Proof with auto with typeclass_instances.
intros x Hx.
split.
now apply round_N_pt.
unfold NE_prop.
set (mx := scaled_mantissa beta fexp x).
set (xr := round beta fexp ZnearestE x).
destruct (Req_dec (mx - Z2R (Zfloor mx)) (/2)) as [Hm|Hm].
(* midpoint *)
left.
exists (Float beta (Ztrunc (scaled_mantissa beta fexp xr)) (canonic_exp beta fexp xr)).
split.
apply round_N_pt...
split.
unfold Fcore_generic_fmt.canonic. simpl.
apply f_equal.
apply round_N_pt...
simpl.
unfold xr, round, Znearest.
fold mx.
rewrite Hm.
rewrite Rcompare_Eq. 2: apply refl_equal.
case_eq (Zeven (Zfloor mx)) ; intros Hmx.
(* . even floor *)
change (Zeven (Ztrunc (scaled_mantissa beta fexp (round beta fexp Zfloor x))) = true).
destruct (Rle_or_lt (round beta fexp Zfloor x) 0) as [Hr|Hr].
rewrite (Rle_antisym _ _ Hr).
unfold scaled_mantissa.
rewrite Rmult_0_l.
change 0%R with (Z2R 0).
now rewrite (Ztrunc_Z2R 0).
rewrite <- (round_0 beta fexp Zfloor).
apply round_le...
now apply Rlt_le.
rewrite scaled_mantissa_DN...
now rewrite Ztrunc_Z2R.
(* . odd floor *)
change (Zeven (Ztrunc (scaled_mantissa beta fexp (round beta fexp Zceil x))) = true).
destruct (ln_beta beta x) as (ex, Hex).
specialize (Hex (Rgt_not_eq _ _ Hx)).
rewrite (Rabs_pos_eq _ (Rlt_le _ _ Hx)) in Hex.
destruct (Z_lt_le_dec (fexp ex) ex) as [He|He].
(* .. large pos *)
assert (Hu := round_bounded_large_pos _ _ Zceil _ _ He Hex).
assert (Hfc: Zceil mx = (Zfloor mx + 1)%Z).
apply Zceil_floor_neq.
intros H.
rewrite H in Hm.
unfold Rminus in Hm.
rewrite Rplus_opp_r in Hm.
elim (Rlt_irrefl 0).
rewrite Hm at 2.
apply Rinv_0_lt_compat.
now apply (Z2R_lt 0 2).
destruct (proj2 Hu) as [Hu'|Hu'].
(* ... u <> bpow *)
unfold scaled_mantissa.
rewrite canonic_exp_fexp_pos with (1 := conj (proj1 Hu) Hu').
unfold round, F2R. simpl.
rewrite canonic_exp_fexp_pos with (1 := Hex).
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
rewrite Ztrunc_Z2R.
fold mx.
rewrite Hfc.
now rewrite Zeven_plus, Hmx.
(* ... u = bpow *)
rewrite Hu'.
unfold scaled_mantissa, canonic_exp.
rewrite ln_beta_bpow.
rewrite <- bpow_plus, <- Z2R_Zpower.
rewrite Ztrunc_Z2R.
case_eq (Zeven beta) ; intros Hr.
destruct exists_NE_ as [Hs|Hs].
now rewrite Hs in Hr.
destruct (Hs ex) as (H,_).
rewrite Zeven_Zpower.
exact Hr.
omega.
assert (Zeven (Zfloor mx) = true). 2: now rewrite H in Hmx.
replace (Zfloor mx) with (Zceil mx + -1)%Z by omega.
rewrite Zeven_plus.
apply eqb_true.
unfold mx.
replace (Zceil (scaled_mantissa beta fexp x)) with (Zpower beta (ex - fexp ex)).
rewrite Zeven_Zpower_odd with (2 := Hr).
easy.
omega.
apply eq_Z2R.
rewrite Z2R_Zpower. 2: omega.
apply Rmult_eq_reg_r with (bpow (fexp ex)).
unfold Zminus.
rewrite bpow_plus.
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_l, Rmult_1_r.
pattern (fexp ex) ; rewrite <- canonic_exp_fexp_pos with (1 := Hex).
now apply sym_eq.
apply Rgt_not_eq.
apply bpow_gt_0.
generalize (proj1 (valid_exp ex) He).
omega.
(* .. small pos *)
assert (Zeven (Zfloor mx) = true). 2: now rewrite H in Hmx.
unfold mx, scaled_mantissa.
rewrite canonic_exp_fexp_pos with (1 := Hex).
now rewrite mantissa_DN_small_pos.
(* not midpoint *)
right.
intros g Hg.
destruct (Req_dec x g) as [Hxg|Hxg].
rewrite <- Hxg.
apply sym_eq.
apply round_generic...
rewrite Hxg.
apply Hg.
set (d := round beta fexp Zfloor x).
set (u := round beta fexp Zceil x).
apply Rnd_N_pt_unicity with (d := d) (u := u) (4 := Hg).
now apply round_DN_pt.
now apply round_UP_pt.
2: now apply round_N_pt.
rewrite <- (scaled_mantissa_mult_bpow beta fexp x).
unfold d, u, round, F2R. simpl. fold mx.
rewrite <- 2!Rmult_minus_distr_r.
intros H.
apply Rmult_eq_reg_r in H.
apply Hm.
apply Rcompare_Eq_inv.
rewrite Rcompare_floor_ceil_mid.
now apply Rcompare_Eq.
contradict Hxg.
apply sym_eq.
apply Rnd_N_pt_idempotent with (1 := Hg).
rewrite <- (scaled_mantissa_mult_bpow beta fexp x).
fold mx.
rewrite <- Hxg.
change (Z2R (Zfloor mx) * bpow (canonic_exp beta fexp x))%R with d.
now eapply round_DN_pt.
apply Rgt_not_eq.
apply bpow_gt_0.
Qed.

Lemma round_NE_abs:
forall x : R,
round beta fexp ZnearestE (Rabs x) = Rabs (round beta fexp ZnearestE x).