Fcore_generic_fmt.v 25.3 KB
Newer Older
1 2 3 4
Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_rnd.
Require Import Fcore_float_prop.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
5 6 7 8 9

Section RND_generic.

Variable beta : radix.

10
Notation bpow e := (bpow beta e).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
11 12 13

Variable fexp : Z -> Z.

14 15 16 17 18 19 20 21
Definition valid_exp :=
  forall k : Z,
  ( (fexp k < k)%Z -> (fexp (k + 1) <= k)%Z ) /\
  ( (k <= fexp k)%Z ->
    (fexp (fexp k + 1) <= fexp k)%Z /\
    forall l : Z, (l <= fexp k)%Z -> fexp l = fexp k ).

Variable prop_exp : valid_exp.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
22

23 24 25 26 27
Definition canonic_exponent x :=
  fexp (projT1 (ln_beta beta x)).

Definition canonic (f : float beta) :=
  Fexp f = canonic_exponent (F2R f).
28

29 30 31
Definition scaled_mantissa x :=
  (x * bpow (- canonic_exponent x))%R.

Guillaume Melquiond's avatar
Guillaume Melquiond committed
32
Definition generic_format (x : R) :=
33
  x = F2R (Float beta (Ztrunc (scaled_mantissa x)) (canonic_exponent x)).
34 35 36 37 38 39 40

(*
Theorem canonic_mantissa_0 :
  canonic_mantissa 0 = Z0.
Proof.
unfold canonic_mantissa.
rewrite Rmult_0_l.
41
exact (Zfloor_Z2R 0).
42 43
Qed.
*)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
44

Guillaume Melquiond's avatar
Guillaume Melquiond committed
45 46 47
Theorem generic_format_0 :
  generic_format 0.
Proof.
48
unfold generic_format, scaled_mantissa.
49 50 51 52 53 54 55 56 57 58 59 60
rewrite Rmult_0_l.
change (Ztrunc 0) with (Ztrunc (Z2R 0)).
now rewrite Ztrunc_Z2R, F2R_0.
Qed.

Theorem canonic_exponent_opp :
  forall x,
  canonic_exponent (-x) = canonic_exponent x.
Proof.
intros x.
unfold canonic_exponent.
now rewrite ln_beta_opp.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
61 62
Qed.

63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
(*
Theorem canonic_mantissa_opp :
  forall x,
  generic_format x ->
  canonic_mantissa (-x) = (- canonic_mantissa x)%Z.
Proof.
unfold generic_format, canonic_mantissa.
intros x Hx.
rewrite canonic_exponent_opp.
rewrite Hx at 1 3.
generalize (canonic_exponent x).
intros e.
clear.
unfold F2R. simpl.
rewrite Ropp_mult_distr_l_reverse.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r.
rewrite Rmult_1_r.
rewrite <- opp_Z2R.
81
now rewrite 2!Zfloor_Z2R.
82 83 84
Qed.
*)

85 86 87 88 89
Theorem generic_format_bpow :
  forall e, (fexp (e + 1) <= e)%Z ->
  generic_format (bpow e).
Proof.
intros e H.
90
unfold generic_format, scaled_mantissa, canonic_exponent.
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
rewrite ln_beta_bpow.
rewrite <- bpow_add.
rewrite <- (Z2R_Zpower beta (e + - fexp (e + 1))).
rewrite Ztrunc_Z2R.
rewrite <- F2R_bpow.
rewrite F2R_change_exp with (1 := H).
now rewrite Zmult_1_l.
omega.
Qed.

Theorem generic_format_canonic_exponent :
  forall m e,
  (canonic_exponent (F2R (Float beta m e)) <= e)%Z ->
  generic_format (F2R (Float beta m e)).
Proof.
intros m e.
107
unfold generic_format, scaled_mantissa.
108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149
set (e' := canonic_exponent (F2R (Float beta m e))).
intros He.
unfold F2R at 3. simpl.
assert (H: (Z2R m * bpow e * bpow (- e') = Z2R (m * Zpower (radix_val beta) (e + -e')))%R).
rewrite Rmult_assoc, <- bpow_add, mult_Z2R.
rewrite Z2R_Zpower.
apply refl_equal.
now apply Zle_left.
rewrite H, Ztrunc_Z2R.
unfold F2R. simpl.
rewrite <- H.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_l.
now rewrite Rmult_1_r.
Qed.

Theorem canonic_opp :
  forall m e,
  canonic (Float beta m e) ->
  canonic (Float beta (-m) e).
Proof.
intros m e H.
unfold canonic.
now rewrite <- opp_F2R, canonic_exponent_opp.
Qed.

Theorem canonic_unicity :
  forall f1 f2,
  canonic f1 ->
  canonic f2 ->
  F2R f1 = F2R f2 ->
  f1 = f2.
Proof.
intros (m1, e1) (m2, e2).
unfold canonic. simpl.
intros H1 H2 H.
rewrite H in H1.
rewrite <- H2 in H1. clear H2.
rewrite H1 in H |- *.
apply (f_equal (fun m => Float beta m e2)).
apply F2R_eq_reg with (1 := H).
Qed.

150
Theorem scaled_mantissa_generic :
151 152
  forall x,
  generic_format x ->
153
  scaled_mantissa x = Z2R (Ztrunc (scaled_mantissa x)).
154 155
Proof.
intros x Hx.
156
unfold scaled_mantissa.
157 158 159 160 161 162
pattern x at 1 3 ; rewrite Hx.
unfold F2R. simpl.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r, Rmult_1_r.
now rewrite Ztrunc_Z2R.
Qed.

