| blob | e4102ddf92b5aa311d2eedf40ba6a753e5a5fc1a |
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
3 * Copyright 2012-2013 Ecole Normale Superieure
4 * Copyright 2014-2015 INRIA Rocquencourt
5 * Copyright 2016 Sven Verdoolaege
6 *
7 * Use of this software is governed by the MIT license
8 *
9 * Written by Sven Verdoolaege, K.U.Leuven, Departement
10 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
11 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
12 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
13 * B.P. 105 - 78153 Le Chesnay, France
14 */
16 #include <isl_ctx_private.h>
17 #include <isl_map_private.h>
19 #include <isl/map.h>
20 #include <isl_seq.h>
22 #include <isl_space_private.h>
23 #include <isl_mat_private.h>
24 #include <isl_vec_private.h>
27 {
31 }
34 {
39 }
40 }
43 {
46 }
48 /* Drop n dimensions starting at first.
49 *
50 * In principle, this frees up some extra variables as the number
51 * of columns remains constant, but we would have to extend
52 * the div array too as the number of rows in this array is assumed
53 * to be equal to extra.
54 */
57 {
91 error:
94 }
98 {
119 }
123 error:
126 }
128 /* Move "n" divs starting at "first" to the end of the list of divs.
129 */
132 {
150 error:
153 }
155 /* Drop "n" dimensions of type "type" starting at "first".
156 *
157 * In principle, this frees up some extra variables as the number
158 * of columns remains constant, but we would have to extend
159 * the div array too as the number of rows in this array is assumed
160 * to be equal to extra.
161 */
164 {
207 error:
210 }
214 {
217 }
221 {
223 }
227 {
248 }
252 error:
255 }
259 {
261 }
265 {
267 }
269 /*
270 * We don't cow, as the div is assumed to be redundant.
271 */
274 {
293 }
295 }
308 }
313 error:
316 }
320 {
322 isl_int gcd;
335 }
338 }
346 }
348 }
356 }
359 }
366 }
370 }
374 {
377 }
379 /* Assuming the variable at position "pos" has an integer coefficient
380 * in integer division "div", extract it from this integer division.
381 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
382 * corresponds to the constant term.
383 *
384 * That is, the integer division is of the form
385 *
386 * floor((... + c * d * x_pos + ...)/d)
387 *
388 * Replace it by
389 *
390 * floor((... + 0 * x_pos + ...)/d) + c * x_pos
391 */
394 {
395 isl_int shift;
404 }
406 /* Check if integer division "div" has any integral coefficient
407 * (or constant term). If so, extract them from the integer division.
408 */
411 {
424 }
427 }
429 /* Check if any known integer division has any integral coefficient
430 * (or constant term). If so, extract them from the integer division.
431 */
434 {
448 }
451 }
453 /* Remove any common factor in numerator and denominator of the div expression,
454 * not taking into account the constant term.
455 * That is, if the div is of the form
456 *
457 * floor((a + m f(x))/(m d))
458 *
459 * then replace it by
460 *
461 * floor((floor(a/m) + f(x))/d)
462 *
463 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
464 * and can therefore not influence the result of the floor.
465 */
467 {
483 }
485 /* Remove any common factor in numerator and denominator of a div expression,
486 * not taking into account the constant term.
487 * That is, look for any div of the form
488 *
489 * floor((a + m f(x))/(m d))
490 *
491 * and replace it by
492 *
493 * floor((floor(a/m) + f(x))/d)
494 *
495 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
496 * and can therefore not influence the result of the floor.
497 */
500 {
512 }
514 /* Assumes divs have been ordered if keep_divs is set.
515 */
518 {
536 }
546 }
555 /* We need to be careful about circular definitions,
556 * so for now we just remove the definition of div k
557 * if the equality contains any divs.
558 * If keep_divs is set, then the divs have been ordered
559 * and we can keep the definition as long as the result
560 * is still ordered.
561 */
569 }
570 }
572 /* Assumes divs have been ordered if keep_divs is set.
573 */
576 {
584 }
586 /* Check if elimination of div "div" using equality "eq" would not
587 * result in a div depending on a later div.
588 */
591 {
606 }
609 }
611 /* Elimininate divs based on equalities
612 */
615 {
641 }
642 }
646 }
648 /* Elimininate divs based on inequalities
649 */
652 {
682 }
684 }
688 {
711 }
720 progress);
732 }
733 }
740 }
743 }
747 {
750 }
754 {
760 }
762 }
764 /* Hash table of inequalities in a basic map.
765 * "index" is an array of addresses of inequalities in the basic map, some
766 * of which are NULL. The inequalities are hashed on the coefficients
767 * except the constant term.
768 * "size" is the number of elements in the array and is always a power of two
769 * "bits" is the number of bits need to represent an index into the array.
770 * "total" is the total dimension of the basic map.
771 */
777 };
779 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
780 */
783 {
800 }
802 /* Free the memory allocated by create_constraint_index.
803 */
805 {
807 }
809 /* Return the position in ci->index that contains the address of
810 * an inequality that is equal to *ineq up to the constant term,
811 * provided this address is not identical to "ineq".
812 * If there is no such inequality, then return the position where
813 * such an inequality should be inserted.
814 */
816 {
824 }
826 /* Return the position in ci->index that contains the address of
827 * an inequality that is equal to the k'th inequality of "bmap"
828 * up to the constant term, provided it does not point to the very
829 * same inequality.
830 * If there is no such inequality, then return the position where
831 * such an inequality should be inserted.
832 */
835 {
837 }
841 {
843 }
845 /* Fill in the "ci" data structure with the inequalities of "bset".
846 */
849 {
858 }
861 }
863 /* Is the inequality ineq (obviously) redundant with respect
864 * to the constraints in "ci"?
865 *
866 * Look for an inequality in "ci" with the same coefficients and then
867 * check if the contant term of "ineq" is greater than or equal
868 * to the constant term of that inequality. If so, "ineq" is clearly
869 * redundant.
870 *
871 * Note that hash_index_ineq ignores a stored constraint if it has
872 * the same address as the passed inequality. It is ok to pass
873 * the address of a local variable here since it will never be
874 * the same as the address of a constraint in "ci".
875 */
878 {
885 }
887 /* If we can eliminate more than one div, then we need to make
888 * sure we do it from last div to first div, in order not to
889 * change the position of the other divs that still need to
890 * be removed.
891 */
894 {
948 }
950 }
962 }
965 out:
969 }
972 {
984 }
986 }
988 /* Normalize divs that appear in equalities.
989 *
990 * In particular, we assume that bmap contains some equalities
991 * of the form
992 *
993 * a x = m * e_i
994 *
995 * and we want to replace the set of e_i by a minimal set and
996 * such that the new e_i have a canonical representation in terms
997 * of the vector x.
