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                 if (!bmap)
1275                         break;
1276                 if (isl_basic_map_plain_is_empty(bmap))
1277                         break;
1278                 bmap = isl_basic_map_normalize_constraints(bmap);
1279                 bmap = normalize_div_expressions(bmap);
1280                 bmap = remove_duplicate_divs(bmap, &progress);
1281                 bmap = eliminate_unit_divs(bmap, &progress);
1282                 bmap = eliminate_divs_eq(bmap, &progress);
1283                 bmap = eliminate_divs_ineq(bmap, &progress);
1284                 bmap = isl_basic_map_gauss(bmap, &progress);
1285                 /* requires equalities in normal form */
1286                 bmap = normalize_divs(bmap, &progress);
1287                 bmap = remove_duplicate_constraints(bmap, &progress, 1);
1288         }
1289         return bmap;
1290 }
1291
1292 struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset)
1293 {
1294         return (struct isl_basic_set *)
1295                 isl_basic_map_simplify((struct isl_basic_map *)bset);
1296 }
1297
1298
1299 int isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
1300         isl_int *constraint, unsigned div)
1301 {
1302         unsigned pos;
1303
1304         if (!bmap)
1305                 return -1;
1306
1307         pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
1308
1309         if (isl_int_eq(constraint[pos], bmap->div[div][0])) {
1310                 int neg;
1311                 isl_int_sub(bmap->div[div][1],
1312                                 bmap->div[div][1], bmap->div[div][0]);
1313                 isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
1314                 neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos);
1315                 isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
1316                 isl_int_add(bmap->div[div][1],
1317                                 bmap->div[div][1], bmap->div[div][0]);
1318                 if (!neg)
1319                         return 0;
1320                 if (isl_seq_first_non_zero(constraint+pos+1,
1321                                             bmap->n_div-div-1) != -1)
1322                         return 0;
1323         } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
1324                 if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
1325                         return 0;
1326                 if (isl_seq_first_non_zero(constraint+pos+1,
1327                                             bmap->n_div-div-1) != -1)
1328                         return 0;
1329         } else
1330                 return 0;
1331
1332         return 1;
1333 }
1334
1335 int isl_basic_set_is_div_constraint(__isl_keep isl_basic_set *bset,
1336         isl_int *constraint, unsigned div)
1337 {
1338         return isl_basic_map_is_div_constraint(bset, constraint, div);
1339 }
1340
1341
1342 /* If the only constraints a div d=floor(f/m)
1343  * appears in are its two defining constraints
1344  *
1345  *      f - m d >=0
1346  *      -(f - (m - 1)) + m d >= 0
1347  *
1348  * then it can safely be removed.
1349  */
1350 static int div_is_redundant(struct isl_basic_map *bmap, int div)
1351 {
1352         int i;
1353         unsigned pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;
1354
1355         for (i = 0; i < bmap->n_eq; ++i)
1356                 if (!isl_int_is_zero(bmap->eq[i][pos]))
1357                         return 0;
1358
1359         for (i = 0; i < bmap->n_ineq; ++i) {
1360                 if (isl_int_is_zero(bmap->ineq[i][pos]))
1361                         continue;
1362                 if (!isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div))
1363                         return 0;
1364         }
1365
1366         for (i = 0; i < bmap->n_div; ++i)
1367                 if (!isl_int_is_zero(bmap->div[i][1+pos]))
1368                         return 0;
1369
1370         return 1;
1371 }
1372
1373 /*
1374  * Remove divs that don't occur in any of the constraints or other divs.
1375  * These can arise when dropping some of the variables in a quast
1376  * returned by piplib.
1377  */
1378 static struct isl_basic_map *remove_redundant_divs(struct isl_basic_map *bmap)
1379 {
1380         int i;
1381
1382         if (!bmap)
1383                 return NULL;
1384
1385         for (i = bmap->n_div-1; i >= 0; --i) {
1386                 if (!div_is_redundant(bmap, i))
1387                         continue;
1388                 bmap = isl_basic_map_drop_div(bmap, i);
1389         }
1390         return bmap;
1391 }
1392
1393 struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
1394 {
1395         bmap = remove_redundant_divs(bmap);
1396         if (!bmap)
1397                 return NULL;
1398         ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
1399         return bmap;
1400 }
1401
1402 struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
1403 {
1404         return (struct isl_basic_set *)
1405                 isl_basic_map_finalize((struct isl_basic_map *)bset);
1406 }
1407
1408 struct isl_set *isl_set_finalize(struct isl_set *set)
1409 {
1410         int i;
1411
1412         if (!set)
1413                 return NULL;
1414         for (i = 0; i < set->n; ++i) {
1415                 set->p[i] = isl_basic_set_finalize(set->p[i]);
1416                 if (!set->p[i])
1417                         goto error;
1418         }
1419         return set;
1420 error:
1421         isl_set_free(set);
1422         return NULL;
1423 }
1424
1425 struct isl_map *isl_map_finalize(struct isl_map *map)
1426 {
1427         int i;
1428
1429         if (!map)
1430                 return NULL;
1431         for (i = 0; i < map->n; ++i) {
1432                 map->p[i] = isl_basic_map_finalize(map->p[i]);
1433                 if (!map->p[i])
1434                         goto error;
1435         }
1436         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
1437         return map;
1438 error:
1439         isl_map_free(map);
1440         return NULL;
1441 }
1442
1443
1444 /* Remove definition of any div that is defined in terms of the given variable.
1445  * The div itself is not removed.  Functions such as
1446  * eliminate_divs_ineq depend on the other divs remaining in place.
1447  */
1448 static struct isl_basic_map *remove_dependent_vars(struct isl_basic_map *bmap,
1449                                                                         int pos)
1450 {
1451         int i;
1452
1453         if (!bmap)
1454                 return NULL;
1455
1456         for (i = 0; i < bmap->n_div; ++i) {
1457                 if (isl_int_is_zero(bmap->div[i][0]))
1458                         continue;
1459                 if (isl_int_is_zero(bmap->div[i][1+1+pos]))
1460                         continue;
1461                 isl_int_set_si(bmap->div[i][0], 0);
1462         }
1463         return bmap;
1464 }
1465
1466 /* Eliminate the specified variables from the constraints using
1467  * Fourier-Motzkin.  The variables themselves are not removed.
1468  */
1469 struct isl_basic_map *isl_basic_map_eliminate_vars(
1470         struct isl_basic_map *bmap, unsigned pos, unsigned n)
1471 {
1472         int d;
1473         int i, j, k;
1474         unsigned total;
1475         int need_gauss = 0;
1476
1477         if (n == 0)
1478                 return bmap;
1479         if (!bmap)
1480                 return NULL;
1481         total = isl_basic_map_total_dim(bmap);
1482
1483         bmap = isl_basic_map_cow(bmap);
1484         for (d = pos + n - 1; d >= 0 && d >= pos; --d)
1485                 bmap = remove_dependent_vars(bmap, d);
1486         if (!bmap)
1487                 return NULL;
1488
1489         for (d = pos + n - 1;
1490              d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
1491                 isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
1492         for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
1493                 int n_lower, n_upper;
1494                 if (!bmap)
1495                         return NULL;
1496                 for (i = 0; i < bmap->n_eq; ++i) {
1497                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1498                                 continue;
1499                         eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL);
1500                         isl_basic_map_drop_equality(bmap, i);
1501                         need_gauss = 1;
1502                         break;
1503                 }
1504                 if (i < bmap->n_eq)
1505                         continue;
1506                 n_lower = 0;
1507                 n_upper = 0;
1508                 for (i = 0; i < bmap->n_ineq; ++i) {
1509                         if (isl_int_is_pos(bmap->ineq[i][1+d]))
1510                                 n_lower++;
1511                         else if (isl_int_is_neg(bmap->ineq[i][1+d]))
1512                                 n_upper++;
1513                 }
1514                 bmap = isl_basic_map_extend_constraints(bmap,
1515                                 0, n_lower * n_upper);
1516                 if (!bmap)
1517                         goto error;
1518                 for (i = bmap->n_ineq - 1; i >= 0; --i) {
1519                         int last;
1520                         if (isl_int_is_zero(bmap->ineq[i][1+d]))
1521                                 continue;
1522                         last = -1;
1523                         for (j = 0; j < i; ++j) {
1524                                 if (isl_int_is_zero(bmap->ineq[j][1+d]))
1525                                         continue;
1526                                 last = j;
1527                                 if (isl_int_sgn(bmap->ineq[i][1+d]) ==
1528                                     isl_int_sgn(bmap->ineq[j][1+d]))
1529                                         continue;
1530                                 k = isl_basic_map_alloc_inequality(bmap);
1531                                 if (k < 0)
1532                                         goto error;
1533                                 isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
1534                                                 1+total);
1535                                 isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
1536                                                 1+d, 1+total, NULL);
1537                         }
1538                         isl_basic_map_drop_inequality(bmap, i);
1539                         i = last + 1;
1540                 }
1541                 if (n_lower > 0 && n_upper > 0) {
1542                         bmap = isl_basic_map_normalize_constraints(bmap);
1543                         bmap = remove_duplicate_constraints(bmap, NULL, 0);
1544                         bmap = isl_basic_map_gauss(bmap, NULL);
1545                         bmap = isl_basic_map_remove_redundancies(bmap);
1546                         need_gauss = 0;
1547                         if (!bmap)
1548                                 goto error;
1549                         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1550                                 break;
1551                 }
1552         }
1553         ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
1554         if (need_gauss)
1555                 bmap = isl_basic_map_gauss(bmap, NULL);
1556         return bmap;
1557 error:
1558         isl_basic_map_free(bmap);
1559         return NULL;
1560 }
1561
1562 struct isl_basic_set *isl_basic_set_eliminate_vars(
1563         struct isl_basic_set *bset, unsigned pos, unsigned n)
1564 {
1565         return (struct isl_basic_set *)isl_basic_map_eliminate_vars(
1566                         (struct isl_basic_map *)bset, pos, n);
1567 }
1568
1569 /* Eliminate the specified n dimensions starting at first from the
1570  * constraints, without removing the dimensions from the space.