163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182
Theorem scaled_mantissa_bpow :
  forall x,
  (scaled_mantissa x * bpow (canonic_exponent x))%R = x.
Proof.
intros x.
unfold scaled_mantissa.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_l.
apply Rmult_1_r.
Qed.

Theorem scaled_mantissa_opp :
  forall x,
  scaled_mantissa (-x) = (-scaled_mantissa x)%R.
Proof.
intros x.
unfold scaled_mantissa.
rewrite canonic_exponent_opp.
now rewrite Ropp_mult_distr_l_reverse.
Qed.

183 184 185 186 187
Theorem generic_format_opp :
  forall x, generic_format x -> generic_format (-x).
Proof.
intros x Hx.
unfold generic_format.
188 189 190 191
rewrite scaled_mantissa_opp, canonic_exponent_opp.
rewrite Ztrunc_opp.
rewrite <- opp_F2R.
now apply f_equal.
192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238
Qed.

Theorem canonic_exponent_fexp_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  canonic_exponent x = fexp ex.
Proof.
intros x ex Hx.
unfold canonic_exponent.
rewrite <- (Rabs_pos_eq x) in Hx.
now rewrite ln_beta_unique with (1 := Hx).
apply Rle_trans with (2 := proj1 Hx).
apply bpow_ge_0.
Qed.

Theorem canonic_exponent_fexp_neg :
  forall x ex,
  (bpow (ex - 1) <= -x < bpow ex)%R ->
  canonic_exponent x = fexp ex.
Proof.
intros x ex Hx.
unfold canonic_exponent.
rewrite <- (Rabs_left1 x) in Hx.
now rewrite ln_beta_unique with (1 := Hx).
apply Ropp_le_cancel.
rewrite Ropp_0.
apply Rle_trans with (2 := proj1 Hx).
apply bpow_ge_0.
Qed.

Theorem canonic_exponent_fexp :
  forall x ex,
  (bpow (ex - 1) <= Rabs x < bpow ex)%R ->
  canonic_exponent x = fexp ex.
Proof.
intros x ex Hx.
unfold canonic_exponent.
now rewrite ln_beta_unique with (1 := Hx).
Qed.

Theorem mantissa_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  (0 < x * bpow (- fexp ex) < 1)%R.
Proof.
intros x ex Hx He.
239
split.
240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
apply Rmult_lt_0_compat.
apply Rlt_le_trans with (2 := proj1 Hx).
apply bpow_gt_0.
apply bpow_gt_0.
apply Rmult_lt_reg_r with (bpow (fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_l.
rewrite Rmult_1_r, Rmult_1_l.
apply Rlt_le_trans with (1 := proj2 Hx).
now apply -> bpow_le.
Qed.

Theorem mantissa_DN_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  Zfloor (x * bpow (- fexp ex)) = Z0.
Proof.
intros x ex Hx He.
apply Zfloor_imp. simpl.
assert (H := mantissa_small_pos x ex Hx He).
split ; try apply Rlt_le ; apply H.
Qed.

Theorem mantissa_UP_small_pos :
  forall x ex,
  (bpow (ex - 1) <= x < bpow ex)%R ->
  (ex <= fexp ex)%Z ->
  Zceil (x * bpow (- fexp ex)) = 1%Z.
Proof.
intros x ex Hx He.
apply Zceil_imp. simpl.
assert (H := mantissa_small_pos x ex Hx He).
split ; try apply Rlt_le ; apply H.
274 275
Qed.

276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
Theorem generic_format_discrete :
  forall x m,
  let e := canonic_exponent x in
  (F2R (Float beta m e) < x < F2R (Float beta (m + 1) e))%R ->
  ~ generic_format x.
Proof.
intros x m e (Hx,Hx2) Hf.
apply Rlt_not_le with (1 := Hx2). clear Hx2.
rewrite Hf.
fold e.
apply F2R_le_compat.
apply Zlt_le_succ.
apply lt_Z2R.
rewrite <- scaled_mantissa_generic with (1 := Hf).
apply Rmult_lt_reg_r with (bpow e).
apply bpow_gt_0.
now rewrite scaled_mantissa_bpow.
Qed.

Guillaume Melquiond's avatar
Guillaume Melquiond committed
295
Theorem generic_DN_pt_large_pos_ge_pow_aux :
Guillaume Melquiond's avatar
Guillaume Melquiond committed
296 297
  forall x ex,
  (fexp ex < ex)%Z ->
Guillaume Melquiond's avatar
Guillaume Melquiond committed
298 299
  (bpow (ex - 1) <= x)%R ->
  (bpow (ex - 1) <= F2R (Float beta (Zfloor (x * bpow (- fexp ex))) (fexp ex)))%R.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
300
Proof.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
301
intros x ex He1 Hx1.
302
unfold F2R. simpl.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
303
replace (ex - 1)%Z with ((ex - 1 - fexp ex) + fexp ex)%Z by ring.
304
rewrite bpow_add.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
305
apply Rmult_le_compat_r.
306
apply bpow_ge_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
307
assert (Hx2 : bpow (ex - 1 - fexp ex) = Z2R (Zpower (radix_val beta) (ex - 1 - fexp ex))).
308 309 310 311 312 313 314 315
apply sym_eq.
apply Z2R_Zpower.
omega.
rewrite Hx2.
apply Z2R_le.
apply Zfloor_lub.
rewrite <- Hx2.
unfold Zminus at 1.
316
rewrite bpow_add.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
317
apply Rmult_le_compat_r.
318
apply bpow_ge_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
319
exact Hx1.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
320 321
Qed.

