Fcore_Raux.v 32.1 KB
 BOLDO Sylvie committed Jan 20, 2009 1 2 3 ``````Require Export Reals. Require Export ZArith. `````` Guillaume Melquiond committed Jan 22, 2009 4 5 ``````Section Rmissing. `````` Guillaume Melquiond committed Oct 29, 2009 6 ``````Theorem Rle_0_minus : `````` BOLDO Sylvie committed Jan 21, 2009 7 8 9 10 11 12 13 14 `````` forall x y, (x <= y)%R -> (0 <= y - x)%R. Proof. intros. apply Rge_le. apply Rge_minus. now apply Rle_ge. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 15 ``````Theorem Rabs_eq_Rabs : `````` Guillaume Melquiond committed Jan 22, 2009 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 `````` forall x y : R, Rabs x = Rabs y -> x = y \/ x = Ropp y. Proof. intros x y H. unfold Rabs in H. destruct (Rcase_abs x) as [_|_]. assert (H' := f_equal Ropp H). rewrite Ropp_involutive in H'. rewrite H'. destruct (Rcase_abs y) as [_|_]. left. apply Ropp_involutive. now right. rewrite H. now destruct (Rcase_abs y) as [_|_] ; [right|left]. Qed. `````` BOLDO Sylvie committed Sep 14, 2010 33 34 35 36 37 38 39 40 41 42 43 44 45 `````` Theorem Rabs_Rminus_pos: forall x y : R, (0 <= y)%R -> (y <= 2*x)%R -> (Rabs (x-y) <= x)%R. intros x y Hx Hy. unfold Rabs; case (Rcase_abs (x - y)); intros H. apply Rplus_le_reg_l with x; ring_simplify; assumption. apply Rplus_le_reg_l with (-x)%R; ring_simplify. auto with real. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 46 ``````Theorem Rplus_le_reg_r : `````` Guillaume Melquiond committed Apr 09, 2009 47 48 49 50 51 52 53 54 `````` forall r r1 r2 : R, (r1 + r <= r2 + r)%R -> (r1 <= r2)%R. Proof. intros. apply Rplus_le_reg_l with r. now rewrite 2!(Rplus_comm r). Qed. `````` Guillaume Melquiond committed Oct 29, 2009 55 ``````Theorem Rmult_lt_reg_r : `````` Guillaume Melquiond committed Jan 22, 2009 56 57 58 59 60 61 62 63 64 `````` forall r r1 r2 : R, (0 < r)%R -> (r1 * r < r2 * r)%R -> (r1 < r2)%R. Proof. intros. apply Rmult_lt_reg_l with r. exact H. now rewrite 2!(Rmult_comm r). Qed. `````` Guillaume Melquiond committed Oct 29, 2009 65 ``````Theorem Rmult_le_reg_r : `````` Guillaume Melquiond committed Mar 26, 2009 66 67 68 69 70 71 72 73 74 `````` forall r r1 r2 : R, (0 < r)%R -> (r1 * r <= r2 * r)%R -> (r1 <= r2)%R. Proof. intros. apply Rmult_le_reg_l with r. exact H. now rewrite 2!(Rmult_comm r). Qed. `````` Guillaume Melquiond committed Oct 29, 2009 75 ``````Theorem Rmult_eq_reg_r : `````` Guillaume Melquiond committed Sep 18, 2009 76 77 78 79 80 81 82 83 84 `````` forall r r1 r2 : R, (r1 * r)%R = (r2 * r)%R -> r <> 0%R -> r1 = r2. Proof. intros r r1 r2 H1 H2. apply Rmult_eq_reg_l with r. now rewrite 2!(Rmult_comm r). exact H2. Qed. `````` Guillaume Melquiond committed Apr 13, 2010 85 86 87 88 89 90 91 92 93 ``````Theorem Rmult_minus_distr_r : forall r r1 r2 : R, ((r1 - r2) * r = r1 * r - r2 * r)%R. Proof. intros r r1 r2. rewrite <- 3!(Rmult_comm r). apply Rmult_minus_distr_l. Qed. `````` Guillaume Melquiond committed Apr 13, 2010 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 ``````Theorem Rmult_min_distr_r : forall r r1 r2 : R, (0 <= r)%R -> (Rmin r1 r2 * r)%R = Rmin (r1 * r) (r2 * r). Proof. intros r r1 r2 [Hr|Hr]. unfold Rmin. destruct (Rle_dec r1 r2) as [H1|H1] ; destruct (Rle_dec (r1 * r) (r2 * r)) as [H2|H2] ; try easy. apply (f_equal (fun x => Rmult x r)). apply Rle_antisym. exact H1. apply Rmult_le_reg_r with (1 := Hr). apply Rlt_le. now apply Rnot_le_lt. apply Rle_antisym. apply Rmult_le_compat_r. now apply Rlt_le. apply Rlt_le. now apply Rnot_le_lt. exact H2. rewrite <- Hr. rewrite 3!Rmult_0_r. unfold Rmin. destruct (Rle_dec 0 0) as [H0|H0]. easy. elim H0. apply Rle_refl. Qed. Theorem Rmult_min_distr_l : forall r r1 r2 : R, (0 <= r)%R -> (r * Rmin r1 r2)%R = Rmin (r * r1) (r * r2). Proof. intros r r1 r2 Hr. rewrite 3!(Rmult_comm r). now apply Rmult_min_distr_r. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 135 ``````Theorem exp_increasing_weak : `````` Guillaume Melquiond committed Jan 26, 2009 136 137 138 139 140 141 142 143 144 145 `````` forall x y : R, (x <= y)%R -> (exp x <= exp y)%R. Proof. intros x y [H|H]. apply Rlt_le. now apply exp_increasing. rewrite H. apply Rle_refl. Qed. `````` Guillaume Melquiond committed Sep 16, 2010 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 ``````Theorem Rinv_lt : forall x y, (0 < x)%R -> (x < y)%R -> (/y < /x)%R. Proof. intros x y Hx Hxy. apply Rinv_lt_contravar. apply Rmult_lt_0_compat. exact Hx. now apply Rlt_trans with x. exact Hxy. Qed. Theorem Rinv_le : forall x y, (0 < x)%R -> (x <= y)%R -> (/y <= /x)%R. Proof. intros x y Hx Hxy. apply Rle_Rinv. exact Hx. now apply Rlt_le_trans with x. exact Hxy. Qed. `````` Guillaume Melquiond committed May 17, 2010 169 170 171 172 173 174 175 176 177 178 179 ``````Theorem sqrt_ge_0 : forall x : R, (0 <= sqrt x)%R. Proof. intros x. unfold sqrt. destruct (Rcase_abs x) as [_|H]. apply Rle_refl. apply Rsqrt_positivity. Qed. `````` Guillaume Melquiond committed Sep 22, 2010 180 ``````Theorem Rabs_le : `````` Guillaume Melquiond committed Sep 14, 2010 181 182 183 184 185 186 187 188 189 190 191 `````` forall x y, (-y <= x <= y)%R -> (Rabs x <= y)%R. Proof. intros x y (Hyx,Hxy). unfold Rabs. case Rcase_abs ; intros Hx. apply Ropp_le_cancel. now rewrite Ropp_involutive. exact Hxy. Qed. `````` Guillaume Melquiond committed Sep 22, 2010 192 ``````Theorem Rabs_le_inv : `````` Guillaume Melquiond committed Sep 14, 2010 193 194 195 196 197 198 199 200 201 202 203 204 205 206 `````` forall x y, (Rabs x <= y)%R -> (-y <= x <= y)%R. Proof. intros x y Hxy. split. apply Rle_trans with (- Rabs x)%R. now apply Ropp_le_contravar. apply Ropp_le_cancel. rewrite Ropp_involutive, <- Rabs_Ropp. apply RRle_abs. apply Rle_trans with (2 := Hxy). apply RRle_abs. Qed. `````` Guillaume Melquiond committed Sep 22, 2010 207 ``````Theorem Rabs_ge : `````` Guillaume Melquiond committed Sep 14, 2010 208 209 210 211 212 213 214 215 216 217 218 219 220 `````` forall x y, (y <= -x \/ x <= y)%R -> (x <= Rabs y)%R. Proof. intros x y [Hyx|Hxy]. apply Rle_trans with (-y)%R. apply Ropp_le_cancel. now rewrite Ropp_involutive. rewrite <- Rabs_Ropp. apply RRle_abs. apply Rle_trans with (1 := Hxy). apply RRle_abs. Qed. `````` Guillaume Melquiond committed Sep 22, 2010 221 ``````Theorem Rabs_ge_inv : `````` Guillaume Melquiond committed Sep 14, 2010 222 223 224 225 `````` forall x y, (x <= Rabs y)%R -> (y <= -x \/ x <= y)%R. Proof. intros x y. `````` Guillaume Melquiond committed Sep 22, 2010 226 ``````unfold Rabs. `````` Guillaume Melquiond committed Sep 14, 2010 227 228 229 230 231 232 233 ``````case Rcase_abs ; intros Hy Hxy. left. apply Ropp_le_cancel. now rewrite Ropp_involutive. now right. Qed. `````` Guillaume Melquiond committed Sep 22, 2010 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 ``````Theorem Rabs_lt : forall x y, (-y < x < y)%R -> (Rabs x < y)%R. Proof. intros x y (Hyx,Hxy). now apply Rabs_def1. Qed. Theorem Rabs_lt_inv : forall x y, (Rabs x < y)%R -> (-y < x < y)%R. Proof. intros x y H. now split ; eapply Rabs_def2. Qed. Theorem Rabs_gt : forall x y, (y < -x \/ x < y)%R -> (x < Rabs y)%R. Proof. intros x y [Hyx|Hxy]. rewrite <- Rabs_Ropp. apply Rlt_le_trans with (Ropp y). apply Ropp_lt_cancel. now rewrite Ropp_involutive. apply RRle_abs. apply Rlt_le_trans with (1 := Hxy). apply RRle_abs. Qed. Theorem Rabs_gt_inv : forall x y, (x < Rabs y)%R -> (y < -x \/ x < y)%R. Proof. intros x y. unfold Rabs. case Rcase_abs ; intros Hy Hxy. left. apply Ropp_lt_cancel. now rewrite Ropp_involutive. now right. Qed. `````` Guillaume Melquiond committed Jan 22, 2009 277 278 ``````End Rmissing. `````` Guillaume Melquiond committed Apr 09, 2009 279 280 ``````Section Zmissing. `````` Guillaume Melquiond committed Oct 29, 2009 281 ``````Theorem Zopp_le_cancel : `````` Guillaume Melquiond committed Apr 09, 2009 282 283 284 285 286 287 288 289 `````` forall x y : Z, (-y <= -x)%Z -> Zle x y. Proof. intros x y Hxy. apply Zplus_le_reg_r with (-x - y)%Z. now ring_simplify. Qed. `````` Guillaume Melquiond committed Sep 17, 2010 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 ``````Theorem Zmin_left : forall x y : Z, (x <= y)%Z -> Zmin x y = x. Proof. intros x y. generalize (Zmin_spec x y). omega. Qed. Theorem Zmin_right : forall x y : Z, (y <= x)%Z -> Zmin x y = y. Proof. intros x y. generalize (Zmin_spec x y). omega. Qed. `````` Guillaume Melquiond committed Apr 09, 2009 308 309 ``````End Zmissing. `````` BOLDO Sylvie committed Jan 20, 2009 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 ``````Section Z2R. Fixpoint P2R (p : positive) := match p with | xH => 1%R | xO xH => 2%R | xO t => (2 * P2R t)%R | xI xH => 3%R | xI t => (1 + 2 * P2R t)%R end. Definition Z2R n := match n with | Zpos p => P2R p | Zneg p => Ropp (P2R p) | Z0 => R0 end. `````` Guillaume Melquiond committed Oct 29, 2009 328 ``````Theorem P2R_INR : `````` BOLDO Sylvie committed Jan 20, 2009 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 `````` forall n, P2R n = INR (nat_of_P n). Proof. induction n ; simpl ; try ( rewrite IHn ; rewrite <- (mult_INR 2) ; rewrite <- (nat_of_P_mult_morphism 2) ; change (2 * n)%positive with (xO n)). (* xI *) rewrite (Rplus_comm 1). change (nat_of_P (xO n)) with (Pmult_nat n 2). case n ; intros ; simpl ; try apply refl_equal. case (Pmult_nat p 4) ; intros ; try apply refl_equal. rewrite Rplus_0_l. apply refl_equal. apply Rplus_comm. (* xO *) case n ; intros ; apply refl_equal. (* xH *) apply refl_equal. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 350 ``````Theorem Z2R_IZR : `````` BOLDO Sylvie committed Jan 20, 2009 351 352 353 354 355 356 357 358 359 360 `````` forall n, Z2R n = IZR n. Proof. intro. case n ; intros ; simpl. apply refl_equal. apply P2R_INR. apply Ropp_eq_compat. apply P2R_INR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 361 ``````Theorem opp_Z2R : `````` Guillaume Melquiond committed Jan 22, 2009 362 363 364 365 366 367 368 `````` forall n, Z2R (-n) = (- Z2R n)%R. Proof. intros. repeat rewrite Z2R_IZR. apply Ropp_Ropp_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 369 ``````Theorem plus_Z2R : `````` BOLDO Sylvie committed Jan 20, 2009 370 371 372 373 374 375 376 `````` forall m n, (Z2R (m + n) = Z2R m + Z2R n)%R. Proof. intros. repeat rewrite Z2R_IZR. apply plus_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 377 ``````Theorem minus_IZR : `````` BOLDO Sylvie committed Jan 20, 2009 378 379 380 381 382 383 384 385 386 387 `````` forall n m : Z, IZR (n - m) = (IZR n - IZR m)%R. Proof. intros. unfold Zminus. rewrite plus_IZR. rewrite Ropp_Ropp_IZR. apply refl_equal. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 388 ``````Theorem minus_Z2R : `````` BOLDO Sylvie committed Jan 20, 2009 389 390 391 392 393 394 395 `````` forall m n, (Z2R (m - n) = Z2R m - Z2R n)%R. Proof. intros. repeat rewrite Z2R_IZR. apply minus_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 396 ``````Theorem mult_Z2R : `````` BOLDO Sylvie committed Jan 20, 2009 397 398 399 400 401 402 403 `````` forall m n, (Z2R (m * n) = Z2R m * Z2R n)%R. Proof. intros. repeat rewrite Z2R_IZR. apply mult_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 404 ``````Theorem le_Z2R : `````` BOLDO Sylvie committed Jan 20, 2009 405 406 407 408 409 410 411 `````` forall m n, (Z2R m <= Z2R n)%R -> (m <= n)%Z. Proof. intros m n. repeat rewrite Z2R_IZR. apply le_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 412 ``````Theorem Z2R_le : `````` BOLDO Sylvie committed Jan 20, 2009 413 414 415 416 417 418 419 `````` forall m n, (m <= n)%Z -> (Z2R m <= Z2R n)%R. Proof. intros m n. repeat rewrite Z2R_IZR. apply IZR_le. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 420 ``````Theorem lt_Z2R : `````` BOLDO Sylvie committed Jan 20, 2009 421 422 423 424 425 426 427 `````` forall m n, (Z2R m < Z2R n)%R -> (m < n)%Z. Proof. intros m n. repeat rewrite Z2R_IZR. apply lt_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 428 ``````Theorem Z2R_lt : `````` BOLDO Sylvie committed Jan 20, 2009 429 430 431 432 433 434 435 `````` forall m n, (m < n)%Z -> (Z2R m < Z2R n)%R. Proof. intros m n. repeat rewrite Z2R_IZR. apply IZR_lt. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 436 ``````Theorem Z2R_le_lt : `````` Guillaume Melquiond committed Sep 16, 2009 437 438 439 440 441 442 443 444 `````` forall m n p, (m <= n < p)%Z -> (Z2R m <= Z2R n < Z2R p)%R. Proof. intros m n p (H1, H2). split. now apply Z2R_le. now apply Z2R_lt. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 445 ``````Theorem le_lt_Z2R : `````` Guillaume Melquiond committed Sep 16, 2009 446 447 448 449 450 `````` forall m n p, (Z2R m <= Z2R n < Z2R p)%R -> (m <= n < p)%Z. Proof. intros m n p (H1, H2). split. now apply le_Z2R. `````` BOLDO Sylvie committed Mar 06, 2009 451 452 ``````now apply lt_Z2R. Qed. `````` BOLDO Sylvie committed Jan 20, 2009 453 `````` `````` Guillaume Melquiond committed Oct 29, 2009 454 ``````Theorem eq_Z2R : `````` Guillaume Melquiond committed Sep 18, 2009 455 456 457 458 459 460 461 `````` forall m n, (Z2R m = Z2R n)%R -> (m = n)%Z. Proof. intros m n H. apply eq_IZR. now rewrite <- 2!Z2R_IZR. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 462 ``````Theorem neq_Z2R : `````` Guillaume Melquiond committed Sep 18, 2009 463 464 465 466 467 468 469 `````` forall m n, (Z2R m <> Z2R n)%R -> (m <> n)%Z. Proof. intros m n H H'. apply H. now apply f_equal. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 470 ``````Theorem Z2R_neq : `````` Guillaume Melquiond committed Sep 16, 2009 471 `````` forall m n, (m <> n)%Z -> (Z2R m <> Z2R n)%R. `````` BOLDO Sylvie committed Jan 20, 2009 472 473 474 475 476 477 ``````Proof. intros m n. repeat rewrite Z2R_IZR. apply IZR_neq. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 478 479 ``````Theorem abs_Z2R : forall z, Z2R (Zabs z) = Rabs (Z2R z). `````` Guillaume Melquiond committed Sep 16, 2009 480 ``````Proof. `````` BOLDO Sylvie committed Mar 06, 2009 481 482 ``````intros. repeat rewrite Z2R_IZR. `````` Guillaume Melquiond committed Oct 29, 2009 483 ``````now rewrite Rabs_Zabs. `````` BOLDO Sylvie committed Mar 06, 2009 484 485 ``````Qed. `````` BOLDO Sylvie committed Jan 20, 2009 486 487 ``````End Z2R. `````` Guillaume Melquiond committed Apr 13, 2010 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 ``````Section Zcompare. Inductive Zeq_bool_prop (x y : Z) : bool -> Prop := | Zeq_bool_true : x = y -> Zeq_bool_prop x y true | Zeq_bool_false : x <> y -> Zeq_bool_prop x y false. Theorem Zeq_bool_spec : forall x y, Zeq_bool_prop x y (Zeq_bool x y). Proof. intros x y. generalize (Zeq_is_eq_bool x y). case (Zeq_bool x y) ; intros (H1, H2) ; constructor. now apply H2. intros H. specialize (H1 H). discriminate H1. Qed. Inductive Zcompare_prop (x y : Z) : comparison -> Prop := | Zcompare_Lt_ : (x < y)%Z -> Zcompare_prop x y Lt | Zcompare_Eq_ : x = y -> Zcompare_prop x y Eq | Zcompare_Gt_ : (y < x)%Z -> Zcompare_prop x y Gt. Theorem Zcompare_spec : forall x y, Zcompare_prop x y (Zcompare x y). Proof. intros x y. destruct (Z_dec x y) as [[H|H]|H]. generalize (Zlt_compare _ _ H). case (Zcompare x y) ; try easy. now constructor. generalize (Zgt_compare _ _ H). case (Zcompare x y) ; try easy. constructor. now apply Zgt_lt. generalize (proj2 (Zcompare_Eq_iff_eq _ _) H). case (Zcompare x y) ; try easy. now constructor. Qed. Theorem Zcompare_Lt : forall x y, (x < y)%Z -> Zcompare x y = Lt. Proof. easy. Qed. Theorem