1571  * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1572  * Otherwise, they are projected out and the original space is restored.
1573  */
1574 __isl_give isl_basic_map *isl_basic_map_eliminate(
1575         __isl_take isl_basic_map *bmap,
1576         enum isl_dim_type type, unsigned first, unsigned n)
1577 {
1578         isl_space *space;
1579
1580         if (!bmap)
1581                 return NULL;
1582         if (n == 0)
1583                 return bmap;
1584
1585         if (first + n > isl_basic_map_dim(bmap, type) || first + n < first)
1586                 isl_die(bmap->ctx, isl_error_invalid,
1587                         "index out of bounds", goto error);
1588
1589         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
1590                 first += isl_basic_map_offset(bmap, type) - 1;
1591                 bmap = isl_basic_map_eliminate_vars(bmap, first, n);
1592                 return isl_basic_map_finalize(bmap);
1593         }
1594
1595         space = isl_basic_map_get_space(bmap);
1596         bmap = isl_basic_map_project_out(bmap, type, first, n);
1597         bmap = isl_basic_map_insert_dims(bmap, type, first, n);
1598         bmap = isl_basic_map_reset_space(bmap, space);
1599         return bmap;
1600 error:
1601         isl_basic_map_free(bmap);
1602         return NULL;
1603 }
1604
1605 __isl_give isl_basic_set *isl_basic_set_eliminate(
1606         __isl_take isl_basic_set *bset,
1607         enum isl_dim_type type, unsigned first, unsigned n)
1608 {
1609         return isl_basic_map_eliminate(bset, type, first, n);
1610 }
1611
1612 /* Don't assume equalities are in order, because align_divs
1613  * may have changed the order of the divs.
1614  */
1615 static void compute_elimination_index(struct isl_basic_map *bmap, int *elim)
1616 {
1617         int d, i;
1618         unsigned total;
1619
1620         total = isl_space_dim(bmap->dim, isl_dim_all);
1621         for (d = 0; d < total; ++d)
1622                 elim[d] = -1;
1623         for (i = 0; i < bmap->n_eq; ++i) {
1624                 for (d = total - 1; d >= 0; --d) {
1625                         if (isl_int_is_zero(bmap->eq[i][1+d]))
1626                                 continue;
1627                         elim[d] = i;
1628                         break;
1629                 }
1630         }
1631 }
1632
1633 static void set_compute_elimination_index(struct isl_basic_set *bset, int *elim)
1634 {
1635         compute_elimination_index((struct isl_basic_map *)bset, elim);
1636 }
1637
1638 static int reduced_using_equalities(isl_int *dst, isl_int *src,
1639         struct isl_basic_map *bmap, int *elim)
1640 {
1641         int d;
1642         int copied = 0;
1643         unsigned total;
1644
1645         total = isl_space_dim(bmap->dim, isl_dim_all);
1646         for (d = total - 1; d >= 0; --d) {
1647                 if (isl_int_is_zero(src[1+d]))
1648                         continue;
1649                 if (elim[d] == -1)
1650                         continue;
1651                 if (!copied) {
1652                         isl_seq_cpy(dst, src, 1 + total);
1653                         copied = 1;
1654                 }
1655                 isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
1656         }
1657         return copied;
1658 }
1659
1660 static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
1661         struct isl_basic_set *bset, int *elim)
1662 {
1663         return reduced_using_equalities(dst, src,
1664                                         (struct isl_basic_map *)bset, elim);
1665 }
1666
1667 static struct isl_basic_set *isl_basic_set_reduce_using_equalities(
1668         struct isl_basic_set *bset, struct isl_basic_set *context)
1669 {
1670         int i;
1671         int *elim;
1672
1673         if (!bset || !context)
1674                 goto error;
1675
1676         if (context->n_eq == 0) {
1677                 isl_basic_set_free(context);
1678                 return bset;
1679         }
1680
1681         bset = isl_basic_set_cow(bset);
1682         if (!bset)
1683                 goto error;
1684
1685         elim = isl_alloc_array(bset->ctx, int, isl_basic_set_n_dim(bset));
1686         if (!elim)
1687                 goto error;
1688         set_compute_elimination_index(context, elim);
1689         for (i = 0; i < bset->n_eq; ++i)
1690                 set_reduced_using_equalities(bset->eq[i], bset->eq[i],
1691                                                         context, elim);
1692         for (i = 0; i < bset->n_ineq; ++i)
1693                 set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
1694                                                         context, elim);
1695         isl_basic_set_free(context);
1696         free(elim);
1697         bset = isl_basic_set_simplify(bset);
1698         bset = isl_basic_set_finalize(bset);
1699         return bset;
1700 error:
1701         isl_basic_set_free(bset);
1702         isl_basic_set_free(context);
1703         return NULL;
1704 }
1705
1706 static struct isl_basic_set *remove_shifted_constraints(
1707         struct isl_basic_set *bset, struct isl_basic_set *context)
1708 {
1709         unsigned int size;
1710         isl_int ***index;
1711         int bits;
1712         int k, h, l;
1713         isl_ctx *ctx;
1714
1715         if (!bset)
1716                 return NULL;
1717
1718         size = round_up(4 * (context->n_ineq+1) / 3 - 1);
1719         bits = ffs(size) - 1;
1720         ctx = isl_basic_set_get_ctx(bset);
1721         index = isl_calloc_array(ctx, isl_int **, size);
1722         if (!index)
1723                 return bset;
1724
1725         for (k = 0; k < context->n_ineq; ++k) {
1726                 h = set_hash_index(index, size, bits, context, k);
1727                 index[h] = &context->ineq[k];
1728         }
1729         for (k = 0; k < bset->n_ineq; ++k) {
1730                 h = set_hash_index(index, size, bits, bset, k);
1731                 if (!index[h])
1732                         continue;
1733                 l = index[h] - &context->ineq[0];
1734                 if (isl_int_lt(bset->ineq[k][0], context->ineq[l][0]))
1735                         continue;
1736                 bset = isl_basic_set_cow(bset);
1737                 if (!bset)
1738                         goto error;
1739                 isl_basic_set_drop_inequality(bset, k);
1740                 --k;
1741         }
1742         free(index);
1743         return bset;
1744 error:
1745         free(index);
1746         return bset;
1747 }
1748
1749 /* Does the (linear part of a) constraint "c" involve any of the "len"
1750  * "relevant" dimensions?