322 323 324 325 326 327
Theorem generic_format_canonic :
  forall f, canonic f ->
  generic_format (F2R f).
Proof.
intros (m, e) Hf.
unfold canonic in Hf. simpl in Hf.
328
unfold generic_format, scaled_mantissa.
329 330 331 332 333 334 335
rewrite <- Hf.
apply (f_equal (fun m => F2R (Float beta m e))).
unfold F2R. simpl.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r, Rmult_1_r.
now rewrite Ztrunc_Z2R.
Qed.

336
Section Fcore_generic_rounding_pos.
337

338 339 340 341 342 343 344 345 346 347
Record Zrounding := mkZrounding {
  Zrnd : R -> Z ;
  Zrnd_monotone : forall x y, (x <= y)%R -> (Zrnd x <= Zrnd y)%Z ;
  Zrnd_Z2R : forall n, Zrnd (Z2R n) = n
}.

Variable rnd : Zrounding.
Let Zrnd := Zrnd rnd.
Let Zrnd_monotone := Zrnd_monotone rnd.
Let Zrnd_Z2R := Zrnd_Z2R rnd.
348

349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371
Theorem Zrnd_DN_or_UP :
  forall x, Zrnd x = Zfloor x \/ Zrnd x = Zceil x.
Proof.
intros x.
destruct (Zle_or_lt (Zrnd x) (Zfloor x)) as [Hx|Hx].
left.
apply Zle_antisym with (1 := Hx).
rewrite <- (Zrnd_Z2R (Zfloor x)).
apply Zrnd_monotone.
apply Zfloor_lb.
right.
apply Zle_antisym.
rewrite <- (Zrnd_Z2R (Zceil x)).
apply Zrnd_monotone.
apply Zceil_ub.
rewrite Zceil_floor_neq.
omega.
intros H.
rewrite <- H in Hx.
rewrite Zfloor_Z2R, Zrnd_Z2R in Hx.
apply Zlt_irrefl with (1 := Hx).
Qed.

372 373 374
Definition rounding x :=
  F2R (Float beta (Zrnd (scaled_mantissa x)) (canonic_exponent x)).

375 376
Theorem rounding_monotone_pos :
  forall x y, (0 < x)%R -> (x <= y)%R -> (rounding x <= rounding y)%R.
377
Proof.
378
intros x y Hx Hxy.
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465
unfold rounding, scaled_mantissa, canonic_exponent.
destruct (ln_beta beta x) as (ex, Hex). simpl.
destruct (ln_beta beta y) as (ey, Hey). simpl.
specialize (Hex (Rgt_not_eq _ _ Hx)).
specialize (Hey (Rgt_not_eq _ _ (Rlt_le_trans _ _ _ Hx Hxy))).
rewrite Rabs_pos_eq in Hex.
2: now apply Rlt_le.
rewrite Rabs_pos_eq in Hey.
2: apply Rle_trans with (2:=Hxy); now apply Rlt_le.
assert (He: (ex <= ey)%Z).
cut (ex - 1 < ey)%Z. omega.
apply <- bpow_lt.
apply Rle_lt_trans with (1 := proj1 Hex).
apply Rle_lt_trans with (1 := Hxy).
apply Hey.
destruct (Zle_or_lt ey (fexp ey)) as [Hy1|Hy1].
rewrite (proj2 (proj2 (prop_exp ey) Hy1) ex).
apply F2R_le_compat.
apply Zrnd_monotone.
apply Rmult_le_compat_r.
apply bpow_ge_0.
exact Hxy.
now apply Zle_trans with ey.
destruct (Zle_lt_or_eq _ _ He) as [He'|He'].
destruct (Zle_or_lt ey (fexp ex)) as [Hx2|Hx2].
rewrite (proj2 (proj2 (prop_exp ex) (Zle_trans _ _ _ He Hx2)) ey Hx2).
apply F2R_le_compat.
apply Zrnd_monotone.
apply Rmult_le_compat_r.
apply bpow_ge_0.
exact Hxy.
apply Rle_trans with (F2R (Float beta (Zrnd (bpow (ey - 1) * bpow (- fexp ey))%R) (fexp ey))).
rewrite <- bpow_add.
rewrite <- (Z2R_Zpower beta (ey - 1 + -fexp ey)). 2: omega.
rewrite Zrnd_Z2R.
destruct (Zle_or_lt ex (fexp ex)) as [Hx1|Hx1].
apply Rle_trans with (F2R (Float beta 1 (fexp ex))).
apply F2R_le_compat.
rewrite <- (Zrnd_Z2R 1).
apply Zrnd_monotone.
apply Rlt_le.
exact (proj2 (mantissa_small_pos _ _ Hex Hx1)).
unfold F2R. simpl.
rewrite Z2R_Zpower. 2: omega.
rewrite <- bpow_add, Rmult_1_l.
apply -> bpow_le.
omega.
apply Rle_trans with (F2R (Float beta (Zrnd (bpow ex * bpow (- fexp ex))%R) (fexp ex))).
apply F2R_le_compat.
apply Zrnd_monotone.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Rlt_le.
apply Hex.
rewrite <- bpow_add.
rewrite <- Z2R_Zpower. 2: omega.
rewrite Zrnd_Z2R.
unfold F2R. simpl.
rewrite 2!Z2R_Zpower ; try omega.
rewrite <- 2!bpow_add.
apply -> bpow_le.
omega.
apply F2R_le_compat.
apply Zrnd_monotone.
apply Rmult_le_compat_r.
apply bpow_ge_0.
apply Hey.
rewrite He'.
apply F2R_le_compat.
apply Zrnd_monotone.
apply Rmult_le_compat_r.
apply bpow_ge_0.
exact Hxy.
Qed.