Zcompare_Eq : forall x y, (x = y)%Z -> Zcompare x y = Eq. Proof. intros x y. apply <- Zcompare_Eq_iff_eq. Qed. Theorem Zcompare_Gt : forall x y, (y < x)%Z -> Zcompare x y = Gt. Proof. intros x y. apply Zlt_gt. Qed. End Zcompare. Section Rcompare. Definition Rcompare x y := match total_order_T x y with | inleft (left _) => Lt | inleft (right _) => Eq | inright _ => Gt end. Inductive Rcompare_prop (x y : R) : comparison -> Prop := | Rcompare_Lt_ : (x < y)%R -> Rcompare_prop x y Lt | Rcompare_Eq_ : x = y -> Rcompare_prop x y Eq | Rcompare_Gt_ : (y < x)%R -> Rcompare_prop x y Gt. Theorem Rcompare_spec : forall x y, Rcompare_prop x y (Rcompare x y). Proof. intros x y. unfold Rcompare. now destruct (total_order_T x y) as [[H|H]|H] ; constructor. Qed. `````` Guillaume Melquiond committed May 11, 2010 575 ``````Global Opaque Rcompare. `````` Guillaume Melquiond committed Apr 13, 2010 576 577 578 579 580 581 582 583 584 585 586 587 588 589 `````` Theorem Rcompare_Lt : forall x y, (x < y)%R -> Rcompare x y = Lt. Proof. intros x y H. case Rcompare_spec ; intro H'. easy. rewrite H' in H. elim (Rlt_irrefl _ H). elim (Rlt_irrefl x). now apply Rlt_trans with y. Qed. `````` Guillaume Melquiond committed May 17, 2010 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 ``````Theorem Rcompare_Lt_inv : forall x y, Rcompare x y = Lt -> (x < y)%R. Proof. intros x y. now case Rcompare_spec. Qed. Theorem Rcompare_not_Lt : forall x y, (y <= x)%R -> Rcompare x y <> Lt. Proof. intros x y H1 H2. apply Rle_not_lt with (1 := H1). now apply Rcompare_Lt_inv. Qed. Theorem Rcompare_not_Lt_inv : forall x y, Rcompare x y <> Lt -> (y <= x)%R. Proof. intros x y H. apply Rnot_lt_le. contradict H. now apply Rcompare_Lt. Qed. `````` Guillaume Melquiond committed Apr 13, 2010 617 618 619 620 621 622 623 624 625 ``````Theorem Rcompare_Eq : forall x y, x = y -> Rcompare x y = Eq. Proof. intros x y H. rewrite H. now case Rcompare_spec ; intro H' ; try elim (Rlt_irrefl _ H'). Qed. `````` Guillaume Melquiond committed May 11, 2010 626 627 628 629 630 631 632 633 ``````Theorem Rcompare_Eq_inv : forall x y, Rcompare x y = Eq -> x = y. Proof. intros x y. now case Rcompare_spec. Qed. `````` Guillaume Melquiond committed Apr 13, 2010 634 635 636 637 638 639 640 641 642 643 644 645 646 ``````Theorem Rcompare_Gt : forall x y, (y < x)%R -> Rcompare x y = Gt. Proof. intros x y H. case Rcompare_spec ; intro H'. elim (Rlt_irrefl x). now apply Rlt_trans with y. rewrite H' in H. elim (Rlt_irrefl _ H). easy. Qed. `````` Guillaume Melquiond committed May 17, 2010 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 ``````Theorem Rcompare_Gt_inv : forall x y, Rcompare x y = Gt -> (y < x)%R. Proof. intros x y. now case Rcompare_spec. Qed. Theorem Rcompare_not_Gt : forall x y, (x <= y)%R -> Rcompare x y <> Gt. Proof. intros x y H1 H2. apply Rle_not_lt with (1 := H1). now apply Rcompare_Gt_inv. Qed. Theorem Rcompare_not_Gt_inv : forall x y, Rcompare x y <> Gt -> (x <= y)%R. Proof. intros x y H. apply Rnot_lt_le. contradict H. now apply Rcompare_Gt. Qed. `````` Guillaume Melquiond committed Apr 13, 2010 674 675 676 677 678 679 680 681 682 683 684 685 686 ``````Theorem Rcompare_Z2R : forall x y, Rcompare (Z2R x) (Z2R y) = Zcompare x y. Proof. intros x y. case Rcompare_spec ; intros H ; apply sym_eq. apply Zcompare_Lt. now apply lt_Z2R. apply Zcompare_Eq. now apply eq_Z2R. apply Zcompare_Gt. now apply lt_Z2R. Qed. `````` Guillaume Melquiond committed May 11, 2010 687 688 689 690 691 692 693 694 695 696 697 ``````Theorem Rcompare_sym : forall x y, Rcompare x y = CompOpp (Rcompare y x). Proof. intros x y. destruct (Rcompare_spec y x) as [H|H|H]. now apply Rcompare_Gt. now apply Rcompare_Eq. now apply Rcompare_Lt. Qed. `````` Guillaume Melquiond committed May 17, 2010 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 ``````Theorem Rcompare_plus_r : forall z x y, Rcompare (x + z) (y + z) = Rcompare x y. Proof. intros z x y. destruct (Rcompare_spec x y) as [H|H|H]. apply Rcompare_Lt. now apply Rplus_lt_compat_r. apply Rcompare_Eq. now rewrite H. apply Rcompare_Gt. now apply Rplus_lt_compat_r. Qed. Theorem Rcompare_plus_l : forall z x y, Rcompare (z + x) (z + y) = Rcompare x y. Proof. intros z x y. rewrite 2!(Rplus_comm z). apply Rcompare_plus_r. Qed. Theorem Rcompare_mult_r : forall z x y, (0 < z)%R -> Rcompare (x * z) (y * z) = Rcompare x y. Proof. intros z x y Hz. destruct (Rcompare_spec x y) as [H|H|H]. apply Rcompare_Lt. now apply Rmult_lt_compat_r. apply Rcompare_Eq. now rewrite H. apply Rcompare_Gt. now apply Rmult_lt_compat_r. Qed. Theorem Rcompare_mult_l : forall z x y, (0 < z)%R -> Rcompare (z * x) (z * y) = Rcompare x y. Proof. intros z x y. rewrite 2!