| blob | 5b9bb09bed1e6ccb1365b418ee8a395e9dcdd269 |
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>
26 #include <bset_to_bmap.c>
27 #include <bset_from_bmap.c>
28 #include <set_to_map.c>
29 #include <set_from_map.c>
32 {
36 }
39 {
44 }
45 }
49 {
51 isl_int gcd;
64 }
67 }
75 }
77 }
85 }
88 }
95 }
99 }
103 {
106 }
108 /* Reduce the coefficient of the variable at position "pos"
109 * in integer division "div", such that it lies in the half-open
110 * interval (1/2,1/2], extracting any excess value from this integer division.
111 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
112 * corresponds to the constant term.
113 *
114 * That is, the integer division is of the form
115 *
116 * floor((... + (c * d + r) * x_pos + ...)/d)
117 *
118 * with -d < 2 * r <= d.
119 * Replace it by
120 *
121 * floor((... + r * x_pos + ...)/d) + c * x_pos
122 *
123 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
124 * Otherwise, c = floor((c * d + r)/d) + 1.
125 *
126 * This is the same normalization that is performed by isl_aff_floor.
127 */
130 {
131 isl_int shift;
146 }
148 /* Does the coefficient of the variable at position "pos"
149 * in integer division "div" need to be reduced?
150 * That is, does it lie outside the half-open interval (1/2,1/2]?
151 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
152 * 2 * c != d.
153 */
156 {
157 isl_bool r;
169 }
171 /* Reduce the coefficients (including the constant term) of
172 * integer division "div", if needed.
173 * In particular, make sure all coefficients lie in
174 * the half-open interval (1/2,1/2].
175 */
178 {
180 isl_size total;
186 isl_bool reduce;
196 }
199 }
201 /* Reduce the coefficients (including the constant term) of
202 * the known integer divisions, if needed
203 * In particular, make sure all coefficients lie in
204 * the half-open interval (1/2,1/2].
205 */
208 {
222 }
225 }
227 /* Remove any common factor in numerator and denominator of the div expression,
228 * not taking into account the constant term.
229 * That is, if the div is of the form
230 *
231 * floor((a + m f(x))/(m d))
232 *
233 * then replace it by
234 *
235 * floor((floor(a/m) + f(x))/d)
236 *
237 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
238 * and can therefore not influence the result of the floor.
239 */
242 {
262 }
264 /* Remove any common factor in numerator and denominator of a div expression,
265 * not taking into account the constant term.
266 * That is, look for any div of the form
267 *
268 * floor((a + m f(x))/(m d))
269 *
270 * and replace it by
271 *
272 * floor((floor(a/m) + f(x))/d)
273 *
274 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
275 * and can therefore not influence the result of the floor.
276 */
279 {
291 }
293 /* Assumes divs have been ordered if keep_divs is set.
294 */
298 {
299 isl_size total;
318 }
329 }
338 /* We need to be careful about circular definitions,
339 * so for now we just remove the definition of div k
340 * if the equality contains any divs.
341 * If keep_divs is set, then the divs have been ordered
342 * and we can keep the definition as long as the result
343 * is still ordered.
344 */
353 }
356 }
358 /* Assumes divs have been ordered if keep_divs is set.
359 */
362 {
375 }
377 /* Check if elimination of div "div" using equality "eq" would not
378 * result in a div depending on a later div.
379 */
382 {
402 }
405 }
407 /* Eliminate divs based on equalities
408 */
411 {
426 isl_bool ok;
442 }
443 }
447 }
449 /* Eliminate divs based on inequalities
450 */
453 {
483 }
485 }
487 /* Does the equality constraint at position "eq" in "bmap" involve
488 * any local variables in the range [first, first + n)
489 * that are not marked as having an explicit representation?
490 */
493 {
502 isl_bool unknown;
511 }
514 }
516 /* The last local variable involved in the equality constraint
517 * at position "eq" in "bmap" is the local variable at position "div".
518 * It can therefore be used to extract an explicit representation
519 * for that variable.
