isl_map_simplify.c

   1 /*
   2  * Copyright 2008-2009 Katholieke Universiteit Leuven
   3  * Copyright 2012      Ecole Normale Superieure
   4  *
   5  * Use of this software is governed by the MIT license
   6  *
   7  * Written by Sven Verdoolaege, K.U.Leuven, Departement
   8  * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
   9  * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
  10  */
  11
  12 #include <strings.h>
  13 #include <isl_ctx_private.h>
  14 #include <isl_map_private.h>
  15 #include "isl_equalities.h"
  16 #include <isl/map.h>
  17 #include <isl/seq.h>
  18 #include "isl_tab.h"
  19 #include <isl_space_private.h>
  20 #include <isl_mat_private.h>
  21
  22 static void swap_equality(struct isl_basic_map *bmap, int a, int b)
  23 {
  24         isl_int *t = bmap->eq[a];
  25         bmap->eq[a] = bmap->eq[b];
  26         bmap->eq[b] = t;
  27 }
  28
  29 static void swap_inequality(struct isl_basic_map *bmap, int a, int b)
  30 {
  31         if (a != b) {
  32                 isl_int *t = bmap->ineq[a];
  33                 bmap->ineq[a] = bmap->ineq[b];
  34                 bmap->ineq[b] = t;
  35         }
  36 }
  37
  38 static void constraint_drop_vars(isl_int *c, unsigned n, unsigned rem)
  39 {
  40         isl_seq_cpy(c, c + n, rem);
  41         isl_seq_clr(c + rem, n);
  42 }
  43
  44 /* Drop n dimensions starting at first.
  45  *
  46  * In principle, this frees up some extra variables as the number
  47  * of columns remains constant, but we would have to extend
  48  * the div array too as the number of rows in this array is assumed
  49  * to be equal to extra.
  50  */
  51 struct isl_basic_set *isl_basic_set_drop_dims(
  52                 struct isl_basic_set *bset, unsigned first, unsigned n)
  53 {
  54         int i;
  55
  56         if (!bset)
  57                 goto error;
  58
  59         isl_assert(bset->ctx, first + n <= bset->dim->n_out, goto error);
  60
  61         if (n == 0 && !isl_space_get_tuple_name(bset->dim, isl_dim_set))
  62                 return bset;
  63
  64         bset = isl_basic_set_cow(bset);
  65         if (!bset)
  66                 return NULL;
  67
  68         for (i = 0; i < bset->n_eq; ++i)
  69                 constraint_drop_vars(bset->eq[i]+1+bset->dim->nparam+first, n,
  70                                      (bset->dim->n_out-first-n)+bset->extra);
  71
  72         for (i = 0; i < bset->n_ineq; ++i)
  73                 constraint_drop_vars(bset->ineq[i]+1+bset->dim->nparam+first, n,
  74                                      (bset->dim->n_out-first-n)+bset->extra);
  75
  76         for (i = 0; i < bset->n_div; ++i)
  77                 constraint_drop_vars(bset->div[i]+1+1+bset->dim->nparam+first, n,
  78                                      (bset->dim->n_out-first-n)+bset->extra);
  79
  80         bset->dim = isl_space_drop_outputs(bset->dim, first, n);
  81         if (!bset->dim)
  82                 goto error;
  83
  84         ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
  85         bset = isl_basic_set_simplify(bset);
  86         return isl_basic_set_finalize(bset);
  87 error:
  88         isl_basic_set_free(bset);
  89         return NULL;
  90 }
  91
  92 struct isl_set *isl_set_drop_dims(
  93                 struct isl_set *set, unsigned first, unsigned n)
  94 {
  95         int i;
  96
  97         if (!set)
  98                 goto error;
  99
 100         isl_assert(set->ctx, first + n <= set->dim->n_out, goto error);
 101
 102         if (n == 0 && !isl_space_get_tuple_name(set->dim, isl_dim_set))
 103                 return set;
 104         set = isl_set_cow(set);
 105         if (!set)
 106                 goto error;
 107         set->dim = isl_space_drop_outputs(set->dim, first, n);
 108         if (!set->dim)
 109                 goto error;
 110
 111         for (i = 0; i < set->n; ++i) {
 112                 set->p[i] = isl_basic_set_drop_dims(set->p[i], first, n);
 113                 if (!set->p[i])
 114                         goto error;
 115         }
 116
 117         ISL_F_CLR(set, ISL_SET_NORMALIZED);
 118         return set;
 119 error:
 120         isl_set_free(set);
 121         return NULL;
 122 }
 123
 124 /* Move "n" divs starting at "first" to the end of the list of divs.
 125  */
 126 static struct isl_basic_map *move_divs_last(struct isl_basic_map *bmap,
 127         unsigned first, unsigned n)
 128 {
 129         isl_int **div;
 130         int i;
 131
 132         if (first + n == bmap->n_div)
 133                 return bmap;
 134
 135         div = isl_alloc_array(bmap->ctx, isl_int *, n);
 136         if (!div)
 137                 goto error;
 138         for (i = 0; i < n; ++i)
 139                 div[i] = bmap->div[first + i];
 140         for (i = 0; i < bmap->n_div - first - n; ++i)
 141                 bmap->div[first + i] = bmap->div[first + n + i];
 142         for (i = 0; i < n; ++i)
 143                 bmap->div[bmap->n_div - n + i] = div[i];
 144         free(div);
 145         return bmap;
 146 error:
 147         isl_basic_map_free(bmap);
 148         return NULL;
 149 }
 150
 151 /* Drop "n" dimensions of type "type" starting at "first".
 152  *
 153  * In principle, this frees up some extra variables as the number
 154  * of columns remains constant, but we would have to extend
 155  * the div array too as the number of rows in this array is assumed
 156  * to be equal to extra.
 157  */
 158 struct isl_basic_map *isl_basic_map_drop(struct isl_basic_map *bmap,
 159         enum isl_dim_type type, unsigned first, unsigned n)
 160 {
 161         int i;
 162         unsigned dim;
 163         unsigned offset;
 164         unsigned left;
 165
 166         if (!bmap)
 167                 goto error;
 168
 169         dim = isl_basic_map_dim(bmap, type);
 170         isl_assert(bmap->ctx, first + n <= dim, goto error);
 171
 172         if (n == 0 && !isl_space_is_named_or_nested(bmap->dim, type))
 173                 return bmap;
 174
 175         bmap = isl_basic_map_cow(bmap);
 176         if (!bmap)
 177                 return NULL;
 178
 179         offset = isl_basic_map_offset(bmap, type) + first;
 180         left = isl_basic_map_total_dim(bmap) - (offset - 1) - n;
 181         for (i = 0; i < bmap->n_eq; ++i)
 182                 constraint_drop_vars(bmap->eq[i]+offset, n, left);
 183
 184         for (i = 0; i < bmap->n_ineq; ++i)
 185                 constraint_drop_vars(bmap->ineq[i]+offset, n, left);
 186
 187         for (i = 0; i < bmap->n_div; ++i)
 188                 constraint_drop_vars(bmap->div[i]+1+offset, n, left);
 189
 190         if (type == isl_dim_div) {
 191                 bmap = move_divs_last(bmap, first, n);
 192                 if (!bmap)
 193                         goto error;
 194                 isl_basic_map_free_div(bmap, n);
 195         } else
 196                 bmap->dim = isl_space_drop_dims(bmap->dim, type, first, n);
 197         if (!bmap->dim)
 198                 goto error;
 199
 200         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 201         bmap = isl_basic_map_simplify(bmap);
 202         return isl_basic_map_finalize(bmap);
 203 error:
 204         isl_basic_map_free(bmap);
 205         return NULL;
 206 }
 207
 208 __isl_give isl_basic_set *isl_basic_set_drop(__isl_take isl_basic_set *bset,
 209         enum isl_dim_type type, unsigned first, unsigned n)
 210 {
 211         return (isl_basic_set *)isl_basic_map_drop((isl_basic_map *)bset,
 212                                                         type, first, n);
 213 }
 214
 215 struct isl_basic_map *isl_basic_map_drop_inputs(
 216                 struct isl_basic_map *bmap, unsigned first, unsigned n)
 217 {
 218         return isl_basic_map_drop(bmap, isl_dim_in, first, n);
 219 }
 220
 221 struct isl_map *isl_map_drop(struct isl_map *map,
 222         enum isl_dim_type type, unsigned first, unsigned n)
 223 {
 224         int i;
 225
 226         if (!map)
 227                 goto error;
 228
 229         isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
 230
 231         if (n == 0 && !isl_space_get_tuple_name(map->dim, type))
 232                 return map;
 233         map = isl_map_cow(map);
 234         if (!map)
 235                 goto error;
 236         map->dim = isl_space_drop_dims(map->dim, type, first, n);
 237         if (!map->dim)
 238                 goto error;
 239
 240         for (i = 0; i < map->n; ++i) {
 241                 map->p[i] = isl_basic_map_drop(map->p[i], type, first, n);
 242                 if (!map->p[i])
 243                         goto error;
 244         }
 245         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
 246
 247         return map;
 248 error:
 249         isl_map_free(map);
 250         return NULL;
 251 }
 252
 253 struct isl_set *isl_set_drop(struct isl_set *set,
 254         enum isl_dim_type type, unsigned first, unsigned n)
 255 {
 256         return (isl_set *)isl_map_drop((isl_map *)set, type, first, n);
 257 }
 258
 259 struct isl_map *isl_map_drop_inputs(
 260                 struct isl_map *map, unsigned first, unsigned n)
 261 {
 262         return isl_map_drop(map, isl_dim_in, first, n);
 263 }
 264
 265 /*
 266  * We don't cow, as the div is assumed to be redundant.
 267  */
 268 static struct isl_basic_map *isl_basic_map_drop_div(
 269                 struct isl_basic_map *bmap, unsigned div)
 270 {
 271         int i;
 272         unsigned pos;
 273
 274         if (!bmap)
 275                 goto error;
 276
 277         pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
 278
 279         isl_assert(bmap->ctx, div < bmap->n_div, goto error);
 280
 281         for (i = 0; i < bmap->n_eq; ++i)
 282                 constraint_drop_vars(bmap->eq[i]+pos, 1, bmap->extra-div-1);
 283
 284         for (i = 0; i < bmap->n_ineq; ++i) {
 285                 if (!isl_int_is_zero(bmap->ineq[i][pos])) {
 286                         isl_basic_map_drop_inequality(bmap, i);
 287                         --i;
 288                         continue;
 289                 }
 290                 constraint_drop_vars(bmap->ineq[i]+pos, 1, bmap->extra-div-1);
 291         }
 292
 293         for (i = 0; i < bmap->n_div; ++i)
 294                 constraint_drop_vars(bmap->div[i]+1+pos, 1, bmap->extra-div-1);
 295
 296         if (div != bmap->n_div - 1) {
 297                 int j;
 298                 isl_int *t = bmap->div[div];
 299
 300                 for (j = div; j < bmap->n_div - 1; ++j)
 301                         bmap->div[j] = bmap->div[j+1];
 302
 303                 bmap->div[bmap->n_div - 1] = t;
 304         }
 305         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 306         isl_basic_map_free_div(bmap, 1);
 307
 308         return bmap;
 309 error:
 310         isl_basic_map_free(bmap);
 311         return NULL;
 312 }
 313
 314 struct isl_basic_map *isl_basic_map_normalize_constraints(
 315         struct isl_basic_map *bmap)
 316 {
 317         int i;
 318         isl_int gcd;
 319         unsigned total = isl_basic_map_total_dim(bmap);
 320
 321         if (!bmap)
 322                 return NULL;
 323
 324         isl_int_init(gcd);
 325         for (i = bmap->n_eq - 1; i >= 0; --i) {
 326                 isl_seq_gcd(bmap->eq[i]+1, total, &gcd);
 327                 if (isl_int_is_zero(gcd)) {
 328                         if (!isl_int_is_zero(bmap->eq[i][0])) {
 329                                 bmap = isl_basic_map_set_to_empty(bmap);
 330                                 break;
 331                         }
 332                         isl_basic_map_drop_equality(bmap, i);
 333                         continue;
 334                 }
 335                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 336                         isl_int_gcd(gcd, gcd, bmap->eq[i][0]);
 337                 if (isl_int_is_one(gcd))
 338                         continue;
 339                 if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) {
 340                         bmap = isl_basic_map_set_to_empty(bmap);
 341                         break;
 342                 }
 343                 isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total);
 344         }
 345
 346         for (i = bmap->n_ineq - 1; i >= 0; --i) {
 347                 isl_seq_gcd(bmap->ineq[i]+1, total, &gcd);
 348                 if (isl_int_is_zero(gcd)) {
 349                         if (isl_int_is_neg(bmap->ineq[i][0])) {
 350                                 bmap = isl_basic_map_set_to_empty(bmap);
 351                                 break;
 352                         }
 353                         isl_basic_map_drop_inequality(bmap, i);
 354                         continue;
 355                 }
 356                 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
 357                         isl_int_gcd(gcd, gcd, bmap->ineq[i][0]);
 358                 if (isl_int_is_one(gcd))
 359                         continue;
 360                 isl_int_fdiv_q(bmap->ineq[i][0], bmap->ineq[i][0], gcd);
 361                 isl_seq_scale_down(bmap->ineq[i]+1, bmap->ineq[i]+1, gcd, total);
 362         }
 363         isl_int_clear(gcd);
 364
 365         return bmap;
 366 }
 367
 368 struct isl_basic_set *isl_basic_set_normalize_constraints(
 369         struct isl_basic_set *bset)
 370 {
 371         return (struct isl_basic_set *)isl_basic_map_normalize_constraints(
 372                 (struct isl_basic_map *)bset);
 373 }
 374
 375 /* Remove any common factor in numerator and denominator of the div expression,
 376  * not taking into account the constant term.