(Rmult_comm z). apply Rcompare_mult_r. Qed. Theorem Rcompare_middle : forall x d u, Rcompare (x - d) (u - x) = Rcompare x ((d + u) / 2). Proof. intros x d u. rewrite <- (Rcompare_plus_r (- x / 2 - d / 2) x). rewrite <- (Rcompare_mult_r (/2) (x - d)). field_simplify (x + (- x / 2 - d / 2))%R. now field_simplify ((d + u) / 2 + (- x / 2 - d / 2))%R. apply Rinv_0_lt_compat. now apply (Z2R_lt 0 2). Qed. Theorem Rcompare_half_l : forall x y, Rcompare (x / 2) y = Rcompare x (2 * y). Proof. intros x y. rewrite <- (Rcompare_mult_r 2%R). unfold Rdiv. rewrite Rmult_assoc, Rinv_l, Rmult_1_r. now rewrite Rmult_comm. now apply (Z2R_neq 2 0). now apply (Z2R_lt 0 2). Qed. Theorem Rcompare_half_r : forall x y, Rcompare x (y / 2) = Rcompare (2 * x) y. Proof. intros x y. rewrite <- (Rcompare_mult_r 2%R). unfold Rdiv. rewrite Rmult_assoc, Rinv_l, Rmult_1_r. now rewrite Rmult_comm. now apply (Z2R_neq 2 0). now apply (Z2R_lt 0 2). Qed. `````` Guillaume Melquiond committed May 17, 2010 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 ``````Theorem Rcompare_sqr : forall x y, (0 <= x)%R -> (0 <= y)%R -> Rcompare (x * x) (y * y) = Rcompare x y. Proof. intros x y Hx Hy. destruct (Rcompare_spec x y) as [H|H|H]. apply Rcompare_Lt. now apply Rsqr_incrst_1. rewrite H. now apply Rcompare_Eq. apply Rcompare_Gt. now apply Rsqr_incrst_1. Qed. `````` Guillaume Melquiond committed Apr 13, 2010 798 799 ``````End Rcompare. `````` Guillaume Melquiond committed Sep 14, 2010 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 ``````Section Rle_bool. Definition Rle_bool x y := match Rcompare x y with | Gt => false | _ => true end. Inductive Rle_bool_prop (x y : R) : bool -> Prop := | Rle_bool_true_ : (x <= y)%R -> Rle_bool_prop x y true | Rle_bool_false_ : (y < x)%R -> Rle_bool_prop x y false. Theorem Rle_bool_spec : forall x y, Rle_bool_prop x y (Rle_bool x y). Proof. intros x y. unfold Rle_bool. case Rcompare_spec ; constructor. now apply Rlt_le. rewrite H. apply Rle_refl. exact H. Qed. Theorem Rle_bool_true : forall x y, (x <= y)%R -> Rle_bool x y = true. Proof. intros x y Hxy. case Rle_bool_spec ; intros H. apply refl_equal. elim (Rlt_irrefl x). now apply Rle_lt_trans with y. Qed. Theorem Rle_bool_false : forall x y, (y < x)%R -> Rle_bool x y = false. Proof. intros x y Hxy. case Rle_bool_spec ; intros H. elim (Rlt_irrefl x). now apply Rle_lt_trans with y. apply refl_equal. Qed. End Rle_bool. `````` Guillaume Melquiond committed Apr 09, 2009 848 849 850 851 ``````Section Floor_Ceil. Definition Zfloor (x : R) := (up x - 1)%Z. `````` Guillaume Melquiond committed Oct 29, 2009 852 ``````Theorem Zfloor_lb : `````` Guillaume Melquiond committed Apr 09, 2009 853 854 855 856 857 858 859 860 861 862 863 864 865 `````` forall x : R, (Z2R (Zfloor x) <= x)%R. Proof. intros x. unfold Zfloor. rewrite minus_Z2R. simpl. rewrite Z2R_IZR. apply Rplus_le_reg_r with (1 - x)%R. ring_simplify. exact (proj2 (archimed x)). Qed. `````` Guillaume Melquiond committed Oct 29, 2009 866 ``````Theorem Zfloor_ub : `````` Guillaume Melquiond committed Apr 10, 2009 867 868 869 870 871 872 873 874 875 876 877 878 879 `````` forall x : R, (x < Z2R (Zfloor x) + 1)%R. Proof. intros x. unfold Zfloor. rewrite minus_Z2R. unfold Rminus. rewrite Rplus_assoc. rewrite Rplus_opp_l, Rplus_0_r. rewrite Z2R_IZR. exact (proj1 (archimed x)). Qed. `````` Guillaume Melquiond committed Oct 29, 2009 880 ``````Theorem Zfloor_lub : `````` Guillaume Melquiond committed Apr 09, 2009 881 882 883 884 885 886 887 888 `````` forall n x, (Z2R n <= x)%R -> (n <= Zfloor x)%Z. Proof. intros n x Hnx. apply Zlt_succ_le. apply lt_Z2R. apply Rle_lt_trans with (1 := Hnx). `````` Guillaume Melquiond committed Apr 10, 2009 889 890 891 ``````unfold Zsucc. rewrite plus_Z2R. apply Zfloor_ub. `````` Guillaume Melquiond committed Apr 09, 2009 892 893 ``````Qed. `````` Guillaume Melquiond committed Oct 29, 2009 894 ``````Theorem Zfloor_imp : `````` Guillaume Melquiond committed Apr 09, 2009 895 896 897 898 899 900 901 902 903 904 905 906 907 `````` forall n x, (Z2R n <= x < Z2R (n + 1))%R -> Zfloor x = n. Proof. intros n x Hnx. apply Zle_antisym. apply Zlt_succ_le. apply lt_Z2R. apply Rle_lt_trans with (2 := proj2 Hnx). apply Zfloor_lb. now apply Zfloor_lub. Qed. `````` Guillaume Melquiond committed Mar 04, 2010 908 ``````Theorem Zfloor_Z2R : `````` 909 910 911 912 913 914 915 916 917 918 919 `````` forall n, Zfloor (Z2R n) = n. Proof. intros n. apply Zfloor_imp. split. apply Rle_refl. apply Z2R_lt. apply Zlt_succ. Qed. `````` Guillaume Melquiond committed Mar 10, 2010 920 921 922 923 924 925 926 927 928 929 ``````Theorem Zfloor_le : forall x y, (x <= y)%R -> (Zfloor x <= Zfloor y)%Z. Proof. intros x y Hxy. apply Zfloor_lub. apply Rle_trans with (2 := Hxy). apply Zfloor_lb. Qed. `````` Guillaume Melquiond committed Apr 09, 2009 930 931 ``````Definition Zceil (x : R) := (- Zfloor (- x))%Z. `````` Guillaume Melquiond committed Oct 29, 2009 932 ``````Theorem Zceil_ub : `````` Guillaume Melquiond committed Apr 09, 2009 933 934 935 936 937 938 939 940 941 942 943 `````` forall x : R, (x <= Z2R (Zceil x))%R. Proof. intros x. unfold Zceil. rewrite opp_Z2R. apply Ropp_le_cancel. rewrite Ropp_involutive. apply Zfloor_lb. Qed. `````` Guillaume Melquiond committed Mar 10, 2010 944 ``````Theorem Zceil_glb : `````` Guillaume Melquiond committed Apr 09, 2009 945 946 947 948 949 950 951 952 953 954 955 956 957 `````` forall n x, (x <= Z2R n)%R -> (Zceil x <= n)%Z. Proof. intros n x Hnx. unfold Zceil. apply Zopp_le_cancel. rewrite Zopp_involutive. apply Zfloor_lub. rewrite opp_Z2R. now apply Ropp_le_contravar. