isl_map_simplify.c

   1 /*
   2  * Copyright 2008-2009 Katholieke Universiteit Leuven
   3  *
   4  * Use of this software is governed by the GNU LGPLv2.1 license
   5  *
   6  * Written by Sven Verdoolaege, K.U.Leuven, Departement
   7  * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
   8  */
   9
  10 #include "isl_equalities.h"
  11 #include "isl_map.h"
  12 #include "isl_map_private.h"
  13 #include "isl_seq.h"
  14 #include "isl_tab.h"
  15
  16 static void swap_equality(struct isl_basic_map *bmap, int a, int b)
  17 {
  18         isl_int *t = bmap->eq[a];
  19         bmap->eq[a] = bmap->eq[b];
  20         bmap->eq[b] = t;
  21 }
  22
  23 static void swap_inequality(struct isl_basic_map *bmap, int a, int b)
  24 {
  25         if (a != b) {
  26                 isl_int *t = bmap->ineq[a];
  27                 bmap->ineq[a] = bmap->ineq[b];
  28                 bmap->ineq[b] = t;
  29         }
  30 }
  31
  32 static void set_swap_inequality(struct isl_basic_set *bset, int a, int b)
  33 {
  34         swap_inequality((struct isl_basic_map *)bset, a, b);
  35 }
  36
  37 static void constraint_drop_vars(isl_int *c, unsigned n, unsigned rem)
  38 {
  39         isl_seq_cpy(c, c + n, rem);
  40         isl_seq_clr(c + rem, n);
  41 }
  42
  43 /* Drop n dimensions starting at first.
  44  *
  45  * In principle, this frees up some extra variables as the number
  46  * of columns remains constant, but we would have to extend
  47  * the div array too as the number of rows in this array is assumed
  48  * to be equal to extra.
  49  */
  50 struct isl_basic_set *isl_basic_set_drop_dims(
  51                 struct isl_basic_set *bset, unsigned first, unsigned n)
  52 {
  53         int i;
  54
  55         if (!bset)
  56                 goto error;
  57
  58         isl_assert(bset->ctx, first + n <= bset->dim->n_out, goto error);
  59
  60         if (n == 0)
  61                 return bset;
  62
  63         bset = isl_basic_set_cow(bset);
  64         if (!bset)
  65                 return NULL;
  66
  67         for (i = 0; i < bset->n_eq; ++i)
  68                 constraint_drop_vars(bset->eq[i]+1+bset->dim->nparam+first, n,
  69                                      (bset->dim->n_out-first-n)+bset->extra);
  70
  71         for (i = 0; i < bset->n_ineq; ++i)
  72                 constraint_drop_vars(bset->ineq[i]+1+bset->dim->nparam+first, n,
  73                                      (bset->dim->n_out-first-n)+bset->extra);
  74
  75         for (i = 0; i < bset->n_div; ++i)
  76                 constraint_drop_vars(bset->div[i]+1+1+bset->dim->nparam+first, n,
  77                                      (bset->dim->n_out-first-n)+bset->extra);
  78
  79         bset->dim = isl_dim_drop_outputs(bset->dim, first, n);
  80         if (!bset->dim)
  81                 goto error;
  82
  83         ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
  84         bset = isl_basic_set_simplify(bset);
  85         return isl_basic_set_finalize(bset);
  86 error:
  87         isl_basic_set_free(bset);
  88         return NULL;
  89 }
  90
  91 struct isl_set *isl_set_drop_dims(
  92                 struct isl_set *set, unsigned first, unsigned n)
  93 {
  94         int i;
  95
  96         if (!set)
  97                 goto error;
  98
  99         isl_assert(set->ctx, first + n <= set->dim->n_out, goto error);
 100
 101         if (n == 0)
 102                 return set;
 103         set = isl_set_cow(set);
 104         if (!set)
 105                 goto error;
 106         set->dim = isl_dim_drop_outputs(set->dim, first, n);
 107         if (!set->dim)
 108                 goto error;
 109
 110         for (i = 0; i < set->n; ++i) {
 111                 set->p[i] = isl_basic_set_drop_dims(set->p[i], first, n);
 112                 if (!set->p[i])
 113                         goto error;
 114         }
 115
 116         ISL_F_CLR(set, ISL_SET_NORMALIZED);
 117         return set;
 118 error:
 119         isl_set_free(set);
 120         return NULL;
 121 }
 122
 123 /* Move "n" divs starting at "first" to the end of the list of divs.
 124  */
 125 static struct isl_basic_map *move_divs_last(struct isl_basic_map *bmap,
 126         unsigned first, unsigned n)
 127 {
 128         isl_int **div;
 129         int i;
 130
 131         if (first + n == bmap->n_div)
 132                 return bmap;
 133
 134         div = isl_alloc_array(bmap->ctx, isl_int *, n);
 135         if (!div)
 136                 goto error;
 137         for (i = 0; i < n; ++i)
 138                 div[i] = bmap->div[first + i];
 139         for (i = 0; i < bmap->n_div - first - n; ++i)
 140                 bmap->div[first + i] = bmap->div[first + n + i];
 141         for (i = 0; i < n; ++i)
 142                 bmap->div[bmap->n_div - n + i] = div[i];
 143         free(div);
 144         return bmap;
 145 error:
 146         isl_basic_map_free(bmap);
 147         return NULL;
 148 }
 149
 150 /* Drop "n" dimensions of type "type" starting at "first".
 151  *
 152  * In principle, this frees up some extra variables as the number
 153  * of columns remains constant, but we would have to extend
 154  * the div array too as the number of rows in this array is assumed
 155  * to be equal to extra.
 156  */
 157 struct isl_basic_map *isl_basic_map_drop(struct isl_basic_map *bmap,
 158         enum isl_dim_type type, unsigned first, unsigned n)
 159 {
 160         int i;
 161         unsigned dim;
 162         unsigned offset;
 163         unsigned left;
 164
 165         if (!bmap)
 166                 goto error;
 167
 168         dim = isl_basic_map_dim(bmap, type);
 169         isl_assert(bmap->ctx, first + n <= dim, goto error);
 170
 171         if (n == 0)
 172                 return bmap;
 173
 174         bmap = isl_basic_map_cow(bmap);
 175         if (!bmap)
 176                 return NULL;
 177
 178         offset = isl_basic_map_offset(bmap, type) + first;
 179         left = isl_basic_map_total_dim(bmap) - (offset - 1) - n;
 180         for (i = 0; i < bmap->n_eq; ++i)
 181                 constraint_drop_vars(bmap->eq[i]+offset, n, left);
 182
 183         for (i = 0; i < bmap->n_ineq; ++i)
 184                 constraint_drop_vars(bmap->ineq[i]+offset, n, left);
 185
 186         for (i = 0; i < bmap->n_div; ++i)
 187                 constraint_drop_vars(bmap->div[i]+1+offset, n, left);
 188
 189         if (type == isl_dim_div) {
 190                 bmap = move_divs_last(bmap, first, n);
 191                 if (!bmap)
 192                         goto error;
 193                 isl_basic_map_free_div(bmap, n);
 194         } else
 195                 bmap->dim = isl_dim_drop(bmap->dim, type, first, n);
 196         if (!bmap->dim)
 197                 goto error;
 198
 199         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 200         bmap = isl_basic_map_simplify(bmap);
 201         return isl_basic_map_finalize(bmap);
 202 error:
 203         isl_basic_map_free(bmap);
 204         return NULL;
 205 }
 206
 207 struct isl_basic_map *isl_basic_map_drop_inputs(
 208                 struct isl_basic_map *bmap, unsigned first, unsigned n)
 209 {
 210         return isl_basic_map_drop(bmap, isl_dim_in, first, n);
 211 }
 212
 213 struct isl_map *isl_map_drop(struct isl_map *map,
 214         enum isl_dim_type type, unsigned first, unsigned n)
 215 {
 216         int i;
 217
 218         if (!map)
 219                 goto error;
 220
 221         isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
 222
 223         if (n == 0)
 224                 return map;
 225         map = isl_map_cow(map);
 226         if (!map)
 227                 goto error;
 228         map->dim = isl_dim_drop(map->dim, type, first, n);
 229         if (!map->dim)
 230                 goto error;
 231
 232         for (i = 0; i < map->n; ++i) {
 233                 map->p[i] = isl_basic_map_drop(map->p[i], type, first, n);
 234                 if (!map->p[i])
 235                         goto error;
 236         }
 237         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
 238
 239         return map;
 240 error:
 241         isl_map_free(map);
 242         return NULL;
 243 }
 244
 245 struct isl_set *isl_set_drop(struct isl_set *set,
 246         enum isl_dim_type type, unsigned first, unsigned n)
 247 {
 248         return (isl_set *)isl_map_drop((isl_map *)set, type, first, n);
 249 }
 250
 251 struct isl_map *isl_map_drop_inputs(
 252                 struct isl_map *map, unsigned first, unsigned n)
 253 {
 254         return isl_map_drop(map, isl_dim_in, first, n);
 255 }
 256
 257 /*
 258  * We don't cow, as the div is assumed to be redundant.
 259  */
 260 static struct isl_basic_map *isl_basic_map_drop_div(
 261                 struct isl_basic_map *bmap, unsigned div)
 262 {
 263         int i;
 264         unsigned pos;
 265
 266         if (!bmap)
 267                 goto error;
 268
 269         pos = 1 + isl_dim_total(bmap->dim) + div;
 270
 271         isl_assert(bmap->ctx, div < bmap->n_div, goto error);
 272
 273         for (i = 0; i < bmap->n_eq; ++i)
 274                 constraint_drop_vars(bmap->eq[i]+pos, 1, bmap->extra-div-1);
 275
 276         for (i = 0; i < bmap->n_ineq; ++i) {
 277                 if (!isl_int_is_zero(bmap->ineq[i][pos])) {
 278                         isl_basic_map_drop_inequality(bmap, i);
 279                         --i;
 280                         continue;
 281                 }
 282                 constraint_drop_vars(bmap->ineq[i]+pos, 1, bmap->extra-div-1);
 283         }
 284
 285         for (i = 0; i < bmap->n_div; ++i)
 286                 constraint_drop_vars(bmap->div[i]+1+pos, 1, bmap->extra-div-1);
 287
 288         if (div != bmap->n_div - 1) {
 289                 int j;
 290                 isl_int *t = bmap->div[div];
 291
 292                 for (j = div; j < bmap->n_div - 1; ++j)
 293                         bmap->div[j] = bmap->div[j+1];
 294
 295                 bmap->div[bmap->n_div - 1] = t;
 296         }
 297         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 298         isl_basic_map_free_div(bmap, 1);
 299
 300         return bmap;
 301 error:
 302         isl_basic_map_free(bmap);
 303         return NULL;
 304 }
 305
 306 struct isl_basic_map *isl_basic_map_normalize_constraints(
 307         struct isl_basic_map *bmap)
 308 {
 309         int i;
 310         isl_int gcd;
 311         unsigned total = isl_basic_map_total_dim(bmap);
 312
 313         isl_int_init(gcd);
 314         for (i = bmap->n_eq - 1; i >= 0; --i) {
 315                 isl_seq_gcd(bmap->eq[i]+1, total, &gcd);
 316                 if (isl_int_is_zero(gcd)) {
 317                         if (!isl_int_is_zero(bmap->eq[i][0])) {
 318                                 bmap = isl_basic_map_set_to_empty(bmap);
 319                                 break;
 320                         }
 321                         isl_basic_map_drop_equality(bmap, i);
 322                         continue;
 323                 }
 324                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 325                         isl_int_gcd(gcd, gcd, bmap->eq[i][0]);
 326                 if (isl_int_is_one(gcd))
 327                         continue;
 328                 if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) {
 329                         bmap = isl_basic_map_set_to_empty(bmap);
 330                         break;
 331                 }
 332                 isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total);
 333         }
 334
 335         for (i = bmap->n_ineq - 1; i >= 0; --i) {
 336                 isl_seq_gcd(bmap->ineq[i]+1, total, &gcd);
 337                 if (isl_int_is_zero(gcd)) {
 338                         if (isl_int_is_neg(bmap->ineq[i][0])) {
 339                                 bmap = isl_basic_map_set_to_empty(bmap);
 340                                 break;
 341                         }
 342                         isl_basic_map_drop_inequality(bmap, i);
 343                         continue;
 344                 }
 345                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 346                         isl_int_gcd(gcd, gcd, bmap->ineq[i][0]);
 347                 if (isl_int_is_one(gcd))
 348                         continue;
 349                 isl_int_fdiv_q(bmap->ineq[i][0], bmap->ineq[i][0], gcd);
 350                 isl_seq_scale_down(bmap->ineq[i]+1, bmap->ineq[i]+1, gcd, total);
 351         }
 352         isl_int_clear(gcd);
 353
 354         return bmap;
 355 }
 356
 357 struct isl_basic_set *isl_basic_set_normalize_constraints(
 358         struct isl_basic_set *bset)
 359 {
 360         return (struct isl_basic_set *)isl_basic_map_normalize_constraints(
 361                 (struct isl_basic_map *)bset);
 362 }
 363
 364 /* Assumes divs have been ordered if keep_divs is set.