 377  * That is, if the div is of the form
 378  *
 379  *      floor((a + m f(x))/(m d))
 380  *
 381  * then replace it by
 382  *
 383  *      floor((floor(a/m) + f(x))/d)
 384  *
 385  * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
 386  * and can therefore not influence the result of the floor.
 387  */
 388 static void normalize_div_expression(__isl_keep isl_basic_map *bmap, int div)
 389 {
 390         unsigned total = isl_basic_map_total_dim(bmap);
 391         isl_ctx *ctx = bmap->ctx;
 392
 393         if (isl_int_is_zero(bmap->div[div][0]))
 394                 return;
 395         isl_seq_gcd(bmap->div[div] + 2, total, &ctx->normalize_gcd);
 396         isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, bmap->div[div][0]);
 397         if (isl_int_is_one(ctx->normalize_gcd))
 398                 return;
 399         isl_int_fdiv_q(bmap->div[div][1], bmap->div[div][1],
 400                         ctx->normalize_gcd);
 401         isl_int_divexact(bmap->div[div][0], bmap->div[div][0],
 402                         ctx->normalize_gcd);
 403         isl_seq_scale_down(bmap->div[div] + 2, bmap->div[div] + 2,
 404                         ctx->normalize_gcd, total);
 405 }
 406
 407 /* Remove any common factor in numerator and denominator of a div expression,
 408  * not taking into account the constant term.
 409  * That is, look for any div of the form
 410  *
 411  *      floor((a + m f(x))/(m d))
 412  *
 413  * and replace it by
 414  *
 415  *      floor((floor(a/m) + f(x))/d)
 416  *
 417  * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
 418  * and can therefore not influence the result of the floor.
 419  */
 420 static __isl_give isl_basic_map *normalize_div_expressions(
 421         __isl_take isl_basic_map *bmap)
 422 {
 423         int i;
 424
 425         if (!bmap)
 426                 return NULL;
 427         if (bmap->n_div == 0)
 428                 return bmap;
 429
 430         for (i = 0; i < bmap->n_div; ++i)
 431                 normalize_div_expression(bmap, i);
 432
 433         return bmap;
 434 }
 435
 436 /* Assumes divs have been ordered if keep_divs is set.
 437  */
 438 static void eliminate_var_using_equality(struct isl_basic_map *bmap,
 439         unsigned pos, isl_int *eq, int keep_divs, int *progress)
 440 {
 441         unsigned total;
 442         unsigned space_total;
 443         int k;
 444         int last_div;
 445
 446         total = isl_basic_map_total_dim(bmap);
 447         space_total = isl_space_dim(bmap->dim, isl_dim_all);
 448         last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
 449         for (k = 0; k < bmap->n_eq; ++k) {
 450                 if (bmap->eq[k] == eq)
 451                         continue;
 452                 if (isl_int_is_zero(bmap->eq[k][1+pos]))
 453                         continue;
 454                 if (progress)
 455                         *progress = 1;
 456                 isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL);
 457                 isl_seq_normalize(bmap->ctx, bmap->eq[k], 1 + total);
 458         }
 459
 460         for (k = 0; k < bmap->n_ineq; ++k) {
 461                 if (isl_int_is_zero(bmap->ineq[k][1+pos]))
 462                         continue;
 463                 if (progress)
 464                         *progress = 1;
 465                 isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL);
 466                 isl_seq_normalize(bmap->ctx, bmap->ineq[k], 1 + total);
 467                 ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 468         }
 469
 470         for (k = 0; k < bmap->n_div; ++k) {
 471                 if (isl_int_is_zero(bmap->div[k][0]))
 472                         continue;
 473                 if (isl_int_is_zero(bmap->div[k][1+1+pos]))
 474                         continue;
 475                 if (progress)
 476                         *progress = 1;
 477                 /* We need to be careful about circular definitions,
 478                  * so for now we just remove the definition of div k
 479                  * if the equality contains any divs.
 480                  * If keep_divs is set, then the divs have been ordered
 481                  * and we can keep the definition as long as the result
 482                  * is still ordered.
 483                  */
 484                 if (last_div == -1 || (keep_divs && last_div < k)) {
 485                         isl_seq_elim(bmap->div[k]+1, eq,
 486                                         1+pos, 1+total, &bmap->div[k][0]);
 487                         normalize_div_expression(bmap, k);
 488                 } else
 489                         isl_seq_clr(bmap->div[k], 1 + total);
 490                 ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 491         }
 492 }
 493
 494 /* Assumes divs have been ordered if keep_divs is set.
 495  */
 496 static void eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
 497         unsigned div, int keep_divs)
 498 {
 499         unsigned pos = isl_space_dim(bmap->dim, isl_dim_all) + div;
 500
 501         eliminate_var_using_equality(bmap, pos, eq, keep_divs, NULL);
 502
 503         isl_basic_map_drop_div(bmap, div);
 504 }
 505
 506 /* Check if elimination of div "div" using equality "eq" would not
 507  * result in a div depending on a later div.
 508  */
 509 static int ok_to_eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
 510         unsigned div)
 511 {
 512         int k;
 513         int last_div;
 514         unsigned space_total = isl_space_dim(bmap->dim, isl_dim_all);
 515         unsigned pos = space_total + div;
 516
 517         last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
 518         if (last_div < 0 || last_div <= div)
 519                 return 1;
 520
 521         for (k = 0; k <= last_div; ++k) {
 522                 if (isl_int_is_zero(bmap->div[k][0]))
 523                         return 1;
 524                 if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos]))
 525                         return 0;
 526         }
 527
 528         return 1;
 529 }
 530
 531 /* Elimininate divs based on equalities
 532  */
 533 static struct isl_basic_map *eliminate_divs_eq(
 534                 struct isl_basic_map *bmap, int *progress)
 535 {
 536         int d;
 537         int i;
 538         int modified = 0;
 539         unsigned off;
 540
 541         bmap = isl_basic_map_order_divs(bmap);
 542
 543         if (!bmap)
 544                 return NULL;
 545
 546         off = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 547
 548         for (d = bmap->n_div - 1; d >= 0 ; --d) {
 549                 for (i = 0; i < bmap->n_eq; ++i) {
 550                         if (!isl_int_is_one(bmap->eq[i][off + d]) &&
 551                             !isl_int_is_negone(bmap->eq[i][off + d]))
 552                                 continue;
 553                         if (!ok_to_eliminate_div(bmap, bmap->eq[i], d))
 554                                 continue;
 555                         modified = 1;
 556                         *progress = 1;
 557                         eliminate_div(bmap, bmap->eq[i], d, 1);
 558                         isl_basic_map_drop_equality(bmap, i);
 559                         break;
 560                 }
 561         }
 562         if (modified)
 563                 return eliminate_divs_eq(bmap, progress);
 564         return bmap;
 565 }
 566
 567 /* Elimininate divs based on inequalities
 568  */
 569 static struct isl_basic_map *eliminate_divs_ineq(
 570                 struct isl_basic_map *bmap, int *progress)
 571 {
 572         int d;
 573         int i;
 574         unsigned off;
 575         struct isl_ctx *ctx;
 576
 577         if (!bmap)
 578                 return NULL;
 579
 580         ctx = bmap->ctx;
 581         off = 1 + isl_space_dim(bmap->dim, isl_dim_all);
 582
 583         for (d = bmap->n_div - 1; d >= 0 ; --d) {
 584                 for (i = 0; i < bmap->n_eq; ++i)
 585                         if (!isl_int_is_zero(bmap->eq[i][off + d]))
 586                                 break;
 587                 if (i < bmap->n_eq)
 588                         continue;
 589                 for (i = 0; i < bmap->n_ineq; ++i)
 590                         if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one))
 591                                 break;
 592                 if (i < bmap->n_ineq)
 593                         continue;
 594                 *progress = 1;
 595                 bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1);
 596                 if (!bmap || ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
 597                         break;
 598                 bmap = isl_basic_map_drop_div(bmap, d);
 599                 if (!bmap)
 600                         break;
 601         }
 602         return bmap;
 603 }
 604
 605 struct isl_basic_map *isl_basic_map_gauss(
 606         struct isl_basic_map *bmap, int *progress)
 607 {
 608         int k;
 609         int done;
 610         int last_var;
 611         unsigned total_var;
 612         unsigned total;
 613
 614         bmap = isl_basic_map_order_divs(bmap);
 615
 616         if (!bmap)
 617                 return NULL;
 618
 619         total = isl_basic_map_total_dim(bmap);
 620         total_var = total - bmap->n_div;
 621
 622         last_var = total - 1;
 623         for (done = 0; done < bmap->n_eq; ++done) {
 624                 for (; last_var >= 0; --last_var) {
 625                         for (k = done; k < bmap->n_eq; ++k)
 626                                 if (!isl_int_is_zero(bmap->eq[k][1+last_var]))
 627                                         break;
 628                         if (k < bmap->n_eq)
 629                                 break;
 630                 }
 631                 if (last_var < 0)
 632                         break;
 633                 if (k != done)
 634                         swap_equality(bmap, k, done);
 635                 if (isl_int_is_neg(bmap->eq[done][1+last_var]))
 636                         isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total);
 637
 638                 eliminate_var_using_equality(bmap, last_var, bmap->eq[done], 1,
 639                                                 progress);
 640
 641                 if (last_var >= total_var &&
 642                     isl_int_is_zero(bmap->div[last_var - total_var][0])) {
 643                         unsigned div = last_var - total_var;
 644                         isl_seq_neg(bmap->div[div]+1, bmap->eq[done], 1+total);
 645                         isl_int_set_si(bmap->div[div][1+1+last_var], 0);
 646                         isl_int_set(bmap->div[div][0],
 647                                     bmap->eq[done][1+last_var]);
 648                         if (progress)
 649                                 *progress = 1;
 650                         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
 651                 }
 652         }
 653         if (done == bmap->n_eq)
 654                 return bmap;
 655         for (k = done; k < bmap->n_eq; ++k) {
 656                 if (isl_int_is_zero(bmap->eq[k][0]))
 657                         continue;
 658                 return isl_basic_map_set_to_empty(bmap);
 659         }
 660         isl_basic_map_free_equality(bmap, bmap->n_eq-done);
 661         return bmap;
 662 }
 663
 664 struct isl_basic_set *isl_basic_set_gauss(
 665         struct isl_basic_set *bset, int *progress)
 666 {
 667         return (struct isl_basic_set*)isl_basic_map_gauss(
 668                         (struct isl_basic_map *)bset, progress);
 669 }
 670
 671
 672 static unsigned int round_up(unsigned int v)
 673 {
 674         int old_v = v;
 675
 676         while (v) {
 677                 old_v = v;
 678                 v ^= v & -v;
 679         }
 680         return old_v << 1;
 681 }
 682
 683 static int hash_index(isl_int ***index, unsigned int size, int bits,
 684                         struct isl_basic_map *bmap, int k)
 685 {
 686         int h;
 687         unsigned total = isl_basic_map_total_dim(bmap);
 688         uint32_t hash = isl_seq_get_hash_bits(bmap->ineq[k]+1, total, bits);
 689         for (h = hash; index[h]; h = (h+1) % size)
 690                 if (&bmap->ineq[k] != index[h] &&
 691                     isl_seq_eq(bmap->ineq[k]+1, index[h][0]+1, total))
 692                         break;
 693         return h;
 694 }
 695
 696 static int set_hash_index(isl_int ***index, unsigned int size, int bits,
 697                           struct isl_basic_set *bset, int k)
 698 {
 699         return hash_index(index, size, bits, (struct isl_basic_map *)bset, k);
 700 }
 701
 702 /* If we can eliminate more than one div, then we need to make
 703  * sure we do it from last div to first div, in order not to
 704  * change the position of the other divs that still need to
 705  * be removed.
 706  */
 707 static struct isl_basic_map *remove_duplicate_divs(
 708         struct isl_basic_map *bmap, int *progress)
 709 {
 710         unsigned int size;
 711         int *index;
 712         int *elim_for;
 713         int k, l, h;
 714         int bits;
 715         struct isl_blk eq;
 716         unsigned total_var;
 717         unsigned total;
 718         struct isl_ctx *ctx;
 719
 720         bmap = isl_basic_map_order_divs(bmap);
 721         if (!bmap || bmap->n_div <= 1)
 722                 return bmap;
 723
 724         total_var = isl_space_dim(bmap->dim, isl_dim_all);
 725         total = total_var + bmap->n_div;
 726
 727         ctx = bmap->ctx;
 728         for (k = bmap->n_div - 1; k >= 0; --k)
 729                 if (!isl_int_is_zero(bmap->div[k][0]))
 730                         break;
 731         if (k <= 0)
 732                 return bmap;
 733
 734         elim_for = isl_calloc_array(ctx, int, bmap->n_div);
 735         size = round_up(4 * bmap->n_div / 3 - 1);
 736         bits = ffs(size) - 1;
 737         index = isl_calloc_array(ctx, int, size);
 738         if (!index)
 739                 return bmap;
 740         eq = isl_blk_alloc(ctx, 1+total);
 741         if (isl_blk_is_error(eq))
 742                 goto out;
 743
 744         isl_seq_clr(eq.data, 1+total);
 745         index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1;
 746         for (--k; k >= 0; --k) {
 747                 uint32_t hash;
 748
 749                 if (isl_int_is_zero(bmap->div[k][0]))
 750                         continue;
 751
 752                 hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits);
 753                 for (h = hash; index[h]; h = (h+1) % size)
 754                         if (isl_seq_eq(bmap->div[k],
 755                                        bmap->div[index[h]-1], 2+total))
 756                                 break;
 757                 if (index[h]) {
 758                         *progress = 1;
 759                         l = index[h] - 1;
 760                         elim_for[l] = k + 1;
 761                 }
 762                 index[h] = k+1;
 763         }
 764         for (l = bmap->n_div - 1; l >= 0; --l) {
 765                 if (!elim_for[l])
 766                         continue;
 767                 k = elim_for[l] - 1;
 768                 isl_int_set_si(eq.data[1+total_var+k], -1);
 769                 isl_int_set_si(eq.data[1+total_var+l], 1);
 770                 eliminate_div(bmap, eq.data, l, 1);
 771                 isl_int_set_si(eq.data[1+total_var+k], 0);
 772                 isl_int_set_si(eq.data[1+total_var+l], 0);
 773         }
 774
 775         isl_blk_free(ctx, eq);
 776 out:
 777         free(index);
 778         free(elim_for);
 779         return bmap;
 780 }
 781
 782 static int n_pure_div_eq(struct isl_basic_map *bmap)
 783 {
 784         int i, j;
 785         unsigned total;
 786
 787         total = isl_space_dim(bmap->dim, isl_dim_all);
 788         for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) {
 789                 while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
 790                         --j;
 791                 if (j < 0)
 792                         break;
 793                 if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total, j) != -1)
 794                         return 0;
 795         }
 796         return i;
 797 }
 798
 799 /* Normalize divs that appear in equalities.