998 * If any of the equalities involves more than one divs, then
999 * we currently simply bail out.
1000 *
1001 * Let us first additionally assume that all equalities involve
1002 * a div. The equalities then express modulo constraints on the
1003 * remaining variables and we can use "parameter compression"
1004 * to find a minimal set of constraints. The result is a transformation
1005 *
1006 * x = T(x') = x_0 + G x'
1007 *
1008 * with G a lower-triangular matrix with all elements below the diagonal
1009 * non-negative and smaller than the diagonal element on the same row.
1010 * We first normalize x_0 by making the same property hold in the affine
1011 * T matrix.
1012 * The rows i of G with a 1 on the diagonal do not impose any modulo
1013 * constraint and simply express x_i = x'_i.
1014 * For each of the remaining rows i, we introduce a div and a corresponding
1015 * equality. In particular
1016 *
1017 * g_ii e_j = x_i - g_i(x')
1018 *
1019 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
1020 * corresponding div (if g_kk != 1).
1021 *
1022 * If there are any equalities not involving any div, then we
1023 * first apply a variable compression on the variables x:
1024 *
1025 * x = C x'' x'' = C_2 x
1026 *
1027 * and perform the above parameter compression on A C instead of on A.
1028 * The resulting compression is then of the form
1029 *
1030 * x'' = T(x') = x_0 + G x'
1031 *
1032 * and in constructing the new divs and the corresponding equalities,
1033 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
1034 * by the corresponding row from C_2.
1035 */
1038 {
1047 isl_int v;
1079 }
1080 }
1089 }
1095 }
1105 }
1112 }
1117 /* We have to be careful because dropping equalities may reorder them */
1127 }
1128 }
1134 else
1135 needed++;
1136 }
1142 }
1152 else
1160 else
1163 }
1167 }
1174 done:
1178 error:
1183 }
1187 {
1197 }
1199 /* Check whether it is ok to define a div based on an inequality.
1200 * To avoid the introduction of circular definitions of divs, we
1201 * do not allow such a definition if the resulting expression would refer to
1202 * any other undefined divs or if any known div is defined in
1203 * terms of the unknown div.
1204 */
1207 {
1211 /* Not defined in terms of unknown divs */
1219 }
1221 /* No other div defined in terms of this one => avoid loops */
1229 }
1232 }
1234 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1235 * be a better expression than the current one?
1236 *
1237 * If we do not have any expression yet, then any expression would be better.
1238 * Otherwise we check if the last variable involved in the inequality
1239 * (disregarding the div that it would define) is in an earlier position
1240 * than the last variable involved in the current div expression.
1241 */
1244 {
1261 }
1263 /* Given two constraints "k" and "l" that are opposite to each other,
1264 * except for the constant term, check if we can use them
1265 * to obtain an expression for one of the hitherto unknown divs or
1266 * a "better" expression for a div for which we already have an expression.
1267 * "sum" is the sum of the constant terms of the constraints.
1268 * If this sum is strictly smaller than the coefficient of one
1269 * of the divs, then this pair can be used define the div.
1270 * To avoid the introduction of circular definitions of divs, we
1271 * do not use the pair if the resulting expression would refer to
1272 * any other undefined divs or if any known div is defined in
1273 * terms of the unknown div.
1274 */
1277 {
1292 else
1297 }
1299 }
1303 {
1307 isl_int sum;
1322 }
1330 }
1345 }
1347 /* We need to break out of the loop after these
1348 * changes since the contents of the hash
1349 * will no longer be valid.
1350 * Plus, we probably we want to regauss first.
1351 */
1359 }
1364 }
1366 /* Detect all pairs of inequalities that form an equality.
1367 *
1368 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1369 * Call it repeatedly while it is making progress.
1370 */
1373 {
1385 }
1387 /* Eliminate knowns divs from constraints where they appear with
1388 * a (positive or negative) unit coefficient.
1389 *
1390 * That is, replace
1391 *
1392 * floor(e/m) + f >= 0
1393 *
1394 * by
1395 *
1396 * e + m f >= 0
1397 *
1398 * and
1399 *
1400 * -floor(e/m) + f >= 0
1401 *
1402 * by
1403 *
1404 * -e + m f + m - 1 >= 0
1405 *
1406 * The first conversion is valid because floor(e/m) >= -f is equivalent
1407 * to e/m >= -f because -f is an integral expression.
1408 * The second conversion follows from the fact that
1409 *
1410 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1411 *
1412 *
1413 * Note that one of the div constraints may have been eliminated
1414 * due to being redundant with respect to the constraint that is
1415 * being modified by this function. The modified constraint may
1416 * no longer imply this div constraint, so we add it back to make
1417 * sure we do not lose any information.
1418 *
1419 * We skip integral divs, i.e., those with denominator 1, as we would
1420 * risk eliminating the div from the div constraints. We do not need
1421 * to handle those divs here anyway since the div constraints will turn
1422 * out to form an equality and this equality can then be use to eliminate
1423 * the div from all constraints.
1424 */
1427 {
1459 else
1469 }
1474 }
1475 }
1478 }
1481 {
1499 /* requires equalities in normal form */
1505 }
1507 }
1510 {
1513 }
1518 {
1550 }
1554 {
1556 }
1559 /* If the only constraints a div d=floor(f/m)
1560 * appears in are its two defining constraints
1561 *
1562 * f - m d >=0
1563 * -(f - (m - 1)) + m d >= 0
1564 *
1565 * then it can safely be removed.
1566 */
1568 {
1581 }
1588 }
1591 }
1593 /*
1594 * Remove divs that don't occur in any of the constraints or other divs.
1595 * These can arise when dropping constraints from a basic map or
1596 * when the divs of a basic map have been temporarily aligned
1597 * with the divs of another basic map.
1598 */
1600 {
1610 }
1612 }
1614 /* Mark "bmap" as final, without checking for obviously redundant
1615 * integer divisions. This function should be used when "bmap"
1616 * is known not to involve any such integer divisions.
1617 */
1620 {
1625 }
1627 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1628 */
1630 {
1634 }
1637 {
1640 }
1643 {
1652 }
1654 error:
1657 }
1660 {
1669 }
1672 error:
1675 }
1678 /* Remove definition of any div that is defined in terms of the given variable.
1679 * The div itself is not removed. Functions such as
1680 * eliminate_divs_ineq depend on the other divs remaining in place.
1681 */
1684 {
1698 }
1700 }
1702 /* Eliminate the specified variables from the constraints using
1703 * Fourier-Motzkin. The variables themselves are not removed.