 365  */
 366 static void eliminate_var_using_equality(struct isl_basic_map *bmap,
 367         unsigned pos, isl_int *eq, int keep_divs, int *progress)
 368 {
 369         unsigned total;
 370         int k;
 371         int last_div;
 372
 373         total = isl_basic_map_total_dim(bmap);
 374         last_div = isl_seq_last_non_zero(eq + 1 + isl_dim_total(bmap->dim),
 375                                                 bmap->n_div);
 376         for (k = 0; k < bmap->n_eq; ++k) {
 377                 if (bmap->eq[k] == eq)
 378                         continue;
 379                 if (isl_int_is_zero(bmap->eq[k][1+pos]))
 380                         continue;
 381                 if (progress)
 382                         *progress = 1;
 383                 isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL);
 384         }
 385
 386         for (k = 0; k < bmap->n_ineq; ++k) {
 387                 if (isl_int_is_zero(bmap->ineq[k][1+pos]))
 388                         continue;
 389                 if (progress)
 390                         *progress = 1;
 391                 isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL);
 392                 ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 393         }
 394
 395         for (k = 0; k < bmap->n_div; ++k) {
 396                 if (isl_int_is_zero(bmap->div[k][0]))
 397                         continue;
 398                 if (isl_int_is_zero(bmap->div[k][1+1+pos]))
 399                         continue;
 400                 if (progress)
 401                         *progress = 1;
 402                 /* We need to be careful about circular definitions,
 403                  * so for now we just remove the definition of div k
 404                  * if the equality contains any divs.
 405                  * If keep_divs is set, then the divs have been ordered
 406                  * and we can keep the definition as long as the result
 407                  * is still ordered.
 408                  */
 409                 if (last_div == -1 || (keep_divs && last_div < k))
 410                         isl_seq_elim(bmap->div[k]+1, eq,
 411                                         1+pos, 1+total, &bmap->div[k][0]);
 412                 else
 413                         isl_seq_clr(bmap->div[k], 1 + total);
 414                 ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 415         }
 416 }
 417
 418 /* Assumes divs have been ordered if keep_divs is set.
 419  */
 420 static void eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
 421         unsigned div, int keep_divs)
 422 {
 423         unsigned pos = isl_dim_total(bmap->dim) + div;
 424
 425         eliminate_var_using_equality(bmap, pos, eq, keep_divs, NULL);
 426
 427         isl_basic_map_drop_div(bmap, div);
 428 }
 429
 430 /* Check if elimination of div "div" using equality "eq" would not
 431  * result in a div depending on a later div.
 432  */
 433 static int ok_to_eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
 434         unsigned div)
 435 {
 436         int k;
 437         int last_div;
 438         unsigned pos = isl_dim_total(bmap->dim) + div;
 439
 440         last_div = isl_seq_last_non_zero(eq + 1 + isl_dim_total(bmap->dim),
 441                                                 bmap->n_div);
 442         if (last_div < 0 || last_div <= div)
 443                 return 1;
 444
 445         for (k = 0; k <= last_div; ++k) {
 446                 if (isl_int_is_zero(bmap->div[k][0]))
 447                         return 1;
 448                 if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos]))
 449                         return 0;
 450         }
 451
 452         return 1;
 453 }
 454
 455 /* Elimininate divs based on equalities
 456  */
 457 static struct isl_basic_map *eliminate_divs_eq(
 458                 struct isl_basic_map *bmap, int *progress)
 459 {
 460         int d;
 461         int i;
 462         int modified = 0;
 463         unsigned off;
 464
 465         bmap = isl_basic_map_order_divs(bmap);
 466
 467         if (!bmap)
 468                 return NULL;
 469
 470         off = 1 + isl_dim_total(bmap->dim);
 471
 472         for (d = bmap->n_div - 1; d >= 0 ; --d) {
 473                 for (i = 0; i < bmap->n_eq; ++i) {
 474                         if (!isl_int_is_one(bmap->eq[i][off + d]) &&
 475                             !isl_int_is_negone(bmap->eq[i][off + d]))
 476                                 continue;
 477                         if (!ok_to_eliminate_div(bmap, bmap->eq[i], d))
 478                                 continue;
 479                         modified = 1;
 480                         *progress = 1;
 481                         eliminate_div(bmap, bmap->eq[i], d, 1);
 482                         isl_basic_map_drop_equality(bmap, i);
 483                         break;
 484                 }
 485         }
 486         if (modified)
 487                 return eliminate_divs_eq(bmap, progress);
 488         return bmap;
 489 }
 490
 491 /* Elimininate divs based on inequalities
 492  */
 493 static struct isl_basic_map *eliminate_divs_ineq(
 494                 struct isl_basic_map *bmap, int *progress)
 495 {
 496         int d;
 497         int i;
 498         unsigned off;
 499         struct isl_ctx *ctx;
 500
 501         if (!bmap)
 502                 return NULL;
 503
 504         ctx = bmap->ctx;
 505         off = 1 + isl_dim_total(bmap->dim);
 506
 507         for (d = bmap->n_div - 1; d >= 0 ; --d) {
 508                 for (i = 0; i < bmap->n_eq; ++i)
 509                         if (!isl_int_is_zero(bmap->eq[i][off + d]))
 510                                 break;
 511                 if (i < bmap->n_eq)
 512                         continue;
 513                 for (i = 0; i < bmap->n_ineq; ++i)
 514                         if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one))
 515                                 break;
 516                 if (i < bmap->n_ineq)
 517                         continue;
 518                 *progress = 1;
 519                 bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1);
 520                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 521                         break;
 522                 bmap = isl_basic_map_drop_div(bmap, d);
 523                 if (!bmap)
 524                         break;
 525         }
 526         return bmap;
 527 }
 528
 529 struct isl_basic_map *isl_basic_map_gauss(
 530         struct isl_basic_map *bmap, int *progress)
 531 {
 532         int k;
 533         int done;
 534         int last_var;
 535         unsigned total_var;
 536         unsigned total;
 537
 538         bmap = isl_basic_map_order_divs(bmap);
 539
 540         if (!bmap)
 541                 return NULL;
 542
 543         total = isl_basic_map_total_dim(bmap);
 544         total_var = total - bmap->n_div;
 545
 546         last_var = total - 1;
 547         for (done = 0; done < bmap->n_eq; ++done) {
 548                 for (; last_var >= 0; --last_var) {
 549                         for (k = done; k < bmap->n_eq; ++k)
 550                                 if (!isl_int_is_zero(bmap->eq[k][1+last_var]))
 551                                         break;
 552                         if (k < bmap->n_eq)
 553                                 break;
 554                 }
 555                 if (last_var < 0)
 556                         break;
 557                 if (k != done)
 558                         swap_equality(bmap, k, done);
 559                 if (isl_int_is_neg(bmap->eq[done][1+last_var]))
 560                         isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total);
 561
 562                 eliminate_var_using_equality(bmap, last_var, bmap->eq[done], 1,
 563                                                 progress);
 564
 565                 if (last_var >= total_var &&
 566                     isl_int_is_zero(bmap->div[last_var - total_var][0])) {
 567                         unsigned div = last_var - total_var;
 568                         isl_seq_neg(bmap->div[div]+1, bmap->eq[done], 1+total);
 569                         isl_int_set_si(bmap->div[div][1+1+last_var], 0);
 570                         isl_int_set(bmap->div[div][0],
 571                                     bmap->eq[done][1+last_var]);
 572                         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 573                 }
 574         }
 575         if (done == bmap->n_eq)
 576                 return bmap;
 577         for (k = done; k < bmap->n_eq; ++k) {
 578                 if (isl_int_is_zero(bmap->eq[k][0]))
 579                         continue;
 580                 return isl_basic_map_set_to_empty(bmap);
 581         }
 582         isl_basic_map_free_equality(bmap, bmap->n_eq-done);
 583         return bmap;
 584 }
 585
 586 struct isl_basic_set *isl_basic_set_gauss(
 587         struct isl_basic_set *bset, int *progress)
 588 {
 589         return (struct isl_basic_set*)isl_basic_map_gauss(
 590                         (struct isl_basic_map *)bset, progress);
 591 }
 592
 593
 594 static unsigned int round_up(unsigned int v)
 595 {
 596         int old_v = v;
 597
 598         while (v) {
 599                 old_v = v;
 600                 v ^= v & -v;
 601         }
 602         return old_v << 1;
 603 }
 604
 605 static int hash_index(isl_int ***index, unsigned int size, int bits,
 606                         struct isl_basic_map *bmap, int k)
 607 {
 608         int h;
 609         unsigned total = isl_basic_map_total_dim(bmap);
 610         uint32_t hash = isl_seq_get_hash_bits(bmap->ineq[k]+1, total, bits);
 611         for (h = hash; index[h]; h = (h+1) % size)
 612                 if (&bmap->ineq[k] != index[h] &&
 613                     isl_seq_eq(bmap->ineq[k]+1, index[h][0]+1, total))
 614                         break;
 615         return h;
 616 }
 617
 618 static int set_hash_index(isl_int ***index, unsigned int size, int bits,
 619                           struct isl_basic_set *bset, int k)
 620 {
 621         return hash_index(index, size, bits, (struct isl_basic_map *)bset, k);
 622 }
 623
 624 /* If we can eliminate more than one div, then we need to make
 625  * sure we do it from last div to first div, in order not to
 626  * change the position of the other divs that still need to
 627  * be removed.