 800  *
 801  * In particular, we assume that bmap contains some equalities
 802  * of the form
 803  *
 804  *      a x = m * e_i
 805  *
 806  * and we want to replace the set of e_i by a minimal set and
 807  * such that the new e_i have a canonical representation in terms
 808  * of the vector x.
 809  * If any of the equalities involves more than one divs, then
 810  * we currently simply bail out.
 811  *
 812  * Let us first additionally assume that all equalities involve
 813  * a div.  The equalities then express modulo constraints on the
 814  * remaining variables and we can use "parameter compression"
 815  * to find a minimal set of constraints.  The result is a transformation
 816  *
 817  *      x = T(x') = x_0 + G x'
 818  *
 819  * with G a lower-triangular matrix with all elements below the diagonal
 820  * non-negative and smaller than the diagonal element on the same row.
 821  * We first normalize x_0 by making the same property hold in the affine
 822  * T matrix.
 823  * The rows i of G with a 1 on the diagonal do not impose any modulo
 824  * constraint and simply express x_i = x'_i.
 825  * For each of the remaining rows i, we introduce a div and a corresponding
 826  * equality.  In particular
 827  *
 828  *      g_ii e_j = x_i - g_i(x')
 829  *
 830  * where each x'_k is replaced either by x_k (if g_kk = 1) or the
 831  * corresponding div (if g_kk != 1).
 832  *
 833  * If there are any equalities not involving any div, then we
 834  * first apply a variable compression on the variables x:
 835  *
 836  *      x = C x''       x'' = C_2 x
 837  *
 838  * and perform the above parameter compression on A C instead of on A.
 839  * The resulting compression is then of the form
 840  *
 841  *      x'' = T(x') = x_0 + G x'
 842  *
 843  * and in constructing the new divs and the corresponding equalities,
 844  * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
 845  * by the corresponding row from C_2.
 846  */
 847 static struct isl_basic_map *normalize_divs(
 848         struct isl_basic_map *bmap, int *progress)
 849 {
 850         int i, j, k;
 851         int total;
 852         int div_eq;
 853         struct isl_mat *B;
 854         struct isl_vec *d;
 855         struct isl_mat *T = NULL;
 856         struct isl_mat *C = NULL;
 857         struct isl_mat *C2 = NULL;
 858         isl_int v;
 859         int *pos;
 860         int dropped, needed;
 861
 862         if (!bmap)
 863                 return NULL;
 864
 865         if (bmap->n_div == 0)
 866                 return bmap;
 867
 868         if (bmap->n_eq == 0)
 869                 return bmap;
 870
 871         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS))
 872                 return bmap;
 873
 874         total = isl_space_dim(bmap->dim, isl_dim_all);
 875         div_eq = n_pure_div_eq(bmap);
 876         if (div_eq == 0)
 877                 return bmap;
 878
 879         if (div_eq < bmap->n_eq) {
 880                 B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, div_eq,
 881                                         bmap->n_eq - div_eq, 0, 1 + total);
 882                 C = isl_mat_variable_compression(B, &C2);
 883                 if (!C || !C2)
 884                         goto error;
 885                 if (C->n_col == 0) {
 886                         bmap = isl_basic_map_set_to_empty(bmap);
 887                         isl_mat_free(C);
 888                         isl_mat_free(C2);
 889                         goto done;
 890                 }
 891         }
 892
 893         d = isl_vec_alloc(bmap->ctx, div_eq);
 894         if (!d)
 895                 goto error;
 896         for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) {
 897                 while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
 898                         --j;
 899                 isl_int_set(d->block.data[i], bmap->eq[i][1 + total + j]);
 900         }
 901         B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + total);
 902
 903         if (C) {
 904                 B = isl_mat_product(B, C);
 905                 C = NULL;
 906         }
 907
 908         T = isl_mat_parameter_compression(B, d);
 909         if (!T)
 910                 goto error;
 911         if (T->n_col == 0) {
 912                 bmap = isl_basic_map_set_to_empty(bmap);
 913                 isl_mat_free(C2);
 914                 isl_mat_free(T);
 915                 goto done;
 916         }
 917         isl_int_init(v);
 918         for (i = 0; i < T->n_row - 1; ++i) {
 919                 isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]);
 920                 if (isl_int_is_zero(v))
 921                         continue;
 922                 isl_mat_col_submul(T, 0, v, 1 + i);
 923         }
 924         isl_int_clear(v);
 925         pos = isl_alloc_array(bmap->ctx, int, T->n_row);
 926         if (!pos)
 927                 goto error;
 928         /* We have to be careful because dropping equalities may reorder them */
 929         dropped = 0;
 930         for (j = bmap->n_div - 1; j >= 0; --j) {
 931                 for (i = 0; i < bmap->n_eq; ++i)
 932                         if (!isl_int_is_zero(bmap->eq[i][1 + total + j]))
 933                                 break;
 934                 if (i < bmap->n_eq) {
 935                         bmap = isl_basic_map_drop_div(bmap, j);
 936                         isl_basic_map_drop_equality(bmap, i);
 937                         ++dropped;
 938                 }
 939         }
 940         pos[0] = 0;
 941         needed = 0;
 942         for (i = 1; i < T->n_row; ++i) {
 943                 if (isl_int_is_one(T->row[i][i]))
 944                         pos[i] = i;
 945                 else
 946                         needed++;
 947         }
 948         if (needed > dropped) {
 949                 bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
 950                                 needed, needed, 0);
 951                 if (!bmap)
 952                         goto error;
 953         }
 954         for (i = 1; i < T->n_row; ++i) {
 955                 if (isl_int_is_one(T->row[i][i]))
 956                         continue;
 957                 k = isl_basic_map_alloc_div(bmap);
 958                 pos[i] = 1 + total + k;
 959                 isl_seq_clr(bmap->div[k] + 1, 1 + total + bmap->n_div);
 960                 isl_int_set(bmap->div[k][0], T->row[i][i]);
 961                 if (C2)
 962                         isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + total);
 963                 else
 964                         isl_int_set_si(bmap->div[k][1 + i], 1);
 965                 for (j = 0; j < i; ++j) {
 966                         if (isl_int_is_zero(T->row[i][j]))
 967                                 continue;
 968                         if (pos[j] < T->n_row && C2)
 969                                 isl_seq_submul(bmap->div[k] + 1, T->row[i][j],
 970                                                 C2->row[pos[j]], 1 + total);
 971                         else
 972                                 isl_int_neg(bmap->div[k][1 + pos[j]],
 973                                                                 T->row[i][j]);
 974                 }
 975                 j = isl_basic_map_alloc_equality(bmap);
 976                 isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+total+bmap->n_div);
 977                 isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]);
 978         }
 979         free(pos);
 980         isl_mat_free(C2);
 981         isl_mat_free(T);
 982
 983         if (progress)
 984                 *progress = 1;
 985 done:
 986         ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
 987
 988         return bmap;
 989 error:
 990         isl_mat_free(C);
 991         isl_mat_free(C2);
 992         isl_mat_free(T);
 993         return bmap;
 994 }
 995
 996 static struct isl_basic_map *set_div_from_lower_bound(
 997         struct isl_basic_map *bmap, int div, int ineq)
 998 {
 999         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
1000
1001         isl_seq_neg(bmap->div[div] + 1, bmap->ineq[ineq], total + bmap->n_div);
1002         isl_int_set(bmap->div[div][0], bmap->ineq[ineq][total + div]);
1003         isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]);
1004         isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
1005         isl_int_set_si(bmap->div[div][1 + total + div], 0);
1006
1007         return bmap;
1008 }
1009
1010 /* Check whether it is ok to define a div based on an inequality.
1011  * To avoid the introduction of circular definitions of divs, we
1012  * do not allow such a definition if the resulting expression would refer to
1013  * any other undefined divs or if any known div is defined in
1014  * terms of the unknown div.
1015  */
1016 static int ok_to_set_div_from_bound(struct isl_basic_map *bmap,
1017         int div, int ineq)
1018 {
1019         int j;
1020         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
1021
1022         /* Not defined in terms of unknown divs */
1023         for (j = 0; j < bmap->n_div; ++j) {
1024                 if (div == j)
1025                         continue;
1026                 if (isl_int_is_zero(bmap->ineq[ineq][total + j]))
1027                         continue;
1028                 if (isl_int_is_zero(bmap->div[j][0]))
1029                         return 0;
1030         }
1031
1032         /* No other div defined in terms of this one => avoid loops */
1033         for (j = 0; j < bmap->n_div; ++j) {
1034                 if (div == j)
1035                         continue;
1036                 if (isl_int_is_zero(bmap->div[j][0]))
1037                         continue;
1038                 if (!isl_int_is_zero(bmap->div[j][1 + total + div]))
1039                         return 0;
1040         }
1041
1042         return 1;
1043 }
1044
1045 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1046  * be a better expression than the current one?
1047  *
1048  * If we do not have any expression yet, then any expression would be better.
1049  * Otherwise we check if the last variable involved in the inequality
1050  * (disregarding the div that it would define) is in an earlier position
1051  * than the last variable involved in the current div expression.
1052  */
1053 static int better_div_constraint(__isl_keep isl_basic_map *bmap,
1054         int div, int ineq)
1055 {
1056         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
1057         int last_div;
1058         int last_ineq;
1059
1060         if (isl_int_is_zero(bmap->div[div][0]))
1061                 return 1;
1062
1063         if (isl_seq_last_non_zero(bmap->ineq[ineq] + total + div + 1,
1064                                   bmap->n_div - (div + 1)) >= 0)
1065                 return 0;
1066
1067         last_ineq = isl_seq_last_non_zero(bmap->ineq[ineq], total + div);
1068         last_div = isl_seq_last_non_zero(bmap->div[div] + 1,
1069                                          total + bmap->n_div);
1070
1071         return last_ineq < last_div;
1072 }
1073
1074 /* Given two constraints "k" and "l" that are opposite to each other,
1075  * except for the constant term, check if we can use them
1076  * to obtain an expression for one of the hitherto unknown divs or
1077  * a "better" expression for a div for which we already have an expression.
1078  * "sum" is the sum of the constant terms of the constraints.
1079  * If this sum is strictly smaller than the coefficient of one
1080  * of the divs, then this pair can be used define the div.
1081  * To avoid the introduction of circular definitions of divs, we
1082  * do not use the pair if the resulting expression would refer to
1083  * any other undefined divs or if any known div is defined in
1084  * terms of the unknown div.
1085  */
1086 static struct isl_basic_map *check_for_div_constraints(
1087         struct isl_basic_map *bmap, int k, int l, isl_int sum, int *progress)
1088 {
1089         int i;
1090         unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
1091
1092         for (i = 0; i < bmap->n_div; ++i) {
1093                 if (isl_int_is_zero(bmap->ineq[k][total + i]))
1094                         continue;
1095                 if (isl_int_abs_ge(sum, bmap->ineq[k][total + i]))
1096                         continue;
1097                 if (!better_div_constraint(bmap, i, k))
1098                         continue;
1099                 if (!ok_to_set_div_from_bound(bmap, i, k))
1100                         break;
1101                 if (isl_int_is_pos(bmap->ineq[k][total + i]))
1102                         bmap = set_div_from_lower_bound(bmap, i, k);
1103                 else
1104                         bmap = set_div_from_lower_bound(bmap, i, l);
1105                 if (progress)
1106                         *progress = 1;
1107                 break;
1108         }
1109         return bmap;
1110 }
1111
1112 static struct isl_basic_map *remove_duplicate_constraints(
1113         struct isl_basic_map *bmap, int *progress, int detect_divs)
1114 {
1115         unsigned int size;
1116         isl_int ***index;
1117         int k, l, h;
1118         int bits;
1119         unsigned total = isl_basic_map_total_dim(bmap);
1120         isl_int sum;
1121         isl_ctx *ctx;
1122
1123         if (!bmap || bmap->n_ineq <= 1)
1124                 return bmap;
1125
1126         size = round_up(4 * (bmap->n_ineq+1) / 3 - 1);
1127         bits = ffs(size) - 1;
1128         ctx = isl_basic_map_get_ctx(bmap);
1129         index = isl_calloc_array(ctx, isl_int **, size);
1130         if (!index)
1131                 return bmap;
1132
1133         index[isl_seq_get_hash_bits(bmap->ineq[0]+1, total, bits)] = &bmap->ineq[0];
1134         for (k = 1; k < bmap->n_ineq; ++k) {
1135                 h = hash_index(index, size, bits, bmap, k);
1136                 if (!index[h]) {
1137                         index[h] = &bmap->ineq[k];
1138                         continue;
1139                 }
1140                 if (progress)
1141                         *progress = 1;
1142                 l = index[h] - &bmap->ineq[0];
1143                 if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0]))
1144                         swap_inequality(bmap, k, l);
1145                 isl_basic_map_drop_inequality(bmap, k);
1146                 --k;
1147         }
1148         isl_int_init(sum);
1149         for (k = 0; k < bmap->n_ineq-1; ++k) {
1150                 isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
1151                 h = hash_index(index, size, bits, bmap, k);
1152                 isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
1153                 if (!index[h])
1154                         continue;
1155                 l = index[h] - &bmap->ineq[0];
1156                 isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]);
1157                 if (isl_int_is_pos(sum)) {
1158                         if (detect_divs)
1159                                 bmap = check_for_div_constraints(bmap, k, l,
1160                                                                  sum, progress);
1161                         continue;
1162                 }
1163                 if (isl_int_is_zero(sum)) {
1164                         /* We need to break out of the loop after these
1165                          * changes since the contents of the hash
1166                          * will no longer be valid.
