Flocq_rnd_generic.v 13.7 KB
Newer Older
Guillaume Melquiond's avatar
Guillaume Melquiond committed
1 2 3
Require Import Flocq_Raux.
Require Import Flocq_defs.
Require Import Flocq_rnd_ex.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
4
Require Import Flocq_float_prop.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
5 6 7 8 9 10 11 12 13

Section RND_generic.

Variable beta : radix.

Notation bpow := (epow beta).

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

Definition generic_format (x : R) :=
  exists f : float beta,
25 26
  x = F2R f /\ forall (H : x <> R0),
  Fexp f = fexp (projT1 (ln_beta beta _ (Rabs_pos_lt _ H))).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
27 28 29 30 31 32 33 34

Theorem generic_format_satisfies_any :
  satisfies_any generic_format.
Proof.
refine ((fun D => Satisfies_any _ _ _ (projT1 D) (projT2 D)) _).
(* symmetric set *)
exists (Float beta 0 0).
split.
35
unfold F2R. simpl.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
36
now rewrite Rmult_0_l.
37 38
intros H.
now elim H.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
39 40 41 42
intros x ((m,e),(H1,H2)).
exists (Float beta (-m) e).
split.
rewrite H1.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
43
apply opp_F2R.
44 45 46 47 48 49 50 51 52 53 54 55
intros H3.
simpl in H2.
assert (H4 := Ropp_neq_0_compat _ H3).
rewrite Ropp_involutive in H4.
rewrite (H2 H4).
clear H2.
destruct (ln_beta beta (Rabs x)) as (ex, H5).
simpl.
apply f_equal.
apply sym_eq.
apply ln_beta_unique.
now rewrite Rabs_Ropp.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
(* rounding down *)
assert (Hxx : forall x, (0 > x)%R -> (0 < -x)%R).
intros.
now apply Ropp_0_gt_lt_contravar.
exists (fun x =>
  match total_order_T 0 x with
  | inleft (left Hx) =>
    let e := fexp (projT1 (ln_beta beta _ Hx)) in
    F2R (Float beta (up (x * bpow (Zopp e)) - 1) e)
  | inleft (right _) => R0
  | inright Hx =>
    let e := fexp (projT1 (ln_beta beta _ (Hxx _ Hx))) in
    F2R (Float beta (up (x * bpow (Zopp e)) - 1) e)
  end).
intros x.
destruct (total_order_T 0 x) as [[Hx|Hx]|Hx].
(* positive *)
clear Hxx.
destruct (ln_beta beta x Hx) as (ex, (Hx1, Hx2)).
simpl.
destruct (Z_lt_le_dec (fexp ex) ex) as [He1|He1].
(* - positive big enough *)
assert (Hbl : (bpow (ex - 1)%Z <= F2R (Float beta (up (x * bpow (- fexp ex)%Z) - 1) (fexp ex)))%R).
(* - . bounded left *)
clear Hx2.
unfold F2R.
simpl.
replace (ex - 1)%Z with ((ex - 1 - fexp ex) + fexp ex)%Z by ring.
rewrite epow_add.
apply Rmult_le_compat_r.
apply epow_ge_0.
assert (bpow (ex - 1 - fexp ex)%Z < Z2R (up (x * bpow (- fexp ex)%Z)))%R.
rewrite Z2R_IZR.
apply Rle_lt_trans with (2 := proj1 (archimed _)).
unfold Zminus.
rewrite epow_add.
apply Rmult_le_compat_r.
apply epow_ge_0.
exact Hx1.
case_eq (ex - 1 - fexp ex)%Z.
intros He2.
change (bpow 0%Z) with (Z2R 1).
apply Z2R_le.
change 1%Z at 1 with (1 + 1 - 1)%Z.
apply Zplus_le_compat_r.
apply (Zlt_le_succ 1).
apply lt_Z2R.
now rewrite He2 in H.
