Flocq_rnd_generic.v 13.7 KB
 Guillaume Melquiond committed Mar 26, 2009 1 2 3 ``````Require Import Flocq_Raux. Require Import Flocq_defs. Require Import Flocq_rnd_ex. `````` Guillaume Melquiond committed Mar 27, 2009 4 ``````Require Import Flocq_float_prop. `````` Guillaume Melquiond committed Mar 26, 2009 5 6 7 8 9 10 11 12 13 `````` Section RND_generic. Variable beta : radix. Notation bpow := (epow beta). Variable fexp : Z -> Z. `````` Guillaume Melquiond committed Apr 08, 2009 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 committed Mar 26, 2009 22 23 24 `````` Definition generic_format (x : R) := exists f : float beta, `````` Guillaume Melquiond committed Mar 27, 2009 25 26 `````` x = F2R f /\ forall (H : x <> R0), Fexp f = fexp (projT1 (ln_beta beta _ (Rabs_pos_lt _ H))). `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 35 ``````unfold F2R. simpl. `````` Guillaume Melquiond committed Mar 26, 2009 36 ``````now rewrite Rmult_0_l. `````` Guillaume Melquiond committed Mar 27, 2009 37 38 ``````intros H. now elim H. `````` Guillaume Melquiond committed Mar 26, 2009 39 40 41 42 ``````intros x ((m,e),(H1,H2)). exists (Float beta (-m) e). split. rewrite H1. `````` Guillaume Melquiond committed Mar 27, 2009 43 ``````apply opp_F2R. `````` Guillaume Melquiond committed Mar 27, 2009 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 committed Mar 26, 2009 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 *) `````` Guillaume Melquiond committed Mar 27, 2009 121 ``````eexists ; split ; [ reflexivity | idtac ]. `````` Guillaume Melquiond committed Mar 26, 2009 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). `````` Guillaume Melquiond committed Mar 27, 2009 134 ``````unfold F2R. simpl. `````` Guillaume Melquiond committed Mar 26, 2009 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 *) `````` Guillaume Melquiond committed Mar 27, 2009 158 ``````unfold F2R. simpl. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 182 ``````specialize (Hg2 (Rgt_not_eq _ _ Hg3)). `````` Guillaume Melquiond committed Mar 26, 2009 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). `````` Guillaume Melquiond committed Mar 27, 2009 190 ``````rewrite <- (Rabs_pos_eq g (Rlt_le _ _ Hg3)) in H. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 224 225 ``````intros H. now elim H. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 232 ``````specialize (Hg2 (Rgt_not_eq _ _ Hg3)). `````` Guillaume Melquiond committed Mar 26, 2009 233 ``````destruct (ln_beta beta g Hg3) as (ge', Hg4). `````` Guillaume Melquiond committed Mar 27, 2009 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 committed Mar 26, 2009 237 238 239 ``````apply (Rlt_not_le _ _ (Rle_lt_trans _ _ _ Hgx Hx2)). apply Rle_trans with (bpow ge). apply -> epow_le. `````` Guillaume Melquiond committed Mar 27, 2009 240 ``````simpl in Hg2. `````` Guillaume Melquiond committed Mar 26, 2009 241 ``````rewrite Hg2. `````` Guillaume Melquiond committed Apr 08, 2009 242 ``````rewrite (proj2 (proj2 (prop_exp ex) He1) ge'). `````` Guillaume Melquiond committed Mar 26, 2009 243 244 ``````exact He1. apply Zle_trans with (2 := He1). `````` Guillaume Melquiond committed Mar 27, 2009 245 ``````cut (ge' - 1 < ex)%Z. omega. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 284 285 ``````intros H. now elim H. `````` Guillaume Melquiond committed Mar 26, 2009 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 committed Mar 27, 2009 347 348 ``````apply Ropp_le_cancel. rewrite Ropp_involutive. