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 <strings.h>
  11 #include <isl_ctx_private.h>
  12 #include <isl_map_private.h>
  13 #include "isl_equalities.h"
  14 #include <isl/map.h>
  15 #include <isl/seq.h>
  16 #include "isl_tab.h"
  17 #include <isl_space_private.h>
  18 #include <isl_mat_private.h>
  19
  20 static void swap_equality(struct isl_basic_map *bmap, int a, int b)
  21 {
  22         isl_int *t = bmap->eq[a];
  23         bmap->eq[a] = bmap->eq[b];
  24         bmap->eq[b] = t;
  25 }
  26
  27 static void swap_inequality(struct isl_basic_map *bmap, int a, int b)
  28 {
  29         if (a != b) {
  30                 isl_int *t = bmap->ineq[a];
  31                 bmap->ineq[a] = bmap->ineq[b];
  32                 bmap->ineq[b] = t;
  33         }
  34 }
  35
  36 static void constraint_drop_vars(isl_int *c, unsigned n, unsigned rem)
  37 {
  38         isl_seq_cpy(c, c + n, rem);
  39         isl_seq_clr(c + rem, n);
  40 }
  41
  42 /* Drop n dimensions starting at first.
  43  *
  44  * In principle, this frees up some extra variables as the number
  45  * of columns remains constant, but we would have to extend
  46  * the div array too as the number of rows in this array is assumed
  47  * to be equal to extra.
  48  */
  49 struct isl_basic_set *isl_basic_set_drop_dims(
  50                 struct isl_basic_set *bset, unsigned first, unsigned n)
  51 {
  52         int i;
  53
  54         if (!bset)
  55                 goto error;
  56
  57         isl_assert(bset->ctx, first + n <= bset->dim->n_out, goto error);
  58
  59         if (n == 0 && !isl_space_get_tuple_name(bset->dim, isl_dim_set))
  60                 return bset;
  61
  62         bset = isl_basic_set_cow(bset);
  63         if (!bset)
  64                 return NULL;
  65
  66         for (i = 0; i < bset->n_eq; ++i)
  67                 constraint_drop_vars(bset->eq[i]+1+bset->dim->nparam+first, n,
  68                                      (bset->dim->n_out-first-n)+bset->extra);
  69
  70         for (i = 0; i < bset->n_ineq; ++i)
  71                 constraint_drop_vars(bset->ineq[i]+1+bset->dim->nparam+first, n,
  72                                      (bset->dim->n_out-first-n)+bset->extra);
  73
  74         for (i = 0; i < bset->n_div; ++i)
  75                 constraint_drop_vars(bset->div[i]+1+1+bset->dim->nparam+first, n,
  76                                      (bset->dim->n_out-first-n)+bset->extra);
  77
  78         bset->dim = isl_space_drop_outputs(bset->dim, first, n);
  79         if (!bset->dim)
  80                 goto error;
  81
  82         ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
  83         bset = isl_basic_set_simplify(bset);
  84         return isl_basic_set_finalize(bset);
  85 error:
  86         isl_basic_set_free(bset);
  87         return NULL;
  88 }
  89
  90 struct isl_set *isl_set_drop_dims(
  91                 struct isl_set *set, unsigned first, unsigned n)
  92 {
  93         int i;
  94
  95         if (!set)
  96                 goto error;
  97
  98         isl_assert(set->ctx, first + n <= set->dim->n_out, goto error);
  99
 100         if (n == 0 && !isl_space_get_tuple_name(set->dim, isl_dim_set))
 101                 return set;
 102         set = isl_set_cow(set);
 103         if (!set)
 104                 goto error;
 105         set->dim = isl_space_drop_outputs(set->dim, first, n);
 106         if (!set->dim)
 107                 goto error;
 108
 109         for (i = 0; i < set->n; ++i) {
 110                 set->p[i] = isl_basic_set_drop_dims(set->p[i], first, n);
 111                 if (!set->p[i])
 112                         goto error;
 113         }
 114
 115         ISL_F_CLR(set, ISL_SET_NORMALIZED);
 116         return set;
 117 error:
 118         isl_set_free(set);
 119         return NULL;
 120 }
 121
 122 /* Move "n" divs starting at "first" to the end of the list of divs.
 123  */
 124 static struct isl_basic_map *move_divs_last(struct isl_basic_map *bmap,
 125         unsigned first, unsigned n)
 126 {
 127         isl_int **div;
 128         int i;
 129
 130         if (first + n == bmap->n_div)
 131                 return bmap;
 132
 133         div = isl_alloc_array(bmap->ctx, isl_int *, n);
 134         if (!div)
 135                 goto error;
 136         for (i = 0; i < n; ++i)
 137                 div[i] = bmap->div[first + i];
 138         for (i = 0; i < bmap->n_div - first - n; ++i)
 139                 bmap->div[first + i] = bmap->div[first + n + i];
 140         for (i = 0; i < n; ++i)
 141                 bmap->div[bmap->n_div - n + i] = div[i];
 142         free(div);
 143         return bmap;
 144 error:
 145         isl_basic_map_free(bmap);
 146         return NULL;
 147 }
 148
 149 /* Drop "n" dimensions of type "type" starting at "first".
 150  *
 151  * In principle, this frees up some extra variables as the number
 152  * of columns remains constant, but we would have to extend
 153  * the div array too as the number of rows in this array is assumed
 154  * to be equal to extra.
 155  */
 156 struct isl_basic_map *isl_basic_map_drop(struct isl_basic_map *bmap,
 157         enum isl_dim_type type, unsigned first, unsigned n)
 158 {
 159         int i;
 160         unsigned dim;
 161         unsigned offset;
 162         unsigned left;
 163
 164         if (!bmap)
 165                 goto error;
 166
 167         dim = isl_basic_map_dim(bmap, type);
 168         isl_assert(bmap->ctx, first + n <= dim, goto error);
 169
 170         if (n == 0 && !isl_space_is_named_or_nested(bmap->dim, type))
 171                 return bmap;
 172
 173         bmap = isl_basic_map_cow(bmap);
 174         if (!bmap)
 175                 return NULL;
 176
 177         offset = isl_basic_map_offset(bmap, type) + first;
 178         left = isl_basic_map_total_dim(bmap) - (offset - 1) - n;
 179         for (i = 0; i < bmap->n_eq; ++i)
 180                 constraint_drop_vars(bmap->eq[i]+offset, n, left);
 181
 182         for (i = 0; i < bmap->n_ineq; ++i)
 183                 constraint_drop_vars(bmap->ineq[i]+offset, n, left);
 184
 185         for (i = 0; i < bmap->n_div; ++i)
 186                 constraint_drop_vars(bmap->div[i]+1+offset, n, left);
 187
 188         if (type == isl_dim_div) {
 189                 bmap = move_divs_last(bmap, first, n);
 190                 if (!bmap)
 191                         goto error;
 192                 isl_basic_map_free_div(bmap, n);
 193         } else
 194                 bmap->dim = isl_space_drop_dims(bmap->dim, type, first, n);
 195         if (!bmap->dim)
 196                 goto error;
 197
 198         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 199         bmap = isl_basic_map_simplify(bmap);
 200         return isl_basic_map_finalize(bmap);
 201 error:
 202         isl_basic_map_free(bmap);
 203         return NULL;
 204 }
 205
 206 __isl_give isl_basic_set *isl_basic_set_drop(__isl_take isl_basic_set *bset,
 207         enum isl_dim_type type, unsigned first, unsigned n)
 208 {
 209         return (isl_basic_set *)isl_basic_map_drop((isl_basic_map *)bset,
 210                                                         type, first, n);
 211 }
 212
 213 struct isl_basic_map *isl_basic_map_drop_inputs(
 214                 struct isl_basic_map *bmap, unsigned first, unsigned n)
 215 {
 216         return isl_basic_map_drop(bmap, isl_dim_in, first, n);
 217 }
 218
 219 struct isl_map *isl_map_drop(struct isl_map *map,
 220         enum isl_dim_type type, unsigned first, unsigned n)
 221 {
 222         int i;
 223
 224         if (!map)
 225                 goto error;
 226
 227         isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
 228
 229         if (n == 0 && !isl_space_get_tuple_name(map->dim, type))
 230                 return map;
 231         map = isl_map_cow(map);
 232         if (!map)
 233                 goto error;
 234         map->dim = isl_space_drop_dims(map->dim, type, first, n);
 235         if (!map->dim)
 236                 goto error;
 237
 238         for (i = 0; i < map->n; ++i) {
 239                 map->p[i] = isl_basic_map_drop(map->p[i], type, first, n);
 240                 if (!map->p[i])
 241                         goto error;
 242         }
 243         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
 244
 245         return map;
 246 error:
 247         isl_map_free(map);
 248         return NULL;
 249 }
 250
 251 struct isl_set *isl_set_drop(struct isl_set *set,
 252         enum isl_dim_type type, unsigned first, unsigned n)
 253 {
 254         return (isl_set *)isl_map_drop((isl_map *)set, type, first, n);
 255 }
 256
 257 struct isl_map *isl_map_drop_inputs(
 258                 struct isl_map *map, unsigned first, unsigned n)
 259 {
 260         return isl_map_drop(map, isl_dim_in, first, n);
 261 }
 262
 263 /*
 264  * We don't cow, as the div is assumed to be redundant.