Theorem rounding_generic :
  forall x,
  generic_format x ->
  rounding x = x.
Proof.
intros x Hx.
unfold rounding.
rewrite scaled_mantissa_generic with (1 := Hx).
rewrite Zrnd_Z2R.
now apply sym_eq.
Qed.

466 467 468 469 470 471 472 473 474 475 476 477
Theorem rounding_0 :
  rounding 0 = R0.
Proof.
unfold rounding, scaled_mantissa.
rewrite Rmult_0_l.
fold (Z2R 0).
rewrite Zrnd_Z2R.
apply F2R_0.
Qed.

End Fcore_generic_rounding_pos.

478 479
Definition ZrndDN := mkZrounding Zfloor Zfloor_le Zfloor_Z2R.
Definition ZrndUP := mkZrounding Zceil Zceil_le Zceil_Z2R.
480

481 482 483 484 485 486 487 488 489 490 491 492
Theorem rounding_DN_or_UP :
  forall rnd x,
  rounding rnd x = rounding ZrndDN x \/ rounding rnd x = rounding ZrndUP x.
Proof.
intros rnd x.
unfold rounding.
unfold Zrnd at 2 4. simpl.
destruct (Zrnd_DN_or_UP rnd (scaled_mantissa x)) as [Hx|Hx].
left. now rewrite Hx.
right. now rewrite Hx.
Qed.

493
Section Fcore_generic_rounding.
494 495

Theorem rounding_monotone :
496
  forall rnd x y, (x <= y)%R -> (rounding rnd x <= rounding rnd y)%R.
497
Proof.
498
intros rnd x y Hxy.
499 500 501 502 503 504 505 506 507 508 509
destruct (total_order_T x 0) as [[Hx|Hx]|Hx].
3: now apply rounding_monotone_pos.
(* x < 0 *)
unfold rounding.
destruct (Rlt_or_le y 0) as [Hy|Hy].
(* . y < 0 *)
rewrite <- (Ropp_involutive x), <- (Ropp_involutive y).
rewrite (scaled_mantissa_opp (-x)), (scaled_mantissa_opp (-y)).
rewrite (canonic_exponent_opp (-x)), (canonic_exponent_opp (-y)).
apply Ropp_le_cancel.
rewrite 2!opp_F2R.
510 511 512
assert (Hrnd_monotone : forall x y, (x <= y)%R -> (- Zrnd rnd (-x) <= - Zrnd rnd (-y))%Z).
clear.
intros x y Hxy.
513 514 515 516
apply Zopp_le_cancel.
rewrite 2!Zopp_involutive.
apply Zrnd_monotone.
now apply Ropp_le_contravar.
517
assert (Hrnd_Z2R : forall n, (- Zrnd rnd (- Z2R n))%Z = n).
518 519 520
intros n.
rewrite <- opp_Z2R, Zrnd_Z2R.
apply Zopp_involutive.
521
apply (rounding_monotone_pos (mkZrounding (fun m => (- Zrnd rnd (- m))%Z) Hrnd_monotone Hrnd_Z2R)).
522 523 524 525 526 527
rewrite <- Ropp_0.
now apply Ropp_lt_contravar.
now apply Ropp_le_contravar.
(* . 0 <= y *)
apply Rle_trans with R0.
apply F2R_le_0_compat. simpl.
528
rewrite <- (Zrnd_Z2R rnd 0).
529 530 531 532 533 534 535
apply Zrnd_monotone.
simpl.
rewrite <- (Rmult_0_l (bpow (- fexp (projT1 (ln_beta beta x))))).
apply Rmult_le_compat_r.
apply bpow_ge_0.
now apply Rlt_le.
apply F2R_ge_0_compat. simpl.
536
rewrite <- (Zrnd_Z2R rnd 0).
537 538 539 540 541 542
apply Zrnd_monotone.
apply Rmult_le_pos.
exact Hy.
apply bpow_ge_0.
(* x = 0 *)
rewrite Hx.
543
rewrite rounding_0.
544 545
apply F2R_ge_0_compat.
simpl.
546
rewrite <- (Zrnd_Z2R rnd 0).
547 548 549 550 551 552
apply Zrnd_monotone.
apply Rmult_le_pos.
now rewrite <- Hx.
apply bpow_ge_0.
Qed.

553 554
End Fcore_generic_rounding.