intros ep He2.
simpl.
apply Z2R_le.
replace (Zpower_pos (radix_val beta) ep) with (Zpower_pos (radix_val beta) ep + 1 - 1)%Z by ring.
apply Zplus_le_compat_r.
apply Zlt_le_succ.
apply lt_Z2R.
change (bpow (Zpos ep) < Z2R (up (x * bpow (- fexp ex)%Z)))%R.
now rewrite <- He2.
clear H Hx1.
intros.
assert (ex - 1 - fexp ex < 0)%Z.
now rewrite H.
apply False_ind.
omega.
split.
(* - . rounded *)
121
eexists ; split ; [ reflexivity | idtac ].
Guillaume Melquiond's avatar
Guillaume Melquiond committed
122 123 124 125 126 127 128 129 130 131 132 133
intros He9.
simpl.
apply f_equal.
apply sym_eq.
apply ln_beta_unique.
clear He9.
rewrite Rabs_right.
split.
exact Hbl.
(* - . . bounded right *)
clear Hbl.
apply Rle_lt_trans with (2 := Hx2).
134
unfold F2R. simpl.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157
pattern x at 2 ; replace x with ((x * bpow (- fexp ex)%Z) * bpow (fexp ex))%R.
generalize (x * bpow (- fexp ex)%Z)%R.
clear.
intros x.
apply Rmult_le_compat_r.
apply epow_ge_0.
rewrite minus_Z2R.
rewrite Z2R_IZR.
simpl.
apply Rplus_le_reg_l with (- x + 1)%R.
ring_simplify.
rewrite Rplus_comm.
exact (proj2 (archimed x)).
rewrite Rmult_assoc.
rewrite <- epow_add.
rewrite Zplus_opp_l.
apply Rmult_1_r.
(* - . . *)
apply Rle_ge.
apply Rle_trans with (2 := Hbl).
apply epow_ge_0.
split.
(* - . smaller *)
158
unfold F2R. simpl.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181
generalize (fexp ex).
clear.
intros e.
pattern x at 2 ; rewrite <- Rmult_1_r.
change R1 with (bpow Z0).
rewrite <- (Zplus_opp_l e).
rewrite epow_add, <- Rmult_assoc.
apply Rmult_le_compat_r.
apply epow_ge_0.
rewrite minus_Z2R.
rewrite Z2R_IZR.
simpl.
apply Rplus_le_reg_l with (1 - x * bpow (-e)%Z)%R.
ring_simplify.
rewrite Rplus_comm.
rewrite Ropp_mult_distr_l_reverse.
exact (proj2 (archimed _)).
(* - . biggest *)
intros g ((gm, ge), (Hg1, Hg2)) Hgx.
destruct (Rle_or_lt g R0) as [Hg3|Hg3].
apply Rle_trans with (2 := Hbl).
apply Rle_trans with (1 := Hg3).
apply epow_ge_0.
182
specialize (Hg2 (Rgt_not_eq _ _ Hg3)).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
183 184 185 186 187 188 189
apply Rnot_lt_le.
intros Hrg.
assert (bpow (ex - 1)%Z <= g < bpow ex)%R.
split.
apply Rle_trans with (1 := Hbl).
now apply Rlt_le.
now apply Rle_lt_trans with (1 := Hgx).
190
rewrite <- (Rabs_pos_eq g (Rlt_le _ _ Hg3)) in H.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223
rewrite ln_beta_unique with (1 := H) in Hg2.
simpl in Hg2.
apply Rlt_not_le with (1 := Hrg).
rewrite Hg1, Hg2.
unfold F2R. simpl.
apply Rmult_le_compat_r.
apply epow_ge_0.
apply Z2R_le.
cut (gm < up (x * bpow (- fexp ex)%Z))%Z.
omega.
apply lt_IZR.
apply Rle_lt_trans with (2 := proj1 (archimed _)).
apply Rmult_le_reg_r with (bpow (fexp ex)).
apply epow_gt_0.
rewrite <- Hg2 at 1.
rewrite <- Z2R_IZR.
rewrite Rmult_assoc.
rewrite <- epow_add.
rewrite Zplus_opp_l.
rewrite Rmult_1_r.
unfold F2R in Hg1.
simpl in Hg1.
now rewrite <- Hg1.