1751  */
1752 static int is_related(isl_int *c, int len, int *relevant)
1753 {
1754         int i;
1755
1756         for (i = 0; i < len; ++i) {
1757                 if (!relevant[i])
1758                         continue;
1759                 if (!isl_int_is_zero(c[i]))
1760                         return 1;
1761         }
1762
1763         return 0;
1764 }
1765
1766 /* Drop constraints from "bset" that do not involve any of
1767  * the dimensions marked "relevant".
1768  */
1769 static __isl_give isl_basic_set *drop_unrelated_constraints(
1770         __isl_take isl_basic_set *bset, int *relevant)
1771 {
1772         int i, dim;
1773
1774         dim = isl_basic_set_dim(bset, isl_dim_set);
1775         for (i = 0; i < dim; ++i)
1776                 if (!relevant[i])
1777                         break;
1778         if (i >= dim)
1779                 return bset;
1780
1781         for (i = bset->n_eq - 1; i >= 0; --i)
1782                 if (!is_related(bset->eq[i] + 1, dim, relevant))
1783                         isl_basic_set_drop_equality(bset, i);
1784
1785         for (i = bset->n_ineq - 1; i >= 0; --i)
1786                 if (!is_related(bset->ineq[i] + 1, dim, relevant))
1787                         isl_basic_set_drop_inequality(bset, i);
1788
1789         return bset;
1790 }
1791
1792 /* Update the groups in "group" based on the (linear part of a) constraint "c".
1793  *
1794  * In particular, for any variable involved in the constraint,
1795  * find the actual group id from before and replace the group
1796  * of the corresponding variable by the minimal group of all
1797  * the variables involved in the constraint considered so far
1798  * (if this minimum is smaller) or replace the minimum by this group
1799  * (if the minimum is larger).
1800  *
1801  * At the end, all the variables in "c" will (indirectly) point
1802  * to the minimal of the groups that they referred to originally.
1803  */
1804 static void update_groups(int dim, int *group, isl_int *c)
1805 {
1806         int j;
1807         int min = dim;
1808
1809         for (j = 0; j < dim; ++j) {
1810                 if (isl_int_is_zero(c[j]))
1811                         continue;
1812                 while (group[j] >= 0 && group[group[j]] != group[j])
1813                         group[j] = group[group[j]];
1814                 if (group[j] == min)
1815                         continue;
1816                 if (group[j] < min) {
1817                         if (min >= 0 && min < dim)
1818                                 group[min] = group[j];
1819                         min = group[j];
1820                 } else
1821                         group[group[j]] = min;
1822         }
1823 }
1824
1825 /* Drop constraints from "context" that are irrelevant for computing
1826  * the gist of "bset".
1827  *
1828  * In particular, drop constraints in variables that are not related
1829  * to any of the variables involved in the constraints of "bset"
1830  * in the sense that there is no sequence of constraints that connects them.
1831  *
1832  * We construct groups of variables that collect variables that
1833  * (indirectly) appear in some common constraint of "context".
1834  * Each group is identified by the first variable in the group,
1835  * except for the special group of variables that appear in "bset"
1836  * (or are related to those variables), which is identified by -1.
1837  * If group[i] is equal to i (or -1), then the group of i is i (or -1),
1838  * otherwise the group of i is the group of group[i].
1839  *
1840  * We first initialize the -1 group with the variables that appear in "bset".
1841  * Then we initialize groups for the remaining variables.
1842  * Then we iterate over the constraints of "context" and update the
1843  * group of the variables in the constraint by the smallest group.
1844  * Finally, we resolve indirect references to groups by running over
1845  * the variables.
1846  *
1847  * After computing the groups, we drop constraints that do not involve
1848  * any variables in the -1 group.
1849  */
1850 static __isl_give isl_basic_set *drop_irrelevant_constraints(
1851         __isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset)
1852 {
1853         isl_ctx *ctx;
1854         int *group;
1855         int dim;
1856         int i, j;
1857         int last;
1858
1859         if (!context || !bset)
1860                 return isl_basic_set_free(context);
1861
1862         dim = isl_basic_set_dim(bset, isl_dim_set);
1863         ctx = isl_basic_set_get_ctx(bset);
1864         group = isl_calloc_array(ctx, int, dim);
1865
1866         if (!group)
1867                 goto error;
1868
1869         for (i = 0; i < dim; ++i) {
1870                 for (j = 0; j < bset->n_eq; ++j)
1871                         if (!isl_int_is_zero(bset->eq[j][1 + i]))
1872                                 break;
1873                 if (j < bset->n_eq) {
1874                         group[i] = -1;
1875                         continue;
1876                 }
1877                 for (j = 0; j < bset->n_ineq; ++j)
1878                         if (!isl_int_is_zero(bset->ineq[j][1 + i]))
1879                                 break;
1880                 if (j < bset->n_ineq)
1881                         group[i] = -1;
1882         }
1883
1884         last = -1;
1885         for (i = 0; i < dim; ++i)
1886                 if (group[i] >= 0)
1887                         last = group[i] = i;
1888         if (last < 0) {
1889                 free(group);
1890                 return context;
1891         }
1892
1893         for (i = 0; i < context->n_eq; ++i)
1894                 update_groups(dim, group, context->eq[i] + 1);
1895         for (i = 0; i < context->n_ineq; ++i)
1896                 update_groups(dim, group, context->ineq[i] + 1);
1897
1898         for (i = 0; i < dim; ++i)
1899                 if (group[i] >= 0)
1900                         group[i] = group[group[i]];
1901
1902         for (i = 0; i < dim; ++i)
1903                 group[i] = group[i] == -1;
1904
1905         context = drop_unrelated_constraints(context, group);
1906
1907         free(group);
1908         return context;
1909 error:
1910         free(group);
1911         return isl_basic_set_free(context);
1912 }
1913
1914 /* Remove all information from bset that is redundant in the context
1915  * of context.  Both bset and context are assumed to be full-dimensional.
1916  *
1917  * We first remove the inequalities from "bset"
1918  * that are obviously redundant with respect to some inequality in "context".
1919  * Then we remove those constraints from "context" that have become
1920  * irrelevant for computing the gist of "bset".
1921  * Note that this removal of constraints cannot be replaced by
1922  * a factorization because factors in "bset" may still be connected
1923  * to each other through constraints in "context".
1924  *
1925  * If there are any inequalities left, we construct a tableau for
1926  * the context and then add the inequalities of "bset".
1927  * Before adding these inequalities, we freeze all constraints such that
1928  * they won't be considered redundant in terms of the constraints of "bset".
1929  * Then we detect all redundant constraints (among the
1930  * constraints that weren't frozen), first by checking for redundancy in the
1931  * the tableau and then by checking if replacing a constraint by its negation
1932  * would lead to an empty set.  This last step is fairly expensive
1933  * and could be optimized by more reuse of the tableau.
1934  * Finally, we update bset according to the results.
