Section Fcore_rnd_NE.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Notation format := (generic_format beta fexp).
Notation canonical := (canonical beta fexp).
Definition NE_prop (_ : R) f :=
exists g : float beta, f = F2R g /\ canonical g /\ Z.even (Fnum g) = true.
Definition Rnd_NE_pt :=
Rnd_NG_pt format NE_prop.
Definition DN_UP_parity_pos_prop :=
forall x xd xu,
(0 < x)%R ->
~ format x ->
canonical xd ->
canonical xu ->
F2R xd = round beta fexp Zfloor x ->
F2R xu = round beta fexp Zceil x ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd)).
Definition DN_UP_parity_prop :=
forall x xd xu,
~ format x ->
canonical xd ->
canonical xu ->
F2R xd = round beta fexp Zfloor x ->
F2R xu = round beta fexp Zceil x ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd)).
Class Exists_NE :=
exists_NE : Z.even beta = false \/ forall e,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\ ((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e).
Context { exists_NE_ : Exists_NE }.
Proof with auto with typeclass_instances.
intros x xd xu H0x Hfx Hd Hu Hxd Hxu.
destruct (mag 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 eq_0_F2R 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 canonical_unique with (1 := Hu).
apply (f_equal fexp).
rewrite <- F2R_change_exp.
now rewrite F2R_bpow, mag_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 canonical_unique with (1 := Hu).
apply (f_equal fexp).
rewrite <- F2R_change_exp.
now rewrite F2R_bpow, mag_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 minus_IZR.
unfold Rminus.
rewrite Rmult_plus_distr_r.
rewrite IZR_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 cexp.
rewrite mag_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 canonical_unique with (1 := Hd) (3 := H).
apply (f_equal fexp).
rewrite <- H.
apply sym_eq.
now apply mag_unique.
rewrite Hd3, Hu3.
unfold Fnum.
rewrite Z.even_mul. simpl.
unfold Zminus at 2.
rewrite Z.even_add.
rewrite eqb_sym. simpl.
fold (negb (Z.even (beta ^ (ex - fexp ex)))).
rewrite Bool.negb_involutive.
rewrite (Z.even_pow 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 Z.even_pow.
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 cexp.
rewrite mag_unique with beta (F2R xu) ex.
rewrite mag_unique with (1 := Hd4).
rewrite mag_unique with (1 := Hexa).
intros H.
replace (Fnum xu) with (Fnum xd + 1)%Z.
rewrite Z.even_add.
now apply eqb_sym.
apply sym_eq.
apply eq_IZR.
rewrite plus_IZR.
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.
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 - IZR (Zfloor mx)) (/2)) as [Hm|Hm].
(* midpoint *)
left.
exists (Float beta (Ztrunc (scaled_mantissa beta fexp xr)) (cexp beta fexp xr)).
split.
apply round_N_pt...
split.
unfold Generic_fmt.canonical. 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 (Z.even (Zfloor mx)) ; intros Hmx.
(* . even floor *)
change (Z.even (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.
now rewrite Ztrunc_IZR.
rewrite <- (round_0 beta fexp Zfloor).
apply round_le...
now apply Rlt_le.
rewrite scaled_mantissa_DN...
now rewrite Ztrunc_IZR.
(* . odd floor *)
change (Z.even (Ztrunc (scaled_mantissa beta fexp (round beta fexp Zceil x))) = true).
destruct (mag 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 IZR_lt.
destruct (proj2 Hu) as [Hu'|Hu'].
(* ... u <> bpow *)
unfold scaled_mantissa.
rewrite cexp_fexp_pos with (1 := conj (proj1 Hu) Hu').
unfold round, F2R. simpl.
rewrite cexp_fexp_pos with (1 := Hex).
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.
rewrite Ztrunc_IZR.
fold mx.
rewrite Hfc.
now rewrite Z.even_add, Hmx.
(* ... u = bpow *)
rewrite Hu'.
unfold scaled_mantissa, cexp.
rewrite mag_bpow.
rewrite <- bpow_plus, <- IZR_Zpower.
rewrite Ztrunc_IZR.
case_eq (Z.even beta) ; intros Hr.
destruct exists_NE_ as [Hs|Hs].
now rewrite Hs in Hr.
destruct (Hs ex) as (H,_).
rewrite Z.even_pow.
exact Hr.
omega.
assert (Z.even (Zfloor mx) = true). 2: now rewrite H in Hmx.
replace (Zfloor mx) with (Zceil mx + -1)%Z by omega.
rewrite Z.even_add.
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_IZR.
rewrite IZR_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 <- cexp_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 (Z.even (Zfloor mx) = true). 2: now rewrite H in Hmx.
unfold mx, scaled_mantissa.
rewrite cexp_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_unique 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_middle.
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 (IZR (Zfloor mx) * bpow (cexp 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).