1167                          * Plus, we probably we want to regauss first.
1168                          */
1169                         if (progress)
1170                                 *progress = 1;
1171                         isl_basic_map_drop_inequality(bmap, l);
1172                         isl_basic_map_inequality_to_equality(bmap, k);
1173                 } else
1174                         bmap = isl_basic_map_set_to_empty(bmap);
1175                 break;
1176         }
1177         isl_int_clear(sum);
1178
1179         free(index);
1180         return bmap;
1181 }
1182
1183
1184 /* Eliminate knowns divs from constraints where they appear with
1185  * a (positive or negative) unit coefficient.
1186  *
1187  * That is, replace
1188  *
1189  *      floor(e/m) + f >= 0
1190  *
1191  * by
1192  *
1193  *      e + m f >= 0
1194  *
1195  * and
1196  *
1197  *      -floor(e/m) + f >= 0
1198  *
1199  * by
1200  *
1201  *      -e + m f + m - 1 >= 0
1202  *
1203  * The first conversion is valid because floor(e/m) >= -f is equivalent
1204  * to e/m >= -f because -f is an integral expression.
1205  * The second conversion follows from the fact that
1206  *
1207  *      -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1208  *
1209  *
1210  * We skip integral divs, i.e., those with denominator 1, as we would
1211  * risk eliminating the div from the div constraints.  We do not need
1212  * to handle those divs here anyway since the div constraints will turn
1213  * out to form an equality and this equality can then be use to eliminate
1214  * the div from all constraints.
1215  */
1216 static __isl_give isl_basic_map *eliminate_unit_divs(
1217         __isl_take isl_basic_map *bmap, int *progress)
1218 {
1219         int i, j;
1220         isl_ctx *ctx;
1221         unsigned total;
1222
1223         if (!bmap)
1224                 return NULL;
1225
1226         ctx = isl_basic_map_get_ctx(bmap);
1227         total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
1228
1229         for (i = 0; i < bmap->n_div; ++i) {
1230                 if (isl_int_is_zero(bmap->div[i][0]))
1231                         continue;
1232                 if (isl_int_is_one(bmap->div[i][0]))
1233                         continue;
1234                 for (j = 0; j < bmap->n_ineq; ++j) {
1235                         int s;
1236
1237                         if (!isl_int_is_one(bmap->ineq[j][total + i]) &&
1238                             !isl_int_is_negone(bmap->ineq[j][total + i]))
1239                                 continue;
1240
1241                         *progress = 1;
1242
1243                         s = isl_int_sgn(bmap->ineq[j][total + i]);
1244                         isl_int_set_si(bmap->ineq[j][total + i], 0);
1245                         if (s < 0)
1246                                 isl_seq_combine(bmap->ineq[j],
1247                                         ctx->negone, bmap->div[i] + 1,
1248                                         bmap->div[i][0], bmap->ineq[j],
1249                                         total + bmap->n_div);
1250                         else
1251                                 isl_seq_combine(bmap->ineq[j],
1252                                         ctx->one, bmap->div[i] + 1,
1253                                         bmap->div[i][0], bmap->ineq[j],
1254                                         total + bmap->n_div);
1255                         if (s < 0) {
1256                                 isl_int_add(bmap->ineq[j][0],
1257                                         bmap->ineq[j][0], bmap->div[i][0]);
1258                                 isl_int_sub_ui(bmap->ineq[j][0],
1259                                         bmap->ineq[j][0], 1);
1260                         }
1261                 }
1262         }
1263
1264         return bmap;
1265 }
1266
1267 struct isl_basic_map *isl_basic_map_simplify(struct isl_basic_map *bmap)
1268 {
1269         int progress = 1;
1270         if (!bmap)
1271                 return NULL;
1272         while (progress) {
1273                 progress = 0;
1274                 bmap = isl_basic_map_normalize_constraints(bmap);
1275                 bmap = normalize_div_expressions(bmap);
1276                 bmap = remove_duplicate_divs(bmap, &progress);
1277                 bmap = eliminate_unit_divs(bmap, &progress);
1278                 bmap = eliminate_divs_eq(bmap, &progress);
1279                 bmap = eliminate_divs_ineq(bmap, &progress);
1280                 bmap = isl_basic_map_gauss(bmap, &progress);
1281                 /* requires equalities in normal form */
1282                 bmap = normalize_divs(bmap, &progress);
1283                 bmap = remove_duplicate_constraints(bmap, &progress, 1);
1284         }
1285         return bmap;
1286 }
1287
1288 struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset)
1289 {
1290         return (struct isl_basic_set *)
1291                 isl_basic_map_simplify((struct isl_basic_map *)bset);
1292 }
1293
1294
1295 int isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
1296         isl_int *constraint, unsigned div)
1297 {
1298         unsigned pos;
1299
1300         if (!bmap)
1301                 return -1;
1302
1303         pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
1304
1305         if (isl_int_eq(constraint[pos], bmap->div[div][0])) {
1306                 int neg;
1307                 isl_int_sub(bmap->div[div][1],
1308                                 bmap->div[div][1], bmap->div[div][0]);
1309                 isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
1310                 neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos);
1311                 isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
1312                 isl_int_add(bmap->div[div][1],
1313                                 bmap->div[div][1], bmap->div[div][0]);
1314                 if (!neg)
1315                         return 0;
1316                 if (isl_seq_first_non_zero(constraint+pos+1,
1317                                             bmap->n_div-div-1) != -1)
1318                         return 0;
1319         } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
1320                 if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
1321                         return 0;
1322                 if (isl_seq_first_non_zero(constraint+pos+1,
1323                                             bmap->n_div-div-1) != -1)
1324                         return 0;
1325         } else
1326                 return 0;
1327
1328         return 1;
1329 }
1330
1331 int isl_basic_set_is_div_constraint(__isl_keep isl_basic_set *bset,
1332         isl_int *constraint, unsigned div)
1333 {
1334         return isl_basic_map_is_div_constraint(bset, constraint, div);
1335 }
1336
1337
1338 /* If the only constraints a div d=floor(f/m)
1339  * appears in are its two defining constraints
1340  *
1341  *      f - m d >=0
1342  *      -(f - (m - 1)) + m d >= 0
1343  *
1344  * then it can safely be removed.
1345  */
1346 static int div_is_redundant(struct isl_basic_map *bmap, int div)
1347 {
1348         int i;
1349         unsigned pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
1350
1351         for (i = 0; i < bmap->n_eq; ++i)
1352                 if (!isl_int_is_zero(bmap->eq[i][pos]))
1353                         return 0;
1354
1355         for (i = 0; i < bmap->n_ineq; ++i) {
1356                 if (isl_int_is_zero(bmap->ineq[i][pos]))
1357                         continue;
1358                 if (!isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div))
1359                         return 0;
1360         }
1361
1362         for (i = 0; i < bmap->n_div; ++i)
1363                 if (!isl_int_is_zero(bmap->div[i][1+pos]))
1364                         return 0;
1365
1366         return 1;
1367 }
1368
1369 /*
1370  * Remove divs that don't occur in any of the constraints or other divs.
1371  * These can arise when dropping some of the variables in a quast
1372  * returned by piplib.
1373  */
1374 static struct isl_basic_map *remove_redundant_divs(struct isl_basic_map *bmap)
1375 {
1376         int i;
1377
1378         if (!bmap)
1379                 return NULL;
1380
1381         for (i = bmap->n_div-1; i >= 0; --i) {
1382                 if (!div_is_redundant(bmap, i))
1383                         continue;
1384                 bmap = isl_basic_map_drop_div(bmap, i);
1385         }
1386         return bmap;
1387 }
1388
1389 struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
1390 {
1391         bmap = remove_redundant_divs(bmap);
1392         if (!bmap)
1393                 return NULL;
1394         ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
1395         return bmap;
1396 }
1397
1398 struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
1399 {
1400         return (struct isl_basic_set *)
1401                 isl_basic_map_finalize((struct isl_basic_map *)bset);
1402 }
1403
1404 struct isl_set *isl_set_finalize(struct isl_set *set)
1405 {
1406         int i;
1407
1408         if (!set)
1409                 return NULL;
1410         for (i = 0; i < set->n; ++i) {
1411                 set->p[i] = isl_basic_set_finalize(set->p[i]);
1412                 if (!set->p[i])
1413                         goto error;
1414         }
1415         return set;
1416 error:
1417         isl_set_free(set);
1418         return NULL;
1419 }
1420
1421 struct isl_map *isl_map_finalize(struct isl_map *map)
1422 {
1423         int i;
1424
1425         if (!map)
1426                 return NULL;
1427         for (i = 0; i < map->n; ++i) {
1428                 map->p[i] = isl_basic_map_finalize(map->p[i]);
1429                 if (!map->p[i])
1430                         goto error;
1431         }
1432         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
1433         return map;
1434 error:
1435         isl_map_free(map);
1436         return NULL;
1437 }
1438
1439
1440 /* Remove definition of any div that is defined in terms of the given variable.
1441  * The div itself is not removed.  Functions such as
1442  * eliminate_divs_ineq depend on the other divs remaining in place.
1443  */
1444 static struct isl_basic_map *remove_dependent_vars(struct isl_basic_map *bmap,
1445                                                                         int pos)
1446 {
1447         int i;
1448
1449         if (!bmap)
1450                 return NULL;
1451
1452         for (i = 0; i < bmap->n_div; ++i) {
1453                 if (isl_int_is_zero(bmap->div[i][0]))
1454                         continue;
1455                 if (isl_int_is_zero(bmap->div[i][1+1+pos]))
1456                         continue;
1457                 isl_int_set_si(bmap->div[i][0], 0);
1458         }
1459         return bmap;
1460 }
1461
1462 /* Eliminate the specified variables from the constraints using
1463  * Fourier-Motzkin.  The variables themselves are not removed.
1464  */
1465 struct isl_basic_map *isl_basic_map_eliminate_vars(
1466         struct isl_basic_map *bmap, unsigned pos, unsigned n)
1467 {
1468         int d;
1469         int i, j, k;
1470         unsigned total;
1471         int need_gauss = 0;
1472
1473         if (n == 0)
1474                 return bmap;
1475         if (!bmap)
1476                 return NULL;
1477         total = isl_basic_map_total_dim(bmap);
1478
1479         bmap = isl_basic_map_cow(bmap);
1480         for (d = pos + n - 1; d >= 0 && d >= pos; --d)
1481                 bmap = remove_dependent_vars(bmap, d);
1482         if (!bmap)
1483                 return NULL;
1484
1485         for (d = pos + n - 1;
1486              d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
1487                 isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
1488         for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
1489                 int n_lower, n_upper;
1490                 if (!bmap)
1491                         return NULL;
1492                 for (i = 0; i < bmap->n_eq; ++i) {
1493                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1494                                 continue;
1495                         eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL);
1496                         isl_basic_map_drop_equality(bmap, i);
1497                         need_gauss = 1;
1498                         break;
1499                 }
1500                 if (i < bmap->n_eq)
1501                         continue;
1502                 n_lower = 0;
1503                 n_upper = 0;
1504                 for (i = 0; i < bmap->n_ineq; ++i) {
1505                         if (isl_int_is_pos(bmap->ineq[i][1+d]))
1506                                 n_lower++;
1507                         else if (isl_int_is_neg(bmap->ineq[i][1+d]))
1508                                 n_upper++;
1509                 }
1510                 bmap = isl_basic_map_extend_constraints(bmap,
1511                                 0, n_lower * n_upper);
1512                 if (!bmap)
1513                         goto error;
1514                 for (i = bmap->n_ineq - 1; i >= 0; --i) {
1515                         int last;
1516                         if (isl_int_is_zero(bmap->ineq[i][1+d]))
1517                                 continue;
1518                         last = -1;
1519                         for (j = 0; j < i; ++j) {
1520                                 if (isl_int_is_zero(bmap->ineq[j][1+d]))
1521                                         continue;
1522                                 last = j;
1523                                 if (isl_int_sgn(bmap->ineq[i][1+d]) ==
1524                                     isl_int_sgn(bmap->ineq[j][1+d]))
1525                                         continue;
1526                                 k = isl_basic_map_alloc_inequality(bmap);
1527                                 if (k < 0)
1528                                         goto error;
1529                                 isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
1530                                                 1+total);
1531                                 isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
1532                                                 1+d, 1+total, NULL);
1533                         }
1534                         isl_basic_map_drop_inequality(bmap, i);
1535                         i = last + 1;
1536                 }
1537                 if (n_lower > 0 && n_upper > 0) {
1538                         bmap = isl_basic_map_normalize_constraints(bmap);
1539                         bmap = remove_duplicate_constraints(bmap, NULL, 0);
1540                         bmap = isl_basic_map_gauss(bmap, NULL);
1541                         bmap = isl_basic_map_remove_redundancies(bmap);
1542                         need_gauss = 0;
1543                         if (!bmap)
1544                                 goto error;
1545                         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1546                                 break;
1547                 }
1548         }
1549         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
1550         if (need_gauss)
1551                 bmap = isl_basic_map_gauss(bmap, NULL);
1552         return bmap;
1553 error:
1554         isl_basic_map_free(bmap);
1555         return NULL;
1556 }
1557
1558 struct isl_basic_set *isl_basic_set_eliminate_vars(
1559         struct isl_basic_set *bset, unsigned pos, unsigned n)
1560 {
1561         return (struct isl_basic_set *)isl_basic_map_eliminate_vars(
1562                         (struct isl_basic_map *)bset, pos, n);
1563 }
1564
1565 /* Eliminate the specified n dimensions starting at first from the
1566  * constraints, without removing the dimensions from the space.