1704 */
1707 {
1739 }
1746 n_lower++;
1748 n_upper++;
1749 }
1773 }
1776 }
1788 }
1789 }
1794 error:
1797 }
1801 {
1804 }
1806 /* Eliminate the specified n dimensions starting at first from the
1807 * constraints, without removing the dimensions from the space.
1808 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1809 * Otherwise, they are projected out and the original space is restored.
1810 */
1814 {
1830 }
1837 error:
1840 }
1845 {
1847 }
1849 /* Remove all constraints from "bmap" that reference any unknown local
1850 * variables (directly or indirectly).
1851 *
1852 * Dropping all constraints on a local variable will make it redundant,
1853 * so it will get removed implicitly by
1854 * isl_basic_map_drop_constraints_involving_dims. Some other local
1855 * variables may also end up becoming redundant if they only appear
1856 * in constraints together with the unknown local variable.
1857 * Therefore, start over after calling
1858 * isl_basic_map_drop_constraints_involving_dims.
1859 */
1862 {
1863 isl_bool known;
1888 }
1891 }
1893 /* Remove all constraints from "map" that reference any unknown local
1894 * variables (directly or indirectly).
1895 *
1896 * Since constraints may get dropped from the basic maps,
1897 * they may no longer be disjoint from each other.
1898 */
1901 {
1903 isl_bool known;
1921 }
1927 }
1929 /* Don't assume equalities are in order, because align_divs
1930 * may have changed the order of the divs.
1931 */
1933 {
1946 }
1947 }
1948 }
1951 {
1953 }
1957 {
1971 }
1973 }
1975 }
1979 {
1982 }
1986 {
1996 }
2017 error:
2021 }
2023 /* For each inequality in "ineq" that is a shifted (more relaxed)
2024 * copy of an inequality in "context", mark the corresponding entry
2025 * in "row" with -1.
2026 * If an inequality only has a non-negative constant term, then
2027 * mark it as well.
2028 */
2031 {
2048 isl_bool redundant;
2054 }
2061 }
2064 error:
2067 }
2071 {
2084 isl_bool redundant;
2096 }
2099 error:
2102 }
2104 /* Remove constraints from "bmap" that are identical to constraints
2105 * in "context" or that are more relaxed (greater constant term).
2106 *
2107 * We perform the test for shifted copies on the pure constraints
2108 * in remove_shifted_constraints.
2109 */
2112 {
2121 }
2134 error:
2138 }
2140 /* Does the (linear part of a) constraint "c" involve any of the "len"
2141 * "relevant" dimensions?
2142 */
2144 {
2152 }
2155 }
2157 /* Drop constraints from "bmap" that do not involve any of
2158 * the dimensions marked "relevant".
2159 */
2162 {
2177 }
2184 }
2187 }
2189 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2190 *
2191 * In particular, for any variable involved in the constraint,
2192 * find the actual group id from before and replace the group
2193 * of the corresponding variable by the minimal group of all
2194 * the variables involved in the constraint considered so far
2195 * (if this minimum is smaller) or replace the minimum by this group
2196 * (if the minimum is larger).
2197 *
2198 * At the end, all the variables in "c" will (indirectly) point
2199 * to the minimal of the groups that they referred to originally.
2200 */
2202 {
2219 }
2220 }
2222 /* Allocate an array of groups of variables, one for each variable
2223 * in "context", initialized to zero.
2224 */
2226 {
2233 }
2235 /* Drop constraints from "bmap" that only involve variables that are
2236 * not related to any of the variables marked with a "-1" in "group".
2237 *
2238 * We construct groups of variables that collect variables that
2239 * (indirectly) appear in some common constraint of "bmap".
2240 * Each group is identified by the first variable in the group,
2241 * except for the special group of variables that was already identified
2242 * in the input as -1 (or are related to those variables).
2243 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2244 * otherwise the group of i is the group of group[i].
2245 *
2246 * We first initialize groups for the remaining variables.
2247 * Then we iterate over the constraints of "bmap" and update the
2248 * group of the variables in the constraint by the smallest group.
2249 * Finally, we resolve indirect references to groups by running over
2250 * the variables.
2251 *
2252 * After computing the groups, we drop constraints that do not involve
2253 * any variables in the -1 group.
2254 */
2257 {
2274 }
2292 }
2294 /* Drop constraints from "context" that are irrelevant for computing
2295 * the gist of "bset".
2296 *
2297 * In particular, drop constraints in variables that are not related
2298 * to any of the variables involved in the constraints of "bset"
2299 * in the sense that there is no sequence of constraints that connects them.
2300 *
2301 * We first mark all variables that appear in "bset" as belonging
2302 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2303 */
2306 {
2327 }
2333 }
2336 }
2338 /* Drop constraints from "context" that are irrelevant for computing
2339 * the gist of the inequalities "ineq".
2340 * Inequalities in "ineq" for which the corresponding element of row
2341 * is set to -1 have already been marked for removal and should be ignored.
2342 *
2343 * In particular, drop constraints in variables that are not related
2344 * to any of the variables involved in "ineq"
2345 * in the sense that there is no sequence of constraints that connects them.
2346 *
2347 * We first mark all variables that appear in "bset" as belonging
2348 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2349 */
2352 {
2373 }
2376 }
2379 }
2381 /* Do all "n" entries of "row" contain a negative value?
2382 */
2384 {
2392 }
2394 /* Update the inequalities in "bset" based on the information in "row"
2395 * and "tab".
2396 *
2397 * In particular, the array "row" contains either -1, meaning that
2398 * the corresponding inequality of "bset" is redundant, or the index
2399 * of an inequality in "tab".
2400 *
2401 * If the row entry is -1, then drop the inequality.
2402 * Otherwise, if the constraint is marked redundant in the tableau,
2403 * then drop the inequality. Similarly, if it is marked as an equality
2404 * in the tableau, then turn the inequality into an equality and
2405 * perform Gaussian elimination.
2406 */
2409 {
2426 }
2436 }
2437 }
2443 }
2445 /* Update the inequalities in "bset" based on the information in "row"
2446 * and "tab" and free all arguments (other than "bset").
2447 */
2452 {
2461 }
2463 /* Remove all information from bset that is redundant in the context
2464 * of context.
2465 * "ineq" contains the (possibly transformed) inequalities of "bset",
2466 * in the same order.
2467 * The (explicit) equalities of "bset" are assumed to have been taken
2468 * into account by the transformation such that only the inequalities
2469 * are relevant.
2470 * "context" is assumed not to be empty.
2471 *
2472 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2473 * A value of -1 means that the inequality is obviously redundant and may
2474 * not even appear in "tab".
2475 *
2476 * We first mark the inequalities of "bset"
2477 * that are obviously redundant with respect to some inequality in "context".