(* - positive too small *)
cutrewrite (up (x * bpow (- fexp ex)%Z) = 1%Z).
(* - . rounded *)
unfold F2R. simpl.
rewrite Rmult_0_l.
split.
exists (Float beta Z0 (fexp ex)).
split.
unfold F2R. simpl.
now rewrite Rmult_0_l.
224 225
intros H.
now elim H.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
226 227 228 229 230 231
split.
now apply Rlt_le.
(* - . biggest *)
intros g ((gm, ge), (Hg1, Hg2)) Hgx.
apply Rnot_lt_le.
intros Hg3.
232
specialize (Hg2 (Rgt_not_eq _ _ Hg3)).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
233
destruct (ln_beta beta g Hg3) as (ge', Hg4).
234 235 236
generalize Hg4. intros Hg5.
rewrite <- (Rabs_pos_eq g (Rlt_le _ _ Hg3)) in Hg5.
rewrite ln_beta_unique with (1 := Hg5) in Hg2.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
237 238 239
apply (Rlt_not_le _ _ (Rle_lt_trans _ _ _ Hgx Hx2)).
apply Rle_trans with (bpow ge).
apply -> epow_le.
240
simpl in Hg2.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
241
rewrite Hg2.
242
rewrite (proj2 (proj2 (prop_exp ex) He1) ge').
Guillaume Melquiond's avatar
Guillaume Melquiond committed
243 244
exact He1.
apply Zle_trans with (2 := He1).
245
cut (ge' - 1 < ex)%Z. omega.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283
apply <- epow_lt.
apply Rle_lt_trans with (2 := Hx2).
apply Rle_trans with (2 := Hgx).
exact (proj1 Hg4).
rewrite Hg1.
unfold F2R. simpl.
pattern (bpow ge) at 1 ; rewrite <- Rmult_1_l.
apply Rmult_le_compat_r.
apply epow_ge_0.
apply (Z2R_le 1).
apply (Zlt_le_succ 0).
apply lt_Z2R.
apply Rmult_lt_reg_r with (bpow ge).
apply epow_gt_0.
rewrite Rmult_0_l.
unfold F2R in Hg1. simpl in Hg1.
now rewrite <- Hg1.
(* - . . *)
apply sym_eq.
rewrite <- (Zplus_0_l 1).
apply up_tech.
apply Rlt_le.
apply Rmult_lt_0_compat.
exact Hx.
apply epow_gt_0.
change (IZR (0 + 1)) with (bpow Z0).
rewrite <- (Zplus_opp_r (fexp ex)).
rewrite epow_add.
apply Rmult_lt_compat_r.
apply epow_gt_0.
apply Rlt_le_trans with (1 := Hx2).
now apply -> epow_le.
(* zero *)
split.
exists (Float beta 0 0).
split.
unfold F2R.
now rewrite Rmult_0_l.
284 285
intros H.
now elim H.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346
rewrite <- Hx.
split.
apply Rle_refl.
intros g _ H.
exact H.
(* negative *)
destruct (ln_beta beta (- x) (Hxx x Hx)) as (ex, (Hx1, Hx2)).
simpl.
clear Hxx.
assert (Hbr : (F2R (Float beta (up (x * bpow (- fexp ex)%Z) - 1) (fexp ex)) <= x)%R).