1567  * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1568  * Otherwise, they are projected out and the original space is restored.
1569  */
1570 __isl_give isl_basic_map *isl_basic_map_eliminate(
1571         __isl_take isl_basic_map *bmap,
1572         enum isl_dim_type type, unsigned first, unsigned n)
1573 {
1574         isl_space *space;
1575
1576         if (!bmap)
1577                 return NULL;
1578         if (n == 0)
1579                 return bmap;
1580
1581         if (first + n > isl_basic_map_dim(bmap, type) || first + n < first)
1582                 isl_die(bmap->ctx, isl_error_invalid,
1583                         "index out of bounds", goto error);
1584
1585         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
1586                 first += isl_basic_map_offset(bmap, type) - 1;
1587                 bmap = isl_basic_map_eliminate_vars(bmap, first, n);
1588                 return isl_basic_map_finalize(bmap);
1589         }
1590
1591         space = isl_basic_map_get_space(bmap);
1592         bmap = isl_basic_map_project_out(bmap, type, first, n);
1593         bmap = isl_basic_map_insert_dims(bmap, type, first, n);
1594         bmap = isl_basic_map_reset_space(bmap, space);
1595         return bmap;
1596 error:
1597         isl_basic_map_free(bmap);
1598         return NULL;
1599 }
1600
1601 __isl_give isl_basic_set *isl_basic_set_eliminate(
1602         __isl_take isl_basic_set *bset,
1603         enum isl_dim_type type, unsigned first, unsigned n)
1604 {
1605         return isl_basic_map_eliminate(bset, type, first, n);
1606 }
1607
1608 /* Don't assume equalities are in order, because align_divs
1609  * may have changed the order of the divs.
1610  */
1611 static void compute_elimination_index(struct isl_basic_map *bmap, int *elim)
1612 {
1613         int d, i;
1614         unsigned total;
1615
1616         total = isl_space_dim(bmap->dim, isl_dim_all);
1617         for (d = 0; d < total; ++d)
1618                 elim[d] = -1;
1619         for (i = 0; i < bmap->n_eq; ++i) {
1620                 for (d = total - 1; d >= 0; --d) {
1621                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1622                                 continue;
1623                         elim[d] = i;
1624                         break;
1625                 }
1626         }
1627 }
1628
1629 static void set_compute_elimination_index(struct isl_basic_set *bset, int *elim)
1630 {
1631         compute_elimination_index((struct isl_basic_map *)bset, elim);
1632 }
1633
1634 static int reduced_using_equalities(isl_int *dst, isl_int *src,
1635         struct isl_basic_map *bmap, int *elim)
1636 {
1637         int d;
1638         int copied = 0;
1639         unsigned total;
1640
1641         total = isl_space_dim(bmap->dim, isl_dim_all);
1642         for (d = total - 1; d >= 0; --d) {
1643                 if (isl_int_is_zero(src[1+d]))
1644                         continue;
1645                 if (elim[d] == -1)
1646                         continue;
1647                 if (!copied) {
1648                         isl_seq_cpy(dst, src, 1 + total);
1649                         copied = 1;
1650                 }
1651                 isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
1652         }
1653         return copied;
1654 }
1655
1656 static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
1657         struct isl_basic_set *bset, int *elim)
1658 {
1659         return reduced_using_equalities(dst, src,
1660                                         (struct isl_basic_map *)bset, elim);
1661 }
1662
1663 static struct isl_basic_set *isl_basic_set_reduce_using_equalities(
1664         struct isl_basic_set *bset, struct isl_basic_set *context)
1665 {
1666         int i;
1667         int *elim;
1668
1669         if (!bset || !context)
1670                 goto error;
1671
1672         if (context->n_eq == 0) {
1673                 isl_basic_set_free(context);
1674                 return bset;
1675         }
1676
1677         bset = isl_basic_set_cow(bset);
1678         if (!bset)
1679                 goto error;
1680
1681         elim = isl_alloc_array(bset->ctx, int, isl_basic_set_n_dim(bset));
1682         if (!elim)
1683                 goto error;
1684         set_compute_elimination_index(context, elim);
1685         for (i = 0; i < bset->n_eq; ++i)
1686                 set_reduced_using_equalities(bset->eq[i], bset->eq[i],
1687                                                         context, elim);
1688         for (i = 0; i < bset->n_ineq; ++i)
1689                 set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
1690                                                         context, elim);
1691         isl_basic_set_free(context);
1692         free(elim);
1693         bset = isl_basic_set_simplify(bset);
1694         bset = isl_basic_set_finalize(bset);
1695         return bset;
1696 error:
1697         isl_basic_set_free(bset);
1698         isl_basic_set_free(context);
1699         return NULL;
1700 }
1701
1702 static struct isl_basic_set *remove_shifted_constraints(
1703         struct isl_basic_set *bset, struct isl_basic_set *context)
1704 {
1705         unsigned int size;
1706         isl_int ***index;
1707         int bits;
1708         int k, h, l;
1709         isl_ctx *ctx;
1710
1711         if (!bset)
1712                 return NULL;
1713
1714         size = round_up(4 * (context->n_ineq+1) / 3 - 1);
1715         bits = ffs(size) - 1;
1716         ctx = isl_basic_set_get_ctx(bset);
1717         index = isl_calloc_array(ctx, isl_int **, size);
1718         if (!index)
1719                 return bset;
1720
1721         for (k = 0; k < context->n_ineq; ++k) {
1722                 h = set_hash_index(index, size, bits, context, k);
1723                 index[h] = &context->ineq[k];
1724         }
1725         for (k = 0; k < bset->n_ineq; ++k) {
1726                 h = set_hash_index(index, size, bits, bset, k);
1727                 if (!index[h])
1728                         continue;
1729                 l = index[h] - &context->ineq[0];
1730                 if (isl_int_lt(bset->ineq[k][0], context->ineq[l][0]))
1731                         continue;
1732                 bset = isl_basic_set_cow(bset);
1733                 if (!bset)
1734                         goto error;
1735                 isl_basic_set_drop_inequality(bset, k);
1736                 --k;
1737         }
1738         free(index);
1739         return bset;
1740 error:
1741         free(index);
1742         return bset;
1743 }
1744
1745 /* Does the (linear part of a) constraint "c" involve any of the "len"
1746  * "relevant" dimensions?
1747  */
1748 static int is_related(isl_int *c, int len, int *relevant)
1749 {
1750         int i;
1751
1752         for (i = 0; i < len; ++i) {
1753                 if (!relevant[i])
1754                         continue;
1755                 if (!isl_int_is_zero(c[i]))
1756                         return 1;
1757         }
1758
1759         return 0;
1760 }
1761
1762 /* Drop constraints from "bset" that do not involve any of
1763  * the dimensions marked "relevant".
1764  */
1765 static __isl_give isl_basic_set *drop_unrelated_constraints(
1766         __isl_take isl_basic_set *bset, int *relevant)
1767 {
1768         int i, dim;
1769
1770         dim = isl_basic_set_dim(bset, isl_dim_set);
1771         for (i = 0; i < dim; ++i)
1772                 if (!relevant[i])
1773                         break;
1774         if (i >= dim)
1775                 return bset;
1776
1777         for (i = bset->n_eq - 1; i >= 0; --i)
1778                 if (!is_related(bset->eq[i] + 1, dim, relevant))
1779                         isl_basic_set_drop_equality(bset, i);
1780
1781         for (i = bset->n_ineq - 1; i >= 0; --i)
1782                 if (!is_related(bset->ineq[i] + 1, dim, relevant))
1783                         isl_basic_set_drop_inequality(bset, i);
1784
1785         return bset;
1786 }
1787
1788 /* Update the groups in "group" based on the (linear part of a) constraint "c".
1789  *
1790  * In particular, for any variable involved in the constraint,
1791  * find the actual group id from before and replace the group
1792  * of the corresponding variable by the minimal group of all
1793  * the variables involved in the constraint considered so far
1794  * (if this minimum is smaller) or replace the minimum by this group
1795  * (if the minimum is larger).
1796  *
1797  * At the end, all the variables in "c" will (indirectly) point
1798  * to the minimal of the groups that they referred to originally.
1799  */
1800 static void update_groups(int dim, int *group, isl_int *c)
1801 {
1802         int j;
1803         int min = dim;
1804
1805         for (j = 0; j < dim; ++j) {
1806                 if (isl_int_is_zero(c[j]))
1807                         continue;
1808                 while (group[j] >= 0 && group[group[j]] != group[j])
1809                         group[j] = group[group[j]];
1810                 if (group[j] == min)
1811                         continue;
1812                 if (group[j] < min) {
1813                         if (min >= 0 && min < dim)
1814                                 group[min] = group[j];
1815                         min = group[j];
1816                 } else
1817                         group[group[j]] = min;
1818         }
1819 }
1820
1821 /* Drop constraints from "context" that are irrelevant for computing
1822  * the gist of "bset".
1823  *
1824  * In particular, drop constraints in variables that are not related
1825  * to any of the variables involved in the constraints of "bset"
1826  * in the sense that there is no sequence of constraints that connects them.
1827  *
1828  * We construct groups of variables that collect variables that
1829  * (indirectly) appear in some common constraint of "context".
1830  * Each group is identified by the first variable in the group,
1831  * except for the special group of variables that appear in "bset"
1832  * (or are related to those variables), which is identified by -1.
1833  * If group[i] is equal to i (or -1), then the group of i is i (or -1),
1834  * otherwise the group of i is the group of group[i].
1835  *
1836  * We first initialize the -1 group with the variables that appear in "bset".
1837  * Then we initialize groups for the remaining variables.
1838  * Then we iterate over the constraints of "context" and update the
1839  * group of the variables in the constraint by the smallest group.
1840  * Finally, we resolve indirect references to groups by running over
1841  * the variables.
1842  *
1843  * After computing the groups, we drop constraints that do not involve
1844  * any variables in the -1 group.
1845  */
1846 static __isl_give isl_basic_set *drop_irrelevant_constraints(
1847         __isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset)
1848 {
1849         isl_ctx *ctx;
1850         int *group;
1851         int dim;
1852         int i, j;
1853         int last;
1854
1855         if (!context || !bset)
1856                 return isl_basic_set_free(context);
1857
1858         dim = isl_basic_set_dim(bset, isl_dim_set);
1859         ctx = isl_basic_set_get_ctx(bset);
1860         group = isl_calloc_array(ctx, int, dim);
1861
1862         if (!group)
1863                 goto error;
1864
1865         for (i = 0; i < dim; ++i) {
1866                 for (j = 0; j < bset->n_eq; ++j)
1867                         if (!isl_int_is_zero(bset->eq[j][1 + i]))
1868                                 break;
1869                 if (j < bset->n_eq) {
1870                         group[i] = -1;
1871                         continue;
1872                 }
1873                 for (j = 0; j < bset->n_ineq; ++j)
1874                         if (!isl_int_is_zero(bset->ineq[j][1 + i]))
1875                                 break;
1876                 if (j < bset->n_ineq)
1877                         group[i] = -1;
1878         }
1879
1880         last = -1;
1881         for (i = 0; i < dim; ++i)
1882                 if (group[i] >= 0)
1883                         last = group[i] = i;
1884         if (last < 0) {
1885                 free(group);
1886                 return context;
1887         }
1888
1889         for (i = 0; i < context->n_eq; ++i)
1890                 update_groups(dim, group, context->eq[i] + 1);
1891         for (i = 0; i < context->n_ineq; ++i)
1892                 update_groups(dim, group, context->ineq[i] + 1);
1893
1894         for (i = 0; i < dim; ++i)
1895                 if (group[i] >= 0)
1896                         group[i] = group[group[i]];
1897
1898         for (i = 0; i < dim; ++i)
1899                 group[i] = group[i] == -1;
1900
1901         context = drop_unrelated_constraints(context, group);
1902
1903         free(group);
1904         return context;
1905 error:
1906         free(group);
1907         return isl_basic_set_free(context);
1908 }
1909
1910 /* Remove all information from bset that is redundant in the context
1911  * of context.  Both bset and context are assumed to be full-dimensional.
1912  *
1913  * We first remove the inequalities from "bset"
1914  * that are obviously redundant with respect to some inequality in "context".
1915  * Then we remove those constraints from "context" that have become
1916  * irrelevant for computing the gist of "bset".
1917  * Note that this removal of constraints cannot be replaced by
1918  * a factorization because factors in "bset" may still be connected
1919  * to each other through constraints in "context".
1920  *
1921  * If there are any inequalities left, we construct a tableau for
1922  * the context and then add the inequalities of "bset".
1923  * Before adding these inequalities, we freeze all constraints such that
1924  * they won't be considered redundant in terms of the constraints of "bset".
1925  * Then we detect all redundant constraints (among the
1926  * constraints that weren't frozen), first by checking for redundancy in the
1927  * the tableau and then by checking if replacing a constraint by its negation
1928  * would lead to an empty set.  This last step is fairly expensive
1929  * and could be optimized by more reuse of the tableau.
1930  * Finally, we update bset according to the results.