 265  */
 266 static struct isl_basic_map *isl_basic_map_drop_div(
 267                 struct isl_basic_map *bmap, unsigned div)
 268 {
 269         int i;
 270         unsigned pos;
 271
 272         if (!bmap)
 273                 goto error;
 274
 275         pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
 276
 277         isl_assert(bmap->ctx, div < bmap->n_div, goto error);
 278
 279         for (i = 0; i < bmap->n_eq; ++i)
 280                 constraint_drop_vars(bmap->eq[i]+pos, 1, bmap->extra-div-1);
 281
 282         for (i = 0; i < bmap->n_ineq; ++i) {
 283                 if (!isl_int_is_zero(bmap->ineq[i][pos])) {
 284                         isl_basic_map_drop_inequality(bmap, i);
 285                         --i;
 286                         continue;
 287                 }
 288                 constraint_drop_vars(bmap->ineq[i]+pos, 1, bmap->extra-div-1);
 289         }
 290
 291         for (i = 0; i < bmap->n_div; ++i)
 292                 constraint_drop_vars(bmap->div[i]+1+pos, 1, bmap->extra-div-1);
 293
 294         if (div != bmap->n_div - 1) {
 295                 int j;
 296                 isl_int *t = bmap->div[div];
 297
 298                 for (j = div; j < bmap->n_div - 1; ++j)
 299                         bmap->div[j] = bmap->div[j+1];
 300
 301                 bmap->div[bmap->n_div - 1] = t;
 302         }
 303         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 304         isl_basic_map_free_div(bmap, 1);
 305
 306         return bmap;
 307 error:
 308         isl_basic_map_free(bmap);
 309         return NULL;
 310 }
 311
 312 struct isl_basic_map *isl_basic_map_normalize_constraints(
 313         struct isl_basic_map *bmap)
 314 {
 315         int i;
 316         isl_int gcd;
 317         unsigned total = isl_basic_map_total_dim(bmap);
 318
 319         if (!bmap)
 320                 return NULL;
 321
 322         isl_int_init(gcd);
 323         for (i = bmap->n_eq - 1; i >= 0; --i) {
 324                 isl_seq_gcd(bmap->eq[i]+1, total, &gcd);
 325                 if (isl_int_is_zero(gcd)) {
 326                         if (!isl_int_is_zero(bmap->eq[i][0])) {
 327                                 bmap = isl_basic_map_set_to_empty(bmap);
 328                                 break;
 329                         }
 330                         isl_basic_map_drop_equality(bmap, i);
 331                         continue;
 332                 }
 333                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 334                         isl_int_gcd(gcd, gcd, bmap->eq[i][0]);
 335                 if (isl_int_is_one(gcd))
 336                         continue;
 337                 if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) {
 338                         bmap = isl_basic_map_set_to_empty(bmap);
 339                         break;
 340                 }
 341                 isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total);
 342         }
 343
 344         for (i = bmap->n_ineq - 1; i >= 0; --i) {
 345                 isl_seq_gcd(bmap->ineq[i]+1, total, &gcd);
 346                 if (isl_int_is_zero(gcd)) {
 347                         if (isl_int_is_neg(bmap->ineq[i][0])) {
 348                                 bmap = isl_basic_map_set_to_empty(bmap);
 349                                 break;
 350                         }
 351                         isl_basic_map_drop_inequality(bmap, i);
 352                         continue;
 353                 }
 354                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 355                         isl_int_gcd(gcd, gcd, bmap->ineq[i][0]);
 356                 if (isl_int_is_one(gcd))
 357                         continue;
 358                 isl_int_fdiv_q(bmap->ineq[i][0], bmap->ineq[i][0], gcd);
 359                 isl_seq_scale_down(bmap->ineq[i]+1, bmap->ineq[i]+1, gcd, total);
 360         }
 361         isl_int_clear(gcd);
 362
 363         return bmap;
 364 }
 365
 366 struct isl_basic_set *isl_basic_set_normalize_constraints(
 367         struct isl_basic_set *bset)
 368 {
 369         return (struct isl_basic_set *)isl_basic_map_normalize_constraints(
 370                 (struct isl_basic_map *)bset);
 371 }
 372
 373 /* Assumes divs have been ordered if keep_divs is set.
 374  */
 375 static void eliminate_var_using_equality(struct isl_basic_map *bmap,
 376         unsigned pos, isl_int *eq, int keep_divs, int *progress)
 377 {
 378         unsigned total;
 379         unsigned space_total;
 380         int k;
 381         int last_div;
 382
 383         total = isl_basic_map_total_dim(bmap);
 384         space_total = isl_space_dim(bmap->dim, isl_dim_all);
 385         last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
 386         for (k = 0; k < bmap->n_eq; ++k) {
 387                 if (bmap->eq[k] == eq)
 388                         continue;
 389                 if (isl_int_is_zero(bmap->eq[k][1+pos]))
 390                         continue;
 391                 if (progress)
 392                         *progress = 1;
 393                 isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL);
 394                 isl_seq_normalize(bmap->ctx, bmap->eq[k], 1 + total);
 395         }
 396
 397         for (k = 0; k < bmap->n_ineq; ++k) {
 398                 if (isl_int_is_zero(bmap->ineq[k][1+pos]))
 399                         continue;
 400                 if (progress)
 401                         *progress = 1;
 402                 isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL);
 403                 isl_seq_normalize(bmap->ctx, bmap->ineq[k], 1 + total);
 404                 ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 405         }
 406
 407         for (k = 0; k < bmap->n_div; ++k) {
 408                 if (isl_int_is_zero(bmap->div[k][0]))
 409                         continue;
 410                 if (isl_int_is_zero(bmap->div[k][1+1+pos]))
 411                         continue;
 412                 if (progress)
 413                         *progress = 1;
 414                 /* We need to be careful about circular definitions,
 415                  * so for now we just remove the definition of div k
 416                  * if the equality contains any divs.
 417                  * If keep_divs is set, then the divs have been ordered
 418                  * and we can keep the definition as long as the result
 419                  * is still ordered.
 420                  */
 421                 if (last_div == -1 || (keep_divs && last_div < k))
 422                         isl_seq_elim(bmap->div[k]+1, eq,
 423                                         1+pos, 1+total, &bmap->div[k][0]);
 424                 else
 425                         isl_seq_clr(bmap->div[k], 1 + total);
 426                 ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 427         }
 428 }
 429
 430 /* Assumes divs have been ordered if keep_divs is set.
 431  */
 432 static void eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
 433         unsigned div, int keep_divs)
 434 {
 435         unsigned pos = isl_space_dim(bmap->dim, isl_dim_all) + div;
 436
 437         eliminate_var_using_equality(bmap, pos, eq, keep_divs, NULL);
 438
 439         isl_basic_map_drop_div(bmap, div);
 440 }
 441
 442 /* Check if elimination of div "div" using equality "eq" would not
 443  * result in a div depending on a later div.
 444  */
 445 static int ok_to_eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
 446         unsigned div)
 447 {
 448         int k;
 449         int last_div;
 450         unsigned space_total = isl_space_dim(bmap->dim, isl_dim_all);
 451         unsigned pos = space_total + div;
 452
 453         last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
 454         if (last_div < 0 || last_div <= div)
 455                 return 1;
 456
 457         for (k = 0; k <= last_div; ++k) {
 458                 if (isl_int_is_zero(bmap->div[k][0]))
 459                         return 1;
 460                 if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos]))
 461                         return 0;
 462         }
 463
 464         return 1;
 465 }
 466
 467 /* Elimininate divs based on equalities
 468  */
 469 static struct isl_basic_map *eliminate_divs_eq(
 470                 struct isl_basic_map *bmap, int *progress)
 471 {
 472         int d;
 473         int i;
 474         int modified = 0;
 475         unsigned off;
 476
 477         bmap = isl_basic_map_order_divs(bmap);
 478
 479         if (!bmap)
 480                 return NULL;
 481
 482         off = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 483
 484         for (d = bmap->n_div - 1; d >= 0 ; --d) {
 485                 for (i = 0; i < bmap->n_eq; ++i) {
 486                         if (!isl_int_is_one(bmap->eq[i][off + d]) &&
 487                             !isl_int_is_negone(bmap->eq[i][off + d]))
 488                                 continue;
 489                         if (!ok_to_eliminate_div(bmap, bmap->eq[i], d))
 490                                 continue;
 491                         modified = 1;
 492                         *progress = 1;
 493                         eliminate_div(bmap, bmap->eq[i], d, 1);
 494                         isl_basic_map_drop_equality(bmap, i);
 495                         break;
 496                 }
 497         }
 498         if (modified)
 499                 return eliminate_divs_eq(bmap, progress);
 500         return bmap;
 501 }
 502
 503 /* Elimininate divs based on inequalities
 504  */
 505 static struct isl_basic_map *eliminate_divs_ineq(
 506                 struct isl_basic_map *bmap, int *progress)
 507 {
 508         int d;
 509         int i;
 510         unsigned off;
 511         struct isl_ctx *ctx;
 512
 513         if (!bmap)
 514                 return NULL;
 515
 516         ctx = bmap->ctx;
 517         off = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 518
 519         for (d = bmap->n_div - 1; d >= 0 ; --d) {
 520                 for (i = 0; i < bmap->n_eq; ++i)
 521                         if (!isl_int_is_zero(bmap->eq[i][off + d]))
 522                                 break;
 523                 if (i < bmap->n_eq)
 524                         continue;
 525                 for (i = 0; i < bmap->n_ineq; ++i)
 526                         if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one))
 527                                 break;
 528                 if (i < bmap->n_ineq)
 529                         continue;
 530                 *progress = 1;
 531                 bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1);
 532                 if (!bmap || ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 533                         break;
 534                 bmap = isl_basic_map_drop_div(bmap, d);
 535                 if (!bmap)
 536                         break;
 537         }
 538         return bmap;
 539 }
 540
 541 struct isl_basic_map *isl_basic_map_gauss(
 542         struct isl_basic_map *bmap, int *progress)
 543 {
 544         int k;
 545         int done;
 546         int last_var;
 547         unsigned total_var;
 548         unsigned total;
 549
 550         bmap = isl_basic_map_order_divs(bmap);
 551
 552         if (!bmap)
 553                 return NULL;
 554
 555         total = isl_basic_map_total_dim(bmap);
 556         total_var = total - bmap->n_div;
 557
 558         last_var = total - 1;
 559         for (done = 0; done < bmap->n_eq; ++done) {
 560                 for (; last_var >= 0; --last_var) {
 561                         for (k = done; k < bmap->n_eq; ++k)
 562                                 if (!isl_int_is_zero(bmap->eq[k][1+last_var]))
 563                                         break;
 564                         if (k < bmap->n_eq)
 565                                 break;
 566                 }
 567                 if (last_var < 0)
 568                         break;
 569                 if (k != done)
 570                         swap_equality(bmap, k, done);
 571                 if (isl_int_is_neg(bmap->eq[done][1+last_var]))
 572                         isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total);
 573
 574                 eliminate_var_using_equality(bmap, last_var, bmap->eq[done], 1,
 575                                                 progress);
 576
 577                 if (last_var >= total_var &&
 578                     isl_int_is_zero(bmap->div[last_var - total_var][0])) {
 579                         unsigned div = last_var - total_var;
 580                         isl_seq_neg(bmap->div[div]+1, bmap->eq[done], 1+total);
 581                         isl_int_set_si(bmap->div[div][1+1+last_var], 0);
 582                         isl_int_set(bmap->div[div][0],
 583                                     bmap->eq[done][1+last_var]);
 584                         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 585                 }
 586         }
 587         if (done == bmap->n_eq)
 588                 return bmap;
 589         for (k = done; k < bmap->n_eq; ++k) {
 590                 if (isl_int_is_zero(bmap->eq[k][0]))
 591                         continue;
 592                 return isl_basic_map_set_to_empty(bmap);
 593         }
 594         isl_basic_map_free_equality(bmap, bmap->n_eq-done);
 595         return bmap;
 596 }
 597
 598 struct isl_basic_set *isl_basic_set_gauss(
 599         struct isl_basic_set *bset, int *progress)
 600 {
 601         return (struct isl_basic_set*)isl_basic_map_gauss(
 602                         (struct isl_basic_map *)bset, progress);
 603 }
 604
 605
 606 static unsigned int round_up(unsigned int v)
 607 {
 608         int old_v = v;
 609
 610         while (v) {
 611                 old_v = v;
 612                 v ^= v & -v;
 613         }
 614         return old_v << 1;
 615 }
 616
 617 static int hash_index(isl_int ***index, unsigned int size, int bits,
 618                         struct isl_basic_map *bmap, int k)
 619 {
 620         int h;
 621         unsigned total = isl_basic_map_total_dim(bmap);
 622         uint32_t hash = isl_seq_get_hash_bits(bmap->ineq[k]+1, total, bits);
 623         for (h = hash; index[h]; h = (h+1) % size)
 624                 if (&bmap->ineq[k] != index[h] &&
 625                     isl_seq_eq(bmap->ineq[k]+1, index[h][0]+1, total))
 626                         break;
 627         return h;
 628 }
 629
 630 static int set_hash_index(isl_int ***index, unsigned int size, int bits,
 631                           struct isl_basic_set *bset, int k)
 632 {
 633         return hash_index(index, size, bits, (struct isl_basic_map *)bset, k);
 634 }
 635
 636 /* If we can eliminate more than one div, then we need to make
 637  * sure we do it from last div to first div, in order not to
 638  * change the position of the other divs that still need to
 639  * be removed.