2478 * Then we remove those constraints from "context" that have become
2479 * irrelevant for computing the gist of "bset".
2480 * Note that this removal of constraints cannot be replaced by
2481 * a factorization because factors in "bset" may still be connected
2482 * to each other through constraints in "context".
2483 *
2484 * If there are any inequalities left, we construct a tableau for
2485 * the context and then add the inequalities of "bset".
2486 * Before adding these inequalities, we freeze all constraints such that
2487 * they won't be considered redundant in terms of the constraints of "bset".
2488 * Then we detect all redundant constraints (among the
2489 * constraints that weren't frozen), first by checking for redundancy in the
2490 * the tableau and then by checking if replacing a constraint by its negation
2491 * would lead to an empty set. This last step is fairly expensive
2492 * and could be optimized by more reuse of the tableau.
2493 * Finally, we update bset according to the results.
2494 */
2497 {
2513 }
2549 }
2574 }
2579 }
2583 error:
2591 }
2593 /* Extract the inequalities of "bset" as an isl_mat.
2594 */
2596 {
2610 }
2612 /* Remove all information from "bset" that is redundant in the context
2613 * of "context", for the case where both "bset" and "context" are
2614 * full-dimensional.
2615 */
2618 {
2623 }
2625 /* Remove all information from "bset" that is redundant in the context
2626 * of "context", for the case where the combined equalities of
2627 * "bset" and "context" allow for a compression that can be obtained
2628 * by preapplication of "T".
2629 *
2630 * "bset" itself is not transformed by "T". Instead, the inequalities
2631 * are extracted from "bset" and those are transformed by "T".
2632 * uset_gist_full then determines which of the transformed inequalities
2633 * are redundant with respect to the transformed "context" and removes
2634 * the corresponding inequalities from "bset".
2635 *
2636 * After preapplying "T" to the inequalities, any common factor is
2637 * removed from the coefficients. If this results in a tightening
2638 * of the constant term, then the same tightening is applied to
2639 * the corresponding untransformed inequality in "bset".
2640 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2641 *
2642 * g f'(x) + r >= 0
2643 *
2644 * with 0 <= r < g, then it is equivalent to
2645 *
2646 * f'(x) >= 0
2647 *
2648 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2649 * subspace compressed by T since the latter would be transformed to
2650 *
2651 * g f'(x) >= 0
2652 */
2656 {
2660 isl_int rem;
2672 }
2695 }
2699 error:
2704 }
2706 /* Project "bset" onto the variables that are involved in "template".
2707 */
2710 {
2719 isl_bool involved;
2728 }
2731 }
2733 /* Remove all information from bset that is redundant in the context
2734 * of context. In particular, equalities that are linear combinations
2735 * of those in context are removed. Then the inequalities that are
2736 * redundant in the context of the equalities and inequalities of
2737 * context are removed.
2738 *
2739 * First of all, we drop those constraints from "context"
2740 * that are irrelevant for computing the gist of "bset".
2741 * Alternatively, we could factorize the intersection of "context" and "bset".
2742 *
2743 * We first compute the intersection of the integer affine hulls
2744 * of "bset" and "context",
2745 * compute the gist inside this intersection and then reduce
2746 * the constraints with respect to the equalities of the context
2747 * that only involve variables already involved in the input.
2748 *
2749 * If two constraints are mutually redundant, then uset_gist_full
2750 * will remove the second of those constraints. We therefore first
2751 * sort the constraints so that constraints not involving existentially
2752 * quantified variables are given precedence over those that do.
2753 * We have to perform this sorting before the variable compression,
2754 * because that may effect the order of the variables.
2755 */
2758 {
2783 }
2788 }
2798 }
2809 }
2812 error:
2816 }
2818 /* Return the number of equality constraints in "bmap" that involve
2819 * local variables. This function assumes that Gaussian elimination
2820 * has been applied to the equality constraints.
2821 */
2823 {
2843 }
2845 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2846 * The constraints are assumed not to involve any local variables.
2847 */
2850 {
2868 }
2873 error:
2878 }
2880 /* Construct and return a variable compression based on the equality
2881 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2882 * "n1" is the number of (initial) equality constraints in "bmap1"
2883 * that do involve local variables.
2884 * "n2" is the number of (initial) equality constraints in "bmap2"
2885 * that do involve local variables.
2886 * "total" is the total number of other variables.
2887 * This function assumes that Gaussian elimination
2888 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2889 * such that the equality constraints not involving local variables
2890 * are those that start at "n1" or "n2".
2891 *
2892 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2893 * then simply compute the compression based on the equality constraints
2894 * in the other basic map.
2895 * Otherwise, combine the equality constraints from both into a new
2896 * basic map such that Gaussian elimination can be applied to this combination
2897 * and then construct a variable compression from the resulting
2898 * equality constraints.
2899 */
2903 {
2913 }
2918 }
2933 }
2935 /* Extract the stride constraints from "bmap", compressed
2936 * with respect to both the stride constraints in "context" and
2937 * the remaining equality constraints in both "bmap" and "context".
2938 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2939 * "context_n_eq" is the number of (initial) stride constraints in "context".
2940 *
2941 * Let x be all variables in "bmap" (and "context") other than the local
2942 * variables. First compute a variable compression
2943 *
2944 * x = V x'
2945 *
2946 * based on the non-stride equality constraints in "bmap" and "context".
2947 * Consider the stride constraints of "context",
2948 *
2949 * A(x) + B(y) = 0
2950 *
2951 * with y the local variables and plug in the variable compression,
2952 * resulting in
2953 *
2954 * A(V x') + B(y) = 0
2955 *
2956 * Use these constraints to compute a parameter compression on x'
2957 *
2958 * x' = T x''
2959 *
2960 * Now consider the stride constraints of "bmap"
2961 *
2962 * C(x) + D(y) = 0
2963 *
2964 * and plug in x = V*T x''.
2965 * That is, return A = [C*V*T D].
2966 */
2970 {
2999 }
3001 /* Remove the prime factors from *g that have an exponent that
3002 * is strictly smaller than the exponent in "c".
3003 * All exponents in *g are known to be smaller than or equal
3004 * to those in "c".
3005 *
3006 * That is, if *g is equal to
3007 *
3008 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
3009 *
3010 * and "c" is equal to
3011 *
3012 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
3013 *
3014 * then update *g to
3015 *
3016 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
3017 * p_n^{e_n * (e_n = f_n)}
3018 *
3019 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
3020 * neither does the gcd of *g and c / *g.
3021 * If e_i < f_i, then the gcd of *g and c / *g has a positive
3022 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
3023 * Dividing *g by this gcd therefore strictly reduces the exponent
3024 * of the prime factors that need to be removed, while leaving the
3025 * other prime factors untouched.