1931  */
1932 static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset,
1933         __isl_take isl_basic_set *context)
1934 {
1935         int i, k;
1936         isl_basic_set *combined = NULL;
1937         struct isl_tab *tab = NULL;
1938         unsigned context_ineq;
1939         unsigned total;
1940
1941         if (!bset || !context)
1942                 goto error;
1943
1944         if (isl_basic_set_is_universe(bset)) {
1945                 isl_basic_set_free(context);
1946                 return bset;
1947         }
1948
1949         if (isl_basic_set_is_universe(context)) {
1950                 isl_basic_set_free(context);
1951                 return bset;
1952         }
1953
1954         bset = remove_shifted_constraints(bset, context);
1955         if (!bset)
1956                 goto error;
1957         if (bset->n_ineq == 0)
1958                 goto done;
1959
1960         context = drop_irrelevant_constraints(context, bset);
1961         if (!context)
1962                 goto error;
1963         if (isl_basic_set_is_universe(context)) {
1964                 isl_basic_set_free(context);
1965                 return bset;
1966         }
1967
1968         context_ineq = context->n_ineq;
1969         combined = isl_basic_set_cow(isl_basic_set_copy(context));
1970         combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq);
1971         tab = isl_tab_from_basic_set(combined, 0);
1972         for (i = 0; i < context_ineq; ++i)
1973                 if (isl_tab_freeze_constraint(tab, i) < 0)
1974                         goto error;
1975         tab = isl_tab_extend(tab, bset->n_ineq);
1976         for (i = 0; i < bset->n_ineq; ++i)
1977                 if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
1978                         goto error;
1979         bset = isl_basic_set_add_constraints(combined, bset, 0);
1980         combined = NULL;
1981         if (!bset)
1982                 goto error;
1983         if (isl_tab_detect_redundant(tab) < 0)
1984                 goto error;
1985         total = isl_basic_set_total_dim(bset);
1986         for (i = context_ineq; i < bset->n_ineq; ++i) {
1987                 int is_empty;
1988                 if (tab->con[i].is_redundant)
1989                         continue;
1990                 tab->con[i].is_redundant = 1;
1991                 combined = isl_basic_set_dup(bset);
1992                 combined = isl_basic_set_update_from_tab(combined, tab);
1993                 combined = isl_basic_set_extend_constraints(combined, 0, 1);
1994                 k = isl_basic_set_alloc_inequality(combined);
1995                 if (k < 0)
1996                         goto error;
1997                 isl_seq_neg(combined->ineq[k], bset->ineq[i], 1 + total);
1998                 isl_int_sub_ui(combined->ineq[k][0], combined->ineq[k][0], 1);
1999                 is_empty = isl_basic_set_is_empty(combined);
2000                 if (is_empty < 0)
2001                         goto error;
2002                 isl_basic_set_free(combined);
2003                 combined = NULL;
2004                 if (!is_empty)
2005                         tab->con[i].is_redundant = 0;
2006         }
2007         for (i = 0; i < context_ineq; ++i)
2008                 tab->con[i].is_redundant = 1;
2009         bset = isl_basic_set_update_from_tab(bset, tab);
2010         if (bset) {
2011                 ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
2012                 ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
2013         }
2014
2015         isl_tab_free(tab);
2016 done:
2017         bset = isl_basic_set_simplify(bset);
2018         bset = isl_basic_set_finalize(bset);
2019         isl_basic_set_free(context);
2020         return bset;
2021 error:
2022         isl_tab_free(tab);
2023         isl_basic_set_free(combined);
2024         isl_basic_set_free(context);
2025         isl_basic_set_free(bset);
2026         return NULL;
2027 }
2028
2029 /* Remove all information from bset that is redundant in the context
2030  * of context.  In particular, equalities that are linear combinations
2031  * of those in context are removed.  Then the inequalities that are
2032  * redundant in the context of the equalities and inequalities of
2033  * context are removed.
2034  *
2035  * First of all, we drop those constraints from "context"
2036  * that are irrelevant for computing the gist of "bset".
2037  * Alternatively, we could factorize the intersection of "context" and "bset".
2038  *
2039  * We first compute the integer affine hull of the intersection,
2040  * compute the gist inside this affine hull and then add back
2041  * those equalities that are not implied by the context.
2042  *
2043  * If two constraints are mutually redundant, then uset_gist_full
2044  * will remove the second of those constraints.  We therefore first
2045  * sort the constraints so that constraints not involving existentially
2046  * quantified variables are given precedence over those that do.
2047  * We have to perform this sorting before the variable compression,
2048  * because that may effect the order of the variables.
2049  */
2050 static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset,
2051         __isl_take isl_basic_set *context)
2052 {
2053         isl_mat *eq;
2054         isl_mat *T, *T2;
2055         isl_basic_set *aff;
2056         isl_basic_set *aff_context;
2057         unsigned total;
2058
2059         if (!bset || !context)
2060                 goto error;
2061
2062         context = drop_irrelevant_constraints(context, bset);
2063
2064         bset = isl_basic_set_intersect(bset, isl_basic_set_copy(context));
2065         if (isl_basic_set_plain_is_empty(bset)) {
2066                 isl_basic_set_free(context);
2067                 return bset;
2068         }
2069         bset = isl_basic_set_sort_constraints(bset);
2070         aff = isl_basic_set_affine_hull(isl_basic_set_copy(bset));
2071         if (!aff)
2072                 goto error;
2073         if (isl_basic_set_plain_is_empty(aff)) {
2074                 isl_basic_set_free(aff);
2075                 isl_basic_set_free(context);
2076                 return bset;
2077         }
2078         if (aff->n_eq == 0) {
2079                 isl_basic_set_free(aff);
2080                 return uset_gist_full(bset, context);
2081         }
2082         total = isl_basic_set_total_dim(bset);
2083         eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total);
2084         eq = isl_mat_cow(eq);
2085         T = isl_mat_variable_compression(eq, &T2);
2086         if (T && T->n_col == 0) {
2087                 isl_mat_free(T);
2088                 isl_mat_free(T2);
2089                 isl_basic_set_free(context);
2090                 isl_basic_set_free(aff);
2091                 return isl_basic_set_set_to_empty(bset);
2092         }
2093
2094         aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));
2095
2096         bset = isl_basic_set_preimage(bset, isl_mat_copy(T));
2097         context = isl_basic_set_preimage(context, T);
2098
2099         bset = uset_gist_full(bset, context);
2100         bset = isl_basic_set_preimage(bset, T2);
2101         bset = isl_basic_set_intersect(bset, aff);
2102         bset = isl_basic_set_reduce_using_equalities(bset, aff_context);
2103
2104         if (bset) {
2105                 ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
2106                 ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
2107         }
2108
2109         return bset;
2110 error:
2111         isl_basic_set_free(bset);
2112         isl_basic_set_free(context);
2113         return NULL;
2114 }
2115
2116 /* Normalize the divs in "bmap" in the context of the equalities in "context".
2117  * We simply add the equalities in context to bmap and then do a regular
2118  * div normalizations.  Better results can be obtained by normalizing
2119  * only the divs in bmap than do not also appear in context.
2120  * We need to be careful to reduce the divs using the equalities
2121  * so that later calls to isl_basic_map_overlying_set wouldn't introduce
2122  * spurious constraints.
2123  */
2124 static struct isl_basic_map *normalize_divs_in_context(
2125         struct isl_basic_map *bmap, struct isl_basic_map *context)
2126 {
2127         int i;
2128         unsigned total_context;
2129         int div_eq;
2130
2131         div_eq = n_pure_div_eq(bmap);
2132         if (div_eq == 0)
2133                 return bmap;
2134
2135         if (context->n_div > 0)
2136                 bmap = isl_basic_map_align_divs(bmap, context);
2137
2138         total_context = isl_basic_map_total_dim(context);
2139         bmap = isl_basic_map_extend_constraints(bmap, context->n_eq, 0);
2140         for (i = 0; i < context->n_eq; ++i) {
2141                 int k;
2142                 k = isl_basic_map_alloc_equality(bmap);
2143                 isl_seq_cpy(bmap->eq[k], context->eq[i], 1 + total_context);
2144                 isl_seq_clr(bmap->eq[k] + 1 + total_context,
2145                                 isl_basic_map_total_dim(bmap) - total_context);
2146         }
2147         bmap = isl_basic_map_gauss(bmap, NULL);
2148         bmap = normalize_divs(bmap, NULL);
2149         bmap = isl_basic_map_gauss(bmap, NULL);
2150         return bmap;
2151 }
2152
2153 struct isl_basic_map *isl_basic_map_gist(struct isl_basic_map *bmap,
2154         struct isl_basic_map *context)
2155 {
2156         struct isl_basic_set *bset;
2157
2158         if (!bmap || !context)
2159                 goto error;
2160
2161         if (isl_basic_map_is_universe(bmap)) {
2162                 isl_basic_map_free(context);
2163                 return bmap;
2164         }
2165         if (isl_basic_map_plain_is_empty(context)) {
2166                 isl_basic_map_free(bmap);
2167                 return context;
2168         }
2169         if (isl_basic_map_plain_is_empty(bmap)) {
2170                 isl_basic_map_free(context);
2171                 return bmap;
2172         }
2173
2174         bmap = isl_basic_map_remove_redundancies(bmap);
2175         context = isl_basic_map_remove_redundancies(context);
2176
2177         if (context->n_eq)
2178                 bmap = normalize_divs_in_context(bmap, context);
2179
2180         context = isl_basic_map_align_divs(context, bmap);
2181         bmap = isl_basic_map_align_divs(bmap, context);
2182
2183         bset = uset_gist(isl_basic_map_underlying_set(isl_basic_map_copy(bmap)),
2184                          isl_basic_map_underlying_set(context));
2185
2186         return isl_basic_map_overlying_set(bset, bmap);
2187 error:
2188         isl_basic_map_free(bmap);
2189         isl_basic_map_free(context);
2190         return NULL;
2191 }
2192
2193 /*
2194  * Assumes context has no implicit divs.
2195  */
2196 __isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
2197         __isl_take isl_basic_map *context)
2198 {
2199         int i;
2200
2201         if (!map || !context)
2202                 goto error;;
2203
2204         if (isl_basic_map_plain_is_empty(context)) {
2205                 isl_map_free(map);
2206                 return isl_map_from_basic_map(context);
2207         }
2208
2209         context = isl_basic_map_remove_redundancies(context);
2210         map = isl_map_cow(map);
2211         if (!map || !context)
2212                 goto error;;
2213         isl_assert(map->ctx, isl_space_is_equal(map->dim, context->dim), goto error);
2214         map = isl_map_compute_divs(map);
2215         for (i = 0; i < map->n; ++i)
2216                 context = isl_basic_map_align_divs(context, map->p[i]);
2217         for (i = map->n - 1; i >= 0; --i) {
2218                 map->p[i] = isl_basic_map_gist(map->p[i],
2219                                                 isl_basic_map_copy(context));
2220                 if (!map->p[i])
2221                         goto error;
2222                 if (isl_basic_map_plain_is_empty(map->p[i])) {
2223                         isl_basic_map_free(map->p[i]);
2224                         if (i != map->n - 1)
2225                                 map->p[i] = map->p[map->n - 1];
2226                         map->n--;
2227                 }
2228         }
2229         isl_basic_map_free(context);
2230         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
2231         return map;
2232 error:
2233         isl_map_free(map);
2234         isl_basic_map_free(context);
2235         return NULL;
2236 }
2237
2238 /* Return a map that has the same intersection with "context" as "map"
2239  * and that as "simple" as possible.
2240  *
2241  * If "map" is already the universe, then we cannot make it any simpler.
2242  * Similarly, if "context" is the universe, then we cannot exploit it
2243  * to simplify "map"
2244  * If "map" and "context" are identical to each other, then we can
2245  * return the corresponding universe.
2246  *
2247  * If none of these cases apply, we have to work a bit harder.
2248  */
2249 static __isl_give isl_map *map_gist(__isl_take isl_map *map,
2250         __isl_take isl_map *context)
2251 {
2252         int equal;
2253         int is_universe;
2254
2255         is_universe = isl_map_plain_is_universe(map);
2256         if (is_universe >= 0 && !is_universe)
2257                 is_universe = isl_map_plain_is_universe(context);
2258         if (is_universe < 0)
2259                 goto error;
2260         if (is_universe) {
2261                 isl_map_free(context);
2262                 return map;
2263         }
2264
2265         equal = isl_map_plain_is_equal(map, context);
2266         if (equal < 0)
2267                 goto error;
2268         if (equal) {
2269                 isl_map *res = isl_map_universe(isl_map_get_space(map));
2270                 isl_map_free(map);
2271                 isl_map_free(context);
2272                 return res;
2273         }
2274
2275         context = isl_map_compute_divs(context);
2276         return isl_map_gist_basic_map(map, isl_map_simple_hull(context));
2277 error:
2278         isl_map_free(map);
2279         isl_map_free(context);
2280         return NULL;
2281 }
2282
2283 __isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
2284         __isl_take isl_map *context)
2285 {
2286         return isl_map_align_params_map_map_and(map, context, &map_gist);
2287 }
2288
2289 struct isl_basic_set *isl_basic_set_gist(struct isl_basic_set *bset,
2290                                                 struct isl_basic_set *context)
2291 {
2292         return (struct isl_basic_set *)isl_basic_map_gist(
2293                 (struct isl_basic_map *)bset, (struct isl_basic_map *)context);
2294 }
2295
2296 __isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
2297         __isl_take isl_basic_set *context)
2298 {
2299         return (struct isl_set *)isl_map_gist_basic_map((struct isl_map *)set,
2300                                         (struct isl_basic_map *)context);
2301 }
2302
2303 __isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set,
2304         __isl_take isl_basic_set *context)
2305 {
2306         isl_space *space = isl_set_get_space(set);
2307         isl_basic_set *dom_context = isl_basic_set_universe(space);
2308         dom_context = isl_basic_set_intersect_params(dom_context, context);
2309         return isl_set_gist_basic_set(set, dom_context);
2310 }
2311
2312 __isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
2313         __isl_take isl_set *context)
2314 {
2315         return (struct isl_set *)isl_map_gist((struct isl_map *)set,
2316                                         (struct isl_map *)context);
2317 }
2318
2319 __isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map,
2320         __isl_take isl_set *context)
2321 {
2322         isl_map *map_context = isl_map_universe(isl_map_get_space(map));
2323         map_context = isl_map_intersect_domain(map_context, context);
2324         return isl_map_gist(map, map_context);
2325 }
2326
2327 __isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map,
2328         __isl_take isl_set *context)
2329 {
2330         isl_map *map_context = isl_map_universe(isl_map_get_space(map));
2331         map_context = isl_map_intersect_range(map_context, context);
2332         return isl_map_gist(map, map_context);
2333 }
2334
2335 __isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map,
2336         __isl_take isl_set *context)
2337 {
2338         isl_map *map_context = isl_map_universe(isl_map_get_space(map));
2339         map_context = isl_map_intersect_params(map_context, context);
2340         return isl_map_gist(map, map_context);
2341 }
2342
2343 __isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set,
2344         __isl_take isl_set *context)
2345 {
2346         return isl_map_gist_params(set, context);
2347 }
2348
2349 /* Quick check to see if two basic maps are disjoint.
