Proof with auto with typeclass_instances.
intros x xd xu H0x Hfx Hd Hu Hxd Hxu.
destruct (ln_beta beta x) as (ex, Hexa).
specialize (Hexa (Rgt_not_eq _ _ H0x)).
generalize Hexa. intros Hex.
rewrite (Rabs_pos_eq _ (Rlt_le _ _ H0x)) in Hex.
destruct (Zle_or_lt ex (fexp ex)) as [Hxe|Hxe].
(* small x *)
assert (Hd3 : Fnum xd = Z0).
apply F2R_eq_0_reg with beta (Fexp xd).
change (F2R xd = R0).
rewrite Hxd.
apply round_DN_small_pos with (1 := Hex) (2 := Hxe).
assert (Hu3 : xu = Float beta (1 * Zpower beta (fexp ex - fexp (fexp ex + 1))) (fexp (fexp ex + 1))).
apply canonic_unicity with (1 := Hu).
apply (f_equal fexp).
rewrite <- F2R_change_exp.
now rewrite F2R_bpow, ln_beta_bpow.
now apply valid_exp.
rewrite <- F2R_change_exp.
rewrite F2R_bpow.
apply sym_eq.
rewrite Hxu.
apply sym_eq.
apply round_UP_small_pos with (1 := Hex) (2 := Hxe).
now apply valid_exp.
rewrite Hd3, Hu3.
rewrite Zmult_1_l.
simpl.
destruct exists_NE_ as [H|H].
apply Zeven_Zpower_odd with (2 := H).
apply Zle_minus_le_0.
now apply valid_exp.
rewrite (proj2 (H ex)).
now rewrite Zminus_diag.
exact Hxe.
(* large x *)
assert (Hd4: (bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R).
rewrite Rabs_pos_eq.
rewrite Hxd.
split.
apply (round_DN_pt beta fexp x).
apply generic_format_bpow.
ring_simplify (ex - 1 + 1)%Z.
omega.
apply Hex.
apply Rle_lt_trans with (2 := proj2 Hex).
apply (round_DN_pt beta fexp x).
rewrite Hxd.
apply (round_DN_pt beta fexp x).
apply generic_format_0.
now apply Rlt_le.
assert (Hxe2 : (fexp (ex + 1) <= ex)%Z) by now apply valid_exp.
assert (Hud: (F2R xu = F2R xd + ulp beta fexp x)%R).
rewrite Hxu, Hxd.
now apply round_UP_DN_ulp.
destruct (total_order_T (bpow ex) (F2R xu)) as [[Hu2|Hu2]|Hu2].
(* - xu > bpow ex  *)
elim (Rlt_not_le _ _ Hu2).
rewrite Hxu.
apply round_bounded_large_pos...
(* - xu = bpow ex *)
assert (Hu3: xu = Float beta (1 * Zpower beta (ex - fexp (ex + 1))) (fexp (ex + 1))).
apply canonic_unicity with (1 := Hu).
apply (f_equal fexp).
rewrite <- F2R_change_exp.
now rewrite F2R_bpow, ln_beta_bpow.
now apply valid_exp.
rewrite <- Hu2.
apply sym_eq.
rewrite <- F2R_change_exp.
apply F2R_bpow.
exact Hxe2.
assert (Hd3: xd = Float beta (Zpower beta (ex - fexp ex) - 1) (fexp ex)).
assert (H: F2R xd = F2R (Float beta (Zpower beta (ex - fexp ex) - 1) (fexp ex))).
unfold F2R. simpl.
rewrite Z2R_minus.
unfold Rminus.
rewrite Rmult_plus_distr_r.
rewrite Z2R_Zpower, <- bpow_plus.
ring_simplify (ex - fexp ex + fexp ex)%Z.
rewrite Hu2, Hud.
rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq].
unfold canonic_exp.
rewrite ln_beta_unique with beta x ex.
unfold F2R.
simpl. ring.
rewrite Rabs_pos_eq.
exact Hex.
now apply Rlt_le.
apply Zle_minus_le_0.
now apply Zlt_le_weak.
apply canonic_unicity with (1 := Hd) (3 := H).
apply (f_equal fexp).
rewrite <- H.
apply sym_eq.
now apply ln_beta_unique.
rewrite Hd3, Hu3.
unfold Fnum.
rewrite Zeven_mult. simpl.
unfold Zminus at 2.
rewrite Zeven_plus.
rewrite eqb_sym. simpl.
fold (negb (Zeven (beta ^ (ex - fexp ex)))).
rewrite Bool.negb_involutive.
rewrite (Zeven_Zpower beta (ex - fexp ex)). 2: omega.
destruct exists_NE_.
rewrite H.
apply Zeven_Zpower_odd with (2 := H).
now apply Zle_minus_le_0.
apply Zeven_Zpower.
specialize (H ex).
omega.
(* - xu < bpow ex *)
revert Hud.
rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq].
unfold F2R.
rewrite Hd, Hu.
unfold canonic_exp.
rewrite ln_beta_unique with beta (F2R xu) ex.
rewrite ln_beta_unique with (1 := Hd4).
rewrite ln_beta_unique with (1 := Hexa).
intros H.
replace (Fnum xu) with (Fnum xd + 1)%Z.
rewrite Zeven_plus.
now apply eqb_sym.
apply sym_eq.
apply eq_Z2R.
rewrite Z2R_plus.
apply Rmult_eq_reg_r with (bpow (fexp ex)).
rewrite H.
simpl. ring.
apply Rgt_not_eq.
apply bpow_gt_0.
rewrite Rabs_pos_eq.
split.
apply Rle_trans with (1 := proj1 Hex).
rewrite Hxu.
apply (round_UP_pt beta fexp x).
exact Hu2.
apply Rlt_le.
apply Rlt_le_trans with (1 := H0x).
rewrite Hxu.
apply (round_UP_pt beta fexp x).
Qed.

Theorem Rnd_NE_pt_round :
round_pred Rnd_NE_pt.