(* - bounded right *)
unfold F2R. simpl.
pattern x at 2 ; rewrite <- Rmult_1_r.
change R1 with (bpow Z0).
rewrite <- (Zplus_opp_l (fexp ex)).
rewrite epow_add.
rewrite <- Rmult_assoc.
generalize (x * bpow (- fexp ex)%Z)%R.
clear.
intros x.
apply Rmult_le_compat_r.
apply epow_ge_0.
rewrite minus_Z2R.
simpl.
rewrite Z2R_IZR.
apply Rplus_le_reg_l with (-x + 1)%R.
ring_simplify.
rewrite Rplus_comm.
exact (proj2 (archimed x)).
destruct (Z_lt_le_dec (fexp ex) ex) as [He1|He1].
(* - negative big enough *)
assert (Hbl : (- bpow ex <= F2R (Float beta (up (x * bpow (- fexp ex)%Z) - 1) (fexp ex)))%R).
(* - . bounded left *)
unfold F2R. simpl.
pattern ex at 1 ; replace ex with (ex - fexp ex + fexp ex)%Z by ring.
rewrite epow_add.
rewrite <- Ropp_mult_distr_l_reverse.
apply Rmult_le_compat_r.
apply epow_ge_0.
cut (0 < ex - fexp ex)%Z. 2: omega.
case_eq (ex - fexp ex)%Z ; try (intros ; discriminate H0).
intros ep Hp _.
simpl.
rewrite <- opp_Z2R.
apply Z2R_le.
cut (- Zpower_pos (radix_val beta) ep < up (x * bpow (- fexp ex)%Z))%Z.
omega.
apply lt_Z2R.
apply Rle_lt_trans with (x * bpow (- fexp ex)%Z)%R.
rewrite opp_Z2R.
change (- bpow (Zpos ep) <= x * bpow (- fexp ex)%Z)%R.
rewrite <- Hp.
apply Rmult_le_reg_r with (bpow (fexp ex)).
apply epow_gt_0.
rewrite Rmult_assoc.
rewrite <- epow_add.
rewrite Zplus_opp_l.
rewrite Rmult_1_r.
rewrite Ropp_mult_distr_l_reverse.
rewrite <- epow_add.
replace (ex - fexp ex + fexp ex)%Z with ex by ring.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
347 348
apply Ropp_le_cancel.
rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
349 350 351 352 353 354 355 356 357 358 359 360 361
now apply Rlt_le.
rewrite Z2R_IZR.
exact (proj1 (archimed _)).
split.
(* - . rounded *)
destruct (Rle_lt_or_eq_dec _ _ Hbl) as [Hbl2|Hbl2].
(* - . . not a radix power *)
eexists ; split ; [ reflexivity | idtac ].
intros Hr.
simpl.
apply f_equal.
apply sym_eq.
apply ln_beta_unique.
362
rewrite Rabs_left.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
363
split.
364 365
apply Ropp_le_cancel.
rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
366
apply Rle_trans with (1 := Hbr).
367 368 369 370
apply Ropp_le_cancel.
now rewrite Ropp_involutive.
apply Ropp_lt_cancel.
now rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
371 372 373 374
apply Rle_lt_trans with (1 := Hbr).
exact Hx.
(* - . . a radix power *)
rewrite <- Hbl2.
375
generalize (proj1 (prop_exp _) He1).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
376 377 378 379 380 381 382 383 384 385 386 387 388 389 390
clear.
intros He2.
exists (Float beta (- Zpower (radix_val beta) (ex - fexp (ex + 1))) (fexp (ex + 1))).
unfold F2R. simpl.
split.
clear -He2.
pattern ex at 1 ; replace ex with (ex - fexp (ex + 1) + fexp (ex + 1))%Z by ring.
rewrite epow_add.
rewrite <- Ropp_mult_distr_l_reverse.
rewrite opp_Z2R.
apply (f_equal (fun x => (- x * _)%R)).
cut (0 <= ex - fexp (ex + 1))%Z. 2: omega.
case (ex - fexp (ex + 1))%Z ; trivial.
intros ep H.
now elim H.