 628  */
 629 static struct isl_basic_map *remove_duplicate_divs(
 630         struct isl_basic_map *bmap, int *progress)
 631 {
 632         unsigned int size;
 633         int *index;
 634         int *elim_for;
 635         int k, l, h;
 636         int bits;
 637         struct isl_blk eq;
 638         unsigned total_var = isl_dim_total(bmap->dim);
 639         unsigned total = total_var + bmap->n_div;
 640         struct isl_ctx *ctx;
 641
 642         if (bmap->n_div <= 1)
 643                 return bmap;
 644
 645         ctx = bmap->ctx;
 646         for (k = bmap->n_div - 1; k >= 0; --k)
 647                 if (!isl_int_is_zero(bmap->div[k][0]))
 648                         break;
 649         if (k <= 0)
 650                 return bmap;
 651
 652         elim_for = isl_calloc_array(ctx, int, bmap->n_div);
 653         size = round_up(4 * bmap->n_div / 3 - 1);
 654         bits = ffs(size) - 1;
 655         index = isl_calloc_array(ctx, int, size);
 656         if (!index)
 657                 return bmap;
 658         eq = isl_blk_alloc(ctx, 1+total);
 659         if (isl_blk_is_error(eq))
 660                 goto out;
 661
 662         isl_seq_clr(eq.data, 1+total);
 663         index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1;
 664         for (--k; k >= 0; --k) {
 665                 uint32_t hash;
 666
 667                 if (isl_int_is_zero(bmap->div[k][0]))
 668                         continue;
 669
 670                 hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits);
 671                 for (h = hash; index[h]; h = (h+1) % size)
 672                         if (isl_seq_eq(bmap->div[k],
 673                                        bmap->div[index[h]-1], 2+total))
 674                                 break;
 675                 if (index[h]) {
 676                         *progress = 1;
 677                         l = index[h] - 1;
 678                         elim_for[l] = k + 1;
 679                 }
 680                 index[h] = k+1;
 681         }
 682         for (l = bmap->n_div - 1; l >= 0; --l) {
 683                 if (!elim_for[l])
 684                         continue;
 685                 k = elim_for[l] - 1;
 686                 isl_int_set_si(eq.data[1+total_var+k], -1);
 687                 isl_int_set_si(eq.data[1+total_var+l], 1);
 688                 eliminate_div(bmap, eq.data, l, 0);
 689                 isl_int_set_si(eq.data[1+total_var+k], 0);
 690                 isl_int_set_si(eq.data[1+total_var+l], 0);
 691         }
 692
 693         isl_blk_free(ctx, eq);
 694 out:
 695         free(index);
 696         free(elim_for);
 697         return bmap;
 698 }
 699
 700 static int n_pure_div_eq(struct isl_basic_map *bmap)
 701 {
 702         int i, j;
 703         unsigned total;
 704
 705         total = isl_dim_total(bmap->dim);
 706         for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) {
 707                 while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
 708                         --j;
 709                 if (j < 0)
 710                         break;
 711                 if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total, j) != -1)
 712                         return 0;
 713         }
 714         return i;
 715 }
 716
 717 /* Normalize divs that appear in equalities.
 718  *
 719  * In particular, we assume that bmap contains some equalities
 720  * of the form
 721  *
 722  *      a x = m * e_i
 723  *
 724  * and we want to replace the set of e_i by a minimal set and
 725  * such that the new e_i have a canonical representation in terms
 726  * of the vector x.
 727  * If any of the equalities involves more than one divs, then
 728  * we currently simply bail out.
 729  *
 730  * Let us first additionally assume that all equalities involve
 731  * a div.  The equalities then express modulo constraints on the
 732  * remaining variables and we can use "parameter compression"
 733  * to find a minimal set of constraints.  The result is a transformation
 734  *
 735  *      x = T(x') = x_0 + G x'
 736  *
 737  * with G a lower-triangular matrix with all elements below the diagonal
 738  * non-negative and smaller than the diagonal element on the same row.
 739  * We first normalize x_0 by making the same property hold in the affine
 740  * T matrix.
 741  * The rows i of G with a 1 on the diagonal do not impose any modulo
 742  * constraint and simply express x_i = x'_i.
 743  * For each of the remaining rows i, we introduce a div and a corresponding
 744  * equality.  In particular
 745  *
 746  *      g_ii e_j = x_i - g_i(x')
 747  *
 748  * where each x'_k is replaced either by x_k (if g_kk = 1) or the
 749  * corresponding div (if g_kk != 1).
 750  *
 751  * If there are any equalities not involving any div, then we
 752  * first apply a variable compression on the variables x:
 753  *
 754  *      x = C x''       x'' = C_2 x
 755  *
 756  * and perform the above parameter compression on A C instead of on A.
 757  * The resulting compression is then of the form
 758  *
 759  *      x'' = T(x') = x_0 + G x'
 760  *
 761  * and in constructing the new divs and the corresponding equalities,
 762  * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
 763  * by the corresponding row from C_2.
 764  */
 765 static struct isl_basic_map *normalize_divs(
 766         struct isl_basic_map *bmap, int *progress)
 767 {
 768         int i, j, k;
 769         int total;
 770         int div_eq;
 771         struct isl_mat *B;
 772         struct isl_vec *d;
 773         struct isl_mat *T = NULL;
 774         struct isl_mat *C = NULL;
 775         struct isl_mat *C2 = NULL;
 776         isl_int v;
 777         int *pos;
 778         int dropped, needed;
 779
 780         if (!bmap)
 781                 return NULL;
 782
 783         if (bmap->n_div == 0)
 784                 return bmap;
 785
 786         if (bmap->n_eq == 0)
 787                 return bmap;
 788
 789         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS))
 790                 return bmap;
 791
 792         total = isl_dim_total(bmap->dim);
 793         div_eq = n_pure_div_eq(bmap);
 794         if (div_eq == 0)
 795                 return bmap;
 796
 797         if (div_eq < bmap->n_eq) {
 798                 B = isl_mat_sub_alloc(bmap->ctx, bmap->eq, div_eq,
 799                                         bmap->n_eq - div_eq, 0, 1 + total);
 800                 C = isl_mat_variable_compression(B, &C2);
 801                 if (!C || !C2)
 802                         goto error;
 803                 if (C->n_col == 0) {
 804                         bmap = isl_basic_map_set_to_empty(bmap);
 805                         isl_mat_free(C);
 806                         isl_mat_free(C2);
 807                         goto done;
 808                 }
 809         }
 810
 811         d = isl_vec_alloc(bmap->ctx, div_eq);
 812         if (!d)
 813                 goto error;
 814         for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) {
 815                 while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
 816                         --j;
 817                 isl_int_set(d->block.data[i], bmap->eq[i][1 + total + j]);
 818         }
 819         B = isl_mat_sub_alloc(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + total);
 820
 821         if (C) {
 822                 B = isl_mat_product(B, C);
 823                 C = NULL;
 824         }
 825
 826         T = isl_mat_parameter_compression(B, d);
 827         if (!T)
 828                 goto error;
 829         if (T->n_col == 0) {
 830                 bmap = isl_basic_map_set_to_empty(bmap);
 831                 isl_mat_free(C2);
 832                 isl_mat_free(T);
 833                 goto done;
 834         }
 835         isl_int_init(v);
 836         for (i = 0; i < T->n_row - 1; ++i) {
 837                 isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]);
 838                 if (isl_int_is_zero(v))
 839                         continue;
 840                 isl_mat_col_submul(T, 0, v, 1 + i);
 841         }
 842         isl_int_clear(v);
 843         pos = isl_alloc_array(bmap->ctx, int, T->n_row);
 844         /* We have to be careful because dropping equalities may reorder them */
 845         dropped = 0;
 846         for (j = bmap->n_div - 1; j >= 0; --j) {
 847                 for (i = 0; i < bmap->n_eq; ++i)
 848                         if (!isl_int_is_zero(bmap->eq[i][1 + total + j]))
 849                                 break;
 850                 if (i < bmap->n_eq) {
 851                         bmap = isl_basic_map_drop_div(bmap, j);
 852                         isl_basic_map_drop_equality(bmap, i);
 853                         ++dropped;
 854                 }
 855         }
 856         pos[0] = 0;
 857         needed = 0;
 858         for (i = 1; i < T->n_row; ++i) {
 859                 if (isl_int_is_one(T->row[i][i]))
 860                         pos[i] = i;
 861                 else
 862                         needed++;
 863         }
 864         if (needed > dropped) {
 865                 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim),
 866                                 needed, needed, 0);
 867                 if (!bmap)
 868                         goto error;
 869         }
 870         for (i = 1; i < T->n_row; ++i) {
 871                 if (isl_int_is_one(T->row[i][i]))
 872                         continue;
 873                 k = isl_basic_map_alloc_div(bmap);
 874                 pos[i] = 1 + total + k;
 875                 isl_seq_clr(bmap->div[k] + 1, 1 + total + bmap->n_div);
 876                 isl_int_set(bmap->div[k][0], T->row[i][i]);
 877                 if (C2)
 878                         isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + total);
 879                 else
 880                         isl_int_set_si(bmap->div[k][1 + i], 1);
 881                 for (j = 0; j < i; ++j) {
 882                         if (isl_int_is_zero(T->row[i][j]))
 883                                 continue;
 884                         if (pos[j] < T->n_row && C2)
 885                                 isl_seq_submul(bmap->div[k] + 1, T->row[i][j],
 886                                                 C2->row[pos[j]], 1 + total);
 887                         else
 888                                 isl_int_neg(bmap->div[k][1 + pos[j]],
 889                                                                 T->row[i][j]);
 890                 }
 891                 j = isl_basic_map_alloc_equality(bmap);
 892                 isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+total+bmap->n_div);
 893                 isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]);
 894         }
 895         free(pos);
 896         isl_mat_free(C2);
 897         isl_mat_free(T);
 898
 899         if (progress)
 900                 *progress = 1;
 901 done:
 902         ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
 903
 904         return bmap;
 905 error:
 906         isl_mat_free(C);
 907         isl_mat_free(C2);
 908         isl_mat_free(T);
 909         return bmap;
 910 }
 911
 912 static struct isl_basic_map *set_div_from_lower_bound(
 913         struct isl_basic_map *bmap, int div, int ineq)
 914 {
 915         unsigned total = 1 + isl_dim_total(bmap->dim);
 916
 917         isl_seq_neg(bmap->div[div] + 1, bmap->ineq[ineq], total + bmap->n_div);
 918         isl_int_set(bmap->div[div][0], bmap->ineq[ineq][total + div]);
 919         isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]);
 920         isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
 921         isl_int_set_si(bmap->div[div][1 + total + div], 0);
 922
 923         return bmap;
 924 }
 925
 926 /* Check whether it is ok to define a div based on an inequality.
 927  * To avoid the introduction of circular definitions of divs, we
 928  * do not allow such a definition if the resulting expression would refer to
 929  * any other undefined divs or if any known div is defined in
 930  * terms of the unknown div.
