Proof with auto with typeclass_instances.
intros x Zx Fx eps.
case Zx; intros Zx1.
(* . 0 < x *)
intros (Heps1,[Heps2|Heps2]).
assert (Heps: (0 <= eps < ulp x)%R).
split.
now apply Rlt_le.
exact Heps2.
assert (Hd := round_DN_plus_eps_pos x Zx Fx eps Heps).
rewrite round_UP_DN_ulp.
rewrite Hd.
rewrite 2!ulp_neq_0.
unfold canonic_exp.
now rewrite ln_beta_plus_eps.
now apply Rgt_not_eq.
now apply Rgt_not_eq, Rplus_lt_0_compat.
intros Fs.
rewrite round_generic in Hd...
apply Rgt_not_eq with (2 := Hd).
pattern x at 2 ; rewrite <- Rplus_0_r.
now apply Rplus_lt_compat_l.
rewrite Heps2.
apply round_generic...
now apply generic_format_succ_aux1.
(* . x=0 *)
rewrite <- Zx1, 2!Rplus_0_l.
intros Heps.
case (proj2 Heps).
unfold round, scaled_mantissa, canonic_exp.
unfold ulp.
rewrite Req_bool_true; trivial.
case negligible_exp_spec.
intros H2.
intros J; absurd (0 < 0)%R; auto with real.
apply Rlt_trans with eps; try assumption; apply Heps.
intros n Hn H.
assert (fexp (ln_beta beta eps) = fexp n).
apply valid_exp; try assumption.
assert(ln_beta beta eps-1 < fexp n)%Z;[idtac|omega].
apply lt_bpow with beta.
apply Rle_lt_trans with (2:=H).
destruct (ln_beta beta eps) as (e,He).
simpl; rewrite Rabs_pos_eq in He.
now apply He, Rgt_not_eq.
now left.
replace (Zceil (eps * bpow (- fexp (ln_beta beta eps)))) with 1%Z.
unfold F2R; simpl; rewrite H0; ring.
apply sym_eq, Zceil_imp.
split.
simpl; apply Rmult_lt_0_compat.
apply Heps.
apply bpow_gt_0.
apply Rmult_le_reg_r with (bpow (fexp n)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_plus.
rewrite H0; ring_simplify (-fexp n + fexp n)%Z.
simpl; rewrite Rmult_1_l, Rmult_1_r.
now left.
intros P; rewrite P.
apply round_generic...
apply generic_format_ulp_0.
Qed.

Theorem round_DN_minus_eps_pos :
forall x,  (0 < x)%R -> F x ->
forall eps, (0 < eps <= ulp (pred x))%R ->
round beta fexp Zfloor (x - eps) = pred x.