391
intros H.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
392 393 394
apply f_equal.
apply sym_eq.
apply ln_beta_unique.
395 396
rewrite Rabs_Ropp.
rewrite Rabs_right.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
397 398 399 400 401 402 403 404 405 406 407 408 409 410 411
split.
apply -> epow_le.
omega.
apply -> epow_lt.
apply Zlt_succ.
apply Rle_ge.
apply epow_ge_0.
split.
exact Hbr.
(* - . biggest *)
intros g ((gm, ge), (Hg1, Hg2)) Hgx.
apply Rnot_lt_le.
intros Hg3.
assert (Hg4 : (g < 0)%R).
now apply Rle_lt_trans with (1 := Hgx).
412 413
specialize (Hg2 (Rlt_not_eq _ _ Hg4)).
destruct (ln_beta beta (Rabs g) (Rabs_pos_lt g (Rlt_not_eq g 0 Hg4))) as (ge', Hge).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
414 415 416 417 418 419 420 421 422
simpl in Hg2.
apply Rlt_not_le with (1 := Hg3).
rewrite Hg1.
unfold F2R. simpl.
rewrite Hg2.
assert (Hge' : ge' = ex).
apply epow_unique with (1 := Hge).
split.
apply Rle_trans with (1 := Hx1).
423
rewrite Rabs_left with (1 := Hg4).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
424 425
now apply Ropp_le_contravar.
apply Ropp_lt_cancel.
426 427 428
rewrite Rabs_left with (1 := Hg4).
rewrite Ropp_involutive.
now apply Rle_lt_trans with (1 := Hbl).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
429 430 431 432
rewrite Hge'.
apply Rmult_le_compat_r.
apply epow_ge_0.
apply Z2R_le.
433
cut (gm < up (x * bpow (- fexp ex)%Z))%Z. omega.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453
apply lt_IZR.
apply Rle_lt_trans with (2 := proj1 (archimed _)).
rewrite <- Z2R_IZR.
apply Rmult_le_reg_r with (bpow (fexp ex)).
apply epow_gt_0.
rewrite Rmult_assoc.
rewrite <- epow_add.
rewrite Zplus_opp_l.
rewrite Rmult_1_r.
rewrite <- Hge'.
rewrite <- Hg2.
unfold F2R in Hg1. simpl in Hg1.
now rewrite <- Hg1.
(* - negative too small *)
cutrewrite (up (x * bpow (- fexp ex)%Z) = 0%Z).
unfold F2R. simpl.
rewrite Ropp_mult_distr_l_reverse.
rewrite Rmult_1_l.
(* - . rounded *)
split.
454
destruct (proj2 (prop_exp _) He1) as (He2, _).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
455 456 457 458 459 460 461 462 463 464 465 466 467
exists (Float beta (- Zpower (radix_val beta) (fexp ex - fexp (fexp ex + 1))) (fexp (fexp ex + 1))).
unfold F2R. simpl.
split.
rewrite opp_Z2R.
pattern (fexp ex) at 1 ; replace (fexp ex) with (fexp ex - fexp (fexp ex + 1) + fexp (fexp ex + 1))%Z by ring.
rewrite epow_add.
rewrite Ropp_mult_distr_l_reverse.
apply (f_equal (fun x => (- (x * _))%R)).
cut (0 <= fexp ex - fexp (fexp ex + 1))%Z. 2: omega.
clear.
case (fexp ex - fexp (fexp ex + 1))%Z ; trivial.
intros ep Hp.
now elim Hp.
468
intros H.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
469 470 471
apply f_equal.
apply sym_eq.
apply ln_beta_unique.
472 473
rewrite Rabs_left.
rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
474 475 476 477 478
split.
replace (fexp ex + 1 - 1)%Z with (fexp ex) by ring.
apply Rle_refl.
apply -> epow_lt.
apply Zlt_succ.