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 958 ``````Theorem Zceil_imp : `````` Guillaume Melquiond committed Apr 09, 2009 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 `````` forall n x, (Z2R (n - 1) < x <= Z2R n)%R -> Zceil x = n. Proof. intros n x Hnx. unfold Zceil. rewrite <- (Zopp_involutive n). apply f_equal. apply Zfloor_imp. split. rewrite opp_Z2R. now apply Ropp_le_contravar. rewrite <- (Zopp_involutive 1). rewrite <- Zopp_plus_distr. rewrite opp_Z2R. now apply Ropp_lt_contravar. Qed. `````` Guillaume Melquiond committed Mar 04, 2010 977 ``````Theorem Zceil_Z2R : `````` 978 979 980 981 982 `````` forall n, Zceil (Z2R n) = n. Proof. intros n. unfold Zceil. `````` Guillaume Melquiond committed Mar 04, 2010 983 ``````rewrite <- opp_Z2R, Zfloor_Z2R. `````` 984 985 986 ``````apply Zopp_involutive. Qed. `````` Guillaume Melquiond committed Mar 10, 2010 987 988 989 990 991 992 993 994 995 996 ``````Theorem Zceil_le : forall x y, (x <= y)%R -> (Zceil x <= Zceil y)%Z. Proof. intros x y Hxy. apply Zceil_glb. apply Rle_trans with (1 := Hxy). apply Zceil_ub. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 997 ``````Theorem Zceil_floor_neq : `````` Guillaume Melquiond committed Apr 10, 2009 998 `````` forall x : R, `````` Guillaume Melquiond committed Apr 09, 2009 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 `````` (Z2R (Zfloor x) <> x)%R -> (Zceil x = Zfloor x + 1)%Z. Proof. intros x Hx. apply Zceil_imp. split. ring_simplify (Zfloor x + 1 - 1)%Z. apply Rnot_le_lt. intros H. apply Hx. apply Rle_antisym. apply Zfloor_lb. exact H. `````` Guillaume Melquiond committed Apr 10, 2009 1012 1013 1014 ``````apply Rlt_le. rewrite plus_Z2R. apply Zfloor_ub. `````` Guillaume Melquiond committed Apr 09, 2009 1015 1016 ``````Qed. `````` Guillaume Melquiond committed Mar 04, 2010 1017 1018 1019 1020 1021 1022 1023 1024 1025 ``````Definition Ztrunc x := if Rlt_le_dec x 0 then Zceil x else Zfloor x. Theorem Ztrunc_Z2R : forall n, Ztrunc (Z2R n) = n. Proof. intros n. unfold Ztrunc. destruct Rlt_le_dec as [H|H]. `````` Guillaume Melquiond committed Mar 04, 2010 1026 1027 ``````apply Zceil_Z2R. apply Zfloor_Z2R. `````` Guillaume Melquiond committed Mar 04, 2010 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 ``````Qed. Theorem Ztrunc_floor : forall x, (0 <= x)%R -> Ztrunc x = Zfloor x. Proof. intros x Hx. unfold Ztrunc. destruct Rlt_le_dec as [H|_]. elim Rlt_irrefl with (1 := Rle_lt_trans _ _ _ Hx H). easy. Qed. Theorem Ztrunc_ceil : forall x, (x <= 0)%R -> Ztrunc x = Zceil x. Proof. intros x Hx. unfold Ztrunc. destruct Rlt_le_dec as [_|H]. easy. rewrite (Rle_antisym _ _ Hx H). fold (Z2R 0). `````` Guillaume Melquiond committed Mar 04, 2010 1053 1054 ``````rewrite Zceil_Z2R. apply Zfloor_Z2R. `````` Guillaume Melquiond committed Mar 04, 2010 1055 1056 ``````Qed. `````` BOLDO Sylvie committed Sep 03, 2010 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 ``````Theorem Ztrunc_le : forall x y, (x <= y)%R -> (Ztrunc x <= Ztrunc y)%Z. Proof. intros x y Hxy; unfold Ztrunc at 1. destruct Rlt_le_dec as [Hx|Hx]. unfold Ztrunc; destruct Rlt_le_dec as [Hy|Hy]. now apply Zceil_le. apply Zle_trans with 0%Z. apply Zceil_glb. simpl; auto with real. apply Zfloor_lub. now simpl. rewrite Ztrunc_floor. now apply Zfloor_le. now apply Rle_trans with (1:=Hx). Qed. `````` Guillaume Melquiond committed Mar 04, 2010 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 ``````Theorem Ztrunc_opp : forall x, Ztrunc (- x) = Zopp (Ztrunc x). Proof. intros x. destruct (Rlt_le_dec x 0) as [H|H]. rewrite Ztrunc_floor. rewrite Ztrunc_ceil. apply sym_eq. apply Zopp_involutive. now apply Rlt_le. rewrite <- Ropp_0. apply Ropp_le_contravar. now apply Rlt_le. rewrite Ztrunc_ceil. unfold Zceil. rewrite Ropp_involutive. now rewrite Ztrunc_floor. rewrite <- Ropp_0. now apply Ropp_le_contravar. Qed. Theorem Ztrunc_abs : forall x, Ztrunc (Rabs x) = Zabs (Ztrunc x). Proof. intros x. rewrite Ztrunc_floor. 2: apply Rabs_pos. unfold Ztrunc. destruct Rlt_le_dec as [H|H]. rewrite Rabs_left with (1 := H). rewrite Zabs_non_eq. apply sym_eq. apply Zopp_involutive. `````` Guillaume Melquiond committed Mar 10, 2010 1110 ``````apply Zceil_glb. `````` Guillaume Melquiond committed Mar 04, 2010 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 ``````now apply Rlt_le. rewrite Rabs_pos_eq with (1 := H). apply sym_eq. apply Zabs_eq. now apply Zfloor_lub. Qed. Theorem Ztrunc_lub : forall n x, (Z2R n <= Rabs x)%R -> (n <= Zabs (Ztrunc x))%Z. Proof. intros n x H. rewrite <- Ztrunc_abs. rewrite Ztrunc_floor. 