Proof with auto with typeclass_instances.
intros x Fx.
assert (Hx:(round  Zfloor x <= x <= round Zceil x)%R).
split.
apply round_DN_pt.
apply round_UP_pt.
split.
destruct Hx as [Hx _].
apply Rnot_le_lt; intro Hle.
assert (x = round Zfloor x) by now apply Rle_antisym.
rewrite H in Fx.
contradict Fx.
apply generic_format_round...
destruct Hx as [_ Hx].
apply Rnot_le_lt; intro Hle.
assert (x = round Zceil x) by now apply Rle_antisym.
rewrite H in Fx.
contradict Fx.
apply generic_format_round...
Qed.

(* Specifications: *)
Theorem round_UP_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
round Zceil x = (bpow (fexp ex)).
Theorem generic_format_EM :
forall x,
generic_format x \/ ~generic_format x.