479 480 481
rewrite <- Ropp_0.
apply Ropp_lt_contravar.
apply epow_gt_0.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
482 483
split.
(* - . smaller *)
484 485
apply Ropp_le_cancel.
rewrite Ropp_involutive.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
486 487 488 489 490 491 492 493 494
apply Rlt_le.
apply Rlt_le_trans with (1 := Hx2).
now apply -> epow_le.
(* - . biggest *)
intros g ((gm, ge), (Hg1, Hg2)) Hgx.
apply Rnot_lt_le.
intros Hg3.
assert (Hg4 : (g < 0)%R).
now apply Rle_lt_trans with (1 := Hgx).
495 496
specialize (Hg2 (Rlt_not_eq _ _ Hg4)).
destruct (ln_beta beta (Rabs g) (Rabs_pos_lt g (Rlt_not_eq g 0 Hg4))) as (ge', Hge).
Guillaume Melquiond's avatar
Guillaume Melquiond committed
497
simpl in Hg2.
498
rewrite (Rabs_left _ Hg4) in Hge.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
499 500 501 502 503 504
assert (Hge' : (ge' <= fexp ex)%Z).
cut (ge' - 1 < fexp ex)%Z. omega.
apply <- epow_lt.
apply Rle_lt_trans with (1 := proj1 Hge).
apply Ropp_lt_cancel.
now rewrite Ropp_involutive.
505
rewrite (proj2 (proj2 (prop_exp _) He1) _ Hge') in Hg2.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523
rewrite <- Hg2 in Hge'.
apply Rlt_not_le with (1 := proj2 Hge).
rewrite Hg1.
unfold F2R. simpl.
rewrite <- Ropp_mult_distr_l_reverse.
replace ge with (ge - ge' + ge')%Z by ring.
rewrite epow_add.
rewrite <- Rmult_assoc.
pattern (bpow ge') at 1 ; rewrite <- Rmult_1_l.
apply Rmult_le_compat_r.
apply epow_ge_0.
rewrite <- opp_Z2R.
assert (1 <= -gm)%Z.
apply (Zlt_le_succ 0).
apply lt_Z2R.
apply Rmult_lt_reg_r with (bpow ge).
apply epow_gt_0.
rewrite Rmult_0_l.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
524 525
change (0 < F2R (Float beta (-gm) ge))%R.
rewrite <- opp_F2R.
526 527 528
rewrite <- Hg1.
rewrite <- Ropp_0.
now apply Ropp_lt_contravar.
Guillaume Melquiond's avatar
Guillaume Melquiond committed
529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568
apply Rle_trans with (1 * bpow (ge - ge')%Z)%R.
rewrite Rmult_1_l.
cut (0 <= ge - ge')%Z. 2: omega.
clear.
case (ge - ge')%Z.
intros _.
apply Rle_refl.
intros ep _.
simpl.
apply (Z2R_le 1).
apply (Zlt_le_succ 0).
apply Zpower_pos_lt.
now apply Zlt_le_trans with (2 := radix_prop beta).
intros ep Hp. now elim Hp.
apply Rmult_le_compat_r.
apply epow_ge_0.
now apply (Z2R_le 1).
(* - . . *)
apply sym_eq.
apply (up_tech _ (-1)).
apply Ropp_le_cancel.
simpl.
rewrite Ropp_involutive.
change R1 with (bpow Z0).
rewrite <- (Zplus_opp_r (fexp ex)).
rewrite epow_add.
rewrite <- Ropp_mult_distr_l_reverse.
apply Rmult_le_compat_r.
apply epow_ge_0.
apply Rlt_le.
apply Rlt_le_trans with (1 := Hx2).
now apply -> epow_le.
simpl.
rewrite <- (Rmult_0_l (bpow (- fexp ex)%Z)).
apply Rmult_lt_compat_r.
apply epow_gt_0.
exact Hx.
Qed.

End RND_generic.