 931  */
 932 static int ok_to_set_div_from_bound(struct isl_basic_map *bmap,
 933         int div, int ineq)
 934 {
 935         int j;
 936         unsigned total = 1 + isl_dim_total(bmap->dim);
 937
 938         /* Not defined in terms of unknown divs */
 939         for (j = 0; j < bmap->n_div; ++j) {
 940                 if (div == j)
 941                         continue;
 942                 if (isl_int_is_zero(bmap->ineq[ineq][total + j]))
 943                         continue;
 944                 if (isl_int_is_zero(bmap->div[j][0]))
 945                         return 0;
 946         }
 947
 948         /* No other div defined in terms of this one => avoid loops */
 949         for (j = 0; j < bmap->n_div; ++j) {
 950                 if (div == j)
 951                         continue;
 952                 if (isl_int_is_zero(bmap->div[j][0]))
 953                         continue;
 954                 if (!isl_int_is_zero(bmap->div[j][1 + total + div]))
 955                         return 0;
 956         }
 957
 958         return 1;
 959 }
 960
 961 /* Given two constraints "k" and "l" that are opposite to each other,
 962  * except for the constant term, check if we can use them
 963  * to obtain an expression for one of the hitherto unknown divs.
 964  * "sum" is the sum of the constant terms of the constraints.
 965  * If this sum is strictly smaller than the coefficient of one
 966  * of the divs, then this pair can be used define the div.
 967  * To avoid the introduction of circular definitions of divs, we
 968  * do not use the pair if the resulting expression would refer to
 969  * any other undefined divs or if any known div is defined in
 970  * terms of the unknown div.
 971  */
 972 static struct isl_basic_map *check_for_div_constraints(
 973         struct isl_basic_map *bmap, int k, int l, isl_int sum, int *progress)
 974 {
 975         int i;
 976         unsigned total = 1 + isl_dim_total(bmap->dim);
 977
 978         for (i = 0; i < bmap->n_div; ++i) {
 979                 if (!isl_int_is_zero(bmap->div[i][0]))
 980                         continue;
 981                 if (isl_int_is_zero(bmap->ineq[k][total + i]))
 982                         continue;
 983                 if (isl_int_abs_ge(sum, bmap->ineq[k][total + i]))
 984                         continue;
 985                 if (!ok_to_set_div_from_bound(bmap, i, k))
 986                         break;
 987                 if (isl_int_is_pos(bmap->ineq[k][total + i]))
 988                         bmap = set_div_from_lower_bound(bmap, i, k);
 989                 else
 990                         bmap = set_div_from_lower_bound(bmap, i, l);
 991                 if (progress)
 992                         *progress = 1;
 993                 break;
 994         }
 995         return bmap;
 996 }
 997
 998 static struct isl_basic_map *remove_duplicate_constraints(
 999         struct isl_basic_map *bmap, int *progress)
1000 {
1001         unsigned int size;
1002         isl_int ***index;
1003         int k, l, h;
1004         int bits;
1005         unsigned total = isl_basic_map_total_dim(bmap);
1006         isl_int sum;
1007
1008         if (bmap->n_ineq <= 1)
1009                 return bmap;
1010
1011         size = round_up(4 * (bmap->n_ineq+1) / 3 - 1);
1012         bits = ffs(size) - 1;
1013         index = isl_calloc_array(ctx, isl_int **, size);
1014         if (!index)
1015                 return bmap;
1016
1017         index[isl_seq_get_hash_bits(bmap->ineq[0]+1, total, bits)] = &bmap->ineq[0];
1018         for (k = 1; k < bmap->n_ineq; ++k) {
1019                 h = hash_index(index, size, bits, bmap, k);
1020                 if (!index[h]) {
1021                         index[h] = &bmap->ineq[k];
1022                         continue;
1023                 }
1024                 if (progress)
1025                         *progress = 1;
1026                 l = index[h] - &bmap->ineq[0];
1027                 if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0]))
1028                         swap_inequality(bmap, k, l);
1029                 isl_basic_map_drop_inequality(bmap, k);
1030                 --k;
1031         }
1032         isl_int_init(sum);
1033         for (k = 0; k < bmap->n_ineq-1; ++k) {
1034                 isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
1035                 h = hash_index(index, size, bits, bmap, k);
1036                 isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
1037                 if (!index[h])
1038                         continue;
1039                 l = index[h] - &bmap->ineq[0];
1040                 isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]);
1041                 if (isl_int_is_pos(sum)) {
1042                         bmap = check_for_div_constraints(bmap, k, l, sum,
1043                                                          progress);
1044                         continue;
1045                 }
1046                 if (isl_int_is_zero(sum)) {
1047                         /* We need to break out of the loop after these
1048                          * changes since the contents of the hash
1049                          * will no longer be valid.
1050                          * Plus, we probably we want to regauss first.
1051                          */
1052                         if (progress)
1053                                 *progress = 1;
1054                         isl_basic_map_drop_inequality(bmap, l);
1055                         isl_basic_map_inequality_to_equality(bmap, k);
1056                 } else
1057                         bmap = isl_basic_map_set_to_empty(bmap);
1058                 break;
1059         }
1060         isl_int_clear(sum);
1061
1062         free(index);
1063         return bmap;
1064 }
1065
1066
1067 struct isl_basic_map *isl_basic_map_simplify(struct isl_basic_map *bmap)
1068 {
1069         int progress = 1;
1070         if (!bmap)
1071                 return NULL;
1072         while (progress) {
1073                 progress = 0;
1074                 bmap = isl_basic_map_normalize_constraints(bmap);
1075                 bmap = remove_duplicate_divs(bmap, &progress);
1076                 bmap = eliminate_divs_eq(bmap, &progress);
1077                 bmap = eliminate_divs_ineq(bmap, &progress);
1078                 bmap = isl_basic_map_gauss(bmap, &progress);
1079                 /* requires equalities in normal form */
1080                 bmap = normalize_divs(bmap, &progress);
1081                 bmap = remove_duplicate_constraints(bmap, &progress);
1082         }
1083         return bmap;
1084 }
1085
1086 struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset)
1087 {
1088         return (struct isl_basic_set *)
1089                 isl_basic_map_simplify((struct isl_basic_map *)bset);
1090 }
1091
1092
1093 /* If the only constraints a div d=floor(f/m)
1094  * appears in are its two defining constraints
1095  *
1096  *      f - m d >=0
1097  *      -(f - (m - 1)) + m d >= 0
1098  *
1099  * then it can safely be removed.
1100  */
1101 static int div_is_redundant(struct isl_basic_map *bmap, int div)
1102 {
1103         int i;
1104         unsigned pos = 1 + isl_dim_total(bmap->dim) + div;
1105
1106         for (i = 0; i < bmap->n_eq; ++i)
1107                 if (!isl_int_is_zero(bmap->eq[i][pos]))
1108                         return 0;
1109
1110         for (i = 0; i < bmap->n_ineq; ++i) {
1111                 if (isl_int_is_zero(bmap->ineq[i][pos]))
1112                         continue;
1113                 if (isl_int_eq(bmap->ineq[i][pos], bmap->div[div][0])) {
1114                         int neg;
1115                         isl_int_sub(bmap->div[div][1],
1116                                         bmap->div[div][1], bmap->div[div][0]);
1117                         isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
1118                         neg = isl_seq_is_neg(bmap->ineq[i], bmap->div[div]+1, pos);
1119                         isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
1120                         isl_int_add(bmap->div[div][1],
1121                                         bmap->div[div][1], bmap->div[div][0]);
1122                         if (!neg)
1123                                 return 0;
1124                         if (isl_seq_first_non_zero(bmap->ineq[i]+pos+1,
1125                                                     bmap->n_div-div-1) != -1)
1126                                 return 0;
1127                 } else if (isl_int_abs_eq(bmap->ineq[i][pos], bmap->div[div][0])) {
1128                         if (!isl_seq_eq(bmap->ineq[i], bmap->div[div]+1, pos))
1129                                 return 0;
1130                         if (isl_seq_first_non_zero(bmap->ineq[i]+pos+1,
1131                                                     bmap->n_div-div-1) != -1)
1132                                 return 0;
1133                 } else
1134                         return 0;
1135         }
1136
1137         for (i = 0; i < bmap->n_div; ++i)
1138                 if (!isl_int_is_zero(bmap->div[i][1+pos]))
1139                         return 0;
1140
1141         return 1;
1142 }
1143
1144 /*
1145  * Remove divs that don't occur in any of the constraints or other divs.
1146  * These can arise when dropping some of the variables in a quast
1147  * returned by piplib.
1148  */
1149 static struct isl_basic_map *remove_redundant_divs(struct isl_basic_map *bmap)
1150 {
1151         int i;
1152
1153         if (!bmap)
1154                 return NULL;
1155
1156         for (i = bmap->n_div-1; i >= 0; --i) {
1157                 if (!div_is_redundant(bmap, i))
1158                         continue;
1159                 bmap = isl_basic_map_drop_div(bmap, i);
1160         }
1161         return bmap;
1162 }
1163
1164 struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
1165 {
1166         bmap = remove_redundant_divs(bmap);
1167         if (!bmap)
1168                 return NULL;
1169         ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
1170         return bmap;
1171 }
1172
1173 struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
1174 {
1175         return (struct isl_basic_set *)
1176                 isl_basic_map_finalize((struct isl_basic_map *)bset);
1177 }
1178
1179 struct isl_set *isl_set_finalize(struct isl_set *set)
1180 {
1181         int i;
1182
1183         if (!set)
1184                 return NULL;
1185         for (i = 0; i < set->n; ++i) {
1186                 set->p[i] = isl_basic_set_finalize(set->p[i]);
1187                 if (!set->p[i])
1188                         goto error;
1189         }
1190         return set;
1191 error:
1192         isl_set_free(set);
1193         return NULL;
1194 }
1195
1196 struct isl_map *isl_map_finalize(struct isl_map *map)
1197 {
1198         int i;
1199
1200         if (!map)
1201                 return NULL;
1202         for (i = 0; i < map->n; ++i) {
1203                 map->p[i] = isl_basic_map_finalize(map->p[i]);
1204                 if (!map->p[i])
1205                         goto error;
1206         }
1207         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
1208         return map;
1209 error:
1210         isl_map_free(map);
1211         return NULL;
1212 }
1213
1214
1215 /* Remove definition of any div that is defined in terms of the given variable.
1216  * The div itself is not removed.  Functions such as
1217  * eliminate_divs_ineq depend on the other divs remaining in place.
1218  */
1219 static struct isl_basic_map *remove_dependent_vars(struct isl_basic_map *bmap,
1220                                                                         int pos)
1221 {
1222         int i;
1223
1224         for (i = 0; i < bmap->n_div; ++i) {
1225                 if (isl_int_is_zero(bmap->div[i][0]))
1226                         continue;
1227                 if (isl_int_is_zero(bmap->div[i][1+1+pos]))
1228                         continue;
1229                 isl_int_set_si(bmap->div[i][0], 0);
1230         }
1231         return bmap;
1232 }
1233
1234 /* Eliminate the specified variables from the constraints using
1235  * Fourier-Motzkin.  The variables themselves are not removed.