3026 * Repeating this process until gcd(*g, c / *g) = 1 therefore
3027 * removes all undesired factors, without removing any others.
3028 */
3030 {
3031 isl_int t;
3040 }
3042 }
3044 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
3045 * of the same stride constraints in a compressed space that exploits
3046 * all equalities in the context and the other equalities in "bmap".
3047 *
3048 * If the stride constraints of "bmap" are of the form
3049 *
3050 * C(x) + D(y) = 0
3051 *
3052 * then A is of the form
3053 *
3054 * B(x') + D(y) = 0
3055 *
3056 * If any of these constraints involves only a single local variable y,
3057 * then the constraint appears as
3058 *
3059 * f(x) + m y_i = 0
3060 *
3061 * in "bmap" and as
3062 *
3063 * h(x') + m y_i = 0
3064 *
3065 * in "A".
3066 *
3067 * Let g be the gcd of m and the coefficients of h.
3068 * Then, in particular, g is a divisor of the coefficients of h and
3069 *
3070 * f(x) = h(x')
3071 *
3072 * is known to be a multiple of g.
3073 * If some prime factor in m appears with the same exponent in g,
3074 * then it can be removed from m because f(x) is already known
3075 * to be a multiple of g and therefore in particular of this power
3076 * of the prime factors.
3077 * Prime factors that appear with a smaller exponent in g cannot
3078 * be removed from m.
3079 * Let g' be the divisor of g containing all prime factors that
3080 * appear with the same exponent in m and g, then
3081 *
3082 * f(x) + m y_i = 0
3083 *
3084 * can be replaced by
3085 *
3086 * f(x) + m/g' y_i' = 0
3087 *
3088 * Note that (if g' != 1) this changes the explicit representation
3089 * of y_i to that of y_i', so the integer division at position i
3090 * is marked unknown and later recomputed by a call to
3091 * isl_basic_map_gauss.
3092 */
3095 {
3099 isl_int gcd;
3133 }
3140 error:
3144 }
3146 /* Simplify the stride constraints in "bmap" based on
3147 * the remaining equality constraints in "bmap" and all equality
3148 * constraints in "context".
3149 * Only do this if both "bmap" and "context" have stride constraints.
3150 *
3151 * First extract a copy of the stride constraints in "bmap" in a compressed
3152 * space exploiting all the other equality constraints and then
3153 * use this compressed copy to simplify the original stride constraints.
3154 */
3157 {
3179 }
3181 /* Return a basic map that has the same intersection with "context" as "bmap"
3182 * and that is as "simple" as possible.
3183 *
3184 * The core computation is performed on the pure constraints.
3185 * When we add back the meaning of the integer divisions, we need
3186 * to (re)introduce the div constraints. If we happen to have
3187 * discovered that some of these integer divisions are equal to
3188 * some affine combination of other variables, then these div
3189 * constraints may end up getting simplified in terms of the equalities,
3190 * resulting in extra inequalities on the other variables that
3191 * may have been removed already or that may not even have been
3192 * part of the input. We try and remove those constraints of
3193 * this form that are most obviously redundant with respect to
3194 * the context. We also remove those div constraints that are
3195 * redundant with respect to the other constraints in the result.
3196 *
3197 * The stride constraints among the equality constraints in "bmap" are
3198 * also simplified with respecting to the other equality constraints
3199 * in "bmap" and with respect to all equality constraints in "context".
3200 */
3203 {
3214 }
3220 }
3224 }
3246 }
3265 error:
3269 }
3271 /*
3272 * Assumes context has no implicit divs.
3273 */
3276 {
3287 }
3307 }
3308 }
3312 error:
3316 }
3318 /* Drop all inequalities from "bmap" that also appear in "context".
3319 * "context" is assumed to have only known local variables and
3320 * the initial local variables of "bmap" are assumed to be the same
3321 * as those of "context".
3322 * The constraints of both "bmap" and "context" are assumed
3323 * to have been sorted using isl_basic_map_sort_constraints.
3324 *
3325 * Run through the inequality constraints of "bmap" and "context"
3326 * in sorted order.
3327 * If a constraint of "bmap" involves variables not in "context",
3328 * then it cannot appear in "context".
3329 * If a matching constraint is found, it is removed from "bmap".
3330 */
3333 {
3352 }
3358 }
3362 }
3367 }
3370 }
3373 }
3375 /* Drop all equalities from "bmap" that also appear in "context".
3376 * "context" is assumed to have only known local variables and
3377 * the initial local variables of "bmap" are assumed to be the same
3378 * as those of "context".
3379 *
3380 * Run through the equality constraints of "bmap" and "context"
3381 * in sorted order.
3382 * If a constraint of "bmap" involves variables not in "context",
3383 * then it cannot appear in "context".
3384 * If a matching constraint is found, it is removed from "bmap".
3385 */
3388 {
3412 }
3416 }
3421 }
3424 }
3427 }
3429 /* Remove the constraints in "context" from "bmap".
3430 * "context" is assumed to have explicit representations
3431 * for all local variables.
3432 *
3433 * First align the divs of "bmap" to those of "context" and
3434 * sort the constraints. Then drop all constraints from "bmap"
3435 * that appear in "context".
3436 */
3439 {
3454 }
3473 error:
3477 }
3479 /* Replace "map" by the disjunct at position "pos" and free "context".
3480 */
3483 {
3490 }
3492 /* Remove the constraints in "context" from "map".
3493 * If any of the disjuncts in the result turns out to be the universe,
3494 * then return this universe.
3495 * "context" is assumed to have explicit representations
3496 * for all local variables.
3497 */
3500 {
3510 }
3529 }
3536 error:
3540 }
3542 /* Replace "map" by a universe map in the same space and free "drop".
3543 */
3546 {
3553 }
3555 /* Return a map that has the same intersection with "context" as "map"
3556 * and that is as "simple" as possible.
3557 *
3558 * If "map" is already the universe, then we cannot make it any simpler.
3559 * Similarly, if "context" is the universe, then we cannot exploit it
3560 * to simplify "map"
3561 * If "map" and "context" are identical to each other, then we can
3562 * return the corresponding universe.
3563 *
3564 * If either "map" or "context" consists of multiple disjuncts,
3565 * then check if "context" happens to be a subset of "map",
3566 * in which case all constraints can be removed.
3567 * In case of multiple disjuncts, the standard procedure
3568 * may not be able to detect that all constraints can be removed.
3569 *
3570 * If none of these cases apply, we have to work a bit harder.
3571 * During this computation, we make use of a single disjunct context,
3572 * so if the original context consists of more than one disjunct
3573 * then we need to approximate the context by a single disjunct set.