Guillaume Melquiond's avatar
Guillaume Melquiond committed
555
Theorem generic_DN_pt_pos :
556
  forall x, (0 < x)%R ->
557
  Rnd_DN_pt generic_format x (rounding ZrndDN x).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
558
Proof.
559
intros x H0x.
560
unfold rounding, scaled_mantissa, canonic_exponent.
561 562 563 564
destruct (ln_beta beta x) as (ex, He).
simpl.
specialize (He (Rgt_not_eq _ _ H0x)).
rewrite (Rabs_pos_eq _ (Rlt_le _ _ H0x)) in He.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
565 566
destruct (Z_lt_le_dec (fexp ex) ex) as [He1|He1].
(* - positive big enough *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
567
assert (Hbl : (bpow (ex - 1) <= F2R (Float beta (Zfloor (x * bpow (- fexp ex))) (fexp ex)))%R).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
568
now apply generic_DN_pt_large_pos_ge_pow_aux.
569
(* - . smaller *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
570
assert (Hrx : (F2R (Float beta (Zfloor (x * bpow (- fexp ex))) (fexp ex)) <= x)%R).
571
unfold F2R. simpl.
572 573 574
apply Rmult_le_reg_r with (bpow (- fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r, Rmult_1_r.
575
apply Zfloor_lb.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
576 577
split.
(* - . rounded *)
578
apply generic_format_canonic.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
579
apply sym_eq.
580
apply canonic_exponent_fexp_pos.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
581 582
split.
exact Hbl.
583
now apply Rle_lt_trans with (2 := proj2 He).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
584
split.
585
exact Hrx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
586
(* - . biggest *)
587
intros g Hg Hgx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
588 589 590
destruct (Rle_or_lt g R0) as [Hg3|Hg3].
apply Rle_trans with (2 := Hbl).
apply Rle_trans with (1 := Hg3).
591
apply bpow_ge_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
592 593
apply Rnot_lt_le.
intros Hrg.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
594
assert (bpow (ex - 1) <= g < bpow ex)%R.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
595 596 597 598
split.
apply Rle_trans with (1 := Hbl).
now apply Rlt_le.
now apply Rle_lt_trans with (1 := Hgx).
599 600
assert (Hcg: canonic_exponent g = fexp ex).
unfold canonic_exponent.
601
rewrite <- (Rabs_pos_eq g (Rlt_le _ _ Hg3)) in H.
602
now rewrite ln_beta_unique with (1 := H).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
603
apply Rlt_not_le with (1 := Hrg).
604 605
rewrite Hg.
rewrite Hcg.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
606
apply F2R_le_compat.
607
apply Zfloor_lub.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
608
apply Rmult_le_reg_r with (bpow (fexp ex)).
609
apply bpow_gt_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
610
rewrite Rmult_assoc.
611
rewrite <- bpow_add.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
612 613
rewrite Zplus_opp_l.
rewrite Rmult_1_r.
614 615
rewrite <- Hcg.
now rewrite Hg in Hgx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
616
(* - positive too small *)
617
rewrite mantissa_DN_small_pos with (1 := He) (2 := He1).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
618
rewrite F2R_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
619
split.
620
(* - . rounded *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
621
exact generic_format_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
622
split.
623
now apply Rlt_le.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
624
(* - . biggest *)
625
intros g Hg Hgx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
626 627
apply Rnot_lt_le.
intros Hg3.
628 629
destruct (ln_beta beta g) as (ge, Hg4).
simpl in Hg.
630
specialize (Hg4 (Rgt_not_eq _ _ Hg3)).
631 632 633
assert (Hcg: canonic_exponent g = fexp ge).
unfold canonic_exponent.
now rewrite ln_beta_unique with (1 := Hg4).
634
rewrite Rabs_pos_eq in Hg4.
635 636
apply (Rlt_not_le _ _ (Rle_lt_trans _ _ _ Hgx (proj2 He))).
apply Rle_trans with (bpow (fexp ge)).
637
apply -> bpow_le.
638
rewrite (proj2 (proj2 (prop_exp ex) He1) ge).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
639 640
exact He1.
apply Zle_trans with (2 := He1).
641
cut (ge - 1 < ex)%Z. omega.
642
apply <- bpow_lt.
643
apply Rle_lt_trans with (2 := proj2 He).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
644 645
apply Rle_trans with (2 := Hgx).
exact (proj1 Hg4).
646
rewrite Hg.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
647
rewrite <- F2R_bpow.
648
rewrite Hcg.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
649
apply F2R_le_compat.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
650
apply (Zlt_le_succ 0).
651 652 653
apply F2R_lt_reg with beta (fexp ge).
rewrite F2R_0, <- Hcg.
now rewrite Hg in Hg3.
654
now apply Rlt_le.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
655 656 657
Qed.