1935  */
1936 static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset,
1937         __isl_take isl_basic_set *context)
1938 {
1939         int i, k;
1940         isl_basic_set *combined = NULL;
1941         struct isl_tab *tab = NULL;
1942         unsigned context_ineq;
1943         unsigned total;
1944
1945         if (!bset || !context)
1946                 goto error;
1947
1948         if (isl_basic_set_is_universe(bset)) {
1949                 isl_basic_set_free(context);
1950                 return bset;
1951         }
1952
1953         if (isl_basic_set_is_universe(context)) {
1954                 isl_basic_set_free(context);
1955                 return bset;
1956         }
1957
1958         bset = remove_shifted_constraints(bset, context);
1959         if (!bset)
1960                 goto error;
1961         if (bset->n_ineq == 0)
1962                 goto done;
1963
1964         context = drop_irrelevant_constraints(context, bset);
1965         if (!context)
1966                 goto error;
1967         if (isl_basic_set_is_universe(context)) {
1968                 isl_basic_set_free(context);
1969                 return bset;
1970         }
1971
1972         context_ineq = context->n_ineq;
1973         combined = isl_basic_set_cow(isl_basic_set_copy(context));
1974         combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq);
1975         tab = isl_tab_from_basic_set(combined, 0);
1976         for (i = 0; i < context_ineq; ++i)
1977                 if (isl_tab_freeze_constraint(tab, i) < 0)
1978                         goto error;
1979         tab = isl_tab_extend(tab, bset->n_ineq);
1980         for (i = 0; i < bset->n_ineq; ++i)
1981                 if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
1982                         goto error;
1983         bset = isl_basic_set_add_constraints(combined, bset, 0);
1984         combined = NULL;
1985         if (!bset)
1986                 goto error;
1987         if (isl_tab_detect_redundant(tab) < 0)
1988                 goto error;
1989         total = isl_basic_set_total_dim(bset);
1990         for (i = context_ineq; i < bset->n_ineq; ++i) {
1991                 int is_empty;
1992                 if (tab->con[i].is_redundant)
1993                         continue;
1994                 tab->con[i].is_redundant = 1;
1995                 combined = isl_basic_set_dup(bset);
1996                 combined = isl_basic_set_update_from_tab(combined, tab);
1997                 combined = isl_basic_set_extend_constraints(combined, 0, 1);
1998                 k = isl_basic_set_alloc_inequality(combined);
1999                 if (k < 0)
2000                         goto error;
2001                 isl_seq_neg(combined->ineq[k], bset->ineq[i], 1 + total);
2002                 isl_int_sub_ui(combined->ineq[k][0], combined->ineq[k][0], 1);
2003                 is_empty = isl_basic_set_is_empty(combined);
2004                 if (is_empty < 0)
2005                         goto error;
2006                 isl_basic_set_free(combined);
2007                 combined = NULL;
2008                 if (!is_empty)
2009                         tab->con[i].is_redundant = 0;
2010         }
2011         for (i = 0; i < context_ineq; ++i)
2012                 tab->con[i].is_redundant = 1;
2013         bset = isl_basic_set_update_from_tab(bset, tab);
2014         if (bset) {
2015                 ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
2016                 ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
2017         }
2018
2019         isl_tab_free(tab);
2020 done:
2021         bset = isl_basic_set_simplify(bset);
2022         bset = isl_basic_set_finalize(bset);
2023         isl_basic_set_free(context);
2024         return bset;
2025 error:
2026         isl_tab_free(tab);
2027         isl_basic_set_free(combined);
2028         isl_basic_set_free(context);
2029         isl_basic_set_free(bset);
2030         return NULL;
2031 }
2032
2033 /* Remove all information from bset that is redundant in the context
2034  * of context.  In particular, equalities that are linear combinations
2035  * of those in context are removed.  Then the inequalities that are
2036  * redundant in the context of the equalities and inequalities of
2037  * context are removed.
2038  *
2039  * First of all, we drop those constraints from "context"
2040  * that are irrelevant for computing the gist of "bset".
2041  * Alternatively, we could factorize the intersection of "context" and "bset".
2042  *
2043  * We first compute the integer affine hull of the intersection,
2044  * compute the gist inside this affine hull and then add back
2045  * those equalities that are not implied by the context.
2046  *
2047  * If two constraints are mutually redundant, then uset_gist_full
2048  * will remove the second of those constraints.  We therefore first
2049  * sort the constraints so that constraints not involving existentially
2050  * quantified variables are given precedence over those that do.
2051  * We have to perform this sorting before the variable compression,
2052  * because that may effect the order of the variables.
2053  */
2054 static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset,
2055         __isl_take isl_basic_set *context)
2056 {
2057         isl_mat *eq;
2058         isl_mat *T, *T2;
2059         isl_basic_set *aff;
2060         isl_basic_set *aff_context;
2061         unsigned total;
2062
2063         if (!bset || !context)
2064                 goto error;
2065
2066         context = drop_irrelevant_constraints(context, bset);
2067
2068         bset = isl_basic_set_intersect(bset, isl_basic_set_copy(context));
2069         if (isl_basic_set_plain_is_empty(bset)) {
2070                 isl_basic_set_free(context);
2071                 return bset;
2072         }
2073         bset = isl_basic_set_sort_constraints(bset);
2074         aff = isl_basic_set_affine_hull(isl_basic_set_copy(bset));
2075         if (!aff)
2076                 goto error;
2077         if (isl_basic_set_plain_is_empty(aff)) {
2078                 isl_basic_set_free(aff);
2079                 isl_basic_set_free(context);
2080                 return bset;
2081         }
2082         if (aff->n_eq == 0) {
2083                 isl_basic_set_free(aff);
2084                 return uset_gist_full(bset, context);
2085         }
2086         total = isl_basic_set_total_dim(bset);
2087         eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total);
2088         eq = isl_mat_cow(eq);
2089         T = isl_mat_variable_compression(eq, &T2);
2090         if (T && T->n_col == 0) {
2091                 isl_mat_free(T);
2092                 isl_mat_free(T2);
2093                 isl_basic_set_free(context);
2094                 isl_basic_set_free(aff);
2095                 return isl_basic_set_set_to_empty(bset);
2096         }
2097
2098         aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));
2099
2100         bset = isl_basic_set_preimage(bset, isl_mat_copy(T));
2101         context = isl_basic_set_preimage(context, T);
2102
2103         bset = uset_gist_full(bset, context);
2104         bset = isl_basic_set_preimage(bset, T2);
2105         bset = isl_basic_set_intersect(bset, aff);
2106         bset = isl_basic_set_reduce_using_equalities(bset, aff_context);
2107
2108         if (bset) {
2109                 ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
2110                 ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
2111         }
2112
2113         return bset;
2114 error:
2115         isl_basic_set_free(bset);
2116         isl_basic_set_free(context);
2117         return NULL;
2118 }
2119
2120 /* Normalize the divs in "bmap" in the context of the equalities in "context".
2121  * We simply add the equalities in context to bmap and then do a regular
2122  * div normalizations.  Better results can be obtained by normalizing
2123  * only the divs in bmap than do not also appear in context.
2124  * We need to be careful to reduce the divs using the equalities
2125  * so that later calls to isl_basic_map_overlying_set wouldn't introduce
2126  * spurious constraints.
2127  */
2128 static struct isl_basic_map *normalize_divs_in_context(
2129         struct isl_basic_map *bmap, struct isl_basic_map *context)
2130 {
2131         int i;
2132         unsigned total_context;
2133         int div_eq;
2134
2135         div_eq = n_pure_div_eq(bmap);
2136         if (div_eq == 0)
2137                 return bmap;
2138
2139         if (context->n_div > 0)
2140                 bmap = isl_basic_map_align_divs(bmap, context);
2141
2142         total_context = isl_basic_map_total_dim(context);
2143         bmap = isl_basic_map_extend_constraints(bmap, context->n_eq, 0);
2144         for (i = 0; i < context->n_eq; ++i) {
2145                 int k;
2146                 k = isl_basic_map_alloc_equality(bmap);
2147                 isl_seq_cpy(bmap->eq[k], context->eq[i], 1 + total_context);
2148                 isl_seq_clr(bmap->eq[k] + 1 + total_context,
2149                                 isl_basic_map_total_dim(bmap) - total_context);
2150         }
2151         bmap = isl_basic_map_gauss(bmap, NULL);
2152         bmap = normalize_divs(bmap, NULL);
2153         bmap = isl_basic_map_gauss(bmap, NULL);
2154         return bmap;
2155 }
2156
2157 struct isl_basic_map *isl_basic_map_gist(struct isl_basic_map *bmap,
2158         struct isl_basic_map *context)
2159 {
2160         struct isl_basic_set *bset;
2161
2162         if (!bmap || !context)
2163                 goto error;
2164
2165         if (isl_basic_map_is_universe(bmap)) {
2166                 isl_basic_map_free(context);
2167                 return bmap;
2168         }
2169         if (isl_basic_map_plain_is_empty(context)) {
2170                 isl_basic_map_free(bmap);
2171                 return context;
2172         }
2173         if (isl_basic_map_plain_is_empty(bmap)) {
2174                 isl_basic_map_free(context);
2175                 return bmap;
2176         }
2177
2178         bmap = isl_basic_map_remove_redundancies(bmap);
2179         context = isl_basic_map_remove_redundancies(context);
2180
2181         if (context->n_eq)
2182                 bmap = normalize_divs_in_context(bmap, context);
2183
2184         context = isl_basic_map_align_divs(context, bmap);
2185         bmap = isl_basic_map_align_divs(bmap, context);
2186
2187         bset = uset_gist(isl_basic_map_underlying_set(isl_basic_map_copy(bmap)),
2188                          isl_basic_map_underlying_set(context));
2189
2190         return isl_basic_map_overlying_set(bset, bmap);
2191 error:
2192         isl_basic_map_free(bmap);
2193         isl_basic_map_free(context);
2194         return NULL;
2195 }
2196
2197 /*
2198  * Assumes context has no implicit divs.