3574 * Simply taking the simple hull may drop constraints that are
3575 * only implicitly available in each disjunct. We therefore also
3576 * look for constraints among those defining "map" that are valid
3577 * for the context. These can then be used to simplify away
3578 * the corresponding constraints in "map".
3579 */
3582 {
3586 isl_bool subset;
3597 }
3613 }
3629 list);
3630 }
3632 error:
3636 }
3640 {
3642 }
3646 {
3649 }
3653 {
3656 }
3660 {
3665 }
3669 {
3672 }
3674 /* Compute the gist of "bmap" with respect to the constraints "context"
3675 * on the domain.
3676 */
3679 {
3685 }
3689 {
3693 }
3697 {
3701 }
3705 {
3709 }
3713 {
3715 }
3717 /* Quick check to see if two basic maps are disjoint.
3718 * In particular, we reduce the equalities and inequalities of
3719 * one basic map in the context of the equalities of the other
3720 * basic map and check if we get a contradiction.
3721 */
3724 {
3756 }
3764 }
3773 }
3777 disjoint:
3781 error:
3785 }
3789 {
3792 }
3794 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3795 */
3799 {
3810 }
3811 }
3814 }
3816 /* Are "map1" and "map2" obviously disjoint, based on information
3817 * that can be derived without looking at the individual basic maps?
3818 *
3819 * In particular, if one of them is empty or if they live in different spaces
3820 * (ignoring parameters), then they are clearly disjoint.
3821 */
3824 {
3825 isl_bool disjoint;
3826 isl_bool match;
3850 }
3852 /* Are "map1" and "map2" obviously disjoint?
3853 *
3854 * If one of them is empty or if they live in different spaces (ignoring
3855 * parameters), then they are clearly disjoint.
3856 * This is checked by isl_map_plain_is_disjoint_global.
3857 *
3858 * If they have different parameters, then we skip any further tests.
3859 *
3860 * If they are obviously equal, but not obviously empty, then we will
3861 * not be able to detect if they are disjoint.
3862 *
3863 * Otherwise we check if each basic map in "map1" is obviously disjoint
3864 * from each basic map in "map2".
3865 */
3868 {
3869 isl_bool disjoint;
3870 isl_bool intersect;
3871 isl_bool match;
3887 }
3889 /* Are "map1" and "map2" disjoint?
3890 *
3891 * They are disjoint if they are "obviously disjoint" or if one of them
3892 * is empty. Otherwise, they are not disjoint if one of them is universal.
3893 * If the two inputs are (obviously) equal and not empty, then they are
3894 * not disjoint.
3895 * If none of these cases apply, then check if all pairs of basic maps
3896 * are disjoint.
3897 */
3899 {
3900 isl_bool disjoint;
3901 isl_bool intersect;
3928 }
3930 /* Are "bmap1" and "bmap2" disjoint?
3931 *
3932 * They are disjoint if they are "obviously disjoint" or if one of them
3933 * is empty. Otherwise, they are not disjoint if one of them is universal.
3934 * If none of these cases apply, we compute the intersection and see if
3935 * the result is empty.
3936 */
3939 {
3940 isl_bool disjoint;
3941 isl_bool intersect;
3970 }
3972 /* Are "bset1" and "bset2" disjoint?
3973 */
3976 {
3978 }
3982 {
3985 }
3987 /* Are "set1" and "set2" disjoint?
3988 */
3990 {
3992 }
3994 /* Is "v" equal to 0, 1 or -1?
3995 */
3997 {
3999 }
4001 /* Check if we can combine a given div with lower bound l and upper
4002 * bound u with some other div and if so return that other div.
4003 * Otherwise return -1.
4004 *
4005 * We first check that
4006 * - the bounds are opposites of each other (except for the constant
4007 * term)
4008 * - the bounds do not reference any other div
4009 * - no div is defined in terms of this div
4010 *
4011 * Let m be the size of the range allowed on the div by the bounds.
4012 * That is, the bounds are of the form
4013 *
4014 * e <= a <= e + m - 1
4015 *
4016 * with e some expression in the other variables.
4017 * We look for another div b such that no third div is defined in terms
4018 * of this second div b and such that in any constraint that contains
4019 * a (except for the given lower and upper bound), also contains b
4020 * with a coefficient that is m times that of b.
4021 * That is, all constraints (execpt for the lower and upper bound)
4022 * are of the form
4023 *
4024 * e + f (a + m b) >= 0
4025 *
4026 * Furthermore, in the constraints that only contain b, the coefficient
4027 * of b should be equal to 1 or -1.
4028 * If so, we return b so that "a + m b" can be replaced by
4029 * a single div "c = a + m b".
4030 */
4033 {
4055 }
4065 }
4077 }
4088 }
4101 }
4106 }
4110 }
4112 /* Internal data structure used during the construction and/or evaluation of
4113 * an inequality that ensures that a pair of bounds always allows
4114 * for an integer value.
4115 *
4116 * "tab" is the tableau in which the inequality is evaluated. It may
4117 * be NULL until it is actually needed.
4118 * "v" contains the inequality coefficients.
4119 * "g", "fl" and "fu" are temporary scalars used during the construction and
4120 * evaluation.
4121 */
4125 isl_int g;
4126 isl_int fl;
4127 isl_int fu;
4128 };
4130 /* Free all the memory allocated by the fields of "data".
4131 */
4133 {
4139 }
4141 /* Is the inequality stored in data->v satisfied by "bmap"?
4142 * That is, does it only attain non-negative values?
4143 * data->tab is a tableau corresponding to "bmap".
4144 */
4147 {
4158 }
4160 /* Given a lower and an upper bound on div i, do they always allow
4161 * for an integer value of the given div?
4162 * Determine this property by constructing an inequality
4163 * such that the property is guaranteed when the inequality is nonnegative.
4164 * The lower bound is inequality l, while the upper bound is inequality u.
4165 * The constructed inequality is stored in data->v.
4166 *
4167 * Let the upper bound be
4168 *
4169 * -n_u a + e_u >= 0
4170 *
4171 * and the lower bound
4172 *
4173 * n_l a + e_l >= 0
4174 *
4175 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4176 * We have
4177 *
4178 * - f_u e_l <= f_u f_l g a <= f_l e_u
4179 *
4180 * Since all variables are integer valued, this is equivalent to
4181 *
4182 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4183 *
4184 * If this interval is at least f_u f_l g, then it contains at least
4185 * one integer value for a.
4186 * That is, the test constraint is
4187 *
4188 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4189 *
4190 * or
4191 *
4192 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4193 *
4194 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4195 * then the constraint can be scaled down by a factor g',
4196 * with the constant term replaced by
4197 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4198 * Note that the result of applying Fourier-Motzkin to this pair
4199 * of constraints is
4200 *
4201 * f_l e_u + f_u e_l >= 0
4202 *
4203 * If the constant term of the scaled down version of this constraint,
4204 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4205 * term of the scaled down test constraint, then the test constraint
4206 * is known to hold and no explicit evaluation is required.