2: apply Rabs_pos. now apply Zfloor_lub. Qed. `````` Guillaume Melquiond committed Apr 09, 2009 1129 ``````End Floor_Ceil. `````` BOLDO Sylvie committed Jan 20, 2009 1130 `````` `````` Guillaume Melquiond committed May 11, 2010 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 ``````Section Even_Odd. Definition Zeven (n : Z) := match n with | Zpos (xO _) => true | Zneg (xO _) => true | Z0 => true | _ => false end. Theorem Zeven_mult : forall x y, Zeven (x * y) = orb (Zeven x) (Zeven y). Proof. now intros [|[xp|xp|]|[xp|xp|]] [|[yp|yp|]|[yp|yp|]]. Qed. Theorem Zeven_opp : forall x, Zeven (- x) = Zeven x. Proof. now intros [|[n|n|]|[n|n|]]. Qed. Theorem Zeven_ex : forall x, exists p, x = (2 * p + if Zeven x then 0 else 1)%Z. Proof. intros [|[n|n|]|[n|n|]]. now exists Z0. now exists (Zpos n). now exists (Zpos n). now exists Z0. exists (Zneg n - 1)%Z. change (2 * Zneg n - 1 = 2 * (Zneg n - 1) + 1)%Z. ring. now exists (Zneg n). now exists (-1)%Z. Qed. Theorem Zeven_2xp1 : forall n, Zeven (2 * n + 1) = false. Proof. intros n. destruct (Zeven_ex (2 * n + 1)) as (p, Hp). revert Hp. case (Zeven (2 * n + 1)) ; try easy. intros H. apply False_ind. omega. Qed. Theorem Zeven_plus : forall x y, Zeven (x + y) = Bool.eqb (Zeven x) (Zeven y). Proof. intros x y. destruct (Zeven_ex x) as (px, Hx). rewrite Hx at 1. destruct (Zeven_ex y) as (py, Hy). rewrite Hy at 1. replace (2 * px + (if Zeven x then 0 else 1) + (2 * py + (if Zeven y then 0 else 1)))%Z with (2 * (px + py) + ((if Zeven x then 0 else 1) + (if Zeven y then 0 else 1)))%Z by ring. case (Zeven x) ; case (Zeven y). rewrite Zplus_0_r. now rewrite Zeven_mult. apply Zeven_2xp1. apply Zeven_2xp1. replace (2 * (px + py) + (1 + 1))%Z with (2 * (px + py + 1))%Z by ring. now rewrite Zeven_mult. Qed. End Even_Odd. `````` Guillaume Melquiond committed Aug 19, 2010 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 ``````Section Proof_Irrelevance. Scheme eq_dep_elim := Induction for eq Sort Type. Definition eqbool_dep P (h1 : P true) b := match b return P b -> Prop with | true => fun (h2 : P true) => h1 = h2 | false => fun (h2 : P false) => False end. Lemma eqbool_irrelevance : forall (b : bool) (h1 h2 : b = true), h1 = h2. Proof. assert (forall (h : true = true), refl_equal true = h). apply (eq_dep_elim bool true (eqbool_dep _ _) (refl_equal _)). intros b. case b. intros h1 h2. now rewrite <- (H h1). intros h. discriminate h. Qed. End Proof_Irrelevance. `````` BOLDO Sylvie committed Jan 20, 2009 1225 1226 ``````Section pow. `````` Guillaume Melquiond committed Aug 19, 2010 1227 ``````Record radix := { radix_val : Z ; radix_prop : Zle_bool 2 radix_val = true }. `````` BOLDO Sylvie committed Jan 20, 2009 1228 `````` `````` Guillaume Melquiond committed Aug 19, 2010 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 ``````Theorem radix_val_inj : forall r1 r2, radix_val r1 = radix_val r2 -> r1 = r2. Proof. intros (r1, H1) (r2, H2) H. simpl in H. revert H1. rewrite H. intros H1. apply f_equal. apply eqbool_irrelevance. Qed. `````` Guillaume Melquiond committed Aug 19, 2010 1241 ``````Variable r : radix. `````` BOLDO Sylvie committed Jan 20, 2009 1242 `````` `````` Guillaume Melquiond committed Aug 19, 2010 1243 ``````Theorem radix_gt_1 : (1 < radix_val r)%Z. `````` Guillaume Melquiond committed Apr 09, 2009 1244 ``````Proof. `````` Guillaume Melquiond committed Aug 19, 2010 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 ``````destruct r as (v, Hr). simpl. apply Zlt_le_trans with 2%Z. easy. now apply Zle_bool_imp_le. Qed. Theorem radix_pos : (0 < Z2R (radix_val r))%R. Proof. destruct r as (v, Hr). simpl. apply (Z2R_lt 0). apply Zlt_le_trans with 2%Z. easy. now apply Zle_bool_imp_le. `````` BOLDO Sylvie committed Mar 06, 2009 1258 1259 ``````Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1260 ``````Definition bpow e := `````` BOLDO Sylvie committed Jan 20, 2009 1261 1262 1263 1264 1265 1266 `````` match e with | Zpos p => Z2R (Zpower_pos (radix_val r) p) | Zneg p => Rinv (Z2R (Zpower_pos (radix_val r) p)) | Z0 => R1 end. `````` Guillaume Melquiond committed Oct 29, 2009 1267 ``````Theorem Zpower_pos_powerRZ : `````` BOLDO Sylvie committed Jan 20, 2009 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 `````` forall n m, Z2R (Zpower_pos n m) = powerRZ (Z2R n) (Zpos m). Proof. intros. rewrite Zpower_pos_nat. simpl. induction (nat_of_P m). apply refl_equal. unfold Zpower_nat. simpl. rewrite mult_Z2R. apply Rmult_eq_compat_l. exact IHn0. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1283 ``````Theorem bpow_powerRZ : `````` Guillaume Melquiond committed Apr 09, 2009 1284 `````` forall e, `````` Guillaume Melquiond committed Oct 29, 2009 1285 `````` bpow e = powerRZ (Z2R (radix_val r)) e. `````` BOLDO Sylvie committed Jan 20, 2009 1286 ``````Proof. `````` Guillaume Melquiond committed Oct 29, 2009 1287 ``````destruct e ; unfold bpow. `````` BOLDO Sylvie committed Jan 20, 2009 1288 1289 1290 1291 1292 ``````reflexivity. now rewrite Zpower_pos_powerRZ. now rewrite Zpower_pos_powerRZ. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1293 1294 ``````Theorem bpow_ge_0 : forall e : Z, (0 <= bpow e)%R. `````` BOLDO Sylvie committed Jan 20, 2009 1295 1296 ``````Proof. intros. `````` Guillaume Melquiond committed Oct 29, 2009 1297 ``````rewrite bpow_powerRZ. `````` BOLDO Sylvie committed Jan 20, 2009 1298 ``````apply powerRZ_le. `````` Guillaume Melquiond committed Aug 19, 2010 1299 ``````apply radix_pos. `````` BOLDO Sylvie committed Jan 20, 2009 1300 1301 ``````Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1302 1303 ``````Theorem bpow_gt_0 : forall e : Z, (0 < bpow e)%R. `````` BOLDO Sylvie committed Jan 20, 2009 1304 1305 ``````Proof. intros. `````` Guillaume Melquiond committed Oct 29, 2009 1306 ``````rewrite bpow_powerRZ. `````` BOLDO Sylvie committed Jan 20, 2009 1307 ``````apply powerRZ_lt. `````` Guillaume Melquiond committed Aug 19, 2010 1308 ``````apply radix_pos. `````` BOLDO Sylvie committed Jan 20, 2009 1309 1310 ``````Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1311 1312 ``````Theorem bpow_add : forall e1 e2 : Z, (bpow (e1 + e2) = bpow e1 * bpow e2)%R. `````` BOLDO Sylvie committed Jan 20, 2009 1313 1314 ``````Proof. intros. `````` Guillaume Melquiond committed Oct 29, 2009 1315 ``````repeat rewrite bpow_powerRZ. `````` BOLDO Sylvie committed Jan 20, 2009 1316 ``````apply powerRZ_add. `````` Guillaume Melquiond committed Aug 19, 2010 1317 1318 ``````apply Rgt_not_eq. apply radix_pos. `````` BOLDO Sylvie committed Jan 20, 2009 1319 1320 ``````Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1321 1322 ``````Theorem bpow_1 : bpow 1 = Z2R (radix_val r). `````` BOLDO Sylvie committed Jan 20, 2009 1323 ``````Proof. `````` Guillaume Melquiond committed Oct 29, 2009 1324 ``````unfold bpow, Zpower_pos, iter_pos. `````` BOLDO Sylvie committed Jan 20, 2009 1325 1326 1327 ``````now rewrite Zmult_1_r. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1328 1329 ``````Theorem bpow_add1 : forall e : Z, (bpow (e+1) = Z2R (radix_val r) * bpow e)%R. `````` BOLDO Sylvie committed Mar 06, 2009 1330 1331 ``````Proof. intros. `````` Guillaume Melquiond committed Oct 29, 2009 1332 1333 ``````rewrite <- bpow_1. rewrite <- bpow_add. `````` BOLDO Sylvie committed Mar 06, 2009 1334 1335 1336 ``````now rewrite Zplus_comm. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1337 1338 ``````Theorem bpow_opp : forall e : Z, (bpow (-e) = /bpow e)%R. `````` BOLDO Sylvie committed Mar 06, 2009 1339 1340 1341 1342 1343 1344 ``````Proof. intros e; destruct e. simpl; now rewrite Rinv_1. now replace (-Zpos p)%Z with (Zneg p) by reflexivity. replace (-Zneg p)%Z with (Zpos p) by reflexivity. simpl; rewrite Rinv_involutive; trivial. `````` Guillaume Melquiond committed Oct 29, 2009 1345 ``````generalize (bpow_gt_0 (Zpos p)). `````` BOLDO Sylvie committed Mar 06, 2009 1346 1347 1348 ``````simpl; auto with real. Qed. `````` Guillaume Melquiond committed Aug 20, 2010 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 ``````Theorem Z2R_Zpower_nat : forall e : nat, Z2R (Zpower_nat (radix_val r) e) = bpow (Z_of_nat e). Proof. intros [|e]. split. rewrite <- nat_of_P_o_P_of_succ_nat_eq_succ. rewrite <- Zpower_pos_nat. now rewrite <- Zpos_eq_Z_of_nat_o_nat_of_P. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1360 ``````Theorem Z2R_Zpower : `````` Guillaume Melquiond committed Jan 26, 2009 1361 1362 `````` forall e : Z, (0 <= e)%Z -> `````` Guillaume Melquiond committed Oct 29, 2009 1363 `````` Z2R (Zpower (radix_val r) e) = bpow e. `````` Guillaume Melquiond committed Jan 26, 2009 1364 1365 1366 1367 1368 1369 1370 ``````Proof. intros [|e|e] H. split. split. now elim H. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1371 ``````Theorem bpow_lt_aux : `````` Guillaume Melquiond committed Jan 26, 2009 1372 `````` forall e1 e2 : Z, `````` Guillaume Melquiond committed Oct 29, 2009 1373 `````` (e1 < e2)%Z -> (bpow e1 < bpow e2)%R. `````` Guillaume Melquiond committed Jan 26, 2009 1374 1375 1376 ``````Proof. intros e1 e2 H. replace e2 with (e1 + (e2 - e1))%Z by ring. `````` Guillaume Melquiond committed Oct 29, 2009 1377 1378 ``````rewrite <- (Rmult_1_r (bpow e1)). rewrite bpow_add. `````` Guillaume Melquiond committed Jan 26, 2009 1379 ``````apply Rmult_lt_compat_l. `````` Guillaume Melquiond committed Oct 29, 2009 1380 ``````apply bpow_gt_0. `````` Guillaume Melquiond committed Jan 26, 2009 1381 1382 1383 ``````assert (0 < e2 - e1)%Z by omega. destruct (e2 - e1)%Z ; try discriminate H0. clear. `````` Guillaume Melquiond committed Oct 29, 2009 1384 ``````unfold bpow. `````` Guillaume Melquiond committed Jan 26, 2009 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 ``````apply (Z2R_lt 1). rewrite Zpower_pos_nat. case_eq (nat_of_P p). intros H. elim (lt_irrefl 0). pattern O at 2 ; rewrite <- H. apply lt_O_nat_of_P. intros n _. assert (1 < Zpower_nat (radix_val r) 1)%Z. unfold Zpower_nat, iter_nat. rewrite Zmult_1_r. `````` Guillaume Melquiond committed Aug 19, 2010 1396 ``````apply radix_gt_1. `````` Guillaume Melquiond committed Jan 26, 2009 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 ``````induction n. exact H. change (S (S n)) with (1 + (S n))%nat. rewrite Zpower_nat_is_exp. change 1%Z with (1 * 1)%Z. apply Zmult_lt_compat. now split. now split. Qed. `````` Guillaume Melquiond committed Oct 29, 2009 1407 ``````Theorem bpow_lt : `````` Guillaume Melquiond committed Jan 26, 2009 1408 `````` forall e1 e2 : Z, `````` Guillaume Melquiond committed Oct 29, 2009 1409 `````` (e1 < e2)%Z <-> (bpow e1 < bpow e2)%R. `````` Guillaume Melquiond committed Jan 26, 2009 1410 1411 1412 ``````Proof. intros e1 e2. split. `````` Guillaume Melquiond committed Oct 29, 2009 1413 ``````apply bpow_lt_aux. `````` Guillaume Melquiond committed Jan 26, 2009 1414 1415 1416 <