520 * Do so unless the local variable already has an explicit representation or
521 * the explicit representation would involve any other local variables
522 * that in turn do not have an explicit representation.
523 * An equality constraint involving local variables without an explicit
524 * representation can be used in isl_basic_map_drop_redundant_divs
525 * to separate out an independent local variable. Introducing
526 * an explicit representation here would block this transformation,
527 * while the partial explicit representation in itself is not very useful.
528 * Set *progress if anything is changed.
529 *
530 * The equality constraint is of the form
531 *
532 * f(x) + n e >= 0
533 *
534 * with n a positive number. The explicit representation derived from
535 * this constraint is
536 *
537 * floor((-f(x))/n)
538 */
541 {
542 isl_size total;
544 isl_bool involves;
569 }
573 {
578 isl_size total;
596 }
612 }
619 }
622 }
626 {
628 progress));
629 }
633 {
639 }
641 }
643 /* Hash table of inequalities in a basic map.
644 * "index" is an array of addresses of inequalities in the basic map, some
645 * of which are NULL. The inequalities are hashed on the coefficients
646 * except the constant term.
647 * "size" is the number of elements in the array and is always a power of two
648 * "bits" is the number of bits need to represent an index into the array.
649 * "total" is the total dimension of the basic map.
650 */
655 isl_size total;
656 };
658 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
659 */
662 {
681 }
683 /* Free the memory allocated by create_constraint_index.
684 */
686 {
688 }
690 /* Return the position in ci->index that contains the address of
691 * an inequality that is equal to *ineq up to the constant term,
692 * provided this address is not identical to "ineq".
693 * If there is no such inequality, then return the position where
694 * such an inequality should be inserted.
695 */
697 {
705 }
707 /* Return the position in ci->index that contains the address of
708 * an inequality that is equal to the k'th inequality of "bmap"
709 * up to the constant term, provided it does not point to the very
710 * same inequality.
711 * If there is no such inequality, then return the position where
712 * such an inequality should be inserted.
713 */
716 {
718 }
722 {
724 }
726 /* Fill in the "ci" data structure with the inequalities of "bset".
727 */
730 {
739 }
742 }
744 /* Is the inequality ineq (obviously) redundant with respect
745 * to the constraints in "ci"?
746 *
747 * Look for an inequality in "ci" with the same coefficients and then
748 * check if the contant term of "ineq" is greater than or equal
749 * to the constant term of that inequality. If so, "ineq" is clearly
750 * redundant.
751 *
752 * Note that hash_index_ineq ignores a stored constraint if it has
753 * the same address as the passed inequality. It is ok to pass
754 * the address of a local variable here since it will never be
755 * the same as the address of a constraint in "ci".
756 */
759 {
766 }
768 /* If we can eliminate more than one div, then we need to make
769 * sure we do it from last div to first div, in order not to
770 * change the position of the other divs that still need to
771 * be removed.
772 */
775 {
831 }
833 }
845 }
848 out:
852 }
855 {
869 }
871 }
873 /* Normalize divs that appear in equalities.
874 *
875 * In particular, we assume that bmap contains some equalities
876 * of the form
877 *
878 * a x = m * e_i
879 *
880 * and we want to replace the set of e_i by a minimal set and
881 * such that the new e_i have a canonical representation in terms
882 * of the vector x.
883 * If any of the equalities involves more than one divs, then
884 * we currently simply bail out.
885 *
886 * Let us first additionally assume that all equalities involve
887 * a div. The equalities then express modulo constraints on the
888 * remaining variables and we can use "parameter compression"
889 * to find a minimal set of constraints. The result is a transformation
890 *
891 * x = T(x') = x_0 + G x'
892 *
893 * with G a lower-triangular matrix with all elements below the diagonal
894 * non-negative and smaller than the diagonal element on the same row.
895 * We first normalize x_0 by making the same property hold in the affine
896 * T matrix.
897 * The rows i of G with a 1 on the diagonal do not impose any modulo
898 * constraint and simply express x_i = x'_i.