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 362 ``````rewrite Rabs_left. `````` Guillaume Melquiond committed Mar 26, 2009 363 ``````split. `````` Guillaume Melquiond committed Mar 27, 2009 364 365 ``````apply Ropp_le_cancel. rewrite Ropp_involutive. `````` Guillaume Melquiond committed Mar 26, 2009 366 ``````apply Rle_trans with (1 := Hbr). `````` Guillaume Melquiond committed Mar 27, 2009 367 368 369 370 ``````apply Ropp_le_cancel. now rewrite Ropp_involutive. apply Ropp_lt_cancel. now rewrite Ropp_involutive. `````` Guillaume Melquiond committed Mar 26, 2009 371 372 373 374 ``````apply Rle_lt_trans with (1 := Hbr). exact Hx. (* - . . a radix power *) rewrite <- Hbl2. `````` Guillaume Melquiond committed Apr 08, 2009 375 ``````generalize (proj1 (prop_exp _) He1). `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 391 ``````intros H. `````` Guillaume Melquiond committed Mar 26, 2009 392 393 394 ``````apply f_equal. apply sym_eq. apply ln_beta_unique. `````` Guillaume Melquiond committed Mar 27, 2009 395 396 ``````rewrite Rabs_Ropp. rewrite Rabs_right. `````` Guillaume Melquiond committed Mar 26, 2009 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). `````` Guillaume Melquiond committed Mar 27, 2009 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 committed Mar 26, 2009 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). `````` Guillaume Melquiond committed Mar 27, 2009 423 ``````rewrite Rabs_left with (1 := Hg4). `````` Guillaume Melquiond committed Mar 26, 2009 424 425 ``````now apply Ropp_le_contravar. apply Ropp_lt_cancel. `````` Guillaume Melquiond committed Mar 27, 2009 426 427 428 ``````rewrite Rabs_left with (1 := Hg4). rewrite Ropp_involutive. now apply Rle_lt_trans with (1 := Hbl). `````` Guillaume Melquiond committed Mar 26, 2009 429 430 431 432 ``````rewrite Hge'. apply Rmult_le_compat_r. apply epow_ge_0. apply Z2R_le. `````` Guillaume Melquiond committed Mar 27, 2009 433 ``````cut (gm < up (x * bpow (- fexp ex)%Z))%Z. omega. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Apr 08, 2009 454 ``````destruct (proj2 (prop_exp _) He1) as (He2, _). `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 468 ``````intros H. `````` Guillaume Melquiond committed Mar 26, 2009 469 470 471 ``````apply f_equal. apply sym_eq. apply ln_beta_unique. `````` Guillaume Melquiond committed Mar 27, 2009 472 473 ``````rewrite Rabs_left. rewrite Ropp_involutive. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Mar 27, 2009 479 480 481 ``````rewrite <- Ropp_0. apply Ropp_lt_contravar. apply epow_gt_0. `````` Guillaume Melquiond committed Mar 26, 2009 482 483 ``````split. (* - . smaller *) `````` Guillaume Melquiond committed Mar 27, 2009 484 485 ``````apply Ropp_le_cancel. rewrite Ropp_involutive. `````` Guillaume Melquiond committed Mar 26, 2009 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). `````` Guillaume Melquiond committed Mar 27, 2009 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 committed Mar 26, 2009 497 ``````simpl in Hg2. `````` Guillaume Melquiond committed Mar 27, 2009 498 ``````rewrite (Rabs_left _ Hg4) in Hge. `````` Guillaume Melquiond committed Mar 26, 2009 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. `````` Guillaume Melquiond committed Apr 08, 2009 505 ``````rewrite (proj2 (proj2 (prop_exp _) He1) _ Hge') in Hg2. `````` Guillaume Melquiond committed Mar 26, 2009 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 committed Mar 27, 2009 524 525 ``````change (0 < F2R (Float beta (-gm) ge))%R. rewrite <- opp_F2R. `````` Guillaume Melquiond committed Mar 27, 2009 526 527 528 ``````rewrite <- Hg1. rewrite <- Ropp_0. now apply Ropp_lt_contravar. `````` Guillaume Melquiond committed Mar 26, 2009 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.``````