1236  */
1237 struct isl_basic_map *isl_basic_map_eliminate_vars(
1238         struct isl_basic_map *bmap, unsigned pos, unsigned n)
1239 {
1240         int d;
1241         int i, j, k;
1242         unsigned total;
1243
1244         if (n == 0)
1245                 return bmap;
1246         if (!bmap)
1247                 return NULL;
1248         total = isl_basic_map_total_dim(bmap);
1249
1250         bmap = isl_basic_map_cow(bmap);
1251         for (d = pos + n - 1; d >= 0 && d >= pos; --d)
1252                 bmap = remove_dependent_vars(bmap, d);
1253
1254         for (d = pos + n - 1;
1255              d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
1256                 isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
1257         for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
1258                 int n_lower, n_upper;
1259                 if (!bmap)
1260                         return NULL;
1261                 for (i = 0; i < bmap->n_eq; ++i) {
1262                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1263                                 continue;
1264                         eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL);
1265                         isl_basic_map_drop_equality(bmap, i);
1266                         break;
1267                 }
1268                 if (i < bmap->n_eq)
1269                         continue;
1270                 n_lower = 0;
1271                 n_upper = 0;
1272                 for (i = 0; i < bmap->n_ineq; ++i) {
1273                         if (isl_int_is_pos(bmap->ineq[i][1+d]))
1274                                 n_lower++;
1275                         else if (isl_int_is_neg(bmap->ineq[i][1+d]))
1276                                 n_upper++;
1277                 }
1278                 bmap = isl_basic_map_extend_constraints(bmap,
1279                                 0, n_lower * n_upper);
1280                 for (i = bmap->n_ineq - 1; i >= 0; --i) {
1281                         int last;
1282                         if (isl_int_is_zero(bmap->ineq[i][1+d]))
1283                                 continue;
1284                         last = -1;
1285                         for (j = 0; j < i; ++j) {
1286                                 if (isl_int_is_zero(bmap->ineq[j][1+d]))
1287                                         continue;
1288                                 last = j;
1289                                 if (isl_int_sgn(bmap->ineq[i][1+d]) ==
1290                                     isl_int_sgn(bmap->ineq[j][1+d]))
1291                                         continue;
1292                                 k = isl_basic_map_alloc_inequality(bmap);
1293                                 if (k < 0)
1294                                         goto error;
1295                                 isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
1296                                                 1+total);
1297                                 isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
1298                                                 1+d, 1+total, NULL);
1299                         }
1300                         isl_basic_map_drop_inequality(bmap, i);
1301                         i = last + 1;
1302                 }
1303                 if (n_lower > 0 && n_upper > 0) {
1304                         bmap = isl_basic_map_normalize_constraints(bmap);
1305                         bmap = remove_duplicate_constraints(bmap, NULL);
1306                         bmap = isl_basic_map_gauss(bmap, NULL);
1307                         bmap = isl_basic_map_convex_hull(bmap);
1308                         if (!bmap)
1309                                 goto error;
1310                         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1311                                 break;
1312                 }
1313         }
1314         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
1315         return bmap;
1316 error:
1317         isl_basic_map_free(bmap);
1318         return NULL;
1319 }
1320
1321 struct isl_basic_set *isl_basic_set_eliminate_vars(
1322         struct isl_basic_set *bset, unsigned pos, unsigned n)
1323 {
1324         return (struct isl_basic_set *)isl_basic_map_eliminate_vars(
1325                         (struct isl_basic_map *)bset, pos, n);
1326 }
1327
1328 /* Don't assume equalities are in order, because align_divs
1329  * may have changed the order of the divs.
1330  */
1331 static void compute_elimination_index(struct isl_basic_map *bmap, int *elim)
1332 {
1333         int d, i;
1334         unsigned total;
1335
1336         total = isl_dim_total(bmap->dim);
1337         for (d = 0; d < total; ++d)
1338                 elim[d] = -1;
1339         for (i = 0; i < bmap->n_eq; ++i) {
1340                 for (d = total - 1; d >= 0; --d) {
1341                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1342                                 continue;
1343                         elim[d] = i;
1344                         break;
1345                 }
1346         }
1347 }
1348
1349 static void set_compute_elimination_index(struct isl_basic_set *bset, int *elim)
1350 {
1351         compute_elimination_index((struct isl_basic_map *)bset, elim);
1352 }
1353
1354 static int reduced_using_equalities(isl_int *dst, isl_int *src,
1355         struct isl_basic_map *bmap, int *elim)
1356 {
1357         int d;
1358         int copied = 0;
1359         unsigned total;
1360
1361         total = isl_dim_total(bmap->dim);
1362         for (d = total - 1; d >= 0; --d) {
1363                 if (isl_int_is_zero(src[1+d]))
1364                         continue;
1365                 if (elim[d] == -1)
1366                         continue;
1367                 if (!copied) {
1368                         isl_seq_cpy(dst, src, 1 + total);
1369                         copied = 1;
1370                 }
1371                 isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
1372         }
1373         return copied;
1374 }
1375
1376 static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
1377         struct isl_basic_set *bset, int *elim)
1378 {
1379         return reduced_using_equalities(dst, src,
1380                                         (struct isl_basic_map *)bset, elim);
1381 }
1382
1383 static struct isl_basic_set *isl_basic_set_reduce_using_equalities(
1384         struct isl_basic_set *bset, struct isl_basic_set *context)
1385 {
1386         int i;
1387         int *elim;
1388
1389         if (!bset || !context)
1390                 goto error;
1391
1392         bset = isl_basic_set_cow(bset);
1393         if (!bset)
1394                 goto error;
1395
1396         elim = isl_alloc_array(ctx, int, isl_basic_set_n_dim(bset));
1397         if (!elim)
1398                 goto error;
1399         set_compute_elimination_index(context, elim);
1400         for (i = 0; i < bset->n_eq; ++i)
1401                 set_reduced_using_equalities(bset->eq[i], bset->eq[i],
1402                                                         context, elim);
1403         for (i = 0; i < bset->n_ineq; ++i)
1404                 set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
1405                                                         context, elim);
1406         isl_basic_set_free(context);
1407         free(elim);
1408         bset = isl_basic_set_simplify(bset);
1409         bset = isl_basic_set_finalize(bset);
1410         return bset;
1411 error:
1412         isl_basic_set_free(bset);
1413         isl_basic_set_free(context);
1414         return NULL;
1415 }
1416
1417 static struct isl_basic_set *remove_shifted_constraints(
1418         struct isl_basic_set *bset, struct isl_basic_set *context)
1419 {
1420         unsigned int size;
1421         isl_int ***index;
1422         int bits;
1423         int k, h, l;
1424
1425         if (!bset)
1426                 return NULL;
1427
1428         size = round_up(4 * (context->n_ineq+1) / 3 - 1);
1429         bits = ffs(size) - 1;
1430         index = isl_calloc_array(ctx, isl_int **, size);
1431         if (!index)
1432                 return bset;
1433
1434         for (k = 0; k < context->n_ineq; ++k) {
1435                 h = set_hash_index(index, size, bits, context, k);
1436                 index[h] = &context->ineq[k];
1437         }
1438         for (k = 0; k < bset->n_ineq; ++k) {
1439                 h = set_hash_index(index, size, bits, bset, k);
1440                 if (!index[h])
1441                         continue;
1442                 l = index[h] - &context->ineq[0];
1443                 if (isl_int_lt(bset->ineq[k][0], context->ineq[l][0]))
1444                         continue;
1445                 bset = isl_basic_set_cow(bset);
1446                 if (!bset)
1447                         goto error;
1448                 isl_basic_set_drop_inequality(bset, k);
1449                 --k;
1450         }
1451         free(index);
1452         return bset;
1453 error:
1454         free(index);
1455         return bset;
1456 }
1457
1458 /* Tighten (decrease) the constant terms of the inequalities based
1459  * on the equalities, without removing any integer points.
1460  * For example, if there is an equality
1461  *
1462  *              i = 3 * j
1463  *
1464  * and an inequality
1465  *
1466  *              i >= 1
1467  *
1468  * then we want to replace the inequality by
1469  *
1470  *              i >= 3
1471  *
1472  * We do this by computing a variable compression and translating
1473  * the constraints to the compressed space.
1474  * If any constraint has coefficients (except the contant term)
1475  * with a common factor "f", then we can replace the constant term "c"
1476  * by
1477  *
1478  *              f * floor(c/f)
1479  *
1480  * That is, we add
1481  *
1482  *              f * floor(c/f) - c = -fract(c/f)
1483  *
1484  * and we can add the same value to the original constraint.
1485  *
1486  * In the example, the compressed space only contains "j",
1487  * and the inequality translates to
1488  *
1489  *              3 * j - 1 >= 0
1490  *
1491  * We add -fract(-1/3) = -2 to the original constraint to obtain
1492  *
1493  *              i - 3 >= 0
1494  */
1495 static struct isl_basic_set *normalize_constraints_in_compressed_space(
1496         struct isl_basic_set *bset)
1497 {
1498         int i;
1499         unsigned total;
1500         struct isl_mat *B, *C;
1501         isl_int gcd;
1502
1503         if (!bset)
1504                 return NULL;
1505
1506         if (ISL_F_ISSET(bset, ISL_BASIC_SET_RATIONAL))
1507                 return bset;
1508
1509         if (!bset->n_ineq)
1510                 return bset;
1511
1512         bset = isl_basic_set_cow(bset);
1513         if (!bset)
1514                 return NULL;
1515
1516         total = isl_basic_set_total_dim(bset);
1517         B = isl_mat_sub_alloc(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + total);
1518         C = isl_mat_variable_compression(B, NULL);
1519         if (!C)
1520                 return bset;
1521         if (C->n_col == 0) {
1522                 isl_mat_free(C);
1523                 return isl_basic_set_set_to_empty(bset);
1524         }
1525         B = isl_mat_sub_alloc(bset->ctx, bset->ineq,
1526                                                 0, bset->n_ineq, 0, 1 + total);
1527         C = isl_mat_product(B, C);
1528         if (!C)
1529                 return bset;
1530
1531         isl_int_init(gcd);
1532         for (i = 0; i < bset->n_ineq; ++i) {
1533                 isl_seq_gcd(C->row[i] + 1, C->n_col - 1, &gcd);
1534                 if (isl_int_is_one(gcd))
1535                         continue;
1536                 isl_int_fdiv_r(C->row[i][0], C->row[i][0], gcd);
1537                 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], C->row[i][0]);
1538         }
1539         isl_int_clear(gcd);
1540
1541         isl_mat_free(C);
1542
1543         return bset;
1544 }
1545
1546 /* Remove all information from bset that is redundant in the context
1547  * of context.  In particular, equalities that are linear combinations
1548  * of those in context are removed.  Then the inequalities that are
1549  * redundant in the context of the equalities and inequalities of
1550  * context are removed.
1551  *
1552  * We first simplify the constraints of "bset" in the context of the
1553  * equalities of "context".
1554  * Then we simplify the inequalities of the context in the context
1555  * of the equalities of bset and remove the inequalities from "bset"
1556  * that are obviously redundant with respect to some inequality in "context".
1557  *
1558  * If there are any inequalities left, we construct a tableau for
1559  * the context and then add the inequalities of "bset".
1560  * Before adding these equalities, we freeze all constraints such that
1561  * they won't be considered redundant in terms of the constraints of "bset".
