Proof with auto with typeclass_instances.
intros Y.
specialize (Zeven_ex (radix_val beta)); rewrite Even_beta.
intros (b, Hb); rewrite Zplus_0_r in Hb.
destruct u'_eq as (u', (Hu'1,Hu'2)); trivial.
exists (Fmult beta (Float beta b (-1)) (Fplus beta d u'))%R.
split.
rewrite F2R_mult, F2R_plus, Hu'1.
unfold m; rewrite Rmult_comm.
unfold Rdiv; apply f_equal.
unfold F2R; simpl; unfold Z.pow_pos; simpl.
rewrite Zmult_1_r, Hb, Z2R_mult.
simpl; field.
apply Rgt_not_eq, Rmult_lt_reg_l with (1 := Rlt_0_2).
rewrite Rmult_0_r, <- (Z2R_mult 2), <-Hb.
apply radix_pos.
apply trans_eq with (-1+Fexp (Fplus beta d u'))%Z.
unfold Fmult.
destruct  (Fplus beta d u'); reflexivity.
rewrite Zplus_comm; unfold Zminus; apply f_equal2.
2: reflexivity.
rewrite Fexp_Fplus.
rewrite Z.min_l.
now rewrite Fexp_d.
rewrite Hu'2; omega.
Qed.

Lemma m_eq_0: (0 = F2R d)%R ->  exists f:float beta,
F2R f = m /\ (Fexp f = fexp (ln_beta beta (F2R u)) -1)%Z.