2199  */
2200 __isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
2201         __isl_take isl_basic_map *context)
2202 {
2203         int i;
2204
2205         if (!map || !context)
2206                 goto error;;
2207
2208         if (isl_basic_map_plain_is_empty(context)) {
2209                 isl_map_free(map);
2210                 return isl_map_from_basic_map(context);
2211         }
2212
2213         context = isl_basic_map_remove_redundancies(context);
2214         map = isl_map_cow(map);
2215         if (!map || !context)
2216                 goto error;;
2217         isl_assert(map->ctx, isl_space_is_equal(map->dim, context->dim), goto error);
2218         map = isl_map_compute_divs(map);
2219         if (!map)
2220                 goto error;
2221         for (i = 0; i < map->n; ++i)
2222                 context = isl_basic_map_align_divs(context, map->p[i]);
2223         for (i = map->n - 1; i >= 0; --i) {
2224                 map->p[i] = isl_basic_map_gist(map->p[i],
2225                                                 isl_basic_map_copy(context));
2226                 if (!map->p[i])
2227                         goto error;
2228                 if (isl_basic_map_plain_is_empty(map->p[i])) {
2229                         isl_basic_map_free(map->p[i]);
2230                         if (i != map->n - 1)
2231                                 map->p[i] = map->p[map->n - 1];
2232                         map->n--;
2233                 }
2234         }
2235         isl_basic_map_free(context);
2236         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
2237         return map;
2238 error:
2239         isl_map_free(map);
2240         isl_basic_map_free(context);
2241         return NULL;
2242 }
2243
2244 /* Return a map that has the same intersection with "context" as "map"
2245  * and that as "simple" as possible.
2246  *
2247  * If "map" is already the universe, then we cannot make it any simpler.
2248  * Similarly, if "context" is the universe, then we cannot exploit it
2249  * to simplify "map"
2250  * If "map" and "context" are identical to each other, then we can
2251  * return the corresponding universe.
2252  *
2253  * If none of these cases apply, we have to work a bit harder.
2254  */
2255 static __isl_give isl_map *map_gist(__isl_take isl_map *map,
2256         __isl_take isl_map *context)
2257 {
2258         int equal;
2259         int is_universe;
2260
2261         is_universe = isl_map_plain_is_universe(map);
2262         if (is_universe >= 0 && !is_universe)
2263                 is_universe = isl_map_plain_is_universe(context);
2264         if (is_universe < 0)
2265                 goto error;
2266         if (is_universe) {
2267                 isl_map_free(context);
2268                 return map;
2269         }
2270
2271         equal = isl_map_plain_is_equal(map, context);
2272         if (equal < 0)
2273                 goto error;
2274         if (equal) {
2275                 isl_map *res = isl_map_universe(isl_map_get_space(map));
2276                 isl_map_free(map);
2277                 isl_map_free(context);
2278                 return res;
2279         }
2280
2281         context = isl_map_compute_divs(context);
2282         return isl_map_gist_basic_map(map, isl_map_simple_hull(context));
2283 error:
2284         isl_map_free(map);
2285         isl_map_free(context);
2286         return NULL;
2287 }
2288
2289 __isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
2290         __isl_take isl_map *context)
2291 {
2292         return isl_map_align_params_map_map_and(map, context, &map_gist);
2293 }
2294
2295 struct isl_basic_set *isl_basic_set_gist(struct isl_basic_set *bset,
2296                                                 struct isl_basic_set *context)
2297 {
2298         return (struct isl_basic_set *)isl_basic_map_gist(
2299                 (struct isl_basic_map *)bset, (struct isl_basic_map *)context);
2300 }
2301
2302 __isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
2303         __isl_take isl_basic_set *context)
2304 {
2305         return (struct isl_set *)isl_map_gist_basic_map((struct isl_map *)set,
2306                                         (struct isl_basic_map *)context);
2307 }
2308
2309 __isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set,
2310         __isl_take isl_basic_set *context)
2311 {
2312         isl_space *space = isl_set_get_space(set);
2313         isl_basic_set *dom_context = isl_basic_set_universe(space);
2314         dom_context = isl_basic_set_intersect_params(dom_context, context);
2315         return isl_set_gist_basic_set(set, dom_context);
2316 }
2317
2318 __isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
2319         __isl_take isl_set *context)
2320 {
2321         return (struct isl_set *)isl_map_gist((struct isl_map *)set,
2322                                         (struct isl_map *)context);
2323 }
2324
2325 __isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map,
2326         __isl_take isl_set *context)
2327 {
2328         isl_map *map_context = isl_map_universe(isl_map_get_space(map));
2329         map_context = isl_map_intersect_domain(map_context, context);
2330         return isl_map_gist(map, map_context);
2331 }
2332
2333 __isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map,
2334         __isl_take isl_set *context)
2335 {
2336         isl_map *map_context = isl_map_universe(isl_map_get_space(map));
2337         map_context = isl_map_intersect_range(map_context, context);
2338         return isl_map_gist(map, map_context);
2339 }
2340
2341 __isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map,
2342         __isl_take isl_set *context)
2343 {
2344         isl_map *map_context = isl_map_universe(isl_map_get_space(map));
2345         map_context = isl_map_intersect_params(map_context, context);
2346         return isl_map_gist(map, map_context);
2347 }
2348
2349 __isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set,
2350         __isl_take isl_set *context)
2351 {
2352         return isl_map_gist_params(set, context);
2353 }
2354
2355 /* Quick check to see if two basic maps are disjoint.
2356  * In particular, we reduce the equalities and inequalities of
2357  * one basic map in the context of the equalities of the other
2358  * basic map and check if we get a contradiction.
2359  */
2360 int isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1,
2361         __isl_keep isl_basic_map *bmap2)
2362 {
2363         struct isl_vec *v = NULL;
2364         int *elim = NULL;
2365         unsigned total;
2366         int i;
2367
2368         if (!bmap1 || !bmap2)
2369                 return -1;
2370         isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
2371                         return -1);
2372         if (bmap1->n_div || bmap2->n_div)
2373                 return 0;
2374         if (!bmap1->n_eq && !bmap2->n_eq)
2375                 return 0;
2376
2377         total = isl_space_dim(bmap1->dim, isl_dim_all);
2378         if (total == 0)
2379                 return 0;
2380         v = isl_vec_alloc(bmap1->ctx, 1 + total);
2381         if (!v)
2382                 goto error;
2383         elim = isl_alloc_array(bmap1->ctx, int, total);
2384         if (!elim)
2385                 goto error;
2386         compute_elimination_index(bmap1, elim);
2387         for (i = 0; i < bmap2->n_eq; ++i) {
2388                 int reduced;
2389                 reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
2390                                                         bmap1, elim);
2391                 if (reduced && !isl_int_is_zero(v->block.data[0]) &&
2392                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
2393                         goto disjoint;
2394         }
2395         for (i = 0; i < bmap2->n_ineq; ++i) {
2396                 int reduced;
2397                 reduced = reduced_using_equalities(v->block.data,
2398                                                 bmap2->ineq[i], bmap1, elim);
2399                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
2400                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
2401                         goto disjoint;
2402         }
2403         compute_elimination_index(bmap2, elim);
2404         for (i = 0; i < bmap1->n_ineq; ++i) {
2405                 int reduced;
2406                 reduced = reduced_using_equalities(v->block.data,
2407                                                 bmap1->ineq[i], bmap2, elim);
2408                 if (reduced && isl_int_is_neg(v->block.data[0]) &&
2409                     isl_seq_first_non_zero(v->block.data + 1, total) == -1)
2410                         goto disjoint;
2411         }
2412         isl_vec_free(v);
2413         free(elim);
2414         return 0;
2415 disjoint:
2416         isl_vec_free(v);
2417         free(elim);
2418         return 1;
2419 error:
2420         isl_vec_free(v);
2421         free(elim);
2422         return -1;
2423 }
2424
2425 int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1,
2426         __isl_keep isl_basic_set *bset2)
2427 {
2428         return isl_basic_map_plain_is_disjoint((struct isl_basic_map *)bset1,
2429                                               (struct isl_basic_map *)bset2);
2430 }
2431
2432 /* Are "map1" and "map2" obviously disjoint?
2433  *
2434  * If one of them is empty or if they live in different spaces (ignoring
2435  * parameters), then they are clearly disjoint.