 640  */
 641 static struct isl_basic_map *remove_duplicate_divs(
 642         struct isl_basic_map *bmap, int *progress)
 643 {
 644         unsigned int size;
 645         int *index;
 646         int *elim_for;
 647         int k, l, h;
 648         int bits;
 649         struct isl_blk eq;
 650         unsigned total_var;
 651         unsigned total;
 652         struct isl_ctx *ctx;
 653
 654         if (!bmap || bmap->n_div <= 1)
 655                 return bmap;
 656
 657         total_var = isl_space_dim(bmap->dim, isl_dim_all);
 658         total = total_var + bmap->n_div;
 659
 660         ctx = bmap->ctx;
 661         for (k = bmap->n_div - 1; k >= 0; --k)
 662                 if (!isl_int_is_zero(bmap->div[k][0]))
 663                         break;
 664         if (k <= 0)
 665                 return bmap;
 666
 667         elim_for = isl_calloc_array(ctx, int, bmap->n_div);
 668         size = round_up(4 * bmap->n_div / 3 - 1);
 669         bits = ffs(size) - 1;
 670         index = isl_calloc_array(ctx, int, size);
 671         if (!index)
 672                 return bmap;
 673         eq = isl_blk_alloc(ctx, 1+total);
 674         if (isl_blk_is_error(eq))
 675                 goto out;
 676
 677         isl_seq_clr(eq.data, 1+total);
 678         index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1;
 679         for (--k; k >= 0; --k) {
 680                 uint32_t hash;
 681
 682                 if (isl_int_is_zero(bmap->div[k][0]))
 683                         continue;
 684
 685                 hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits);
 686                 for (h = hash; index[h]; h = (h+1) % size)
 687                         if (isl_seq_eq(bmap->div[k],
 688                                        bmap->div[index[h]-1], 2+total))
 689                                 break;
 690                 if (index[h]) {
 691                         *progress = 1;
 692                         l = index[h] - 1;
 693                         elim_for[l] = k + 1;
 694                 }
 695                 index[h] = k+1;
 696         }
 697         for (l = bmap->n_div - 1; l >= 0; --l) {
 698                 if (!elim_for[l])
 699                         continue;
 700                 k = elim_for[l] - 1;
 701                 isl_int_set_si(eq.data[1+total_var+k], -1);
 702                 isl_int_set_si(eq.data[1+total_var+l], 1);
 703                 eliminate_div(bmap, eq.data, l, 0);
 704                 isl_int_set_si(eq.data[1+total_var+k], 0);
 705                 isl_int_set_si(eq.data[1+total_var+l], 0);
 706         }
 707
 708         isl_blk_free(ctx, eq);
 709 out:
 710         free(index);
 711         free(elim_for);
 712         return bmap;
 713 }
 714
 715 static int n_pure_div_eq(struct isl_basic_map *bmap)
 716 {
 717         int i, j;
 718         unsigned total;
 719
 720         total = isl_space_dim(bmap->dim, isl_dim_all);
 721         for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) {
 722                 while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
 723                         --j;
 724                 if (j < 0)
 725                         break;
 726                 if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total, j) != -1)
 727                         return 0;
 728         }
 729         return i;
 730 }
 731
 732 /* Normalize divs that appear in equalities.
 733  *
 734  * In particular, we assume that bmap contains some equalities
 735  * of the form
 736  *
 737  *      a x = m * e_i
 738  *
 739  * and we want to replace the set of e_i by a minimal set and
 740  * such that the new e_i have a canonical representation in terms
 741  * of the vector x.
 742  * If any of the equalities involves more than one divs, then
 743  * we currently simply bail out.
 744  *
 745  * Let us first additionally assume that all equalities involve
 746  * a div.  The equalities then express modulo constraints on the
 747  * remaining variables and we can use "parameter compression"
 748  * to find a minimal set of constraints.  The result is a transformation
 749  *
 750  *      x = T(x') = x_0 + G x'
 751  *
 752  * with G a lower-triangular matrix with all elements below the diagonal
 753  * non-negative and smaller than the diagonal element on the same row.
 754  * We first normalize x_0 by making the same property hold in the affine
 755  * T matrix.
 756  * The rows i of G with a 1 on the diagonal do not impose any modulo
 757  * constraint and simply express x_i = x'_i.
 758  * For each of the remaining rows i, we introduce a div and a corresponding
 759  * equality.  In particular
 760  *
 761  *      g_ii e_j = x_i - g_i(x')
 762  *
 763  * where each x'_k is replaced either by x_k (if g_kk = 1) or the
 764  * corresponding div (if g_kk != 1).
 765  *
 766  * If there are any equalities not involving any div, then we
 767  * first apply a variable compression on the variables x:
 768  *
 769  *      x = C x''       x'' = C_2 x
 770  *
 771  * and perform the above parameter compression on A C instead of on A.
 772  * The resulting compression is then of the form
 773  *
 774  *      x'' = T(x') = x_0 + G x'
 775  *
 776  * and in constructing the new divs and the corresponding equalities,
 777  * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
 778  * by the corresponding row from C_2.
 779  */
 780 static struct isl_basic_map *normalize_divs(
 781         struct isl_basic_map *bmap, int *progress)
 782 {
 783         int i, j, k;
 784         int total;
 785         int div_eq;
 786         struct isl_mat *B;
 787         struct isl_vec *d;
 788         struct isl_mat *T = NULL;
 789         struct isl_mat *C = NULL;
 790         struct isl_mat *C2 = NULL;
 791         isl_int v;
 792         int *pos;
 793         int dropped, needed;
 794
 795         if (!bmap)
 796                 return NULL;
 797
 798         if (bmap->n_div == 0)
 799                 return bmap;
 800
 801         if (bmap->n_eq == 0)
 802                 return bmap;
 803
 804         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS))
 805                 return bmap;
 806
 807         total = isl_space_dim(bmap->dim, isl_dim_all);
 808         div_eq = n_pure_div_eq(bmap);
 809         if (div_eq == 0)
 810                 return bmap;
 811
 812         if (div_eq < bmap->n_eq) {
 813                 B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, div_eq,
 814                                         bmap->n_eq - div_eq, 0, 1 + total);
 815                 C = isl_mat_variable_compression(B, &C2);
 816                 if (!C || !C2)
 817                         goto error;
 818                 if (C->n_col == 0) {
 819                         bmap = isl_basic_map_set_to_empty(bmap);
 820                         isl_mat_free(C);
 821                         isl_mat_free(C2);
 822                         goto done;
 823                 }
 824         }
 825
 826         d = isl_vec_alloc(bmap->ctx, div_eq);
 827         if (!d)
 828                 goto error;
 829         for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) {
 830                 while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
 831                         --j;
 832                 isl_int_set(d->block.data[i], bmap->eq[i][1 + total + j]);
 833         }
 834         B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + total);
 835
 836         if (C) {
 837                 B = isl_mat_product(B, C);
 838                 C = NULL;
 839         }
 840
 841         T = isl_mat_parameter_compression(B, d);
 842         if (!T)
 843                 goto error;
 844         if (T->n_col == 0) {
 845                 bmap = isl_basic_map_set_to_empty(bmap);
 846                 isl_mat_free(C2);
 847                 isl_mat_free(T);
 848                 goto done;
 849         }
 850         isl_int_init(v);
 851         for (i = 0; i < T->n_row - 1; ++i) {
 852                 isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]);
 853                 if (isl_int_is_zero(v))
 854                         continue;
 855                 isl_mat_col_submul(T, 0, v, 1 + i);
 856         }
 857         isl_int_clear(v);
 858         pos = isl_alloc_array(bmap->ctx, int, T->n_row);
 859         if (!pos)
 860                 goto error;
 861         /* We have to be careful because dropping equalities may reorder them */
 862         dropped = 0;
 863         for (j = bmap->n_div - 1; j >= 0; --j) {
 864                 for (i = 0; i < bmap->n_eq; ++i)
 865                         if (!isl_int_is_zero(bmap->eq[i][1 + total + j]))
 866                                 break;
 867                 if (i < bmap->n_eq) {
 868                         bmap = isl_basic_map_drop_div(bmap, j);
 869                         isl_basic_map_drop_equality(bmap, i);
 870                         ++dropped;
 871                 }
 872         }
 873         pos[0] = 0;
 874         needed = 0;
 875         for (i = 1; i < T->n_row; ++i) {
 876                 if (isl_int_is_one(T->row[i][i]))
 877                         pos[i] = i;
 878                 else
 879                         needed++;
 880         }
 881         if (needed > dropped) {
 882                 bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
 883                                 needed, needed, 0);
 884                 if (!bmap)
 885                         goto error;
 886         }
 887         for (i = 1; i < T->n_row; ++i) {
 888                 if (isl_int_is_one(T->row[i][i]))
 889                         continue;
 890                 k = isl_basic_map_alloc_div(bmap);
 891                 pos[i] = 1 + total + k;
 892                 isl_seq_clr(bmap->div[k] + 1, 1 + total + bmap->n_div);
 893                 isl_int_set(bmap->div[k][0], T->row[i][i]);
 894                 if (C2)
 895                         isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + total);
 896                 else
 897                         isl_int_set_si(bmap->div[k][1 + i], 1);
 898                 for (j = 0; j < i; ++j) {
 899                         if (isl_int_is_zero(T->row[i][j]))
 900                                 continue;
 901                         if (pos[j] < T->n_row && C2)
 902                                 isl_seq_submul(bmap->div[k] + 1, T->row[i][j],
 903                                                 C2->row[pos[j]], 1 + total);
 904                         else
 905                                 isl_int_neg(bmap->div[k][1 + pos[j]],
 906                                                                 T->row[i][j]);
 907                 }
 908                 j = isl_basic_map_alloc_equality(bmap);
 909                 isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+total+bmap->n_div);
 910                 isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]);
 911         }
 912         free(pos);
 913         isl_mat_free(C2);
 914         isl_mat_free(T);
 915
 916         if (progress)
 917                 *progress = 1;
 918 done:
 919         ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
 920
 921         return bmap;
 922 error:
 923         isl_mat_free(C);
 924         isl_mat_free(C2);
 925         isl_mat_free(T);
 926         return bmap;
 927 }
 928
 929 static struct isl_basic_map *set_div_from_lower_bound(
 930         struct isl_basic_map *bmap, int div, int ineq)
 931 {
 932         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 933
 934         isl_seq_neg(bmap->div[div] + 1, bmap->ineq[ineq], total + bmap->n_div);
 935         isl_int_set(bmap->div[div][0], bmap->ineq[ineq][total + div]);
 936         isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]);
 937         isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
 938         isl_int_set_si(bmap->div[div][1 + total + div], 0);
 939
 940         return bmap;
 941 }
 942
 943 /* Check whether it is ok to define a div based on an inequality.
 944  * To avoid the introduction of circular definitions of divs, we
 945  * do not allow such a definition if the resulting expression would refer to
 946  * any other undefined divs or if any known div is defined in
 947  * terms of the unknown div.
