Proof with auto with typeclass_instances.
case (d_ge_0); intros Y.
(* *)
destruct m_eq as (g,(Hg1,Hg2)); trivial.
apply exists_even_fexp_lt.
exists g; split; trivial.
rewrite Hg2.
rewrite ln_beta_m; trivial.
rewrite <- Fexp_d; trivial.
rewrite Cd.
unfold canonic_exp.
generalize (fexpe_fexp  (ln_beta beta (F2R d))).
omega.
(* *)
destruct m_eq_0 as (g,(Hg1,Hg2)); trivial.
apply exists_even_fexp_lt.
exists g; split; trivial.
rewrite Hg2.
rewrite ln_beta_m_0; trivial.
apply Zle_lt_trans with (1:=fexpe_fexp _).
assert (fexp (ln_beta beta (F2R u)-1) < fexp (ln_beta beta (F2R u))+1)%Z;[idtac|omega].
now apply fexp_m_eq_0.
Qed.

(* Specifications: *)
Lemma DN_odd_d_aux: forall z, (F2R d<= z< F2R u)%R ->
Rnd_DN_pt (generic_format beta fexp) z (F2R d).