899 * For each of the remaining rows i, we introduce a div and a corresponding
900 * equality. In particular
901 *
902 * g_ii e_j = x_i - g_i(x')
903 *
904 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
905 * corresponding div (if g_kk != 1).
906 *
907 * If there are any equalities not involving any div, then we
908 * first apply a variable compression on the variables x:
909 *
910 * x = C x'' x'' = C_2 x
911 *
912 * and perform the above parameter compression on A C instead of on A.
913 * The resulting compression is then of the form
914 *
915 * x'' = T(x') = x_0 + G x'
916 *
917 * and in constructing the new divs and the corresponding equalities,
918 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
919 * by the corresponding row from C_2.
920 */
923 {
932 isl_int v;
966 }
967 }
976 }
982 }
992 }
999 }
1004 /* We have to be careful because dropping equalities may reorder them */
1014 }
1015 }
1021 else
1022 needed++;
1023 }
1029 }
1039 else
1047 else
1050 }
1054 }
1061 done:
1065 error:
1071 }
1075 {
1085 }
1087 /* Check whether it is ok to define a div based on an inequality.
1088 * To avoid the introduction of circular definitions of divs, we
1089 * do not allow such a definition if the resulting expression would refer to
1090 * any other undefined divs or if any known div is defined in
1091 * terms of the unknown div.
1092 */
1095 {
1099 /* Not defined in terms of unknown divs */
1107 }
1109 /* No other div defined in terms of this one => avoid loops */
1117 }
1120 }
1122 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1123 * be a better expression than the current one?
1124 *
1125 * If we do not have any expression yet, then any expression would be better.
1126 * Otherwise we check if the last variable involved in the inequality
1127 * (disregarding the div that it would define) is in an earlier position
1128 * than the last variable involved in the current div expression.
1129 */
1132 {
1149 }
1151 /* Given two constraints "k" and "l" that are opposite to each other,
1152 * except for the constant term, check if we can use them
1153 * to obtain an expression for one of the hitherto unknown divs or
1154 * a "better" expression for a div for which we already have an expression.
1155 * "sum" is the sum of the constant terms of the constraints.
1156 * If this sum is strictly smaller than the coefficient of one
1157 * of the divs, then this pair can be used define the div.
1158 * To avoid the introduction of circular definitions of divs, we
1159 * do not use the pair if the resulting expression would refer to
1160 * any other undefined divs or if any known div is defined in
1161 * terms of the unknown div.
1162 */
1166 {
1171 isl_bool set_div;
1186 else
1191 }
1193 }
1197 {
1201 isl_int sum;
1216 }
1224 }
1239 }
1241 /* We need to break out of the loop after these
1242 * changes since the contents of the hash
1243 * will no longer be valid.
1244 * Plus, we probably we want to regauss first.
1245 */
1253 }
1258 }
1260 /* Detect all pairs of inequalities that form an equality.
1261 *
1262 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1263 * Call it repeatedly while it is making progress.
1264 */
1267 {
1279 }
1281 /* Eliminate knowns divs from constraints where they appear with
1282 * a (positive or negative) unit coefficient.
1283 *
1284 * That is, replace
1285 *
1286 * floor(e/m) + f >= 0
1287 *
1288 * by
1289 *
1290 * e + m f >= 0
1291 *
1292 * and
1293 *
1294 * -floor(e/m) + f >= 0
1295 *
1296 * by
1297 *
1298 * -e + m f + m - 1 >= 0
1299 *
1300 * The first conversion is valid because floor(e/m) >= -f is equivalent
1301 * to e/m >= -f because -f is an integral expression.
1302 * The second conversion follows from the fact that
1303 *
1304 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1305 *
1306 *
1307 * Note that one of the div constraints may have been eliminated
1308 * due to being redundant with respect to the constraint that is
1309 * being modified by this function. The modified constraint may
1310 * no longer imply this div constraint, so we add it back to make
1311 * sure we do not lose any information.