 948  */
 949 static int ok_to_set_div_from_bound(struct isl_basic_map *bmap,
 950         int div, int ineq)
 951 {
 952         int j;
 953         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 954
 955         /* Not defined in terms of unknown divs */
 956         for (j = 0; j < bmap->n_div; ++j) {
 957                 if (div == j)
 958                         continue;
 959                 if (isl_int_is_zero(bmap->ineq[ineq][total + j]))
 960                         continue;
 961                 if (isl_int_is_zero(bmap->div[j][0]))
 962                         return 0;
 963         }
 964
 965         /* No other div defined in terms of this one => avoid loops */
 966         for (j = 0; j < bmap->n_div; ++j) {
 967                 if (div == j)
 968                         continue;
 969                 if (isl_int_is_zero(bmap->div[j][0]))
 970                         continue;
 971                 if (!isl_int_is_zero(bmap->div[j][1 + total + div]))
 972                         return 0;
 973         }
 974
 975         return 1;
 976 }
 977
 978 /* Given two constraints "k" and "l" that are opposite to each other,
 979  * except for the constant term, check if we can use them
 980  * to obtain an expression for one of the hitherto unknown divs.
 981  * "sum" is the sum of the constant terms of the constraints.
 982  * If this sum is strictly smaller than the coefficient of one
 983  * of the divs, then this pair can be used define the div.
 984  * To avoid the introduction of circular definitions of divs, we
 985  * do not use the pair if the resulting expression would refer to
 986  * any other undefined divs or if any known div is defined in
 987  * terms of the unknown div.
 988  */
 989 static struct isl_basic_map *check_for_div_constraints(
 990         struct isl_basic_map *bmap, int k, int l, isl_int sum, int *progress)
 991 {
 992         int i;
 993         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 994
 995         for (i = 0; i < bmap->n_div; ++i) {
 996                 if (!isl_int_is_zero(bmap->div[i][0]))
 997                         continue;
 998                 if (isl_int_is_zero(bmap->ineq[k][total + i]))
 999                         continue;
1000                 if (isl_int_abs_ge(sum, bmap->ineq[k][total + i]))
1001                         continue;
1002                 if (!ok_to_set_div_from_bound(bmap, i, k))
1003                         break;
1004                 if (isl_int_is_pos(bmap->ineq[k][total + i]))
1005                         bmap = set_div_from_lower_bound(bmap, i, k);
1006                 else
1007                         bmap = set_div_from_lower_bound(bmap, i, l);
1008                 if (progress)
1009                         *progress = 1;
1010                 break;
1011         }
1012         return bmap;
1013 }
1014
1015 static struct isl_basic_map *remove_duplicate_constraints(
1016         struct isl_basic_map *bmap, int *progress, int detect_divs)
1017 {
1018         unsigned int size;
1019         isl_int ***index;
1020         int k, l, h;
1021         int bits;
1022         unsigned total = isl_basic_map_total_dim(bmap);
1023         isl_int sum;
1024         isl_ctx *ctx;
1025
1026         if (!bmap || bmap->n_ineq <= 1)
1027                 return bmap;
1028
1029         size = round_up(4 * (bmap->n_ineq+1) / 3 - 1);
1030         bits = ffs(size) - 1;
1031         ctx = isl_basic_map_get_ctx(bmap);
1032         index = isl_calloc_array(ctx, isl_int **, size);
1033         if (!index)
1034                 return bmap;
1035
1036         index[isl_seq_get_hash_bits(bmap->ineq[0]+1, total, bits)] = &bmap->ineq[0];
1037         for (k = 1; k < bmap->n_ineq; ++k) {
1038                 h = hash_index(index, size, bits, bmap, k);
1039                 if (!index[h]) {
1040                         index[h] = &bmap->ineq[k];
1041                         continue;
1042                 }
1043                 if (progress)
1044                         *progress = 1;
1045                 l = index[h] - &bmap->ineq[0];
1046                 if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0]))
1047                         swap_inequality(bmap, k, l);
1048                 isl_basic_map_drop_inequality(bmap, k);
1049                 --k;
1050         }
1051         isl_int_init(sum);
1052         for (k = 0; k < bmap->n_ineq-1; ++k) {
1053                 isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
1054                 h = hash_index(index, size, bits, bmap, k);
1055                 isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
1056                 if (!index[h])
1057                         continue;
1058                 l = index[h] - &bmap->ineq[0];
1059                 isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]);
1060                 if (isl_int_is_pos(sum)) {
1061                         if (detect_divs)
1062                                 bmap = check_for_div_constraints(bmap, k, l,
1063                                                                  sum, progress);
1064                         continue;
1065                 }
1066                 if (isl_int_is_zero(sum)) {
1067                         /* We need to break out of the loop after these
1068                          * changes since the contents of the hash
1069                          * will no longer be valid.
1070                          * Plus, we probably we want to regauss first.
1071                          */
1072                         if (progress)
1073                                 *progress = 1;
1074                         isl_basic_map_drop_inequality(bmap, l);
1075                         isl_basic_map_inequality_to_equality(bmap, k);
1076                 } else
1077                         bmap = isl_basic_map_set_to_empty(bmap);
1078                 break;
1079         }
1080         isl_int_clear(sum);
1081
1082         free(index);
1083         return bmap;
1084 }
1085
1086
1087 struct isl_basic_map *isl_basic_map_simplify(struct isl_basic_map *bmap)
1088 {
1089         int progress = 1;
1090         if (!bmap)
1091                 return NULL;
1092         while (progress) {
1093                 progress = 0;
1094                 bmap = isl_basic_map_normalize_constraints(bmap);
1095                 bmap = remove_duplicate_divs(bmap, &progress);
1096                 bmap = eliminate_divs_eq(bmap, &progress);
1097                 bmap = eliminate_divs_ineq(bmap, &progress);
1098                 bmap = isl_basic_map_gauss(bmap, &progress);
1099                 /* requires equalities in normal form */
1100                 bmap = normalize_divs(bmap, &progress);
1101                 bmap = remove_duplicate_constraints(bmap, &progress, 1);
1102         }
1103         return bmap;
1104 }
1105
1106 struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset)
1107 {
1108         return (struct isl_basic_set *)
1109                 isl_basic_map_simplify((struct isl_basic_map *)bset);
1110 }
1111
1112
1113 int isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
1114         isl_int *constraint, unsigned div)
1115 {
1116         unsigned pos;
1117
1118         if (!bmap)
1119                 return -1;
1120
1121         pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
1122
1123         if (isl_int_eq(constraint[pos], bmap->div[div][0])) {
1124                 int neg;
1125                 isl_int_sub(bmap->div[div][1],
1126                                 bmap->div[div][1], bmap->div[div][0]);
1127                 isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
1128                 neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos);
1129                 isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
1130                 isl_int_add(bmap->div[div][1],
1131                                 bmap->div[div][1], bmap->div[div][0]);
1132                 if (!neg)
1133                         return 0;
1134                 if (isl_seq_first_non_zero(constraint+pos+1,
1135                                             bmap->n_div-div-1) != -1)
1136                         return 0;
1137         } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
1138                 if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
1139                         return 0;
1140                 if (isl_seq_first_non_zero(constraint+pos+1,
1141                                             bmap->n_div-div-1) != -1)
1142                         return 0;
1143         } else
1144                 return 0;
1145
1146         return 1;
1147 }
1148
1149
1150 /* If the only constraints a div d=floor(f/m)
1151  * appears in are its two defining constraints
1152  *
1153  *      f - m d >=0
1154  *      -(f - (m - 1)) + m d >= 0
1155  *
1156  * then it can safely be removed.
1157  */
1158 static int div_is_redundant(struct isl_basic_map *bmap, int div)
1159 {
1160         int i;
1161         unsigned pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
1162
1163         for (i = 0; i < bmap->n_eq; ++i)
1164                 if (!isl_int_is_zero(bmap->eq[i][pos]))
1165                         return 0;
1166
1167         for (i = 0; i < bmap->n_ineq; ++i) {
1168                 if (isl_int_is_zero(bmap->ineq[i][pos]))
1169                         continue;
1170                 if (!isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div))
1171                         return 0;
1172         }
1173
1174         for (i = 0; i < bmap->n_div; ++i)
1175                 if (!isl_int_is_zero(bmap->div[i][1+pos]))
1176                         return 0;
1177
1178         return 1;
1179 }
1180
1181 /*
1182  * Remove divs that don't occur in any of the constraints or other divs.
1183  * These can arise when dropping some of the variables in a quast
1184  * returned by piplib.
1185  */
1186 static struct isl_basic_map *remove_redundant_divs(struct isl_basic_map *bmap)
1187 {
1188         int i;
1189
1190         if (!bmap)
1191                 return NULL;
1192
1193         for (i = bmap->n_div-1; i >= 0; --i) {
1194                 if (!div_is_redundant(bmap, i))
1195                         continue;
1196                 bmap = isl_basic_map_drop_div(bmap, i);
1197         }
1198         return bmap;
1199 }
1200
1201 struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
1202 {
1203         bmap = remove_redundant_divs(bmap);
1204         if (!bmap)
1205                 return NULL;
1206         ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
1207         return bmap;
1208 }
1209
1210 struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
1211 {
1212         return (struct isl_basic_set *)
1213                 isl_basic_map_finalize((struct isl_basic_map *)bset);
1214 }
1215
1216 struct isl_set *isl_set_finalize(struct isl_set *set)
1217 {
1218         int i;
1219
1220         if (!set)
1221                 return NULL;
1222         for (i = 0; i < set->n; ++i) {
1223                 set->p[i] = isl_basic_set_finalize(set->p[i]);
1224                 if (!set->p[i])
1225                         goto error;
1226         }
1227         return set;
1228 error:
1229         isl_set_free(set);
1230         return NULL;
1231 }
1232
1233 struct isl_map *isl_map_finalize(struct isl_map *map)
1234 {
1235         int i;
1236
1237         if (!map)
1238                 return NULL;
1239         for (i = 0; i < map->n; ++i) {
1240                 map->p[i] = isl_basic_map_finalize(map->p[i]);
1241                 if (!map->p[i])
1242                         goto error;
1243         }
1244         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
1245         return map;
1246 error:
1247         isl_map_free(map);
1248         return NULL;
1249 }
1250
1251
1252 /* Remove definition of any div that is defined in terms of the given variable.
1253  * The div itself is not removed.  Functions such as
1254  * eliminate_divs_ineq depend on the other divs remaining in place.
1255  */
1256 static struct isl_basic_map *remove_dependent_vars(struct isl_basic_map *bmap,
1257                                                                         int pos)
1258 {
1259         int i;
1260
1261         for (i = 0; i < bmap->n_div; ++i) {
1262                 if (isl_int_is_zero(bmap->div[i][0]))
1263                         continue;
1264                 if (isl_int_is_zero(bmap->div[i][1+1+pos]))
1265                         continue;
1266                 isl_int_set_si(bmap->div[i][0], 0);
1267         }
1268         return bmap;
1269 }
1270
1271 /* Eliminate the specified variables from the constraints using
1272  * Fourier-Motzkin.  The variables themselves are not removed.