2350  * In particular, we reduce the equalities and inequalities of
2351  * one basic map in the context of the equalities of the other
2352  * basic map and check if we get a contradiction.
2353  */
2354 int isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1,
2355         __isl_keep isl_basic_map *bmap2)
2356 {
2357         struct isl_vec *v = NULL;
2358         int *elim = NULL;
2359         unsigned total;
2360         int i;
2361
2362         if (!bmap1 || !bmap2)
2363                 return -1;
2364         isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
2365                         return -1);
2366         if (bmap1->n_div || bmap2->n_div)
2367                 return 0;
2368         if (!bmap1->n_eq && !bmap2->n_eq)
2369                 return 0;
2370
2371         total = isl_space_dim(bmap1->dim, isl_dim_all);
2372         if (total == 0)
2373                 return 0;
2374         v = isl_vec_alloc(bmap1->ctx, 1 + total);
2375         if (!v)
2376                 goto error;
2377         elim = isl_alloc_array(bmap1->ctx, int, total);
2378         if (!elim)
2379                 goto error;
2380         compute_elimination_index(bmap1, elim);
2381         for (i = 0; i < bmap2->n_eq; ++i) {
2382                 int reduced;
2383                 reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
2384                                                         bmap1, elim);
2385                 if (reduced && !isl_int_is_zero(v->block.data[0]) &&
2386                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
2387                         goto disjoint;
2388         }
2389         for (i = 0; i < bmap2->n_ineq; ++i) {
2390                 int reduced;
2391                 reduced = reduced_using_equalities(v->block.data,
2392                                                 bmap2->ineq[i], bmap1, elim);
2393                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
2394                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
2395                         goto disjoint;
2396         }
2397         compute_elimination_index(bmap2, elim);
2398         for (i = 0; i < bmap1->n_ineq; ++i) {
2399                 int reduced;
2400                 reduced = reduced_using_equalities(v->block.data,
2401                                                 bmap1->ineq[i], bmap2, elim);
2402                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
2403                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
2404                         goto disjoint;
2405         }
2406         isl_vec_free(v);
2407         free(elim);
2408         return 0;
2409 disjoint:
2410         isl_vec_free(v);
2411         free(elim);
2412         return 1;
2413 error:
2414         isl_vec_free(v);
2415         free(elim);
2416         return -1;
2417 }
2418
2419 int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1,
2420         __isl_keep isl_basic_set *bset2)
2421 {
2422         return isl_basic_map_plain_is_disjoint((struct isl_basic_map *)bset1,
2423                                               (struct isl_basic_map *)bset2);
2424 }
2425
2426 /* Are "map1" and "map2" obviously disjoint?
2427  *
2428  * If one of them is empty or if they live in different spaces (ignoring
2429  * parameters), then they are clearly disjoint.
2430  *
2431  * If they have different parameters, then we skip any further tests.
2432  *
2433  * If they are obviously equal, but not obviously empty, then we will
2434  * not be able to detect if they are disjoint.
2435  *
2436  * Otherwise we check if each basic map in "map1" is obviously disjoint
2437  * from each basic map in "map2".
2438  */
2439 int isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
2440         __isl_keep isl_map *map2)
2441 {
2442         int i, j;
2443         int disjoint;
2444         int intersect;
2445         int match;
2446
2447         if (!map1 || !map2)
2448                 return -1;
2449
2450         disjoint = isl_map_plain_is_empty(map1);
2451         if (disjoint < 0 || disjoint)
2452                 return disjoint;
2453
2454         disjoint = isl_map_plain_is_empty(map2);
2455         if (disjoint < 0 || disjoint)
2456                 return disjoint;
2457
2458         match = isl_space_tuple_match(map1->dim, isl_dim_in,
2459                                 map2->dim, isl_dim_in);
2460         if (match < 0 || !match)
2461                 return match < 0 ? -1 : 1;
2462
2463         match = isl_space_tuple_match(map1->dim, isl_dim_out,
2464                                 map2->dim, isl_dim_out);
2465         if (match < 0 || !match)
2466                 return match < 0 ? -1 : 1;
2467
2468         match = isl_space_match(map1->dim, isl_dim_param,
2469                                 map2->dim, isl_dim_param);
2470         if (match < 0 || !match)
2471                 return match < 0 ? -1 : 0;
2472
2473         intersect = isl_map_plain_is_equal(map1, map2);
2474         if (intersect < 0 || intersect)
2475                 return intersect < 0 ? -1 : 0;
2476
2477         for (i = 0; i < map1->n; ++i) {
2478                 for (j = 0; j < map2->n; ++j) {
2479                         int d = isl_basic_map_plain_is_disjoint(map1->p[i],
2480                                                                map2->p[j]);
2481                         if (d != 1)
2482                                 return d;
2483                 }
2484         }
2485         return 1;
2486 }
2487
2488 /* Are "map1" and "map2" disjoint?
2489  *
2490  * They are disjoint if they are "obviously disjoint" or if one of them
2491  * is empty.  Otherwise, they are not disjoint if one of them is universal.
2492  * If none of these cases apply, we compute the intersection and see if
2493  * the result is empty.
2494  */
2495 int isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
2496 {
2497         int disjoint;
2498         int intersect;
2499         isl_map *test;
2500
2501         disjoint = isl_map_plain_is_disjoint(map1, map2);
2502         if (disjoint < 0 || disjoint)
2503                 return disjoint;
2504
2505         disjoint = isl_map_is_empty(map1);
2506         if (disjoint < 0 || disjoint)
2507                 return disjoint;
2508
2509         disjoint = isl_map_is_empty(map2);
2510         if (disjoint < 0 || disjoint)
2511                 return disjoint;
2512
2513         intersect = isl_map_plain_is_universe(map1);
2514         if (intersect < 0 || intersect)
2515                 return intersect < 0 ? -1 : 0;
2516
2517         intersect = isl_map_plain_is_universe(map2);
2518         if (intersect < 0 || intersect)
2519                 return intersect < 0 ? -1 : 0;
2520
2521         test = isl_map_intersect(isl_map_copy(map1), isl_map_copy(map2));
2522         disjoint = isl_map_is_empty(test);
2523         isl_map_free(test);
2524
2525         return disjoint;
2526 }
2527
2528 int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
2529         __isl_keep isl_set *set2)
2530 {
2531         return isl_map_plain_is_disjoint((struct isl_map *)set1,
2532                                         (struct isl_map *)set2);
2533 }
2534
2535 /* Are "set1" and "set2" disjoint?
2536  */
2537 int isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
2538 {
2539         return isl_map_is_disjoint(set1, set2);
2540 }
2541
2542 int isl_set_fast_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
2543 {
2544         return isl_set_plain_is_disjoint(set1, set2);
2545 }
2546
2547 /* Check if we can combine a given div with lower bound l and upper
2548  * bound u with some other div and if so return that other div.
2549  * Otherwise return -1.
2550  *
2551  * We first check that
2552  *      - the bounds are opposites of each other (except for the constant
2553  *        term)
2554  *      - the bounds do not reference any other div
2555  *      - no div is defined in terms of this div
2556  *
2557  * Let m be the size of the range allowed on the div by the bounds.
2558  * That is, the bounds are of the form
2559  *
2560  *      e <= a <= e + m - 1
2561  *
2562  * with e some expression in the other variables.
2563  * We look for another div b such that no third div is defined in terms
2564  * of this second div b and such that in any constraint that contains
2565  * a (except for the given lower and upper bound), also contains b
2566  * with a coefficient that is m times that of b.
2567  * That is, all constraints (execpt for the lower and upper bound)
2568  * are of the form
2569  *
2570  *      e + f (a + m b) >= 0
2571  *
2572  * If so, we return b so that "a + m b" can be replaced by
2573  * a single div "c = a + m b".
2574  */
2575 static int div_find_coalesce(struct isl_basic_map *bmap, int *pairs,
2576         unsigned div, unsigned l, unsigned u)
2577 {
2578         int i, j;
2579         unsigned dim;
2580         int coalesce = -1;
2581
2582         if (bmap->n_div <= 1)
2583                 return -1;
2584         dim = isl_space_dim(bmap->dim, isl_dim_all);
2585         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim, div) != -1)
2586                 return -1;
2587         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim + div + 1,
2588                                    bmap->n_div - div - 1) != -1)
2589                 return -1;
2590         if (!isl_seq_is_neg(bmap->ineq[l] + 1, bmap->ineq[u] + 1,
2591                             dim + bmap->n_div))
2592                 return -1;
2593
2594         for (i = 0; i < bmap->n_div; ++i) {
2595                 if (isl_int_is_zero(bmap->div[i][0]))
2596                         continue;
2597                 if (!isl_int_is_zero(bmap->div[i][1 + 1 + dim + div]))
2598                         return -1;
2599         }
2600
2601         isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2602         if (isl_int_is_neg(bmap->ineq[l][0])) {
2603                 isl_int_sub(bmap->ineq[l][0],
2604                             bmap->ineq[l][0], bmap->ineq[u][0]);
2605                 bmap = isl_basic_map_copy(bmap);
2606                 bmap = isl_basic_map_set_to_empty(bmap);
2607                 isl_basic_map_free(bmap);
2608                 return -1;
2609         }
2610         isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2611         for (i = 0; i < bmap->n_div; ++i) {
2612                 if (i == div)
2613                         continue;
2614                 if (!pairs[i])
2615                         continue;
2616                 for (j = 0; j < bmap->n_div; ++j) {
2617                         if (isl_int_is_zero(bmap->div[j][0]))
2618                                 continue;
2619                         if (!isl_int_is_zero(bmap->div[j][1 + 1 + dim + i]))
2620                                 break;
2621                 }
2622                 if (j < bmap->n_div)
2623                         continue;
2624                 for (j = 0; j < bmap->n_ineq; ++j) {
2625                         int valid;
2626                         if (j == l || j == u)
2627                                 continue;
2628                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + div]))
2629                                 continue;
2630                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + i]))
2631                                 break;
2632                         isl_int_mul(bmap->ineq[j][1 + dim + div],
2633                                     bmap->ineq[j][1 + dim + div],
2634                                     bmap->ineq[l][0]);
2635                         valid = isl_int_eq(bmap->ineq[j][1 + dim + div],
2636                                            bmap->ineq[j][1 + dim + i]);
2637                         isl_int_divexact(bmap->ineq[j][1 + dim + div],
2638                                          bmap->ineq[j][1 + dim + div],
2639                                          bmap->ineq[l][0]);
2640                         if (!valid)
2641                                 break;
2642                 }
2643                 if (j < bmap->n_ineq)
2644                         continue;
2645                 coalesce = i;
2646                 break;
2647         }
2648         isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2649         isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2650         return coalesce;
2651 }
2652
2653 /* Given a lower and an upper bound on div i, construct an inequality
2654  * that when nonnegative ensures that this pair of bounds always allows
2655  * for an integer value of the given div.
2656  * The lower bound is inequality l, while the upper bound is inequality u.
2657  * The constructed inequality is stored in ineq.
2658  * g, fl, fu are temporary scalars.
2659  *
2660  * Let the upper bound be
2661  *
2662  *      -n_u a + e_u >= 0
2663  *
2664  * and the lower bound
2665  *
2666  *      n_l a + e_l >= 0
2667  *
2668  * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
2669  * We have
2670  *
2671  *      - f_u e_l <= f_u f_l g a <= f_l e_u
2672  *
2673  * Since all variables are integer valued, this is equivalent to
2674  *
2675  *      - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
2676  *
2677  * If this interval is at least f_u f_l g, then it contains at least
2678  * one integer value for a.
2679  * That is, the test constraint is
2680  *
2681  *      f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
2682  */
2683 static void construct_test_ineq(struct isl_basic_map *bmap, int i,
2684         int l, int u, isl_int *ineq, isl_int g, isl_int fl, isl_int fu)
2685 {
2686         unsigned dim;
2687         dim = isl_space_dim(bmap->dim, isl_dim_all);
2688
2689         isl_int_gcd(g, bmap->ineq[l][1 + dim + i], bmap->ineq[u][1 + dim + i]);
2690         isl_int_divexact(fl, bmap->ineq[l][1 + dim + i], g);
2691         isl_int_divexact(fu, bmap->ineq[u][1 + dim + i], g);
2692         isl_int_neg(fu, fu);
2693         isl_seq_combine(ineq, fl, bmap->ineq[u], fu, bmap->ineq[l],
2694                         1 + dim + bmap->n_div);
2695         isl_int_add(ineq[0], ineq[0], fl);
2696         isl_int_add(ineq[0], ineq[0], fu);
2697         isl_int_sub_ui(ineq[0], ineq[0], 1);
2698         isl_int_mul(g, g, fl);
2699         isl_int_mul(g, g, fu);
2700         isl_int_sub(ineq[0], ineq[0], g);
2701 }
2702
2703 /* Remove more kinds of divs that are not strictly needed.
2704  * In particular, if all pairs of lower and upper bounds on a div
2705  * are such that they allow at least one integer value of the div,
2706  * the we can eliminate the div using Fourier-Motzkin without
2707  * introducing any spurious solutions.