Theorem generic_DN_pt_neg :
658
  forall x, (x < 0)%R ->
659
  Rnd_DN_pt generic_format x (rounding ZrndDN x).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
660
Proof.
661
intros x Hx0.
662
unfold rounding, scaled_mantissa, canonic_exponent.
663 664 665 666
destruct (ln_beta beta x) as (ex, He).
simpl.
specialize (He (Rlt_not_eq _ _ Hx0)).
rewrite (Rabs_left _ Hx0) in He.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
667
assert (Hbr : (F2R (Float beta (Zfloor (x * bpow (- fexp ex))) (fexp ex)) <= x)%R).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
668 669
(* - bounded right *)
unfold F2R. simpl.
670 671 672
apply Rmult_le_reg_r with (bpow (-fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r, Rmult_1_r.
673
apply Zfloor_lb.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
674 675
destruct (Z_lt_le_dec (fexp ex) ex) as [He1|He1].
(* - negative big enough *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
676
assert (Hbl : (- bpow ex <= F2R (Float beta (Zfloor (x * bpow (- fexp ex))) (fexp ex)))%R).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
677 678
(* - . bounded left *)
unfold F2R. simpl.
679 680 681 682 683 684
apply Rmult_le_reg_r with (bpow (-fexp ex)).
apply bpow_gt_0.
rewrite Rmult_assoc, <- bpow_add, Zplus_opp_r, Rmult_1_r.
assert (Hp : (- bpow ex * bpow (-fexp ex) = Z2R (- Zpower (radix_val beta) (ex - fexp ex)))%R).
rewrite Ropp_mult_distr_l_reverse.
rewrite <- bpow_add, <- Z2R_Zpower.
685
now rewrite opp_Z2R.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
686
omega.
687 688 689
rewrite Hp.
apply Z2R_le.
apply Zfloor_lub.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
690
rewrite <- Hp.
691
apply Rmult_le_compat_r.
692
apply bpow_ge_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
693 694
apply Ropp_le_cancel.
rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
695 696 697 698 699
now apply Rlt_le.
split.
(* - . rounded *)
destruct (Rle_lt_or_eq_dec _ _ Hbl) as [Hbl2|Hbl2].
(* - . . not a radix power *)
700 701
apply generic_format_canonic.
assert (Hb: (bpow (ex - 1) <= - F2R (Float beta (Zfloor (x * bpow (- fexp ex))) (fexp ex)) < bpow ex)%R).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
702
split.
703 704
apply Rle_trans with (1 := proj1 He).
now apply Ropp_le_contravar.
705 706
apply Ropp_lt_cancel.
now rewrite Ropp_involutive.
707 708
apply sym_eq.
apply canonic_exponent_fexp_neg with (1 := Hb).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
709 710
(* - . . a radix power *)
rewrite <- Hbl2.
711 712 713
apply generic_format_opp.
apply generic_format_bpow.
exact (proj1 (prop_exp _) He1).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
714 715 716
split.
exact Hbr.
(* - . biggest *)
717
intros g Hg Hgx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
718 719 720 721
apply Rnot_lt_le.
intros Hg3.
assert (Hg4 : (g < 0)%R).
now apply Rle_lt_trans with (1 := Hgx).
722
destruct (ln_beta beta g) as (ge, Hge).
723
specialize (Hge (Rlt_not_eq _ _ Hg4)).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
724
apply Rlt_not_le with (1 := Hg3).
725 726
rewrite Hg.
assert (Hge' : ge = ex).
727
apply bpow_unique with (1 := Hge).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
728
split.
729
apply Rle_trans with (1 := proj1 He).
730
rewrite Rabs_left with (1 := Hg4).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
731 732
now apply Ropp_le_contravar.
apply Ropp_lt_cancel.
733 734 735
rewrite Rabs_left with (1 := Hg4).
rewrite Ropp_involutive.
now apply Rle_lt_trans with (1 := Hbl).
736 737 738 739 740
assert (Hcg: canonic_exponent g = fexp ex).
rewrite <- Hge'.
now apply canonic_exponent_fexp.
rewrite Hcg.
apply F2R_le_compat.
741
apply Zfloor_lub.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
742
apply Rmult_le_reg_r with (bpow (fexp ex)).
743
apply bpow_gt_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
744
rewrite Rmult_assoc.
745 746 747
rewrite <- bpow_add, Zplus_opp_l, Rmult_1_r.
rewrite <- Hcg.
now rewrite Hg in Hgx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
748
(* - negative too small *)
749 750 751 752 753
rewrite <- (Zopp_involutive (Zfloor (x * bpow (- fexp ex)))).
rewrite <- (Ropp_involutive x) at 2.
rewrite Ropp_mult_distr_l_reverse.
change (- Zfloor (- (- x * bpow (- fexp ex))))%Z with (Zceil (- x * bpow (- fexp ex)))%Z.
rewrite mantissa_UP_small_pos ; try assumption.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
754 755 756 757 758
unfold F2R. simpl.
rewrite Ropp_mult_distr_l_reverse.
rewrite Rmult_1_l.
(* - . rounded *)
split.
759 760 761
apply generic_format_opp.
apply generic_format_bpow.
exact (proj1 (proj2 (prop_exp _) He1)).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
762 763
split.
(* - . smaller *)
764 765
apply Ropp_le_cancel.
rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
766
apply Rlt_le.
767
apply Rlt_le_trans with (1 := proj2 He).
768
now apply -> bpow_le.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
769
(* - . biggest *)
770
intros g Hg Hgx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
771 772 773 774
apply Rnot_lt_le.
intros Hg3.
assert (Hg4 : (g < 0)%R).
now apply Rle_lt_trans with (1 := Hgx).
775 776
destruct (ln_beta beta g) as (ge, Hge).
simpl in Hg.
777
specialize (Hge (Rlt_not_eq g 0 Hg4)).
778
rewrite (Rabs_left _ Hg4) in Hge.
779 780
assert (Hge' : (ge <= fexp ex)%Z).
cut (ge - 1 < fexp ex)%Z. omega.
781
apply <- bpow_lt.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
782 783 784
apply Rle_lt_trans with (1 := proj1 Hge).
apply Ropp_lt_cancel.
now rewrite Ropp_involutive.
785 786 787 788 789
assert (Hcg: canonic_exponent g = fexp ex).
unfold canonic_exponent.
rewrite <- Rabs_left with (1 := Hg4) in Hge.
rewrite ln_beta_unique with (1 := Hge).
exact (proj2 (proj2 (prop_exp _) He1) _ Hge').
Guillaume Melquiond's avatar
Guillaume Melquiond committed
790
apply Rlt_not_le with (1 := proj2 Hge).
791
rewrite Hg.
792
unfold scaled_mantissa, F2R. simpl.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
793
rewrite <- Ropp_mult_distr_l_reverse.
794 795
rewrite Hcg.
pattern (fexp ex) at 2 ; replace (fexp ex) with (fexp ex - ge + ge)%Z by ring.
796
rewrite bpow_add.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
797
rewrite <- Rmult_assoc.
798
pattern (bpow ge) at 1 ; rewrite <- Rmult_1_l.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
799
apply Rmult_le_compat_r.
800
apply bpow_ge_0.
801 802 803 804 805 806
assert (- Z2R (Ztrunc (g * bpow (- fexp ex))) * bpow (fexp ex - ge) = Z2R (- Ztrunc (g * bpow (-fexp ex)) * Zpower (radix_val beta) (fexp ex - ge)))%R.
rewrite mult_Z2R.
rewrite Z2R_Zpower. 2: omega.
now rewrite opp_Z2R.
rewrite H.
apply (Z2R_le 1).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
807 808
apply (Zlt_le_succ 0).
apply lt_Z2R.
809 810 811 812
rewrite <- H.
unfold Zminus.
rewrite bpow_add, <- Rmult_assoc.
apply Rmult_lt_0_compat.
813
rewrite <- Ropp_0.
814
rewrite Ropp_mult_distr_l_reverse.
815 816 817
apply Ropp_lt_contravar.
rewrite <- Hcg.
now rewrite Hg in Hg4.
818
apply bpow_gt_0.
819 820 821 822 823
Qed.