1273  */
1274 struct isl_basic_map *isl_basic_map_eliminate_vars(
1275         struct isl_basic_map *bmap, unsigned pos, unsigned n)
1276 {
1277         int d;
1278         int i, j, k;
1279         unsigned total;
1280
1281         if (n == 0)
1282                 return bmap;
1283         if (!bmap)
1284                 return NULL;
1285         total = isl_basic_map_total_dim(bmap);
1286
1287         bmap = isl_basic_map_cow(bmap);
1288         for (d = pos + n - 1; d >= 0 && d >= pos; --d)
1289                 bmap = remove_dependent_vars(bmap, d);
1290
1291         for (d = pos + n - 1;
1292              d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
1293                 isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
1294         for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
1295                 int n_lower, n_upper;
1296                 if (!bmap)
1297                         return NULL;
1298                 for (i = 0; i < bmap->n_eq; ++i) {
1299                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1300                                 continue;
1301                         eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL);
1302                         isl_basic_map_drop_equality(bmap, i);
1303                         break;
1304                 }
1305                 if (i < bmap->n_eq)
1306                         continue;
1307                 n_lower = 0;
1308                 n_upper = 0;
1309                 for (i = 0; i < bmap->n_ineq; ++i) {
1310                         if (isl_int_is_pos(bmap->ineq[i][1+d]))
1311                                 n_lower++;
1312                         else if (isl_int_is_neg(bmap->ineq[i][1+d]))
1313                                 n_upper++;
1314                 }
1315                 bmap = isl_basic_map_extend_constraints(bmap,
1316                                 0, n_lower * n_upper);
1317                 if (!bmap)
1318                         goto error;
1319                 for (i = bmap->n_ineq - 1; i >= 0; --i) {
1320                         int last;
1321                         if (isl_int_is_zero(bmap->ineq[i][1+d]))
1322                                 continue;
1323                         last = -1;
1324                         for (j = 0; j < i; ++j) {
1325                                 if (isl_int_is_zero(bmap->ineq[j][1+d]))
1326                                         continue;
1327                                 last = j;
1328                                 if (isl_int_sgn(bmap->ineq[i][1+d]) ==
1329                                     isl_int_sgn(bmap->ineq[j][1+d]))
1330                                         continue;
1331                                 k = isl_basic_map_alloc_inequality(bmap);
1332                                 if (k < 0)
1333                                         goto error;
1334                                 isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
1335                                                 1+total);
1336                                 isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
1337                                                 1+d, 1+total, NULL);
1338                         }
1339                         isl_basic_map_drop_inequality(bmap, i);
1340                         i = last + 1;
1341                 }
1342                 if (n_lower > 0 && n_upper > 0) {
1343                         bmap = isl_basic_map_normalize_constraints(bmap);
1344                         bmap = remove_duplicate_constraints(bmap, NULL, 0);
1345                         bmap = isl_basic_map_gauss(bmap, NULL);
1346                         bmap = isl_basic_map_remove_redundancies(bmap);
1347                         if (!bmap)
1348                                 goto error;
1349                         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1350                                 break;
1351                 }
1352         }
1353         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
1354         return bmap;
1355 error:
1356         isl_basic_map_free(bmap);
1357         return NULL;
1358 }
1359
1360 struct isl_basic_set *isl_basic_set_eliminate_vars(
1361         struct isl_basic_set *bset, unsigned pos, unsigned n)
1362 {
1363         return (struct isl_basic_set *)isl_basic_map_eliminate_vars(
1364                         (struct isl_basic_map *)bset, pos, n);
1365 }
1366
1367 /* Don't assume equalities are in order, because align_divs
1368  * may have changed the order of the divs.
1369  */
1370 static void compute_elimination_index(struct isl_basic_map *bmap, int *elim)
1371 {
1372         int d, i;
1373         unsigned total;
1374
1375         total = isl_space_dim(bmap->dim, isl_dim_all);
1376         for (d = 0; d < total; ++d)
1377                 elim[d] = -1;
1378         for (i = 0; i < bmap->n_eq; ++i) {
1379                 for (d = total - 1; d >= 0; --d) {
1380                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1381                                 continue;
1382                         elim[d] = i;
1383                         break;
1384                 }
1385         }
1386 }
1387
1388 static void set_compute_elimination_index(struct isl_basic_set *bset, int *elim)
1389 {
1390         compute_elimination_index((struct isl_basic_map *)bset, elim);
1391 }
1392
1393 static int reduced_using_equalities(isl_int *dst, isl_int *src,
1394         struct isl_basic_map *bmap, int *elim)
1395 {
1396         int d;
1397         int copied = 0;
1398         unsigned total;
1399
1400         total = isl_space_dim(bmap->dim, isl_dim_all);
1401         for (d = total - 1; d >= 0; --d) {
1402                 if (isl_int_is_zero(src[1+d]))
1403                         continue;
1404                 if (elim[d] == -1)
1405                         continue;
1406                 if (!copied) {
1407                         isl_seq_cpy(dst, src, 1 + total);
1408                         copied = 1;
1409                 }
1410                 isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
1411         }
1412         return copied;
1413 }
1414
1415 static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
1416         struct isl_basic_set *bset, int *elim)
1417 {
1418         return reduced_using_equalities(dst, src,
1419                                         (struct isl_basic_map *)bset, elim);
1420 }
1421
1422 static struct isl_basic_set *isl_basic_set_reduce_using_equalities(
1423         struct isl_basic_set *bset, struct isl_basic_set *context)
1424 {
1425         int i;
1426         int *elim;
1427
1428         if (!bset || !context)
1429                 goto error;
1430
1431         if (context->n_eq == 0) {
1432                 isl_basic_set_free(context);
1433                 return bset;
1434         }
1435
1436         bset = isl_basic_set_cow(bset);
1437         if (!bset)
1438                 goto error;
1439
1440         elim = isl_alloc_array(bset->ctx, int, isl_basic_set_n_dim(bset));
1441         if (!elim)
1442                 goto error;
1443         set_compute_elimination_index(context, elim);
1444         for (i = 0; i < bset->n_eq; ++i)
1445                 set_reduced_using_equalities(bset->eq[i], bset->eq[i],
1446                                                         context, elim);
1447         for (i = 0; i < bset->n_ineq; ++i)
1448                 set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
1449                                                         context, elim);
1450         isl_basic_set_free(context);
1451         free(elim);
1452         bset = isl_basic_set_simplify(bset);
1453         bset = isl_basic_set_finalize(bset);
1454         return bset;
1455 error:
1456         isl_basic_set_free(bset);
1457         isl_basic_set_free(context);
1458         return NULL;
1459 }
1460
1461 static struct isl_basic_set *remove_shifted_constraints(
1462         struct isl_basic_set *bset, struct isl_basic_set *context)
1463 {
1464         unsigned int size;
1465         isl_int ***index;
1466         int bits;
1467         int k, h, l;
1468         isl_ctx *ctx;
1469
1470         if (!bset)
1471                 return NULL;
1472
1473         size = round_up(4 * (context->n_ineq+1) / 3 - 1);
1474         bits = ffs(size) - 1;
1475         ctx = isl_basic_set_get_ctx(bset);
1476         index = isl_calloc_array(ctx, isl_int **, size);
1477         if (!index)
1478                 return bset;
1479
1480         for (k = 0; k < context->n_ineq; ++k) {
1481                 h = set_hash_index(index, size, bits, context, k);
1482                 index[h] = &context->ineq[k];
1483         }
1484         for (k = 0; k < bset->n_ineq; ++k) {
1485                 h = set_hash_index(index, size, bits, bset, k);
1486                 if (!index[h])
1487                         continue;
1488                 l = index[h] - &context->ineq[0];
1489                 if (isl_int_lt(bset->ineq[k][0], context->ineq[l][0]))
1490                         continue;
1491                 bset = isl_basic_set_cow(bset);
1492                 if (!bset)
1493                         goto error;
1494                 isl_basic_set_drop_inequality(bset, k);
1495                 --k;
1496         }
1497         free(index);
1498         return bset;
1499 error:
1500         free(index);
1501         return bset;
1502 }
1503
1504 /* Remove all information from bset that is redundant in the context
1505  * of context.  Both bset and context are assumed to be full-dimensional.
1506  *
1507  * We first * remove the inequalities from "bset"
1508  * that are obviously redundant with respect to some inequality in "context".
1509  *
1510  * If there are any inequalities left, we construct a tableau for
1511  * the context and then add the inequalities of "bset".
1512  * Before adding these inequalities, we freeze all constraints such that
1513  * they won't be considered redundant in terms of the constraints of "bset".
1514  * Then we detect all redundant constraints (among the
1515  * constraints that weren't frozen), first by checking for redundancy in the
1516  * the tableau and then by checking if replacing a constraint by its negation
1517  * would lead to an empty set.  This last step is fairly expensive
1518  * and could be optimized by more reuse of the tableau.
1519  * Finally, we update bset according to the results.
1520  */
1521 static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset,
1522         __isl_take isl_basic_set *context)
1523 {
1524         int i, k;
1525         isl_basic_set *combined = NULL;
1526         struct isl_tab *tab = NULL;
1527         unsigned context_ineq;
1528         unsigned total;
1529
1530         if (!bset || !context)
1531                 goto error;
1532
1533         if (isl_basic_set_is_universe(bset)) {
1534                 isl_basic_set_free(context);
1535                 return bset;
1536         }
1537
1538         if (isl_basic_set_is_universe(context)) {
1539                 isl_basic_set_free(context);
1540                 return bset;
1541         }
1542
1543         bset = remove_shifted_constraints(bset, context);
1544         if (!bset)
1545                 goto error;
1546         if (bset->n_ineq == 0)
1547                 goto done;
1548
1549         context_ineq = context->n_ineq;
1550         combined = isl_basic_set_cow(isl_basic_set_copy(context));
1551         combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq);
1552         tab = isl_tab_from_basic_set(combined);
1553         for (i = 0; i < context_ineq; ++i)
1554                 if (isl_tab_freeze_constraint(tab, i) < 0)
1555                         goto error;
1556         tab = isl_tab_extend(tab, bset->n_ineq);
1557         for (i = 0; i < bset->n_ineq; ++i)
1558                 if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
1559                         goto error;
1560         bset = isl_basic_set_add_constraints(combined, bset, 0);
1561         combined = NULL;
1562         if (!bset)
1563                 goto error;
1564         if (isl_tab_detect_redundant(tab) < 0)
1565                 goto error;
1566         total = isl_basic_set_total_dim(bset);
1567         for (i = context_ineq; i < bset->n_ineq; ++i) {
1568                 int is_empty;
1569                 if (tab->con[i].is_redundant)
1570                         continue;
1571                 tab->con[i].is_redundant = 1;
1572                 combined = isl_basic_set_dup(bset);
1573                 combined = isl_basic_set_update_from_tab(combined, tab);
1574                 combined = isl_basic_set_extend_constraints(combined, 0, 1);
1575                 k = isl_basic_set_alloc_inequality(combined);
1576                 if (k < 0)
1577                         goto error;
1578                 isl_seq_neg(combined->ineq[k], bset->ineq[i], 1 + total);
1579                 isl_int_sub_ui(combined->ineq[k][0], combined->ineq[k][0], 1);
1580                 is_empty = isl_basic_set_is_empty(combined);
1581                 if (is_empty < 0)
1582                         goto error;
1583                 isl_basic_set_free(combined);
1584                 combined = NULL;
1585                 if (!is_empty)
1586                         tab->con[i].is_redundant = 0;
1587         }
1588         for (i = 0; i < context_ineq; ++i)
1589                 tab->con[i].is_redundant = 1;
1590         bset = isl_basic_set_update_from_tab(bset, tab);
1591         if (bset) {
1592                 ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
1593                 ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
1594         }
1595
1596         isl_tab_free(tab);
1597 done:
1598         bset = isl_basic_set_simplify(bset);
1599         bset = isl_basic_set_finalize(bset);
1600         isl_basic_set_free(context);
1601         return bset;
1602 error:
1603         isl_tab_free(tab);
1604         isl_basic_set_free(combined);
1605         isl_basic_set_free(context);
1606         isl_basic_set_free(bset);
1607         return NULL;
1608 }
1609
1610 /* Remove all information from bset that is redundant in the context
1611  * of context.  In particular, equalities that are linear combinations
1612  * of those in context are removed.  Then the inequalities that are
1613  * redundant in the context of the equalities and inequalities of
1614  * context are removed.