2708  */
2709 static struct isl_basic_map *drop_more_redundant_divs(
2710         struct isl_basic_map *bmap, int *pairs, int n)
2711 {
2712         struct isl_tab *tab = NULL;
2713         struct isl_vec *vec = NULL;
2714         unsigned dim;
2715         int remove = -1;
2716         isl_int g, fl, fu;
2717
2718         isl_int_init(g);
2719         isl_int_init(fl);
2720         isl_int_init(fu);
2721
2722         if (!bmap)
2723                 goto error;
2724
2725         dim = isl_space_dim(bmap->dim, isl_dim_all);
2726         vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
2727         if (!vec)
2728                 goto error;
2729
2730         tab = isl_tab_from_basic_map(bmap, 0);
2731
2732         while (n > 0) {
2733                 int i, l, u;
2734                 int best = -1;
2735                 enum isl_lp_result res;
2736
2737                 for (i = 0; i < bmap->n_div; ++i) {
2738                         if (!pairs[i])
2739                                 continue;
2740                         if (best >= 0 && pairs[best] <= pairs[i])
2741                                 continue;
2742                         best = i;
2743                 }
2744
2745                 i = best;
2746                 for (l = 0; l < bmap->n_ineq; ++l) {
2747                         if (!isl_int_is_pos(bmap->ineq[l][1 + dim + i]))
2748                                 continue;
2749                         for (u = 0; u < bmap->n_ineq; ++u) {
2750                                 if (!isl_int_is_neg(bmap->ineq[u][1 + dim + i]))
2751                                         continue;
2752                                 construct_test_ineq(bmap, i, l, u,
2753                                                     vec->el, g, fl, fu);
2754                                 res = isl_tab_min(tab, vec->el,
2755                                                   bmap->ctx->one, &g, NULL, 0);
2756                                 if (res == isl_lp_error)
2757                                         goto error;
2758                                 if (res == isl_lp_empty) {
2759                                         bmap = isl_basic_map_set_to_empty(bmap);
2760                                         break;
2761                                 }
2762                                 if (res != isl_lp_ok || isl_int_is_neg(g))
2763                                         break;
2764                         }
2765                         if (u < bmap->n_ineq)
2766                                 break;
2767                 }
2768                 if (l == bmap->n_ineq) {
2769                         remove = i;
2770                         break;
2771                 }
2772                 pairs[i] = 0;
2773                 --n;
2774         }
2775
2776         isl_tab_free(tab);
2777         isl_vec_free(vec);
2778
2779         isl_int_clear(g);
2780         isl_int_clear(fl);
2781         isl_int_clear(fu);
2782
2783         free(pairs);
2784
2785         if (remove < 0)
2786                 return bmap;
2787
2788         bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1);
2789         return isl_basic_map_drop_redundant_divs(bmap);
2790 error:
2791         free(pairs);
2792         isl_basic_map_free(bmap);
2793         isl_tab_free(tab);
2794         isl_vec_free(vec);
2795         isl_int_clear(g);
2796         isl_int_clear(fl);
2797         isl_int_clear(fu);
2798         return NULL;
2799 }
2800
2801 /* Given a pair of divs div1 and div2 such that, expect for the lower bound l
2802  * and the upper bound u, div1 always occurs together with div2 in the form
2803  * (div1 + m div2), where m is the constant range on the variable div1
2804  * allowed by l and u, replace the pair div1 and div2 by a single
2805  * div that is equal to div1 + m div2.
2806  *
2807  * The new div will appear in the location that contains div2.
2808  * We need to modify all constraints that contain
2809  * div2 = (div - div1) / m
2810  * (If a constraint does not contain div2, it will also not contain div1.)
2811  * If the constraint also contains div1, then we know they appear
2812  * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
2813  * i.e., the coefficient of div is f.
2814  *
2815  * Otherwise, we first need to introduce div1 into the constraint.
2816  * Let the l be
2817  *
2818  *      div1 + f >=0
2819  *
2820  * and u
2821  *
2822  *      -div1 + f' >= 0
2823  *
2824  * A lower bound on div2
2825  *
2826  *      n div2 + t >= 0
2827  *
2828  * can be replaced by
2829  *
2830  *      (n * (m div 2 + div1) + m t + n f)/g >= 0
2831  *
2832  * with g = gcd(m,n).
2833  * An upper bound
2834  *
2835  *      -n div2 + t >= 0
2836  *
2837  * can be replaced by
2838  *
2839  *      (-n * (m div2 + div1) + m t + n f')/g >= 0
2840  *
2841  * These constraint are those that we would obtain from eliminating
2842  * div1 using Fourier-Motzkin.
2843  *
2844  * After all constraints have been modified, we drop the lower and upper
2845  * bound and then drop div1.
2846  */
2847 static struct isl_basic_map *coalesce_divs(struct isl_basic_map *bmap,
2848         unsigned div1, unsigned div2, unsigned l, unsigned u)
2849 {
2850         isl_int a;
2851         isl_int b;
2852         isl_int m;
2853         unsigned dim, total;
2854         int i;
2855
2856         dim = isl_space_dim(bmap->dim, isl_dim_all);
2857         total = 1 + dim + bmap->n_div;
2858
2859         isl_int_init(a);
2860         isl_int_init(b);
2861         isl_int_init(m);
2862         isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
2863         isl_int_add_ui(m, m, 1);
2864
2865         for (i = 0; i < bmap->n_ineq; ++i) {
2866                 if (i == l || i == u)
2867                         continue;
2868                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div2]))
2869                         continue;
2870                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div1])) {
2871                         isl_int_gcd(b, m, bmap->ineq[i][1 + dim + div2]);
2872                         isl_int_divexact(a, m, b);
2873                         isl_int_divexact(b, bmap->ineq[i][1 + dim + div2], b);
2874                         if (isl_int_is_pos(b)) {
2875                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2876                                                 b, bmap->ineq[l], total);
2877                         } else {
2878                                 isl_int_neg(b, b);
2879                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2880                                                 b, bmap->ineq[u], total);
2881                         }
2882                 }
2883                 isl_int_set(bmap->ineq[i][1 + dim + div2],
2884                             bmap->ineq[i][1 + dim + div1]);
2885                 isl_int_set_si(bmap->ineq[i][1 + dim + div1], 0);
2886         }
2887
2888         isl_int_clear(a);
2889         isl_int_clear(b);
2890         isl_int_clear(m);
2891         if (l > u) {
2892                 isl_basic_map_drop_inequality(bmap, l);
2893                 isl_basic_map_drop_inequality(bmap, u);
2894         } else {
2895                 isl_basic_map_drop_inequality(bmap, u);
2896                 isl_basic_map_drop_inequality(bmap, l);
2897         }
2898         bmap = isl_basic_map_drop_div(bmap, div1);
2899         return bmap;
2900 }
2901
2902 /* First check if we can coalesce any pair of divs and
2903  * then continue with dropping more redundant divs.
2904  *
2905  * We loop over all pairs of lower and upper bounds on a div
2906  * with coefficient 1 and -1, respectively, check if there
2907  * is any other div "c" with which we can coalesce the div
2908  * and if so, perform the coalescing.
2909  */
2910 static struct isl_basic_map *coalesce_or_drop_more_redundant_divs(
2911         struct isl_basic_map *bmap, int *pairs, int n)
2912 {
2913         int i, l, u;
2914         unsigned dim;
2915
2916         dim = isl_space_dim(bmap->dim, isl_dim_all);
2917
2918         for (i = 0; i < bmap->n_div; ++i) {
2919                 if (!pairs[i])
2920                         continue;
2921                 for (l = 0; l < bmap->n_ineq; ++l) {
2922                         if (!isl_int_is_one(bmap->ineq[l][1 + dim + i]))
2923                                 continue;
2924                         for (u = 0; u < bmap->n_ineq; ++u) {
2925                                 int c;
2926
2927                                 if (!isl_int_is_negone(bmap->ineq[u][1+dim+i]))
2928                                         continue;
2929                                 c = div_find_coalesce(bmap, pairs, i, l, u);
2930                                 if (c < 0)
2931                                         continue;
2932                                 free(pairs);
2933                                 bmap = coalesce_divs(bmap, i, c, l, u);
2934                                 return isl_basic_map_drop_redundant_divs(bmap);
2935                         }
2936                 }
2937         }
2938
2939         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
2940                 return bmap;
2941
2942         return drop_more_redundant_divs(bmap, pairs, n);
2943 }
2944
2945 /* Remove divs that are not strictly needed.
2946  * In particular, if a div only occurs positively (or negatively)
2947  * in constraints, then it can simply be dropped.
2948  * Also, if a div occurs in only two constraints and if moreover
2949  * those two constraints are opposite to each other, except for the constant
2950  * term and if the sum of the constant terms is such that for any value
2951  * of the other values, there is always at least one integer value of the
2952  * div, i.e., if one plus this sum is greater than or equal to
2953  * the (absolute value) of the coefficent of the div in the constraints,
2954  * then we can also simply drop the div.
2955  *
2956  * We skip divs that appear in equalities or in the definition of other divs.
2957  * Divs that appear in the definition of other divs usually occur in at least
2958  * 4 constraints, but the constraints may have been simplified.
2959  *
2960  * If any divs are left after these simple checks then we move on
2961  * to more complicated cases in drop_more_redundant_divs.
2962  */
2963 struct isl_basic_map *isl_basic_map_drop_redundant_divs(
2964         struct isl_basic_map *bmap)
2965 {
2966         int i, j;
2967         unsigned off;
2968         int *pairs = NULL;
2969         int n = 0;
2970
2971         if (!bmap)
2972                 goto error;
2973
2974         off = isl_space_dim(bmap->dim, isl_dim_all);
2975         pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
2976         if (!pairs)
2977                 goto error;
2978
2979         for (i = 0; i < bmap->n_div; ++i) {
2980                 int pos, neg;
2981                 int last_pos, last_neg;
2982                 int redundant;
2983                 int defined;
2984
2985                 defined = !isl_int_is_zero(bmap->div[i][0]);
2986                 for (j = i; j < bmap->n_div; ++j)
2987                         if (!isl_int_is_zero(bmap->div[j][1 + 1 + off + i]))
2988                                 break;
2989                 if (j < bmap->n_div)
2990                         continue;
2991                 for (j = 0; j < bmap->n_eq; ++j)
2992                         if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
2993                                 break;
2994                 if (j < bmap->n_eq)
2995                         continue;
2996                 ++n;
2997                 pos = neg = 0;
2998                 for (j = 0; j < bmap->n_ineq; ++j) {
2999                         if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
3000                                 last_pos = j;
3001                                 ++pos;
3002                         }
3003                         if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
3004                                 last_neg = j;
3005                                 ++neg;
3006                         }
3007                 }
3008                 pairs[i] = pos * neg;
3009                 if (pairs[i] == 0) {
3010                         for (j = bmap->n_ineq - 1; j >= 0; --j)
3011                                 if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
3012                                         isl_basic_map_drop_inequality(bmap, j);
3013                         bmap = isl_basic_map_drop_div(bmap, i);
3014                         free(pairs);
3015                         return isl_basic_map_drop_redundant_divs(bmap);
3016                 }
3017                 if (pairs[i] != 1)
3018                         continue;
3019                 if (!isl_seq_is_neg(bmap->ineq[last_pos] + 1,
3020                                     bmap->ineq[last_neg] + 1,
3021                                     off + bmap->n_div))
3022                         continue;
3023
3024                 isl_int_add(bmap->ineq[last_pos][0],
3025                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
3026                 isl_int_add_ui(bmap->ineq[last_pos][0],
3027                                bmap->ineq[last_pos][0], 1);
3028                 redundant = isl_int_ge(bmap->ineq[last_pos][0],
3029                                 bmap->ineq[last_pos][1+off+i]);
3030                 isl_int_sub_ui(bmap->ineq[last_pos][0],
3031                                bmap->ineq[last_pos][0], 1);
3032                 isl_int_sub(bmap->ineq[last_pos][0],
3033                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
3034                 if (!redundant) {
3035                         if (defined ||
3036                             !ok_to_set_div_from_bound(bmap, i, last_pos)) {
3037                                 pairs[i] = 0;
3038                                 --n;
3039                                 continue;
3040                         }
3041                         bmap = set_div_from_lower_bound(bmap, i, last_pos);
3042                         bmap = isl_basic_map_simplify(bmap);
3043                         free(pairs);
3044                         return isl_basic_map_drop_redundant_divs(bmap);
3045                 }
3046                 if (last_pos > last_neg) {
3047                         isl_basic_map_drop_inequality(bmap, last_pos);
3048                         isl_basic_map_drop_inequality(bmap, last_neg);
3049                 } else {
3050                         isl_basic_map_drop_inequality(bmap, last_neg);
3051                         isl_basic_map_drop_inequality(bmap, last_pos);
3052                 }
3053                 bmap = isl_basic_map_drop_div(bmap, i);
3054                 free(pairs);
3055                 return isl_basic_map_drop_redundant_divs(bmap);
3056         }
3057
3058         if (n > 0)
3059                 return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);
3060
3061         free(pairs);
3062         return bmap;
3063 error:
3064         free(pairs);
3065         isl_basic_map_free(bmap);
3066         return NULL;
3067 }
3068
3069 struct isl_basic_set *isl_basic_set_drop_redundant_divs(
3070         struct isl_basic_set *bset)
3071 {
3072         return (struct isl_basic_set *)
3073             isl_basic_map_drop_redundant_divs((struct isl_basic_map *)bset);
3074 }
3075
3076 struct isl_map *isl_map_drop_redundant_divs(struct isl_map *map)
3077 {
3078         int i;
3079
3080         if (!map)
3081                 return NULL;
3082         for (i = 0; i < map->n; ++i) {
3083                 map->p[i] = isl_basic_map_drop_redundant_divs(map->p[i]);
3084                 if (!map->p[i])
3085                         goto error;
3086         }
3087         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
3088         return map;
3089 error:
3090         isl_map_free(map);
3091         return NULL;
3092 }
3093
3094 struct isl_set *isl_set_drop_redundant_divs(struct isl_set *set)
3095 {
3096         return (struct isl_set *)
3097             isl_map_drop_redundant_divs((struct isl_map *)set);
3098 }