2436  *
2437  * If they have different parameters, then we skip any further tests.
2438  *
2439  * If they are obviously equal, but not obviously empty, then we will
2440  * not be able to detect if they are disjoint.
2441  *
2442  * Otherwise we check if each basic map in "map1" is obviously disjoint
2443  * from each basic map in "map2".
2444  */
2445 int isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
2446         __isl_keep isl_map *map2)
2447 {
2448         int i, j;
2449         int disjoint;
2450         int intersect;
2451         int match;
2452
2453         if (!map1 || !map2)
2454                 return -1;
2455
2456         disjoint = isl_map_plain_is_empty(map1);
2457         if (disjoint < 0 || disjoint)
2458                 return disjoint;
2459
2460         disjoint = isl_map_plain_is_empty(map2);
2461         if (disjoint < 0 || disjoint)
2462                 return disjoint;
2463
2464         match = isl_space_tuple_match(map1->dim, isl_dim_in,
2465                                 map2->dim, isl_dim_in);
2466         if (match < 0 || !match)
2467                 return match < 0 ? -1 : 1;
2468
2469         match = isl_space_tuple_match(map1->dim, isl_dim_out,
2470                                 map2->dim, isl_dim_out);
2471         if (match < 0 || !match)
2472                 return match < 0 ? -1 : 1;
2473
2474         match = isl_space_match(map1->dim, isl_dim_param,
2475                                 map2->dim, isl_dim_param);
2476         if (match < 0 || !match)
2477                 return match < 0 ? -1 : 0;
2478
2479         intersect = isl_map_plain_is_equal(map1, map2);
2480         if (intersect < 0 || intersect)
2481                 return intersect < 0 ? -1 : 0;
2482
2483         for (i = 0; i < map1->n; ++i) {
2484                 for (j = 0; j < map2->n; ++j) {
2485                         int d = isl_basic_map_plain_is_disjoint(map1->p[i],
2486                                                                map2->p[j]);
2487                         if (d != 1)
2488                                 return d;
2489                 }
2490         }
2491         return 1;
2492 }
2493
2494 /* Are "map1" and "map2" disjoint?
2495  *
2496  * They are disjoint if they are "obviously disjoint" or if one of them
2497  * is empty.  Otherwise, they are not disjoint if one of them is universal.
2498  * If none of these cases apply, we compute the intersection and see if
2499  * the result is empty.
2500  */
2501 int isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
2502 {
2503         int disjoint;
2504         int intersect;
2505         isl_map *test;
2506
2507         disjoint = isl_map_plain_is_disjoint(map1, map2);
2508         if (disjoint < 0 || disjoint)
2509                 return disjoint;
2510
2511         disjoint = isl_map_is_empty(map1);
2512         if (disjoint < 0 || disjoint)
2513                 return disjoint;
2514
2515         disjoint = isl_map_is_empty(map2);
2516         if (disjoint < 0 || disjoint)
2517                 return disjoint;
2518
2519         intersect = isl_map_plain_is_universe(map1);
2520         if (intersect < 0 || intersect)
2521                 return intersect < 0 ? -1 : 0;
2522
2523         intersect = isl_map_plain_is_universe(map2);
2524         if (intersect < 0 || intersect)
2525                 return intersect < 0 ? -1 : 0;
2526
2527         test = isl_map_intersect(isl_map_copy(map1), isl_map_copy(map2));
2528         disjoint = isl_map_is_empty(test);
2529         isl_map_free(test);
2530
2531         return disjoint;
2532 }
2533
2534 int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
2535         __isl_keep isl_set *set2)
2536 {
2537         return isl_map_plain_is_disjoint((struct isl_map *)set1,
2538                                         (struct isl_map *)set2);
2539 }
2540
2541 /* Are "set1" and "set2" disjoint?
2542  */
2543 int isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
2544 {
2545         return isl_map_is_disjoint(set1, set2);
2546 }
2547
2548 int isl_set_fast_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
2549 {
2550         return isl_set_plain_is_disjoint(set1, set2);
2551 }
2552
2553 /* Check if we can combine a given div with lower bound l and upper
2554  * bound u with some other div and if so return that other div.
2555  * Otherwise return -1.
2556  *
2557  * We first check that
2558  *      - the bounds are opposites of each other (except for the constant
2559  *        term)
2560  *      - the bounds do not reference any other div
2561  *      - no div is defined in terms of this div
2562  *
2563  * Let m be the size of the range allowed on the div by the bounds.
2564  * That is, the bounds are of the form
2565  *
2566  *      e <= a <= e + m - 1
2567  *
2568  * with e some expression in the other variables.
2569  * We look for another div b such that no third div is defined in terms
2570  * of this second div b and such that in any constraint that contains
2571  * a (except for the given lower and upper bound), also contains b
2572  * with a coefficient that is m times that of b.
2573  * That is, all constraints (execpt for the lower and upper bound)
2574  * are of the form
2575  *
2576  *      e + f (a + m b) >= 0
2577  *
2578  * If so, we return b so that "a + m b" can be replaced by
2579  * a single div "c = a + m b".
2580  */
2581 static int div_find_coalesce(struct isl_basic_map *bmap, int *pairs,
2582         unsigned div, unsigned l, unsigned u)
2583 {
2584         int i, j;
2585         unsigned dim;
2586         int coalesce = -1;
2587
2588         if (bmap->n_div <= 1)
2589                 return -1;
2590         dim = isl_space_dim(bmap->dim, isl_dim_all);
2591         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim, div) != -1)
2592                 return -1;
2593         if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim + div + 1,
2594                                    bmap->n_div - div - 1) != -1)
2595                 return -1;
2596         if (!isl_seq_is_neg(bmap->ineq[l] + 1, bmap->ineq[u] + 1,
2597                             dim + bmap->n_div))
2598                 return -1;
2599
2600         for (i = 0; i < bmap->n_div; ++i) {
2601                 if (isl_int_is_zero(bmap->div[i][0]))
2602                         continue;
2603                 if (!isl_int_is_zero(bmap->div[i][1 + 1 + dim + div]))
2604                         return -1;
2605         }
2606
2607         isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2608         if (isl_int_is_neg(bmap->ineq[l][0])) {
2609                 isl_int_sub(bmap->ineq[l][0],
2610                             bmap->ineq[l][0], bmap->ineq[u][0]);
2611                 bmap = isl_basic_map_copy(bmap);
2612                 bmap = isl_basic_map_set_to_empty(bmap);
2613                 isl_basic_map_free(bmap);
2614                 return -1;
2615         }
2616         isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2617         for (i = 0; i < bmap->n_div; ++i) {
2618                 if (i == div)
2619                         continue;
2620                 if (!pairs[i])
2621                         continue;
2622                 for (j = 0; j < bmap->n_div; ++j) {
2623                         if (isl_int_is_zero(bmap->div[j][0]))
2624                                 continue;
2625                         if (!isl_int_is_zero(bmap->div[j][1 + 1 + dim + i]))
2626                                 break;
2627                 }
2628                 if (j < bmap->n_div)
2629                         continue;
2630                 for (j = 0; j < bmap->n_ineq; ++j) {
2631                         int valid;
2632                         if (j == l || j == u)
2633                                 continue;
2634                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + div]))
2635                                 continue;
2636                         if (isl_int_is_zero(bmap->ineq[j][1 + dim + i]))
2637                                 break;
2638                         isl_int_mul(bmap->ineq[j][1 + dim + div],
2639                                     bmap->ineq[j][1 + dim + div],
2640                                     bmap->ineq[l][0]);
2641                         valid = isl_int_eq(bmap->ineq[j][1 + dim + div],
2642                                            bmap->ineq[j][1 + dim + i]);
2643                         isl_int_divexact(bmap->ineq[j][1 + dim + div],
2644                                          bmap->ineq[j][1 + dim + div],
2645                                          bmap->ineq[l][0]);
2646                         if (!valid)
2647                                 break;
2648                 }
2649                 if (j < bmap->n_ineq)
2650                         continue;
2651                 coalesce = i;
2652                 break;
2653         }
2654         isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
2655         isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
2656         return coalesce;
2657 }
2658
2659 /* Given a lower and an upper bound on div i, construct an inequality
2660  * that when nonnegative ensures that this pair of bounds always allows
2661  * for an integer value of the given div.
2662  * The lower bound is inequality l, while the upper bound is inequality u.