4207 * This is essentially the Omega test.
4208 *
4209 * If the test constraint consists of only a constant term, then
4210 * it is sufficient to look at the sign of this constant term.
4211 */
4214 {
4239 }
4249 }
4251 /* Remove more kinds of divs that are not strictly needed.
4252 * In particular, if all pairs of lower and upper bounds on a div
4253 * are such that they allow at least one integer value of the div,
4254 * then we can eliminate the div using Fourier-Motzkin without
4255 * introducing any spurious solutions.
4256 *
4257 * If at least one of the two constraints has a unit coefficient for the div,
4258 * then the presence of such a value is guaranteed so there is no need to check.
4259 * In particular, the value attained by the bound with unit coefficient
4260 * can serve as this intermediate value.
4261 */
4264 {
4287 isl_bool has_int;
4295 }
4316 }
4319 }
4323 }
4327 }
4330 }
4341 error:
4346 }
4348 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4349 * and the upper bound u, div1 always occurs together with div2 in the form
4350 * (div1 + m div2), where m is the constant range on the variable div1
4351 * allowed by l and u, replace the pair div1 and div2 by a single
4352 * div that is equal to div1 + m div2.
4353 *
4354 * The new div will appear in the location that contains div2.
4355 * We need to modify all constraints that contain
4356 * div2 = (div - div1) / m
4357 * The coefficient of div2 is known to be equal to 1 or -1.
4358 * (If a constraint does not contain div2, it will also not contain div1.)
4359 * If the constraint also contains div1, then we know they appear
4360 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4361 * i.e., the coefficient of div is f.
4362 *
4363 * Otherwise, we first need to introduce div1 into the constraint.
4364 * Let the l be
4365 *
4366 * div1 + f >=0
4367 *
4368 * and u
4369 *
4370 * -div1 + f' >= 0
4371 *
4372 * A lower bound on div2
4373 *
4374 * div2 + t >= 0
4375 *
4376 * can be replaced by
4377 *
4378 * m div2 + div1 + m t + f >= 0
4379 *
4380 * An upper bound
4381 *
4382 * -div2 + t >= 0
4383 *
4384 * can be replaced by
4385 *
4386 * -(m div2 + div1) + m t + f' >= 0
4387 *
4388 * These constraint are those that we would obtain from eliminating
4389 * div1 using Fourier-Motzkin.
4390 *
4391 * After all constraints have been modified, we drop the lower and upper
4392 * bound and then drop div1.
4393 */
4396 {
4398 isl_int m;
4420 else
4423 }
4427 }
4436 }
4439 }
4441 /* First check if we can coalesce any pair of divs and
4442 * then continue with dropping more redundant divs.
4443 *
4444 * We loop over all pairs of lower and upper bounds on a div
4445 * with coefficient 1 and -1, respectively, check if there
4446 * is any other div "c" with which we can coalesce the div
4447 * and if so, perform the coalescing.
4448 */
4451 {
4474 }
4475 }
4476 }
4482 }
4484 /* Are the "n" coefficients starting at "first" of inequality constraints
4485 * "i" and "j" of "bmap" equal to each other?
4486 */
4489 {
4491 }
4493 /* Are the "n" coefficients starting at "first" of inequality constraints
4494 * "i" and "j" of "bmap" opposite to each other?
4495 */
4498 {
4500 }
4502 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4503 * apart from the constant term?
4504 */
4506 {
4511 }
4513 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4514 * apart from the constant term and the coefficient at position "pos"?
4515 */
4518 {
4524 }
4526 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4527 * apart from the constant term and the coefficient at position "pos"?
4528 */
4531 {
4537 }
4539 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4540 * been modified, simplying it if "simplify" is set.
4541 * Free the temporary data structure "pairs" that was associated
4542 * to the old version of "bmap".
4543 */
4546 {
4551 }
4553 /* Is "div" the single unknown existentially quantified variable
4554 * in inequality constraint "ineq" of "bmap"?
4555 * "div" is known to have a non-zero coefficient in "ineq".
4556 */
4558 {
4575 }
4578 }
4580 /* Does integer division "div" have coefficient 1 in inequality constraint
4581 * "ineq" of "map"?
4582 */
4584 {
4592 }
4594 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4595 * then try and drop redundant divs again,
4596 * freeing the temporary data structure "pairs" that was associated
4597 * to the old version of "bmap".
4598 */
4601 {
4605 }
4607 /* Drop the integer division at position "div", along with the two
4608 * inequality constraints "ineq1" and "ineq2" in which it appears
4609 * from "bmap" and then try and drop redundant divs again,
4610 * freeing the temporary data structure "pairs" that was associated
4611 * to the old version of "bmap".
4612 */
4616 {
4623 }
4626 }
4628 /* Given two inequality constraints
4629 *
4630 * f(x) + n d + c >= 0, (ineq)
4631 *
4632 * with d the variable at position "pos", and
4633 *
4634 * f(x) + c0 >= 0, (lower)
4635 *
4636 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4637 * determined by the first constraint.
4638 * That is, store
4639 *
4640 * ceil((c0 - c)/n)
4641 *
4642 * in *l.
4643 */
4646 {
4650 }
4652 /* Given two inequality constraints
4653 *
4654 * f(x) + n d + c >= 0, (ineq)
4655 *
4656 * with d the variable at position "pos", and
4657 *
4658 * -f(x) - c0 >= 0, (upper)
4659 *
4660 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4661 * determined by the first constraint.
4662 * That is, store
4663 *
4664 * ceil((-c1 - c)/n)
4665 *
4666 * in *u.
4667 */
4670 {
4674 }
4676 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4677 * does the corresponding lower bound have a fixed value in "bmap"?
4678 *
4679 * In particular, "ineq" is of the form
4680 *
4681 * f(x) + n d + c >= 0
4682 *
4683 * with n > 0, c the constant term and
4684 * d the existentially quantified variable "div".
4685 * That is, the lower bound is
4686 *
4687 * ceil((-f(x) - c)/n)
4688 *
4689 * Look for a pair of constraints
4690 *
4691 * f(x) + c0 >= 0
4692 * -f(x) + c1 >= 0
4693 *
4694 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4695 * That is, check that
4696 *
4697 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4698 *
4699 * If so, return the index of inequality f(x) + c0 >= 0.
4700 * Otherwise, return -1.