Theorem generic_format_satisfies_any :
  satisfies_any generic_format.
Proof.
824
split.
825
(* symmetric set *)
826
exact generic_format_0.
827
exact generic_format_opp.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
828
(* rounding down *)
829 830
intros x.
exists (match Req_EM_T x 0 with
831
  | left Hx => R0
832
  | right Hx => F2R (Float beta (Zfloor (x * bpow (- canonic_exponent x))) (canonic_exponent x))
Guillaume Melquiond's avatar
Guillaume Melquiond committed
833
  end).
834 835
destruct (Req_EM_T x 0) as [Hx|Hx].
(* . *)
Guillaume Melquiond's avatar
Guillaume Melquiond committed
836
split.
837 838
apply generic_format_0.
rewrite Hx.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
839 840
split.
apply Rle_refl.
841 842 843
now intros g _ H.
(* . *)
destruct (ln_beta beta x) as (ex, H1).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
844
simpl.
845 846
specialize (H1 Hx).
destruct (Rdichotomy _ _ Hx) as [H2|H2].
847 848 849 850 851 852
now apply generic_DN_pt_neg.
now apply generic_DN_pt_pos.
Qed.

Theorem generic_DN_pt :
  forall x,
853
  Rnd_DN_pt generic_format x (rounding ZrndDN x).
854 855 856 857
Proof.
intros x.
destruct (total_order_T 0 x) as [[Hx|Hx]|Hx].
now apply generic_DN_pt_pos.
858
unfold rounding, scaled_mantissa.
859 860
rewrite <- Hx, Rmult_0_l.
fold (Z2R 0).
861
rewrite Zfloor_Z2R, F2R_0.
862 863 864 865 866
apply Rnd_DN_pt_refl.
apply generic_format_0.
now apply generic_DN_pt_neg.
Qed.

867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894
Theorem generic_DN_opp :
  forall x,
  rounding ZrndDN (-x) = (- rounding ZrndUP x)%R.
Proof.
intros x.
unfold rounding.
rewrite scaled_mantissa_opp.
rewrite opp_F2R.
unfold Zrnd. simpl.
unfold Zceil.
rewrite Zopp_involutive.
now rewrite canonic_exponent_opp.
Qed.

Theorem generic_UP_opp :
  forall x,
  rounding ZrndUP (-x) = (- rounding ZrndDN x)%R.
Proof.
intros x.
unfold rounding.
rewrite scaled_mantissa_opp.
rewrite opp_F2R.
unfold Zrnd. simpl.
unfold Zceil.
rewrite Ropp_involutive.
now rewrite canonic_exponent_opp.
Qed.

895 896
Theorem generic_UP_pt :
  forall x,
897
  Rnd_UP_pt generic_format x (rounding ZrndUP x).
898 899 900 901
Proof.
intros x.
apply Rnd_DN_UP_pt_sym.
apply generic_format_satisfies_any.
902 903 904
pattern x at 2 ; rewrite <- Ropp_involutive.
rewrite generic_UP_opp.
rewrite Ropp_involutive.
905
apply generic_DN_pt.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
906 907
Qed.

908 909 910 911 912 913 914 915 916 917 918
Theorem generic_format_rounding :
  forall rnd x,
  generic_format (rounding rnd x).
Proof.
intros rnd x.
destruct (rounding_DN_or_UP rnd x) as [H|H] ; rewrite H.
apply (generic_DN_pt x).
apply (generic_UP_pt x).
Qed.