1562  * Then we detect all equalities and redundant constraints (among the
1563  * constraints that weren't frozen) and update bset according to the results.
1564  * We have to be careful here because we don't want any of the context
1565  * constraints to remain and because we haven't added the equalities of "bset"
1566  * to the tableau so we temporarily have to pretend that there were no
1567  * equalities.
1568  */
1569 static struct isl_basic_set *uset_gist(struct isl_basic_set *bset,
1570         struct isl_basic_set *context)
1571 {
1572         int i;
1573         struct isl_tab *tab;
1574         unsigned context_ineq;
1575         struct isl_basic_set *combined = NULL;
1576
1577         if (!context || !bset)
1578                 goto error;
1579
1580         if (context->n_eq > 0)
1581                 bset = isl_basic_set_reduce_using_equalities(bset,
1582                                         isl_basic_set_copy(context));
1583         if (!bset)
1584                 goto error;
1585         if (isl_basic_set_fast_is_empty(bset))
1586                 goto done;
1587         if (!bset->n_ineq)
1588                 goto done;
1589
1590         if (bset->n_eq > 0) {
1591                 struct isl_basic_set *affine_hull;
1592                 affine_hull = isl_basic_set_copy(bset);
1593                 affine_hull = isl_basic_set_cow(affine_hull);
1594                 if (!affine_hull)
1595                         goto error;
1596                 isl_basic_set_free_inequality(affine_hull, affine_hull->n_ineq);
1597                 context = isl_basic_set_intersect(context, affine_hull);
1598                 context = isl_basic_set_gauss(context, NULL);
1599                 context = normalize_constraints_in_compressed_space(context);
1600         }
1601         if (!context)
1602                 goto error;
1603         if (ISL_F_ISSET(context, ISL_BASIC_SET_EMPTY)) {
1604                 isl_basic_set_free(bset);
1605                 return context;
1606         }
1607         if (!context->n_ineq)
1608                 goto done;
1609         bset = remove_shifted_constraints(bset, context);
1610         if (!bset->n_ineq)
1611                 goto done;
1612         context_ineq = context->n_ineq;
1613         combined = isl_basic_set_cow(isl_basic_set_copy(context));
1614         if (isl_basic_set_free_equality(combined, context->n_eq) < 0)
1615                 goto error;
1616         combined = isl_basic_set_extend_constraints(combined,
1617                                                     bset->n_eq, bset->n_ineq);
1618         tab = isl_tab_from_basic_set(combined);
1619         if (!tab)
1620                 goto error;
1621         for (i = 0; i < context_ineq; ++i)
1622                 if (isl_tab_freeze_constraint(tab, i) < 0)
1623                         goto error;
1624         tab = isl_tab_extend(tab, bset->n_ineq);
1625         if (!tab)
1626                 goto error;
1627         for (i = 0; i < bset->n_ineq; ++i)
1628                 if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
1629                         goto error;
1630         bset = isl_basic_set_add_constraints(combined, bset, 0);
1631         tab = isl_tab_detect_implicit_equalities(tab);
1632         if (isl_tab_detect_redundant(tab) < 0) {
1633                 isl_tab_free(tab);
1634                 goto error2;
1635         }
1636         for (i = 0; i < context_ineq; ++i) {
1637                 tab->con[i].is_zero = 0;
1638                 tab->con[i].is_redundant = 1;
1639         }
1640         bset = isl_basic_set_update_from_tab(bset, tab);
1641         isl_tab_free(tab);
1642         ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
1643         ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
1644 done:
1645         bset = isl_basic_set_simplify(bset);
1646         bset = isl_basic_set_finalize(bset);
1647         isl_basic_set_free(context);
1648         return bset;
1649 error:
1650         isl_basic_set_free(combined);
1651 error2:
1652         isl_basic_set_free(bset);
1653         isl_basic_set_free(context);
1654         return NULL;
1655 }
1656
1657 /* Normalize the divs in "bmap" in the context of the equalities in "context".
1658  * We simply add the equalities in context to bmap and then do a regular
1659  * div normalizations.  Better results can be obtained by normalizing
1660  * only the divs in bmap than do not also appear in context.
1661  * We need to be careful to reduce the divs using the equalities
1662  * so that later calls to isl_basic_map_overlying_set wouldn't introduce
1663  * spurious constraints.
1664  */
1665 static struct isl_basic_map *normalize_divs_in_context(
1666         struct isl_basic_map *bmap, struct isl_basic_map *context)
1667 {
1668         int i;
1669         unsigned total_context;
1670         int div_eq;
1671
1672         div_eq = n_pure_div_eq(bmap);
1673         if (div_eq == 0)
1674                 return bmap;
1675
1676         if (context->n_div > 0)
1677                 bmap = isl_basic_map_align_divs(bmap, context);
1678
1679         total_context = isl_basic_map_total_dim(context);
1680         bmap = isl_basic_map_extend_constraints(bmap, context->n_eq, 0);
1681         for (i = 0; i < context->n_eq; ++i) {
1682                 int k;
1683                 k = isl_basic_map_alloc_equality(bmap);
1684                 isl_seq_cpy(bmap->eq[k], context->eq[i], 1 + total_context);
1685                 isl_seq_clr(bmap->eq[k] + 1 + total_context,
1686                                 isl_basic_map_total_dim(bmap) - total_context);
1687         }
1688         bmap = isl_basic_map_gauss(bmap, NULL);
1689         bmap = normalize_divs(bmap, NULL);
1690         bmap = isl_basic_map_gauss(bmap, NULL);
1691         return bmap;
1692 }
1693
1694 struct isl_basic_map *isl_basic_map_gist(struct isl_basic_map *bmap,
1695         struct isl_basic_map *context)
1696 {
1697         struct isl_basic_set *bset;
1698
1699         if (!bmap || !context)
1700                 goto error;
1701
1702         if (isl_basic_map_is_universe(context)) {
1703                 isl_basic_map_free(context);
1704                 return bmap;
1705         }
1706         if (isl_basic_map_is_universe(bmap)) {
1707                 isl_basic_map_free(context);
1708                 return bmap;
1709         }
1710         if (isl_basic_map_fast_is_empty(context)) {
1711                 struct isl_dim *dim = isl_dim_copy(bmap->dim);
1712                 isl_basic_map_free(context);
1713                 isl_basic_map_free(bmap);
1714                 return isl_basic_map_universe(dim);
1715         }
1716         if (isl_basic_map_fast_is_empty(bmap)) {
1717                 isl_basic_map_free(context);
1718                 return bmap;
1719         }
1720
1721         bmap = isl_basic_map_convex_hull(bmap);
1722         context = isl_basic_map_convex_hull(context);
1723
1724         if (context->n_eq)
1725                 bmap = normalize_divs_in_context(bmap, context);
1726
1727         context = isl_basic_map_align_divs(context, bmap);
1728         bmap = isl_basic_map_align_divs(bmap, context);
1729
1730         bset = uset_gist(isl_basic_map_underlying_set(isl_basic_map_copy(bmap)),
1731                          isl_basic_map_underlying_set(context));
1732
1733         return isl_basic_map_overlying_set(bset, bmap);
1734 error:
1735         isl_basic_map_free(bmap);
1736         isl_basic_map_free(context);
1737         return NULL;
1738 }
1739
1740 /*
1741  * Assumes context has no implicit divs.
1742  */
1743 __isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
1744         __isl_take isl_basic_map *context)
1745 {
1746         int i;
1747
1748         if (!map || !context)
1749                 goto error;;
1750
1751         if (isl_basic_map_is_universe(context)) {
1752                 isl_basic_map_free(context);
1753                 return map;
1754         }
1755         if (isl_basic_map_fast_is_empty(context)) {
1756                 struct isl_dim *dim = isl_dim_copy(map->dim);
1757                 isl_basic_map_free(context);
1758                 isl_map_free(map);
1759                 return isl_map_universe(dim);
1760         }
1761
1762         context = isl_basic_map_convex_hull(context);
1763         map = isl_map_cow(map);
1764         if (!map || !context)
1765                 goto error;;
1766         isl_assert(map->ctx, isl_dim_equal(map->dim, context->dim), goto error);
1767         map = isl_map_compute_divs(map);
1768         for (i = 0; i < map->n; ++i)
1769                 context = isl_basic_map_align_divs(context, map->p[i]);
1770         for (i = 0; i < map->n; ++i) {
1771                 map->p[i] = isl_basic_map_gist(map->p[i],
1772                                                 isl_basic_map_copy(context));
1773                 if (!map->p[i])
1774                         goto error;
1775         }
1776         isl_basic_map_free(context);
1777         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
1778         return map;
1779 error:
1780         isl_map_free(map);
1781         isl_basic_map_free(context);
1782         return NULL;
1783 }
1784
1785 __isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
1786         __isl_take isl_map *context)
1787 {
1788         return isl_map_gist_basic_map(map, isl_map_convex_hull(context));
1789 }
1790
1791 struct isl_basic_set *isl_basic_set_gist(struct isl_basic_set *bset,
1792                                                 struct isl_basic_set *context)
1793 {
1794         return (struct isl_basic_set *)isl_basic_map_gist(
1795                 (struct isl_basic_map *)bset, (struct isl_basic_map *)context);
1796 }
1797
1798 __isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
1799         __isl_take isl_basic_set *context)
1800 {
1801         return (struct isl_set *)isl_map_gist_basic_map((struct isl_map *)set,
1802                                         (struct isl_basic_map *)context);
1803 }
1804
1805 __isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
1806         __isl_take isl_set *context)
1807 {
1808         return (struct isl_set *)isl_map_gist((struct isl_map *)set,
1809                                         (struct isl_map *)context);
1810 }
1811
1812 /* Quick check to see if two basic maps are disjoint.
1813  * In particular, we reduce the equalities and inequalities of
1814  * one basic map in the context of the equalities of the other
1815  * basic map and check if we get a contradiction.