1312 *
1313 * We skip integral divs, i.e., those with denominator 1, as we would
1314 * risk eliminating the div from the div constraints. We do not need
1315 * to handle those divs here anyway since the div constraints will turn
1316 * out to form an equality and this equality can then be used to eliminate
1317 * the div from all constraints.
1318 */
1321 {
1353 else
1363 }
1369 }
1370 }
1373 }
1376 {
1381 isl_bool empty;
1397 /* requires equalities in normal form */
1403 }
1405 }
1408 {
1410 }
1415 {
1447 }
1451 {
1453 }
1456 /* If the only constraints a div d=floor(f/m)
1457 * appears in are its two defining constraints
1458 *
1459 * f - m d >=0
1460 * -(f - (m - 1)) + m d >= 0
1461 *
1462 * then it can safely be removed.
1463 */
1465 {
1474 isl_bool red;
1481 }
1488 }
1491 }
1493 /*
1494 * Remove divs that don't occur in any of the constraints or other divs.
1495 * These can arise when dropping constraints from a basic map or
1496 * when the divs of a basic map have been temporarily aligned
1497 * with the divs of another basic map.
1498 */
1501 {
1510 isl_bool redundant;
1520 }
1522 }
1524 /* Mark "bmap" as final, without checking for obviously redundant
1525 * integer divisions. This function should be used when "bmap"
1526 * is known not to involve any such integer divisions.
1527 */
1530 {
1535 }
1537 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1538 */
1540 {
1544 }
1547 {
1549 }
1551 /* Remove definition of any div that is defined in terms of the given variable.
1552 * The div itself is not removed. Functions such as
1553 * eliminate_divs_ineq depend on the other divs remaining in place.
1554 */
1557 {
1571 }
1573 }
1575 /* Eliminate the specified variables from the constraints using
1576 * Fourier-Motzkin. The variables themselves are not removed.
1577 */
1580 {
1583 isl_size total;
1614 }
1621 n_lower++;
1623 n_upper++;
1624 }
1648 }
1651 }
1663 }
1664 }
1668 error:
1671 }
1675 {
1678 }
1680 /* Eliminate the specified n dimensions starting at first from the
1681 * constraints, without removing the dimensions from the space.
1682 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1683 * Otherwise, they are projected out and the original space is restored.
1684 */
1688 {
1703 }
1710 }
1715 {
1717 }
1719 /* Remove all constraints from "bmap" that reference any unknown local
1720 * variables (directly or indirectly).
1721 *
1722 * Dropping all constraints on a local variable will make it redundant,
1723 * so it will get removed implicitly by
1724 * isl_basic_map_drop_constraints_involving_dims. Some other local
1725 * variables may also end up becoming redundant if they only appear
1726 * in constraints together with the unknown local variable.
1727 * Therefore, start over after calling
1728 * isl_basic_map_drop_constraints_involving_dims.
1729 */
1732 {
1733 isl_bool known;
1734 isl_size n_div;
1761 }
1764 }
1766 /* Remove all constraints from "map" that reference any unknown local
1767 * variables (directly or indirectly).
1768 *
1769 * Since constraints may get dropped from the basic maps,
1770 * they may no longer be disjoint from each other.
1771 */
1774 {
1776 isl_bool known;
1794 }
1800 }
1802 /* Don't assume equalities are in order, because align_divs
1803 * may have changed the order of the divs.
1804 */
1806 {
1819 }
1820 }
1821 }
1825 {
1827 }
1831 {
1845 }
1847 }
1849 }
1853 {
1856 }
1860 {
1863 isl_size dim;
1871 }
1893 error:
1897 }
1899 /* For each inequality in "ineq" that is a shifted (more relaxed)
1900 * copy of an inequality in "context", mark the corresponding entry
1901 * in "row" with -1.
1902 * If an inequality only has a non-negative constant term, then
1903 * mark it as well.
1904 */
1907 {
1924 isl_bool redundant;
1930 }
1937 }
1940 error:
1943 }
1947 {
1960 isl_bool redundant;
1972 }
1975 error:
1978 }
1980 /* Remove constraints from "bmap" that are identical to constraints
1981 * in "context" or that are more relaxed (greater constant term).