1615  *
1616  * We first compute the integer affine hull of the intersection,
1617  * compute the gist inside this affine hull and then add back
1618  * those equalities that are not implied by the context.
1619  *
1620  * If two constraints are mutually redundant, then uset_gist_full
1621  * will remove the second of those constraints.  We therefore first
1622  * sort the constraints so that constraints not involving existentially
1623  * quantified variables are given precedence over those that do.
1624  * We have to perform this sorting before the variable compression,
1625  * because that may effect the order of the variables.
1626  */
1627 static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset,
1628         __isl_take isl_basic_set *context)
1629 {
1630         isl_mat *eq;
1631         isl_mat *T, *T2;
1632         isl_basic_set *aff;
1633         isl_basic_set *aff_context;
1634         unsigned total;
1635
1636         if (!bset || !context)
1637                 goto error;
1638
1639         bset = isl_basic_set_intersect(bset, isl_basic_set_copy(context));
1640         if (isl_basic_set_plain_is_empty(bset)) {
1641                 isl_basic_set_free(context);
1642                 return bset;
1643         }
1644         bset = isl_basic_set_sort_constraints(bset);
1645         aff = isl_basic_set_affine_hull(isl_basic_set_copy(bset));
1646         if (!aff)
1647                 goto error;
1648         if (isl_basic_set_plain_is_empty(aff)) {
1649                 isl_basic_set_free(aff);
1650                 isl_basic_set_free(context);
1651                 return bset;
1652         }
1653         if (aff->n_eq == 0) {
1654                 isl_basic_set_free(aff);
1655                 return uset_gist_full(bset, context);
1656         }
1657         total = isl_basic_set_total_dim(bset);
1658         eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total);
1659         eq = isl_mat_cow(eq);
1660         T = isl_mat_variable_compression(eq, &T2);
1661         if (T && T->n_col == 0) {
1662                 isl_mat_free(T);
1663                 isl_mat_free(T2);
1664                 isl_basic_set_free(context);
1665                 isl_basic_set_free(aff);
1666                 return isl_basic_set_set_to_empty(bset);
1667         }
1668
1669         aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));
1670
1671         bset = isl_basic_set_preimage(bset, isl_mat_copy(T));
1672         context = isl_basic_set_preimage(context, T);
1673
1674         bset = uset_gist_full(bset, context);
1675         bset = isl_basic_set_preimage(bset, T2);
1676         bset = isl_basic_set_intersect(bset, aff);
1677         bset = isl_basic_set_reduce_using_equalities(bset, aff_context);
1678
1679         if (bset) {
1680                 ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
1681                 ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
1682         }
1683
1684         return bset;
1685 error:
1686         isl_basic_set_free(bset);
1687         isl_basic_set_free(context);
1688         return NULL;
1689 }
1690
1691 /* Normalize the divs in "bmap" in the context of the equalities in "context".
1692  * We simply add the equalities in context to bmap and then do a regular
1693  * div normalizations.  Better results can be obtained by normalizing
1694  * only the divs in bmap than do not also appear in context.
1695  * We need to be careful to reduce the divs using the equalities
1696  * so that later calls to isl_basic_map_overlying_set wouldn't introduce
1697  * spurious constraints.
1698  */
1699 static struct isl_basic_map *normalize_divs_in_context(
1700         struct isl_basic_map *bmap, struct isl_basic_map *context)
1701 {
1702         int i;
1703         unsigned total_context;
1704         int div_eq;
1705
1706         div_eq = n_pure_div_eq(bmap);
1707         if (div_eq == 0)
1708                 return bmap;
1709
1710         if (context->n_div > 0)
1711                 bmap = isl_basic_map_align_divs(bmap, context);
1712
1713         total_context = isl_basic_map_total_dim(context);
1714         bmap = isl_basic_map_extend_constraints(bmap, context->n_eq, 0);
1715         for (i = 0; i < context->n_eq; ++i) {
1716                 int k;
1717                 k = isl_basic_map_alloc_equality(bmap);
1718                 isl_seq_cpy(bmap->eq[k], context->eq[i], 1 + total_context);
1719                 isl_seq_clr(bmap->eq[k] + 1 + total_context,
1720                                 isl_basic_map_total_dim(bmap) - total_context);
1721         }
1722         bmap = isl_basic_map_gauss(bmap, NULL);
1723         bmap = normalize_divs(bmap, NULL);
1724         bmap = isl_basic_map_gauss(bmap, NULL);
1725         return bmap;
1726 }
1727
1728 struct isl_basic_map *isl_basic_map_gist(struct isl_basic_map *bmap,
1729         struct isl_basic_map *context)
1730 {
1731         struct isl_basic_set *bset;
1732
1733         if (!bmap || !context)
1734                 goto error;
1735
1736         if (isl_basic_map_is_universe(bmap)) {
1737                 isl_basic_map_free(context);
1738                 return bmap;
1739         }
1740         if (isl_basic_map_plain_is_empty(context)) {
1741                 isl_space *dim = isl_space_copy(bmap->dim);
1742                 isl_basic_map_free(context);
1743                 isl_basic_map_free(bmap);
1744                 return isl_basic_map_universe(dim);
1745         }
1746         if (isl_basic_map_plain_is_empty(bmap)) {
1747                 isl_basic_map_free(context);
1748                 return bmap;
1749         }
1750
1751         bmap = isl_basic_map_remove_redundancies(bmap);
1752         context = isl_basic_map_remove_redundancies(context);
1753
1754         if (context->n_eq)
1755                 bmap = normalize_divs_in_context(bmap, context);
1756
1757         context = isl_basic_map_align_divs(context, bmap);
1758         bmap = isl_basic_map_align_divs(bmap, context);
1759
1760         bset = uset_gist(isl_basic_map_underlying_set(isl_basic_map_copy(bmap)),
1761                          isl_basic_map_underlying_set(context));
1762
1763         return isl_basic_map_overlying_set(bset, bmap);
1764 error:
1765         isl_basic_map_free(bmap);
1766         isl_basic_map_free(context);
1767         return NULL;
1768 }
1769
1770 /*
1771  * Assumes context has no implicit divs.
1772  */
1773 __isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
1774         __isl_take isl_basic_map *context)
1775 {
1776         int i;
1777
1778         if (!map || !context)
1779                 goto error;;
1780
1781         if (isl_basic_map_plain_is_empty(context)) {
1782                 isl_space *dim = isl_space_copy(map->dim);
1783                 isl_basic_map_free(context);
1784                 isl_map_free(map);
1785                 return isl_map_universe(dim);
1786         }
1787
1788         context = isl_basic_map_remove_redundancies(context);
1789         map = isl_map_cow(map);
1790         if (!map || !context)
1791                 goto error;;
1792         isl_assert(map->ctx, isl_space_is_equal(map->dim, context->dim), goto error);
1793         map = isl_map_compute_divs(map);
1794         for (i = 0; i < map->n; ++i)
1795                 context = isl_basic_map_align_divs(context, map->p[i]);
1796         for (i = map->n - 1; i >= 0; --i) {
1797                 map->p[i] = isl_basic_map_gist(map->p[i],
1798                                                 isl_basic_map_copy(context));
1799                 if (!map->p[i])
1800                         goto error;
1801                 if (isl_basic_map_plain_is_empty(map->p[i])) {
1802                         isl_basic_map_free(map->p[i]);
1803                         if (i != map->n - 1)
1804                                 map->p[i] = map->p[map->n - 1];
1805                         map->n--;
1806                 }
1807         }
1808         isl_basic_map_free(context);
1809         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
1810         return map;
1811 error:
1812         isl_map_free(map);
1813         isl_basic_map_free(context);
1814         return NULL;
1815 }
1816
1817 static __isl_give isl_map *map_gist(__isl_take isl_map *map,
1818         __isl_take isl_map *context)
1819 {
1820         context = isl_map_compute_divs(context);
1821         return isl_map_gist_basic_map(map, isl_map_simple_hull(context));
1822 }
1823
1824 __isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
1825         __isl_take isl_map *context)
1826 {
1827         return isl_map_align_params_map_map_and(map, context, &map_gist);
1828 }
1829
1830 struct isl_basic_set *isl_basic_set_gist(struct isl_basic_set *bset,
1831                                                 struct isl_basic_set *context)
1832 {
1833         return (struct isl_basic_set *)isl_basic_map_gist(
1834                 (struct isl_basic_map *)bset, (struct isl_basic_map *)context);
1835 }
1836
1837 __isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
1838         __isl_take isl_basic_set *context)
1839 {
1840         return (struct isl_set *)isl_map_gist_basic_map((struct isl_map *)set,
1841                                         (struct isl_basic_map *)context);
1842 }
1843
1844 __isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
1845         __isl_take isl_set *context)
1846 {
1847         return (struct isl_set *)isl_map_gist((struct isl_map *)set,
1848                                         (struct isl_map *)context);
1849 }
1850
1851 /* Quick check to see if two basic maps are disjoint.
1852  * In particular, we reduce the equalities and inequalities of
1853  * one basic map in the context of the equalities of the other
1854  * basic map and check if we get a contradiction.