4701 */
4703 {
4721 }
4725 }
4726 }
4743 }
4745 /* Given a lower bound constraint "ineq" on the existentially quantified
4746 * variable "div", such that the corresponding lower bound has
4747 * a fixed value in "bmap", assign this fixed value to the variable and
4748 * then try and drop redundant divs again,
4749 * freeing the temporary data structure "pairs" that was associated
4750 * to the old version of "bmap".
4751 * "lower" determines the constant value for the lower bound.
4752 *
4753 * In particular, "ineq" is of the form
4754 *
4755 * f(x) + n d + c >= 0,
4756 *
4757 * while "lower" is of the form
4758 *
4759 * f(x) + c0 >= 0
4760 *
4761 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4762 * is ceil((c0 - c)/n).
4763 */
4766 {
4767 isl_int c;
4780 }
4782 /* Remove divs that are not strictly needed based on the inequality
4783 * constraints.
4784 * In particular, if a div only occurs positively (or negatively)
4785 * in constraints, then it can simply be dropped.
4786 * Also, if a div occurs in only two constraints and if moreover
4787 * those two constraints are opposite to each other, except for the constant
4788 * term and if the sum of the constant terms is such that for any value
4789 * of the other values, there is always at least one integer value of the
4790 * div, i.e., if one plus this sum is greater than or equal to
4791 * the (absolute value) of the coefficient of the div in the constraints,
4792 * then we can also simply drop the div.
4793 *
4794 * If an existentially quantified variable does not have an explicit
4795 * representation, appears in only a single lower bound that does not
4796 * involve any other such existentially quantified variables and appears
4797 * in this lower bound with coefficient 1,
4798 * then fix the variable to the value of the lower bound. That is,
4799 * turn the inequality into an equality.
4800 * If for any value of the other variables, there is any value
4801 * for the existentially quantified variable satisfying the constraints,
4802 * then this lower bound also satisfies the constraints.
4803 * It is therefore safe to pick this lower bound.
4804 *
4805 * The same reasoning holds even if the coefficient is not one.
4806 * However, fixing the variable to the value of the lower bound may
4807 * in general introduce an extra integer division, in which case
4808 * it may be better to pick another value.
4809 * If this integer division has a known constant value, then plugging
4810 * in this constant value removes the existentially quantified variable
4811 * completely. In particular, if the lower bound is of the form
4812 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4813 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4814 * then the existentially quantified variable can be assigned this
4815 * shared value.
4816 *
4817 * We skip divs that appear in equalities or in the definition of other divs.
4818 * Divs that appear in the definition of other divs usually occur in at least
4819 * 4 constraints, but the constraints may have been simplified.
4820 *
4821 * If any divs are left after these simple checks then we move on
4822 * to more complicated cases in drop_more_redundant_divs.
4823 */
4826 {
4865 }
4869 }
4870 }
4878 }
4888 pairs);
4892 pairs);
4894 }
4912 }
4915 }
4922 error:
4926 }
4928 /* Consider the coefficients at "c" as a row vector and replace
4929 * them with their product with "T". "T" is assumed to be a square matrix.
4930 */
4932 {
4954 }
4956 /* Plug in T for the variables in "bmap" starting at "pos".
4957 * T is a linear unimodular matrix, i.e., without constant term.
4958 */
4961 {
4990 }
4994 error:
4998 }
5000 /* Remove divs that are not strictly needed.
5001 *
5002 * First look for an equality constraint involving two or more
5003 * existentially quantified variables without an explicit
5004 * representation. Replace the combination that appears
5005 * in the equality constraint by a single existentially quantified
5006 * variable such that the equality can be used to derive
5007 * an explicit representation for the variable.
5008 * If there are no more such equality constraints, then continue
5009 * with isl_basic_map_drop_redundant_divs_ineq.
5010 *
5011 * In particular, if the equality constraint is of the form
5012 *
5013 * f(x) + \sum_i c_i a_i = 0
5014 *
5015 * with a_i existentially quantified variable without explicit
5016 * representation, then apply a transformation on the existentially
5017 * quantified variables to turn the constraint into
5018 *
5019 * f(x) + g a_1' = 0
5020 *
5021 * with g the gcd of the c_i.
5022 * In order to easily identify which existentially quantified variables
5023 * have a complete explicit representation, i.e., without being defined
5024 * in terms of other existentially quantified variables without
5025 * an explicit representation, the existentially quantified variables
5026 * are first sorted.
5027 *
5028 * The variable transformation is computed by extending the row
5029 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5030 *
5031 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5032 * [a_2'] [ a_2 ]
5033 * ... = U ....
5034 * [a_n'] [ a_n ]
5035 *
5036 * with [c_1/g ... c_n/g] representing the first row of U.
5037 * The inverse of U is then plugged into the original constraints.
5038 * The call to isl_basic_map_simplify makes sure the explicit
5039 * representation for a_1' is extracted from the equality constraint.
5040 */
5043 {
5078 }
5097 }
5101 {
5104 }
5107 {
5116 }
5119 error:
5122 }
5125 {
5128 }
5130 /* Does "bmap" satisfy any equality that involves more than 2 variables
5131 * and/or has coefficients different from -1 and 1?
5132 */
5134 {
5163 }
5166 }
5168 /* Remove any common factor g from the constraint coefficients in "v".
5169 * The constant term is stored in the first position and is replaced
5170 * by floor(c/g). If any common factor is removed and if this results
5171 * in a tightening of the constraint, then set *tightened.
5172 */
5175 {
5195 }
5197 /* If "bmap" is an integer set that satisfies any equality involving
5198 * more than 2 variables and/or has coefficients different from -1 and 1,
5199 * then use variable compression to reduce the coefficients by removing
5200 * any (hidden) common factor.
5201 * In particular, apply the variable compression to each constraint,
5202 * factor out any common factor in the non-constant coefficients and
5203 * then apply the inverse of the compression.
5204 * At the end, we mark the basic map as having reduced constants.
5205 * If this flag is still set on the next invocation of this function,
5206 * then we skip the computation.
5207 *
5208 * Removing a common factor may result in a tightening of some of
5209 * the constraints. If this happens, then we may end up with two
5210 * opposite inequalities that can be replaced by an equality.
5211 * We therefore call isl_basic_map_detect_inequality_pairs,
5212 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5213 * and isl_basic_map_gauss if such a pair was found.
5214 */
5217 {
5251 }
5262 }
5277 }
5278 }
5281 error:
5286 }
5288 /* Shift the integer division at position "div" of "bmap"
5289 * by "shift" times the variable at position "pos".
5290 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5291 * corresponds to the constant term.
5292 *
5293 * That is, if the integer division has the form
5294 *
5295 * floor(f(x)/d)
5296 *
5297 * then replace it by
5298 *
5299 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5300 */
5303 {
5320 }
5326 }
5334 }
5337 }