2663  * The constructed inequality is stored in ineq.
2664  * g, fl, fu are temporary scalars.
2665  *
2666  * Let the upper bound be
2667  *
2668  *      -n_u a + e_u >= 0
2669  *
2670  * and the lower bound
2671  *
2672  *      n_l a + e_l >= 0
2673  *
2674  * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
2675  * We have
2676  *
2677  *      - f_u e_l <= f_u f_l g a <= f_l e_u
2678  *
2679  * Since all variables are integer valued, this is equivalent to
2680  *
2681  *      - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
2682  *
2683  * If this interval is at least f_u f_l g, then it contains at least
2684  * one integer value for a.
2685  * That is, the test constraint is
2686  *
2687  *      f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
2688  */
2689 static void construct_test_ineq(struct isl_basic_map *bmap, int i,
2690         int l, int u, isl_int *ineq, isl_int g, isl_int fl, isl_int fu)
2691 {
2692         unsigned dim;
2693         dim = isl_space_dim(bmap->dim, isl_dim_all);
2694
2695         isl_int_gcd(g, bmap->ineq[l][1 + dim + i], bmap->ineq[u][1 + dim + i]);
2696         isl_int_divexact(fl, bmap->ineq[l][1 + dim + i], g);
2697         isl_int_divexact(fu, bmap->ineq[u][1 + dim + i], g);
2698         isl_int_neg(fu, fu);
2699         isl_seq_combine(ineq, fl, bmap->ineq[u], fu, bmap->ineq[l],
2700                         1 + dim + bmap->n_div);
2701         isl_int_add(ineq[0], ineq[0], fl);
2702         isl_int_add(ineq[0], ineq[0], fu);
2703         isl_int_sub_ui(ineq[0], ineq[0], 1);
2704         isl_int_mul(g, g, fl);
2705         isl_int_mul(g, g, fu);
2706         isl_int_sub(ineq[0], ineq[0], g);
2707 }
2708
2709 /* Remove more kinds of divs that are not strictly needed.
2710  * In particular, if all pairs of lower and upper bounds on a div
2711  * are such that they allow at least one integer value of the div,
2712  * the we can eliminate the div using Fourier-Motzkin without
2713  * introducing any spurious solutions.
2714  */
2715 static struct isl_basic_map *drop_more_redundant_divs(
2716         struct isl_basic_map *bmap, int *pairs, int n)
2717 {
2718         struct isl_tab *tab = NULL;
2719         struct isl_vec *vec = NULL;
2720         unsigned dim;
2721         int remove = -1;
2722         isl_int g, fl, fu;
2723
2724         isl_int_init(g);
2725         isl_int_init(fl);
2726         isl_int_init(fu);
2727
2728         if (!bmap)
2729                 goto error;
2730
2731         dim = isl_space_dim(bmap->dim, isl_dim_all);
2732         vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
2733         if (!vec)
2734                 goto error;
2735
2736         tab = isl_tab_from_basic_map(bmap, 0);
2737
2738         while (n > 0) {
2739                 int i, l, u;
2740                 int best = -1;
2741                 enum isl_lp_result res;
2742
2743                 for (i = 0; i < bmap->n_div; ++i) {
2744                         if (!pairs[i])
2745                                 continue;
2746                         if (best >= 0 && pairs[best] <= pairs[i])
2747                                 continue;
2748                         best = i;
2749                 }
2750
2751                 i = best;
2752                 for (l = 0; l < bmap->n_ineq; ++l) {
2753                         if (!isl_int_is_pos(bmap->ineq[l][1 + dim + i]))
2754                                 continue;
2755                         for (u = 0; u < bmap->n_ineq; ++u) {
2756                                 if (!isl_int_is_neg(bmap->ineq[u][1 + dim + i]))
2757                                         continue;
2758                                 construct_test_ineq(bmap, i, l, u,
2759                                                     vec->el, g, fl, fu);
2760                                 res = isl_tab_min(tab, vec->el,
2761                                                   bmap->ctx->one, &g, NULL, 0);
2762                                 if (res == isl_lp_error)
2763                                         goto error;
2764                                 if (res == isl_lp_empty) {
2765                                         bmap = isl_basic_map_set_to_empty(bmap);
2766                                         break;
2767                                 }
2768                                 if (res != isl_lp_ok || isl_int_is_neg(g))
2769                                         break;
2770                         }
2771                         if (u < bmap->n_ineq)
2772                                 break;
2773                 }
2774                 if (l == bmap->n_ineq) {
2775                         remove = i;
2776                         break;
2777                 }
2778                 pairs[i] = 0;
2779                 --n;
2780         }
2781
2782         isl_tab_free(tab);
2783         isl_vec_free(vec);
2784
2785         isl_int_clear(g);
2786         isl_int_clear(fl);
2787         isl_int_clear(fu);
2788
2789         free(pairs);
2790
2791         if (remove < 0)
2792                 return bmap;
2793
2794         bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1);
2795         return isl_basic_map_drop_redundant_divs(bmap);
2796 error:
2797         free(pairs);
2798         isl_basic_map_free(bmap);
2799         isl_tab_free(tab);
2800         isl_vec_free(vec);
2801         isl_int_clear(g);
2802         isl_int_clear(fl);
2803         isl_int_clear(fu);
2804         return NULL;
2805 }
2806
2807 /* Given a pair of divs div1 and div2 such that, expect for the lower bound l
2808  * and the upper bound u, div1 always occurs together with div2 in the form
2809  * (div1 + m div2), where m is the constant range on the variable div1
2810  * allowed by l and u, replace the pair div1 and div2 by a single
2811  * div that is equal to div1 + m div2.
2812  *
2813  * The new div will appear in the location that contains div2.
2814  * We need to modify all constraints that contain
2815  * div2 = (div - div1) / m
2816  * (If a constraint does not contain div2, it will also not contain div1.)
2817  * If the constraint also contains div1, then we know they appear
2818  * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
2819  * i.e., the coefficient of div is f.
2820  *
2821  * Otherwise, we first need to introduce div1 into the constraint.
2822  * Let the l be
2823  *
2824  *      div1 + f >=0
2825  *
2826  * and u
2827  *
2828  *      -div1 + f' >= 0
2829  *
2830  * A lower bound on div2
2831  *
2832  *      n div2 + t >= 0
2833  *
2834  * can be replaced by
2835  *
2836  *      (n * (m div 2 + div1) + m t + n f)/g >= 0
2837  *
2838  * with g = gcd(m,n).
2839  * An upper bound
2840  *
2841  *      -n div2 + t >= 0
2842  *
2843  * can be replaced by
2844  *
2845  *      (-n * (m div2 + div1) + m t + n f')/g >= 0
2846  *
2847  * These constraint are those that we would obtain from eliminating
2848  * div1 using Fourier-Motzkin.
2849  *
2850  * After all constraints have been modified, we drop the lower and upper
2851  * bound and then drop div1.
2852  */
2853 static struct isl_basic_map *coalesce_divs(struct isl_basic_map *bmap,
2854         unsigned div1, unsigned div2, unsigned l, unsigned u)
2855 {
2856         isl_int a;
2857         isl_int b;
2858         isl_int m;
2859         unsigned dim, total;
2860         int i;
2861
2862         dim = isl_space_dim(bmap->dim, isl_dim_all);
2863         total = 1 + dim + bmap->n_div;
2864
2865         isl_int_init(a);
2866         isl_int_init(b);
2867         isl_int_init(m);
2868         isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
2869         isl_int_add_ui(m, m, 1);
2870
2871         for (i = 0; i < bmap->n_ineq; ++i) {
2872                 if (i == l || i == u)
2873                         continue;
2874                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div2]))
2875                         continue;
2876                 if (isl_int_is_zero(bmap->ineq[i][1 + dim + div1])) {
2877                         isl_int_gcd(b, m, bmap->ineq[i][1 + dim + div2]);
2878                         isl_int_divexact(a, m, b);
2879                         isl_int_divexact(b, bmap->ineq[i][1 + dim + div2], b);
2880                         if (isl_int_is_pos(b)) {
2881                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2882                                                 b, bmap->ineq[l], total);
2883                         } else {
2884                                 isl_int_neg(b, b);
2885                                 isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
2886                                                 b, bmap->ineq[u], total);
2887                         }
2888                 }
2889                 isl_int_set(bmap->ineq[i][1 + dim + div2],
2890                             bmap->ineq[i][1 + dim + div1]);
2891                 isl_int_set_si(bmap->ineq[i][1 + dim + div1], 0);
2892         }
2893
2894         isl_int_clear(a);
2895         isl_int_clear(b);
2896         isl_int_clear(m);
2897         if (l > u) {
2898                 isl_basic_map_drop_inequality(bmap, l);
2899                 isl_basic_map_drop_inequality(bmap, u);
2900         } else {
2901                 isl_basic_map_drop_inequality(bmap, u);
2902                 isl_basic_map_drop_inequality(bmap, l);
2903         }
2904         bmap = isl_basic_map_drop_div(bmap, div1);
2905         return bmap;
2906 }
2907
2908 /* First check if we can coalesce any pair of divs and
2909  * then continue with dropping more redundant divs.