1816  */
1817 int isl_basic_map_fast_is_disjoint(struct isl_basic_map *bmap1,
1818         struct isl_basic_map *bmap2)
1819 {
1820         struct isl_vec *v = NULL;
1821         int *elim = NULL;
1822         unsigned total;
1823         int i;
1824
1825         if (!bmap1 || !bmap2)
1826                 return -1;
1827         isl_assert(bmap1->ctx, isl_dim_equal(bmap1->dim, bmap2->dim),
1828                         return -1);
1829         if (bmap1->n_div || bmap2->n_div)
1830                 return 0;
1831         if (!bmap1->n_eq && !bmap2->n_eq)
1832                 return 0;
1833
1834         total = isl_dim_total(bmap1->dim);
1835         if (total == 0)
1836                 return 0;
1837         v = isl_vec_alloc(bmap1->ctx, 1 + total);
1838         if (!v)
1839                 goto error;
1840         elim = isl_alloc_array(bmap1->ctx, int, total);
1841         if (!elim)
1842                 goto error;
1843         compute_elimination_index(bmap1, elim);
1844         for (i = 0; i < bmap2->n_eq; ++i) {
1845                 int reduced;
1846                 reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
1847                                                         bmap1, elim);
1848                 if (reduced && !isl_int_is_zero(v->block.data[0]) &&
1849                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
1850                         goto disjoint;
1851         }
1852         for (i = 0; i < bmap2->n_ineq; ++i) {
1853                 int reduced;
1854                 reduced = reduced_using_equalities(v->block.data,
1855                                                 bmap2->ineq[i], bmap1, elim);
1856                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
1857                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
1858                         goto disjoint;
1859         }
1860         compute_elimination_index(bmap2, elim);
1861         for (i = 0; i < bmap1->n_ineq; ++i) {
1862                 int reduced;
1863                 reduced = reduced_using_equalities(v->block.data,
1864                                                 bmap1->ineq[i], bmap2, elim);
1865                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
1866                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
1867                         goto disjoint;
1868         }
1869         isl_vec_free(v);
1870         free(elim);
1871         return 0;
1872 disjoint:
1873         isl_vec_free(v);
1874         free(elim);
1875         return 1;
1876 error:
1877         isl_vec_free(v);
1878         free(elim);
1879         return -1;
1880 }
1881
1882 int isl_basic_set_fast_is_disjoint(struct isl_basic_set *bset1,
1883         struct isl_basic_set *bset2)
1884 {
1885         return isl_basic_map_fast_is_disjoint((struct isl_basic_map *)bset1,
1886                                               (struct isl_basic_map *)bset2);
1887 }
1888
1889 int isl_map_fast_is_disjoint(struct isl_map *map1, struct isl_map *map2)
1890 {
1891         int i, j;
1892
1893         if (!map1 || !map2)
1894                 return -1;
1895
1896         if (isl_map_fast_is_equal(map1, map2))
1897                 return 0;
1898
1899         for (i = 0; i < map1->n; ++i) {
1900                 for (j = 0; j < map2->n; ++j) {
1901                         int d = isl_basic_map_fast_is_disjoint(map1->p[i],
1902                                                                map2->p[j]);
1903                         if (d != 1)
1904                                 return d;
1905                 }
1906         }
1907         return 1;
1908 }
1909
1910 int isl_set_fast_is_disjoint(struct isl_set *set1, struct isl_set *set2)
1911 {
1912         return isl_map_fast_is_disjoint((struct isl_map *)set1,
1913                                         (struct isl_map *)set2);
1914 }
1915
1916 /* Check if we can combine a given div with lower bound l and upper
1917  * bound u with some other div and if so return that other div.
1918  * Otherwise return -1.
1919  *
1920  * We first check that
1921  *      - the bounds are opposites of each other (except for the constant
1922  *        term)
1923  *      - the bounds do not reference any other div
1924  *      - no div is defined in terms of this div
1925  *
1926  * Let m be the size of the range allowed on the div by the bounds.
1927  * That is, the bounds are of the form
1928  *
1929  *      e <= a <= e + m - 1
1930  *
1931  * with e some expression in the other variables.
1932  * We look for another div b such that no third div is defined in terms
1933  * of this second div b and such that in any constraint that contains
1934  * a (except for the given lower and upper bound), also contains b
1935  * with a coefficient that is m times that of b.
1936  * That is, all constraints (execpt for the lower and upper bound)
1937  * are of the form
1938  *
1939  *      e + f (a + m b) >= 0
1940  *
1941  * If so, we return b so that "a + m b" can be replaced by
1942  * a single div "c = a + m b".
1943  */
1944 static int div_find_coalesce(struct isl_basic_map *bmap, int *pairs,
1945         unsigned div, unsigned l, unsigned u)
1946 {
1947         int i, j;
1948         unsigned dim;
1949         int coalesce = -1;
1950
1951         if (bmap->n_div <= 1)
1952                 return -1;
1953         dim = isl_dim_total(bmap->dim);
1954         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim, div) != -1)
1955                 return -1;
1956         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim + div + 1,
1957                                    bmap->n_div - div - 1) != -1)
1958                 return -1;
1959         if (!isl_seq_is_neg(bmap->ineq[l] + 1, bmap->ineq[u] + 1,
1960                             dim + bmap->n_div))
1961                 return -1;
1962
1963         for (i = 0; i < bmap->n_div; ++i) {
1964                 if (isl_int_is_zero(bmap->div[i][0]))
1965                         continue;
1966                 if (!isl_int_is_zero(bmap->div[i][1 + 1 + dim + div]))
1967                         return -1;
1968         }
1969
1970         isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
1971         if (isl_int_is_neg(bmap->ineq[l][0])) {
1972                 isl_int_sub(bmap->ineq[l][0],
1973                             bmap->ineq[l][0], bmap->ineq[u][0]);
1974                 bmap = isl_basic_map_copy(bmap);
1975                 bmap = isl_basic_map_set_to_empty(bmap);
1976                 isl_basic_map_free(bmap);
1977                 return -1;
1978         }
1979         isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
1980         for (i = 0; i < bmap->n_div; ++i) {
1981                 if (i == div)
1982                         continue;
1983                 if (!pairs[i])
1984                         continue;
1985                 for (j = 0; j < bmap->n_div; ++j) {
1986                         if (isl_int_is_zero(bmap->div[j][0]))
1987                                 continue;
1988                         if (!isl_int_is_zero(bmap->div[j][1 + 1 + dim + i]))
1989                                 break;
1990                 }
1991                 if (j < bmap->n_div)
1992                         continue;
1993                 for (j = 0; j < bmap->n_ineq; ++j) {
1994                         int valid;
1995                         if (j == l || j == u)
1996                                 continue;
1997                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + div]))
1998                                 continue;
1999                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + i]))
2000                                 break;
2001                         isl_int_mul(bmap->ineq[j][1 + dim + div],
2002                                     bmap->ineq[j][1 + dim + div],
2003                                     bmap->ineq[l][0]);
2004                         valid = isl_int_eq(bmap->ineq[j][1 + dim + div],
2005                                            bmap->ineq[j][1 + dim + i]);
2006                         isl_int_divexact(bmap->ineq[j][1 + dim + div],
2007                                          bmap->ineq[j][1 + dim + div],
2008                                          bmap->ineq[l][0]);
2009                         if (!valid)
2010                                 break;
2011                 }
2012                 if (j < bmap->n_ineq)
2013                         continue;
2014                 coalesce = i;
2015                 break;
2016         }
2017         isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2018         isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2019         return coalesce;
2020 }
2021
2022 /* Given a lower and an upper bound on div i, construct an inequality
2023  * that when nonnegative ensures that this pair of bounds always allows
2024  * for an integer value of the given div.
2025  * The lower bound is inequality l, while the upper bound is inequality u.
2026  * The constructed inequality is stored in ineq.
2027  * g, fl, fu are temporary scalars.
2028  *
2029  * Let the upper bound be
2030  *
2031  *      -n_u a + e_u >= 0
2032  *
2033  * and the lower bound
2034  *
2035  *      n_l a + e_l >= 0
2036  *
2037  * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
2038  * We have
2039  *
2040  *      - f_u e_l <= f_u f_l g a <= f_l e_u
2041  *
2042  * Since all variables are integer valued, this is equivalent to
2043  *
2044  *      - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
2045  *
2046  * If this interval is at least f_u f_l g, then it contains at least
2047  * one integer value for a.
2048  * That is, the test constraint is
2049  *
2050  *      f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
2051  */
2052 static void construct_test_ineq(struct isl_basic_map *bmap, int i,
2053         int l, int u, isl_int *ineq, isl_int g, isl_int fl, isl_int fu)
2054 {
2055         unsigned dim;
2056         dim = isl_dim_total(bmap->dim);
2057
2058         isl_int_gcd(g, bmap->ineq[l][1 + dim + i], bmap->ineq[u][1 + dim + i]);
2059         isl_int_divexact(fl, bmap->ineq[l][1 + dim + i], g);
2060         isl_int_divexact(fu, bmap->ineq[u][1 + dim + i], g);
2061         isl_int_neg(fu, fu);
2062         isl_seq_combine(ineq, fl, bmap->ineq[u], fu, bmap->ineq[l],
2063                         1 + dim + bmap->n_div);
2064         isl_int_add(ineq[0], ineq[0], fl);
2065         isl_int_add(ineq[0], ineq[0], fu);
2066         isl_int_sub_ui(ineq[0], ineq[0], 1);
2067         isl_int_mul(g, g, fl);
2068         isl_int_mul(g, g, fu);
2069         isl_int_sub(ineq[0], ineq[0], g);
2070 }
2071
2072 /* Remove more kinds of divs that are not strictly needed.
2073  * In particular, if all pairs of lower and upper bounds on a div
2074  * are such that they allow at least one integer value of the div,
2075  * the we can eliminate the div using Fourier-Motzkin without
2076  * introducing any spurious solutions.
2077  */
2078 static struct isl_basic_map *drop_more_redundant_divs(
2079         struct isl_basic_map *bmap, int *pairs, int n)
2080 {
2081         struct isl_tab *tab = NULL;
2082         struct isl_vec *vec = NULL;
2083         unsigned dim;
2084         int remove = -1;
2085         isl_int g, fl, fu;
2086
2087         isl_int_init(g);
2088         isl_int_init(fl);
2089         isl_int_init(fu);
2090
2091         if (!bmap)
2092                 goto error;
2093
2094         dim = isl_dim_total(bmap->dim);
2095         vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
2096         if (!vec)
2097                 goto error;
2098
2099         tab = isl_tab_from_basic_map(bmap);
2100
2101         while (n > 0) {
2102                 int i, l, u;
2103                 int best = -1;
2104                 enum isl_lp_result res;
2105
2106                 for (i = 0; i < bmap->n_div; ++i) {
2107                         if (!pairs[i])
2108                                 continue;
2109                         if (best >= 0 && pairs[best] <= pairs[i])
2110                                 continue;
2111                         best = i;
2112                 }
2113
2114                 i = best;
2115                 for (l = 0; l < bmap->n_ineq; ++l) {
2116                         if (!isl_int_is_pos(bmap->ineq[l][1 + dim + i]))
2117                                 continue;
2118                         for (u = 0; u < bmap->n_ineq; ++u) {
2119                                 if (!isl_int_is_neg(bmap->ineq[u][1 + dim + i]))
2120                                         continue;
2121                                 construct_test_ineq(bmap, i, l, u,
2122                                                     vec->el, g, fl, fu);
2123                                 res = isl_tab_min(tab, vec->el,
2124                                                   bmap->ctx->one, &g, NULL, 0);
2125                                 if (res == isl_lp_error)
2126                                         goto error;
2127                                 if (res == isl_lp_empty) {
2128                                         bmap = isl_basic_map_set_to_empty(bmap);
2129                                         break;
2130                                 }
2131                                 if (res != isl_lp_ok || isl_int_is_neg(g))
2132                                         break;
2133                         }
2134                         if (u < bmap->n_ineq)
2135                                 break;
2136                 }
2137                 if (l == bmap->n_ineq) {
2138                         remove = i;
2139                         break;
2140                 }
2141                 pairs[i] = 0;
2142                 --n;
2143         }
2144
2145         isl_tab_free(tab);
2146         isl_vec_free(vec);
2147
2148         isl_int_clear(g);
2149         isl_int_clear(fl);
2150         isl_int_clear(fu);
2151
2152         free(pairs);
2153
2154         if (remove < 0)
2155                 return bmap;
2156
2157         bmap = isl_basic_map_remove(bmap, isl_dim_div, remove, 1);
2158         return isl_basic_map_drop_redundant_divs(bmap);
2159 error:
2160         free(pairs);
2161         isl_basic_map_free(bmap);
2162         isl_tab_free(tab);
2163         isl_vec_free(vec);
2164         isl_int_clear(g);
2165         isl_int_clear(fl);
2166         isl_int_clear(fu);
2167         return NULL;
2168 }
2169
2170 /* Given a pair of divs div1 and div2 such that, expect for the lower bound l
2171  * and the upper bound u, div1 always occurs together with div2 in the form
2172  * (div1 + m div2), where m is the constant range on the variable div1
2173  * allowed by l and u, replace the pair div1 and div2 by a single
2174  * div that is equal to div1 + m div2.