1855  */
1856 int isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1,
1857         __isl_keep isl_basic_map *bmap2)
1858 {
1859         struct isl_vec *v = NULL;
1860         int *elim = NULL;
1861         unsigned total;
1862         int i;
1863
1864         if (!bmap1 || !bmap2)
1865                 return -1;
1866         isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
1867                         return -1);
1868         if (bmap1->n_div || bmap2->n_div)
1869                 return 0;
1870         if (!bmap1->n_eq && !bmap2->n_eq)
1871                 return 0;
1872
1873         total = isl_space_dim(bmap1->dim, isl_dim_all);
1874         if (total == 0)
1875                 return 0;
1876         v = isl_vec_alloc(bmap1->ctx, 1 + total);
1877         if (!v)
1878                 goto error;
1879         elim = isl_alloc_array(bmap1->ctx, int, total);
1880         if (!elim)
1881                 goto error;
1882         compute_elimination_index(bmap1, elim);
1883         for (i = 0; i < bmap2->n_eq; ++i) {
1884                 int reduced;
1885                 reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
1886                                                         bmap1, elim);
1887                 if (reduced && !isl_int_is_zero(v->block.data[0]) &&
1888                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
1889                         goto disjoint;
1890         }
1891         for (i = 0; i < bmap2->n_ineq; ++i) {
1892                 int reduced;
1893                 reduced = reduced_using_equalities(v->block.data,
1894                                                 bmap2->ineq[i], bmap1, elim);
1895                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
1896                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
1897                         goto disjoint;
1898         }
1899         compute_elimination_index(bmap2, elim);
1900         for (i = 0; i < bmap1->n_ineq; ++i) {
1901                 int reduced;
1902                 reduced = reduced_using_equalities(v->block.data,
1903                                                 bmap1->ineq[i], bmap2, elim);
1904                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
1905                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
1906                         goto disjoint;
1907         }
1908         isl_vec_free(v);
1909         free(elim);
1910         return 0;
1911 disjoint:
1912         isl_vec_free(v);
1913         free(elim);
1914         return 1;
1915 error:
1916         isl_vec_free(v);
1917         free(elim);
1918         return -1;
1919 }
1920
1921 int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1,
1922         __isl_keep isl_basic_set *bset2)
1923 {
1924         return isl_basic_map_plain_is_disjoint((struct isl_basic_map *)bset1,
1925                                               (struct isl_basic_map *)bset2);
1926 }
1927
1928 int isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
1929         __isl_keep isl_map *map2)
1930 {
1931         int i, j;
1932
1933         if (!map1 || !map2)
1934                 return -1;
1935
1936         if (isl_map_plain_is_equal(map1, map2))
1937                 return 0;
1938
1939         for (i = 0; i < map1->n; ++i) {
1940                 for (j = 0; j < map2->n; ++j) {
1941                         int d = isl_basic_map_plain_is_disjoint(map1->p[i],
1942                                                                map2->p[j]);
1943                         if (d != 1)
1944                                 return d;
1945                 }
1946         }
1947         return 1;
1948 }
1949
1950 int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
1951         __isl_keep isl_set *set2)
1952 {
1953         return isl_map_plain_is_disjoint((struct isl_map *)set1,
1954                                         (struct isl_map *)set2);
1955 }
1956
1957 int isl_set_fast_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
1958 {
1959         return isl_set_plain_is_disjoint(set1, set2);
1960 }
1961
1962 /* Check if we can combine a given div with lower bound l and upper
1963  * bound u with some other div and if so return that other div.
1964  * Otherwise return -1.
1965  *
1966  * We first check that
1967  *      - the bounds are opposites of each other (except for the constant
1968  *        term)
1969  *      - the bounds do not reference any other div
1970  *      - no div is defined in terms of this div
1971  *
1972  * Let m be the size of the range allowed on the div by the bounds.
1973  * That is, the bounds are of the form
1974  *
1975  *      e <= a <= e + m - 1
1976  *
1977  * with e some expression in the other variables.
1978  * We look for another div b such that no third div is defined in terms
1979  * of this second div b and such that in any constraint that contains
1980  * a (except for the given lower and upper bound), also contains b
1981  * with a coefficient that is m times that of b.
1982  * That is, all constraints (execpt for the lower and upper bound)
1983  * are of the form
1984  *
1985  *      e + f (a + m b) >= 0
1986  *
1987  * If so, we return b so that "a + m b" can be replaced by
1988  * a single div "c = a + m b".
1989  */
1990 static int div_find_coalesce(struct isl_basic_map *bmap, int *pairs,
1991         unsigned div, unsigned l, unsigned u)
1992 {
1993         int i, j;
1994         unsigned dim;
1995         int coalesce = -1;
1996
1997         if (bmap->n_div <= 1)
1998                 return -1;
1999         dim = isl_space_dim(bmap->dim, isl_dim_all);
2000         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim, div) != -1)
2001                 return -1;
2002         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim + div + 1,
2003                                    bmap->n_div - div - 1) != -1)
2004                 return -1;
2005         if (!isl_seq_is_neg(bmap->ineq[l] + 1, bmap->ineq[u] + 1,
2006                             dim + bmap->n_div))
2007                 return -1;
2008
2009         for (i = 0; i < bmap->n_div; ++i) {
2010                 if (isl_int_is_zero(bmap->div[i][0]))
2011                         continue;
2012                 if (!isl_int_is_zero(bmap->div[i][1 + 1 + dim + div]))
2013                         return -1;
2014         }
2015
2016         isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2017         if (isl_int_is_neg(bmap->ineq[l][0])) {
2018                 isl_int_sub(bmap->ineq[l][0],
2019                             bmap->ineq[l][0], bmap->ineq[u][0]);
2020                 bmap = isl_basic_map_copy(bmap);
2021                 bmap = isl_basic_map_set_to_empty(bmap);
2022                 isl_basic_map_free(bmap);
2023                 return -1;
2024         }
2025         isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2026         for (i = 0; i < bmap->n_div; ++i) {
2027                 if (i == div)
2028                         continue;
2029                 if (!pairs[i])
2030                         continue;
2031                 for (j = 0; j < bmap->n_div; ++j) {
2032                         if (isl_int_is_zero(bmap->div[j][0]))
2033                                 continue;
2034                         if (!isl_int_is_zero(bmap->div[j][1 + 1 + dim + i]))
2035                                 break;
2036                 }
2037                 if (j < bmap->n_div)
2038                         continue;
2039                 for (j = 0; j < bmap->n_ineq; ++j) {
2040                         int valid;
2041                         if (j == l || j == u)
2042                                 continue;
2043                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + div]))
2044                                 continue;
2045                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + i]))
2046                                 break;
2047                         isl_int_mul(bmap->ineq[j][1 + dim + div],
2048                                     bmap->ineq[j][1 + dim + div],
2049                                     bmap->ineq[l][0]);
2050                         valid = isl_int_eq(bmap->ineq[j][1 + dim + div],
2051                                            bmap->ineq[j][1 + dim + i]);
2052                         isl_int_divexact(bmap->ineq[j][1 + dim + div],
2053                                          bmap->ineq[j][1 + dim + div],
2054                                          bmap->ineq[l][0]);
2055                         if (!valid)
2056                                 break;
2057                 }
2058                 if (j < bmap->n_ineq)
2059                         continue;
2060                 coalesce = i;
2061                 break;
2062         }
2063         isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2064         isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2065         return coalesce;
2066 }
2067
2068 /* Given a lower and an upper bound on div i, construct an inequality
2069  * that when nonnegative ensures that this pair of bounds always allows
2070  * for an integer value of the given div.
2071  * The lower bound is inequality l, while the upper bound is inequality u.
2072  * The constructed inequality is stored in ineq.
2073  * g, fl, fu are temporary scalars.
2074  *
2075  * Let the upper bound be
2076  *
2077  *      -n_u a + e_u >= 0
2078  *
2079  * and the lower bound
2080  *
2081  *      n_l a + e_l >= 0
2082  *
2083  * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
2084  * We have
2085  *
2086  *      - f_u e_l <= f_u f_l g a <= f_l e_u
2087  *
2088  * Since all variables are integer valued, this is equivalent to
2089  *
2090  *      - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
2091  *
2092  * If this interval is at least f_u f_l g, then it contains at least
2093  * one integer value for a.
2094  * That is, the test constraint is
2095  *
2096  *      f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
2097  */
2098 static void construct_test_ineq(struct isl_basic_map *bmap, int i,
2099         int l, int u, isl_int *ineq, isl_int g, isl_int fl, isl_int fu)
2100 {
2101         unsigned dim;
2102         dim = isl_space_dim(bmap->dim, isl_dim_all);
2103
2104         isl_int_gcd(g, bmap->ineq[l][1 + dim + i], bmap->ineq[u][1 + dim + i]);
2105         isl_int_divexact(fl, bmap->ineq[l][1 + dim + i], g);
2106         isl_int_divexact(fu, bmap->ineq[u][1 + dim + i], g);
2107         isl_int_neg(fu, fu);
2108         isl_seq_combine(ineq, fl, bmap->ineq[u], fu, bmap->ineq[l],
2109                         1 + dim + bmap->n_div);
2110         isl_int_add(ineq[0], ineq[0], fl);
2111         isl_int_add(ineq[0], ineq[0], fu);
2112         isl_int_sub_ui(ineq[0], ineq[0], 1);
2113         isl_int_mul(g, g, fl);
2114         isl_int_mul(g, g, fu);
2115         isl_int_sub(ineq[0], ineq[0], g);
2116 }
2117
2118 /* Remove more kinds of divs that are not strictly needed.
2119  * In particular, if all pairs of lower and upper bounds on a div
2120  * are such that they allow at least one integer value of the div,
2121  * the we can eliminate the div using Fourier-Motzkin without
2122  * introducing any spurious solutions.
2123  */
2124 static struct isl_basic_map *drop_more_redundant_divs(
2125         struct isl_basic_map *bmap, int *pairs, int n)
2126 {
2127         struct isl_tab *tab = NULL;
2128         struct isl_vec *vec = NULL;
2129         unsigned dim;
2130         int remove = -1;
2131         isl_int g, fl, fu;
2132
2133         isl_int_init(g);
2134         isl_int_init(fl);
2135         isl_int_init(fu);
2136
2137         if (!bmap)
2138                 goto error;
2139
2140         dim = isl_space_dim(bmap->dim, isl_dim_all);
2141         vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
2142         if (!vec)
2143                 goto error;
2144
2145         tab = isl_tab_from_basic_map(bmap);
2146
2147         while (n > 0) {
2148                 int i, l, u;
2149                 int best = -1;
2150                 enum isl_lp_result res;
2151
2152                 for (i = 0; i < bmap->n_div; ++i) {
2153                         if (!pairs[i])
2154                                 continue;
2155                         if (best >= 0 && pairs[best] <= pairs[i])
2156                                 continue;
2157                         best = i;
2158                 }
2159
2160                 i = best;
2161                 for (l = 0; l < bmap->n_ineq; ++l) {
2162                         if (!isl_int_is_pos(bmap->ineq[l][1 + dim + i]))
2163                                 continue;
2164                         for (u = 0; u < bmap->n_ineq; ++u) {
2165                                 if (!isl_int_is_neg(bmap->ineq[u][1 + dim + i]))
2166                                         continue;
2167                                 construct_test_ineq(bmap, i, l, u,
2168                                                     vec->el, g, fl, fu);
2169                                 res = isl_tab_min(tab, vec->el,
2170                                                   bmap->ctx->one, &g, NULL, 0);
2171                                 if (res == isl_lp_error)
2172                                         goto error;
2173                                 if (res == isl_lp_empty) {
2174                                         bmap = isl_basic_map_set_to_empty(bmap);
2175                                         break;
2176                                 }
2177                                 if (res != isl_lp_ok || isl_int_is_neg(g))
2178                                         break;
2179                         }
2180                         if (u < bmap->n_ineq)
2181                                 break;
2182                 }
2183                 if (l == bmap->n_ineq) {
2184                         remove = i;
2185                         break;
2186                 }
2187                 pairs[i] = 0;
2188                 --n;
2189         }
2190
2191         isl_tab_free(tab);
2192         isl_vec_free(vec);
2193
2194         isl_int_clear(g);
2195         isl_int_clear(fl);
2196         isl_int_clear(fu);
2197
2198         free(pairs);
2199
2200         if (remove < 0)
2201                 return bmap;
2202
2203         bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1);
2204         return isl_basic_map_drop_redundant_divs(bmap);
2205 error:
2206         free(pairs);
2207         isl_basic_map_free(bmap);
2208         isl_tab_free(tab);
2209         isl_vec_free(vec);
2210         isl_int_clear(g);
2211         isl_int_clear(fl);
2212         isl_int_clear(fu);
2213         return NULL;
2214 }
2215
2216 /* Given a pair of divs div1 and div2 such that, expect for the lower bound l
2217  * and the upper bound u, div1 always occurs together with div2 in the form
2218  * (div1 + m div2), where m is the constant range on the variable div1
2219  * allowed by l and u, replace the pair div1 and div2 by a single
2220  * div that is equal to div1 + m div2.
