrite succ_pred; trivial.
apply generic_format_round...
Qed.

(* Specifications: *)
(* was betw_eq_DN *)
Theorem round_DN_eq_betw: forall x d, F d
-> (d <= x < succ d)%R
-> round beta fexp Zfloor x = d.
(* was betw_eq_DN *)
Theorem round_DN_eq_betw: forall x d, F d
-> (d <= x < succ d)%R
-> round beta fexp Zfloor x = d.