2175  *
2176  * The new div will appear in the location that contains div2.
2177  * We need to modify all constraints that contain
2178  * div2 = (div - div1) / m
2179  * (If a constraint does not contain div2, it will also not contain div1.)
2180  * If the constraint also contains div1, then we know they appear
2181  * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
2182  * i.e., the coefficient of div is f.
2183  *
2184  * Otherwise, we first need to introduce div1 into the constraint.
2185  * Let the l be
2186  *
2187  *      div1 + f >=0
2188  *
2189  * and u
2190  *
2191  *      -div1 + f' >= 0
2192  *
2193  * A lower bound on div2
2194  *
2195  *      n div2 + t >= 0
2196  *
2197  * can be replaced by
2198  *
2199  *      (n * (m div 2 + div1) + m t + n f)/g >= 0
2200  *
2201  * with g = gcd(m,n).
2202  * An upper bound
2203  *
2204  *      -n div2 + t >= 0
2205  *
2206  * can be replaced by
2207  *
2208  *      (-n * (m div2 + div1) + m t + n f')/g >= 0
2209  *
2210  * These constraint are those that we would obtain from eliminating
2211  * div1 using Fourier-Motzkin.
2212  *
2213  * After all constraints have been modified, we drop the lower and upper
2214  * bound and then drop div1.
2215  */
2216 static struct isl_basic_map *coalesce_divs(struct isl_basic_map *bmap,
2217         unsigned div1, unsigned div2, unsigned l, unsigned u)
2218 {
2219         isl_int a;
2220         isl_int b;
2221         isl_int m;
2222         unsigned dim, total;
2223         int i;
2224
2225         dim = isl_dim_total(bmap->dim);
2226         total = 1 + dim + bmap->n_div;
2227
2228         isl_int_init(a);
2229         isl_int_init(b);
2230         isl_int_init(m);
2231         isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
2232         isl_int_add_ui(m, m, 1);
2233
2234         for (i = 0; i < bmap->n_ineq; ++i) {
2235                 if (i == l || i == u)
2236                         continue;
2237                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div2]))
2238                         continue;
2239                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div1])) {
2240                         isl_int_gcd(b, m, bmap->ineq[i][1 + dim + div2]);
2241                         isl_int_divexact(a, m, b);
2242                         isl_int_divexact(b, bmap->ineq[i][1 + dim + div2], b);
2243                         if (isl_int_is_pos(b)) {
2244                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2245                                                 b, bmap->ineq[l], total);
2246                         } else {
2247                                 isl_int_neg(b, b);
2248                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2249                                                 b, bmap->ineq[u], total);
2250                         }
2251                 }
2252                 isl_int_set(bmap->ineq[i][1 + dim + div2],
2253                             bmap->ineq[i][1 + dim + div1]);
2254                 isl_int_set_si(bmap->ineq[i][1 + dim + div1], 0);
2255         }
2256
2257         isl_int_clear(a);
2258         isl_int_clear(b);
2259         isl_int_clear(m);
2260         if (l > u) {
2261                 isl_basic_map_drop_inequality(bmap, l);
2262                 isl_basic_map_drop_inequality(bmap, u);
2263         } else {
2264                 isl_basic_map_drop_inequality(bmap, u);
2265                 isl_basic_map_drop_inequality(bmap, l);
2266         }
2267         bmap = isl_basic_map_drop_div(bmap, div1);
2268         return bmap;
2269 }
2270
2271 /* First check if we can coalesce any pair of divs and
2272  * then continue with dropping more redundant divs.
2273  *
2274  * We loop over all pairs of lower and upper bounds on a div
2275  * with coefficient 1 and -1, respectively, check if there
2276  * is any other div "c" with which we can coalesce the div
2277  * and if so, perform the coalescing.
2278  */
2279 static struct isl_basic_map *coalesce_or_drop_more_redundant_divs(
2280         struct isl_basic_map *bmap, int *pairs, int n)
2281 {
2282         int i, l, u;
2283         unsigned dim;
2284
2285         dim = isl_dim_total(bmap->dim);
2286
2287         for (i = 0; i < bmap->n_div; ++i) {
2288                 if (!pairs[i])
2289                         continue;
2290                 for (l = 0; l < bmap->n_ineq; ++l) {
2291                         if (!isl_int_is_one(bmap->ineq[l][1 + dim + i]))
2292                                 continue;
2293                         for (u = 0; u < bmap->n_ineq; ++u) {
2294                                 int c;
2295
2296                                 if (!isl_int_is_negone(bmap->ineq[u][1+dim+i]))
2297                                         continue;
2298                                 c = div_find_coalesce(bmap, pairs, i, l, u);
2299                                 if (c < 0)
2300                                         continue;
2301                                 free(pairs);
2302                                 bmap = coalesce_divs(bmap, i, c, l, u);
2303                                 return isl_basic_map_drop_redundant_divs(bmap);
2304                         }
2305                 }
2306         }
2307
2308         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
2309                 return bmap;
2310
2311         return drop_more_redundant_divs(bmap, pairs, n);
2312 }
2313
2314 /* Remove divs that are not strictly needed.
2315  * In particular, if a div only occurs positively (or negatively)
2316  * in constraints, then it can simply be dropped.
2317  * Also, if a div occurs only occurs in two constraints and if moreover
2318  * those two constraints are opposite to each other, except for the constant
2319  * term and if the sum of the constant terms is such that for any value
2320  * of the other values, there is always at least one integer value of the
2321  * div, i.e., if one plus this sum is greater than or equal to
2322  * the (absolute value) of the coefficent of the div in the constraints,
2323  * then we can also simply drop the div.
2324  *
2325  * If any divs are left after these simple checks then we move on
2326  * to more complicated cases in drop_more_redundant_divs.
2327  */
2328 struct isl_basic_map *isl_basic_map_drop_redundant_divs(
2329         struct isl_basic_map *bmap)
2330 {
2331         int i, j;
2332         unsigned off;
2333         int *pairs = NULL;
2334         int n = 0;
2335
2336         if (!bmap)
2337                 goto error;
2338
2339         off = isl_dim_total(bmap->dim);
2340         pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
2341         if (!pairs)
2342                 goto error;
2343
2344         for (i = 0; i < bmap->n_div; ++i) {
2345                 int pos, neg;
2346                 int last_pos, last_neg;
2347                 int redundant;
2348                 int defined;
2349
2350                 defined = !isl_int_is_zero(bmap->div[i][0]);
2351                 for (j = 0; j < bmap->n_eq; ++j)
2352                         if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
2353                                 break;
2354                 if (j < bmap->n_eq)
2355                         continue;
2356                 ++n;
2357                 pos = neg = 0;
2358                 for (j = 0; j < bmap->n_ineq; ++j) {
2359                         if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
2360                                 last_pos = j;
2361                                 ++pos;
2362                         }
2363                         if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
2364                                 last_neg = j;
2365                                 ++neg;
2366                         }
2367                 }
2368                 pairs[i] = pos * neg;
2369                 if (pairs[i] == 0) {
2370                         for (j = bmap->n_ineq - 1; j >= 0; --j)
2371                                 if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
2372                                         isl_basic_map_drop_inequality(bmap, j);
2373                         bmap = isl_basic_map_drop_div(bmap, i);
2374                         free(pairs);
2375                         return isl_basic_map_drop_redundant_divs(bmap);
2376                 }
2377                 if (pairs[i] != 1)
2378                         continue;
2379                 if (!isl_seq_is_neg(bmap->ineq[last_pos] + 1,
2380                                     bmap->ineq[last_neg] + 1,
2381                                     off + bmap->n_div))
2382                         continue;
2383
2384                 isl_int_add(bmap->ineq[last_pos][0],
2385                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
2386                 isl_int_add_ui(bmap->ineq[last_pos][0],
2387                                bmap->ineq[last_pos][0], 1);
2388                 redundant = isl_int_ge(bmap->ineq[last_pos][0],
2389                                 bmap->ineq[last_pos][1+off+i]);
2390                 isl_int_sub_ui(bmap->ineq[last_pos][0],
2391                                bmap->ineq[last_pos][0], 1);
2392                 isl_int_sub(bmap->ineq[last_pos][0],
2393                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
2394                 if (!redundant) {
2395                         if (defined ||
2396                             !ok_to_set_div_from_bound(bmap, i, last_pos)) {
2397                                 pairs[i] = 0;
2398                                 --n;
2399                                 continue;
2400                         }
2401                         bmap = set_div_from_lower_bound(bmap, i, last_pos);
2402                         bmap = isl_basic_map_simplify(bmap);
2403                         free(pairs);
2404                         return isl_basic_map_drop_redundant_divs(bmap);
2405                 }
2406                 if (last_pos > last_neg) {
2407                         isl_basic_map_drop_inequality(bmap, last_pos);
2408                         isl_basic_map_drop_inequality(bmap, last_neg);
2409                 } else {
2410                         isl_basic_map_drop_inequality(bmap, last_neg);
2411                         isl_basic_map_drop_inequality(bmap, last_pos);
2412                 }
2413                 bmap = isl_basic_map_drop_div(bmap, i);
2414                 free(pairs);
2415                 return isl_basic_map_drop_redundant_divs(bmap);
2416         }
2417
2418         if (n > 0)
2419                 return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);
2420
2421         free(pairs);
2422         return bmap;
2423 error:
2424         free(pairs);
2425         isl_basic_map_free(bmap);
2426         return NULL;
2427 }
2428
2429 struct isl_basic_set *isl_basic_set_drop_redundant_divs(
2430         struct isl_basic_set *bset)
2431 {
2432         return (struct isl_basic_set *)
2433             isl_basic_map_drop_redundant_divs((struct isl_basic_map *)bset);
2434 }
2435
2436 struct isl_map *isl_map_drop_redundant_divs(struct isl_map *map)
2437 {
2438         int i;
2439
2440         if (!map)
2441                 return NULL;
2442         for (i = 0; i < map->n; ++i) {
2443                 map->p[i] = isl_basic_map_drop_redundant_divs(map->p[i]);
2444                 if (!map->p[i])
2445                         goto error;
2446         }
2447         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
2448         return map;
2449 error:
2450         isl_map_free(map);
2451         return NULL;
2452 }
2453
2454 struct isl_set *isl_set_drop_redundant_divs(struct isl_set *set)
2455 {
2456         return (struct isl_set *)
2457             isl_map_drop_redundant_divs((struct isl_map *)set);
2458 }