Theorem generic_DN_small_pos :
919
  forall x ex,
Guillaume Melquiond's avatar
Guillaume Melquiond committed
920
  (bpow (ex - 1) <= x < bpow ex)%R ->
921
  (ex <= fexp ex)%Z ->
922
  rounding ZrndDN x = R0.
923 924
Proof.
intros x ex Hx He.
925 926
rewrite <- (F2R_0 beta (canonic_exponent x)).
rewrite <- mantissa_DN_small_pos with (1 := Hx) (2 := He).
927
now rewrite <- canonic_exponent_fexp_pos with (1 := Hx).
928 929
Qed.

930
Theorem generic_UP_small_pos :
931
  forall x ex,
Guillaume Melquiond's avatar
Guillaume Melquiond committed
932
  (bpow (ex - 1) <= x < bpow ex)%R ->
933
  (ex <= fexp ex)%Z ->
934
  rounding ZrndUP x = (bpow (fexp ex)).
935 936
Proof.
intros x ex Hx He.
937 938
rewrite <- F2R_bpow.
rewrite <- mantissa_UP_small_pos with (1 := Hx) (2 := He).
939
now rewrite <- canonic_exponent_fexp_pos with (1 := Hx).
940 941
Qed.

942 943
Theorem generic_UP_large_pos_le_pow :
  forall x ex,
Guillaume Melquiond's avatar
Guillaume Melquiond committed
944
  (bpow (ex - 1) <= x < bpow ex)%R ->
945
  (fexp ex < ex)%Z ->
946
  (rounding ZrndUP x <= bpow ex)%R.
947
Proof.
948 949
intros x ex Hx He.
apply (generic_UP_pt x).
950 951
apply generic_format_bpow.
exact (proj1 (prop_exp _) He).
952 953
apply Rlt_le.
apply Hx.
954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974
Qed.

Theorem generic_format_EM :
  forall x,
  generic_format x \/ ~generic_format x.
Proof.
intros x.
destruct (proj1 (satisfies_any_imp_DN _ generic_format_satisfies_any) x) as (d, Hd).
destruct (Rle_lt_or_eq_dec d x) as [Hxd|Hxd].
apply Hd.
right.
intros Fx.
apply Rlt_not_le with (1 := Hxd).
apply Req_le.
apply sym_eq.
now apply Rnd_DN_pt_idempotent with (1 := Hd).
left.
rewrite <- Hxd.
apply Hd.
Qed.

975 976 977
Theorem generic_DN_large_pos_ge_pow :
  forall x e,
  (0 < rounding ZrndDN x)%R ->
Guillaume Melquiond's avatar
Guillaume Melquiond committed
978
  (bpow e <= x)%R ->
979
  (bpow e <= rounding ZrndDN x)%R.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
980
Proof.
981
intros x e Hd Hex.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
982 983 984
destruct (ln_beta beta x) as (ex, He).
assert (Hx: (0 < x)%R).
apply Rlt_le_trans with (1 := Hd).
985
apply (generic_DN_pt x).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
986 987 988 989 990 991 992
specialize (He (Rgt_not_eq _ _ Hx)).
rewrite Rabs_pos_eq in He. 2: now apply Rlt_le.
apply Rle_trans with (bpow (ex - 1)).
apply -> bpow_le.
cut (e < ex)%Z. omega.
apply <- bpow_lt.
now apply Rle_lt_trans with (2 := proj2 He).
993
apply (generic_DN_pt x) with (2 := proj1 He).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
994 995 996
apply generic_format_bpow.
destruct (Zle_or_lt ex (fexp ex)).
elim Rgt_not_eq with (1 := Hd).
997
now apply generic_DN_small_pos with (1 := He).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
998 999 1000 1001
ring_simplify (ex - 1 + 1)%Z.
omega.
Qed.

1002 1003 1004 1005
Theorem canonic_exponent_DN :
  forall x,
  (0 < rounding ZrndDN x)%R ->
  canonic_exponent (rounding ZrndDN x) = canonic_exponent x.
1006
Proof.
1007
intros x Hd.
1008 1009 1010
unfold canonic_exponent.
apply f_equal.
apply ln_beta_unique.
1011
rewrite (Rabs_pos_eq (rounding ZrndDN x)). 2: now apply Rlt_le.
1012 1013 1014 1015
destruct (ln_beta beta x) as (ex, He).
simpl.
assert (Hx: (0 < x)%R).
apply Rlt_le_trans with (1 := Hd).
1016
apply (generic_DN_pt x).
1017 1018 1019
specialize (He (Rgt_not_eq _ _ Hx)).
rewrite Rabs_pos_eq in He. 2: now apply Rlt_le.
split.
1020 1021
apply generic_DN_large_pos_ge_pow with (1 := Hd).
apply He.
1022
apply Rle_lt_trans with (2 := proj2 He).
1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038
apply (generic_DN_pt x).
Qed.

Theorem generic_N_pt_DN_or_UP :
  forall x f,
  Rnd_N_pt generic_format x f ->
  f = rounding ZrndDN x \/ f = rounding ZrndUP x.
Proof.
intros x f Hxf.
destruct (Rnd_N_pt_DN_or_UP _ _ _ Hxf).
left.
apply Rnd_DN_pt_unicity with (1 := H).
apply generic_DN_pt.
right.
apply Rnd_UP_pt_unicity with (1 := H).
apply generic_UP_pt.
1039 1040
Qed.

1041
End RND_generic.