2910  *
2911  * We loop over all pairs of lower and upper bounds on a div
2912  * with coefficient 1 and -1, respectively, check if there
2913  * is any other div "c" with which we can coalesce the div
2914  * and if so, perform the coalescing.
2915  */
2916 static struct isl_basic_map *coalesce_or_drop_more_redundant_divs(
2917         struct isl_basic_map *bmap, int *pairs, int n)
2918 {
2919         int i, l, u;
2920         unsigned dim;
2921
2922         dim = isl_space_dim(bmap->dim, isl_dim_all);
2923
2924         for (i = 0; i < bmap->n_div; ++i) {
2925                 if (!pairs[i])
2926                         continue;
2927                 for (l = 0; l < bmap->n_ineq; ++l) {
2928                         if (!isl_int_is_one(bmap->ineq[l][1 + dim + i]))
2929                                 continue;
2930                         for (u = 0; u < bmap->n_ineq; ++u) {
2931                                 int c;
2932
2933                                 if (!isl_int_is_negone(bmap->ineq[u][1+dim+i]))
2934                                         continue;
2935                                 c = div_find_coalesce(bmap, pairs, i, l, u);
2936                                 if (c < 0)
2937                                         continue;
2938                                 free(pairs);
2939                                 bmap = coalesce_divs(bmap, i, c, l, u);
2940                                 return isl_basic_map_drop_redundant_divs(bmap);
2941                         }
2942                 }
2943         }
2944
2945         if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
2946                 return bmap;
2947
2948         return drop_more_redundant_divs(bmap, pairs, n);
2949 }
2950
2951 /* Remove divs that are not strictly needed.
2952  * In particular, if a div only occurs positively (or negatively)
2953  * in constraints, then it can simply be dropped.
2954  * Also, if a div occurs in only two constraints and if moreover
2955  * those two constraints are opposite to each other, except for the constant
2956  * term and if the sum of the constant terms is such that for any value
2957  * of the other values, there is always at least one integer value of the
2958  * div, i.e., if one plus this sum is greater than or equal to
2959  * the (absolute value) of the coefficent of the div in the constraints,
2960  * then we can also simply drop the div.
2961  *
2962  * We skip divs that appear in equalities or in the definition of other divs.
2963  * Divs that appear in the definition of other divs usually occur in at least
2964  * 4 constraints, but the constraints may have been simplified.
2965  *
2966  * If any divs are left after these simple checks then we move on
2967  * to more complicated cases in drop_more_redundant_divs.
2968  */
2969 struct isl_basic_map *isl_basic_map_drop_redundant_divs(
2970         struct isl_basic_map *bmap)
2971 {
2972         int i, j;
2973         unsigned off;
2974         int *pairs = NULL;
2975         int n = 0;
2976
2977         if (!bmap)
2978                 goto error;
2979
2980         off = isl_space_dim(bmap->dim, isl_dim_all);
2981         pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
2982         if (!pairs)
2983                 goto error;
2984
2985         for (i = 0; i < bmap->n_div; ++i) {
2986                 int pos, neg;
2987                 int last_pos, last_neg;
2988                 int redundant;
2989                 int defined;
2990
2991                 defined = !isl_int_is_zero(bmap->div[i][0]);
2992                 for (j = i; j < bmap->n_div; ++j)
2993                         if (!isl_int_is_zero(bmap->div[j][1 + 1 + off + i]))
2994                                 break;
2995                 if (j < bmap->n_div)
2996                         continue;
2997                 for (j = 0; j < bmap->n_eq; ++j)
2998                         if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
2999                                 break;
3000                 if (j < bmap->n_eq)
3001                         continue;
3002                 ++n;
3003                 pos = neg = 0;
3004                 for (j = 0; j < bmap->n_ineq; ++j) {
3005                         if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
3006                                 last_pos = j;
3007                                 ++pos;
3008                         }
3009                         if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
3010                                 last_neg = j;
3011                                 ++neg;
3012                         }
3013                 }
3014                 pairs[i] = pos * neg;
3015                 if (pairs[i] == 0) {
3016                         for (j = bmap->n_ineq - 1; j >= 0; --j)
3017                                 if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
3018                                         isl_basic_map_drop_inequality(bmap, j);
3019                         bmap = isl_basic_map_drop_div(bmap, i);
3020                         free(pairs);
3021                         return isl_basic_map_drop_redundant_divs(bmap);
3022                 }
3023                 if (pairs[i] != 1)
3024                         continue;
3025                 if (!isl_seq_is_neg(bmap->ineq[last_pos] + 1,
3026                                     bmap->ineq[last_neg] + 1,
3027                                     off + bmap->n_div))
3028                         continue;
3029
3030                 isl_int_add(bmap->ineq[last_pos][0],
3031                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
3032                 isl_int_add_ui(bmap->ineq[last_pos][0],
3033                                bmap->ineq[last_pos][0], 1);
3034                 redundant = isl_int_ge(bmap->ineq[last_pos][0],
3035                                 bmap->ineq[last_pos][1+off+i]);
3036                 isl_int_sub_ui(bmap->ineq[last_pos][0],
3037                                bmap->ineq[last_pos][0], 1);
3038                 isl_int_sub(bmap->ineq[last_pos][0],
3039                             bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
3040                 if (!redundant) {
3041                         if (defined ||
3042                             !ok_to_set_div_from_bound(bmap, i, last_pos)) {
3043                                 pairs[i] = 0;
3044                                 --n;
3045                                 continue;
3046                         }
3047                         bmap = set_div_from_lower_bound(bmap, i, last_pos);
3048                         bmap = isl_basic_map_simplify(bmap);
3049                         free(pairs);
3050                         return isl_basic_map_drop_redundant_divs(bmap);
3051                 }
3052                 if (last_pos > last_neg) {
3053                         isl_basic_map_drop_inequality(bmap, last_pos);
3054                         isl_basic_map_drop_inequality(bmap, last_neg);
3055                 } else {
3056                         isl_basic_map_drop_inequality(bmap, last_neg);
3057                         isl_basic_map_drop_inequality(bmap, last_pos);
3058                 }
3059                 bmap = isl_basic_map_drop_div(bmap, i);
3060                 free(pairs);
3061                 return isl_basic_map_drop_redundant_divs(bmap);
3062         }
3063
3064         if (n > 0)
3065                 return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);
3066
3067         free(pairs);
3068         return bmap;
3069 error:
3070         free(pairs);
3071         isl_basic_map_free(bmap);
3072         return NULL;
3073 }
3074
3075 struct isl_basic_set *isl_basic_set_drop_redundant_divs(
3076         struct isl_basic_set *bset)
3077 {
3078         return (struct isl_basic_set *)
3079             isl_basic_map_drop_redundant_divs((struct isl_basic_map *)bset);
3080 }
3081
3082 struct isl_map *isl_map_drop_redundant_divs(struct isl_map *map)
3083 {
3084         int i;
3085
3086         if (!map)
3087                 return NULL;
3088         for (i = 0; i < map->n; ++i) {
3089                 map->p[i] = isl_basic_map_drop_redundant_divs(map->p[i]);
3090                 if (!map->p[i])
3091                         goto error;
3092         }
3093         ISL_F_CLR(map, ISL_MAP_NORMALIZED);
3094         return map;
3095 error:
3096         isl_map_free(map);
3097         return NULL;
3098 }
3099
3100 struct isl_set *isl_set_drop_redundant_divs(struct isl_set *set)
3101 {
3102         return (struct isl_set *)
3103             isl_map_drop_redundant_divs((struct isl_map *)set);
3104 }