2221  *
2222  * The new div will appear in the location that contains div2.
2223  * We need to modify all constraints that contain
2224  * div2 = (div - div1) / m
2225  * (If a constraint does not contain div2, it will also not contain div1.)
2226  * If the constraint also contains div1, then we know they appear
2227  * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
2228  * i.e., the coefficient of div is f.
2229  *
2230  * Otherwise, we first need to introduce div1 into the constraint.
2231  * Let the l be
2232  *
2233  *      div1 + f >=0
2234  *
2235  * and u
2236  *
2237  *      -div1 + f' >= 0
2238  *
2239  * A lower bound on div2
2240  *
2241  *      n div2 + t >= 0
2242  *
2243  * can be replaced by
2244  *
2245  *      (n * (m div 2 + div1) + m t + n f)/g >= 0
2246  *
2247  * with g = gcd(m,n).
2248  * An upper bound
2249  *
2250  *      -n div2 + t >= 0
2251  *
2252  * can be replaced by
2253  *
2254  *      (-n * (m div2 + div1) + m t + n f')/g >= 0
2255  *
2256  * These constraint are those that we would obtain from eliminating
2257  * div1 using Fourier-Motzkin.
2258  *
2259  * After all constraints have been modified, we drop the lower and upper
2260  * bound and then drop div1.
2261  */
2262 static struct isl_basic_map *coalesce_divs(struct isl_basic_map *bmap,
2263         unsigned div1, unsigned div2, unsigned l, unsigned u)
2264 {
2265         isl_int a;
2266         isl_int b;
2267         isl_int m;
2268         unsigned dim, total;
2269         int i;
2270
2271         dim = isl_space_dim(bmap->dim, isl_dim_all);
2272         total = 1 + dim + bmap->n_div;
2273
2274         isl_int_init(a);
2275         isl_int_init(b);
2276         isl_int_init(m);
2277         isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
2278         isl_int_add_ui(m, m, 1);
2279
2280         for (i = 0; i < bmap->n_ineq; ++i) {
2281                 if (i == l || i == u)
2282                         continue;
2283                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div2]))
2284                         continue;
2285                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div1])) {
2286                         isl_int_gcd(b, m, bmap->ineq[i][1 + dim + div2]);
2287                         isl_int_divexact(a, m, b);
2288                         isl_int_divexact(b, bmap->ineq[i][1 + dim + div2], b);
2289                         if (isl_int_is_pos(b)) {
2290                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2291                                                 b, bmap->ineq[l], total);
2292                         } else {
2293                                 isl_int_neg(b, b);
2294                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2295                                                 b, bmap->ineq[u], total);
2296                         }
2297                 }
2298                 isl_int_set(bmap->ineq[i][1 + dim + div2],
2299                             bmap->ineq[i][1 + dim + div1]);
2300                 isl_int_set_si(bmap->ineq[i][1 + dim + div1], 0);
2301         }
2302
2303         isl_int_clear(a);
2304         isl_int_clear(b);
2305         isl_int_clear(m);
2306         if (l > u) {
2307                 isl_basic_map_drop_inequality(bmap, l);
2308                 isl_basic_map_drop_inequality(bmap, u);
2309         } else {
2310                 isl_basic_map_drop_inequality(bmap, u);
2311                 isl_basic_map_drop_inequality(bmap, l);
2312         }
2313         bmap = isl_basic_map_drop_div(bmap, div1);
2314         return bmap;
2315 }
2316
2317 /* First check if we can coalesce any pair of divs and
2318  * then continue with dropping more redundant divs.
2319  *
2320  * We loop over all pairs of lower and upper bounds on a div
2321  * with coefficient 1 and -1, respectively, check if there
2322  * is any other div "c" with which we can coalesce the div
2323  * and if so, perform the coalescing.
2324  */
2325 static struct isl_basic_map *coalesce_or_drop_more_redundant_divs(
2326         struct isl_basic_map *bmap, int *pairs, int n)
2327 {
2328         int i, l, u;
2329         unsigned dim;
2330
2331         dim = isl_space_dim(bmap->dim, isl_dim_all);
2332
2333         for (i = 0; i < bmap->n_div; ++i) {
2334                 if (!pairs[i])
2335                         continue;
2336                 for (l = 0; l < bmap->n_ineq; ++l) {
2337                         if (!isl_int_is_one(bmap->ineq[l][1 + dim + i]))
2338                                 continue;
2339                         for (u = 0; u < bmap->n_ineq; ++u) {
2340                                 int c;
2341
2342                                 if (!isl_int_is_negone(bmap->ineq[u][1+dim+i]))
2343                                         continue;
2344                                 c = div_find_coalesce(bmap, pairs, i, l, u);
2345                                 if (c < 0)
2346                                         continue;
2347                                 free(pairs);
2348                                 bmap = coalesce_divs(bmap, i, c, l, u);
2349                                 return isl_basic_map_drop_redundant_divs(bmap);
2350                         }
2351                 }
2352         }
2353
2354         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
2355                 return bmap;
2356
2357         return drop_more_redundant_divs(bmap, pairs, n);
2358 }
2359
2360 /* Remove divs that are not strictly needed.
2361  * In particular, if a div only occurs positively (or negatively)
2362  * in constraints, then it can simply be dropped.
2363  * Also, if a div occurs only occurs in two constraints and if moreover
2364  * those two constraints are opposite to each other, except for the constant
2365  * term and if the sum of the constant terms is such that for any value
2366  * of the other values, there is always at least one integer value of the
2367  * div, i.e., if one plus this sum is greater than or equal to
2368  * the (absolute value) of the coefficent of the div in the constraints,
2369  * then we can also simply drop the div.
2370  *
2371  * If any divs are left after these simple checks then we move on
2372  * to more complicated cases in drop_more_redundant_divs.
2373  */
2374 struct isl_basic_map *isl_basic_map_drop_redundant_divs(
2375         struct isl_basic_map *bmap)
2376 {
2377         int i, j;
2378         unsigned off;
2379         int *pairs = NULL;
2380         int n = 0;
2381
2382         if (!bmap)
2383                 goto error;
2384
2385         off = isl_space_dim(bmap->dim, isl_dim_all);
2386         pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
2387         if (!pairs)
2388                 goto error;
2389
2390         for (i = 0; i < bmap->n_div; ++i) {
2391                 int pos, neg;
2392                 int last_pos, last_neg;
2393                 int redundant;
2394                 int defined;
2395
2396                 defined = !isl_int_is_zero(bmap->div[i][0]);
2397                 for (j = 0; j < bmap->n_eq; ++j)
2398                         if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
2399                                 break;
2400                 if (j < bmap->n_eq)
2401                         continue;
2402                 ++n;
2403                 pos = neg = 0;
2404                 for (j = 0; j < bmap->n_ineq; ++j) {
2405                         if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
2406                                 last_pos = j;
2407                                 ++pos;
2408                         }
2409                         if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
2410                                 last_neg = j;
2411                                 ++neg;
2412                         }
2413                 }
2414                 pairs[i] = pos * neg;
2415                 if (pairs[i] == 0) {
2416                         for (j = bmap->n_ineq - 1; j >= 0; --j)
2417                                 if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
2418                                         isl_basic_map_drop_inequality(bmap, j);
2419                         bmap = isl_basic_map_drop_div(bmap, i);
2420                         free(pairs);
2421                         return isl_basic_map_drop_redundant_divs(bmap);
2422                 }
2423                 if (pairs[i] != 1)
2424                         continue;
2425                 if (!isl_seq_is_neg(bmap->ineq[last_pos] + 1,
2426                                     bmap->ineq[last_neg] + 1,
2427                                     off + bmap->n_div))
2428                         continue;
2429
2430                 isl_int_add(bmap->ineq[last_pos][0],
2431                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
2432                 isl_int_add_ui(bmap->ineq[last_pos][0],
2433                                bmap->ineq[last_pos][0], 1);
2434                 redundant = isl_int_ge(bmap->ineq[last_pos][0],
2435                                 bmap->ineq[last_pos][1+off+i]);
2436                 isl_int_sub_ui(bmap->ineq[last_pos][0],
2437                                bmap->ineq[last_pos][0], 1);
2438                 isl_int_sub(bmap->ineq[last_pos][0],
2439                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
2440                 if (!redundant) {
2441                         if (defined ||
2442                             !ok_to_set_div_from_bound(bmap, i, last_pos)) {
2443                                 pairs[i] = 0;
2444                                 --n;
2445                                 continue;
2446                         }
2447                         bmap = set_div_from_lower_bound(bmap, i, last_pos);
2448                         bmap = isl_basic_map_simplify(bmap);
2449                         free(pairs);
2450                         return isl_basic_map_drop_redundant_divs(bmap);
2451                 }
2452                 if (last_pos > last_neg) {
2453                         isl_basic_map_drop_inequality(bmap, last_pos);
2454                         isl_basic_map_drop_inequality(bmap, last_neg);
2455                 } else {
2456                         isl_basic_map_drop_inequality(bmap, last_neg);
2457                         isl_basic_map_drop_inequality(bmap, last_pos);
2458                 }
2459                 bmap = isl_basic_map_drop_div(bmap, i);
2460                 free(pairs);
2461                 return isl_basic_map_drop_redundant_divs(bmap);
2462         }
2463
2464         if (n > 0)
2465                 return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);
2466
2467         free(pairs);
2468         return bmap;
2469 error:
2470         free(pairs);
2471         isl_basic_map_free(bmap);
2472         return NULL;
2473 }
2474
2475 struct isl_basic_set *isl_basic_set_drop_redundant_divs(
2476         struct isl_basic_set *bset)
2477 {
2478         return (struct isl_basic_set *)
2479             isl_basic_map_drop_redundant_divs((struct isl_basic_map *)bset);
2480 }
2481
2482 struct isl_map *isl_map_drop_redundant_divs(struct isl_map *map)
2483 {
2484         int i;
2485
2486         if (!map)
2487                 return NULL;
2488         for (i = 0; i < map->n; ++i) {
2489                 map->p[i] = isl_basic_map_drop_redundant_divs(map->p[i]);
2490                 if (!map->p[i])
2491                         goto error;
2492         }
2493         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
2494         return map;
2495 error:
2496         isl_map_free(map);
2497         return NULL;
2498 }
2499
2500 struct isl_set *isl_set_drop_redundant_divs(struct isl_set *set)
2501 {
2502         return (struct isl_set *)
2503             isl_map_drop_redundant_divs((struct isl_map *)set);
2504 }