1982 *
1983 * We perform the test for shifted copies on the pure constraints
1984 * in remove_shifted_constraints.
1985 */
1988 {
1997 }
2010 error:
2014 }
2016 /* Does the (linear part of a) constraint "c" involve any of the "len"
2017 * "relevant" dimensions?
2018 */
2020 {
2028 }
2031 }
2033 /* Drop constraints from "bmap" that do not involve any of
2034 * the dimensions marked "relevant".
2035 */
2038 {
2040 isl_size dim;
2056 }
2063 }
2066 }
2068 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2069 *
2070 * In particular, for any variable involved in the constraint,
2071 * find the actual group id from before and replace the group
2072 * of the corresponding variable by the minimal group of all
2073 * the variables involved in the constraint considered so far
2074 * (if this minimum is smaller) or replace the minimum by this group
2075 * (if the minimum is larger).
2076 *
2077 * At the end, all the variables in "c" will (indirectly) point
2078 * to the minimal of the groups that they referred to originally.
2079 */
2081 {
2098 }
2099 }
2101 /* Allocate an array of groups of variables, one for each variable
2102 * in "context", initialized to zero.
2103 */
2105 {
2107 isl_size dim;
2114 }
2116 /* Drop constraints from "bmap" that only involve variables that are
2117 * not related to any of the variables marked with a "-1" in "group".
2118 *
2119 * We construct groups of variables that collect variables that
2120 * (indirectly) appear in some common constraint of "bmap".
2121 * Each group is identified by the first variable in the group,
2122 * except for the special group of variables that was already identified
2123 * in the input as -1 (or are related to those variables).
2124 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2125 * otherwise the group of i is the group of group[i].
2126 *
2127 * We first initialize groups for the remaining variables.
2128 * Then we iterate over the constraints of "bmap" and update the
2129 * group of the variables in the constraint by the smallest group.
2130 * Finally, we resolve indirect references to groups by running over
2131 * the variables.
2132 *
2133 * After computing the groups, we drop constraints that do not involve
2134 * any variables in the -1 group.
2135 */
2138 {
2139 isl_size dim;
2154 }
2172 }
2174 /* Drop constraints from "context" that are irrelevant for computing
2175 * the gist of "bset".
2176 *
2177 * In particular, drop constraints in variables that are not related
2178 * to any of the variables involved in the constraints of "bset"
2179 * in the sense that there is no sequence of constraints that connects them.
2180 *
2181 * We first mark all variables that appear in "bset" as belonging
2182 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2183 */
2186 {
2188 isl_size dim;
2207 }
2213 }
2216 }
2218 /* Drop constraints from "context" that are irrelevant for computing
2219 * the gist of the inequalities "ineq".
2220 * Inequalities in "ineq" for which the corresponding element of row
2221 * is set to -1 have already been marked for removal and should be ignored.
2222 *
2223 * In particular, drop constraints in variables that are not related
2224 * to any of the variables involved in "ineq"
2225 * in the sense that there is no sequence of constraints that connects them.
2226 *
2227 * We first mark all variables that appear in "bset" as belonging
2228 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2229 */
2232 {
2234 isl_size dim;
2253 }
2256 }
2259 }
2261 /* Do all "n" entries of "row" contain a negative value?
2262 */
2264 {
2272 }
2274 /* Update the inequalities in "bset" based on the information in "row"
2275 * and "tab".
2276 *
2277 * In particular, the array "row" contains either -1, meaning that
2278 * the corresponding inequality of "bset" is redundant, or the index
2279 * of an inequality in "tab".
2280 *
2281 * If the row entry is -1, then drop the inequality.
2282 * Otherwise, if the constraint is marked redundant in the tableau,
2283 * then drop the inequality. Similarly, if it is marked as an equality
2284 * in the tableau, then turn the inequality into an equality and
2285 * perform Gaussian elimination.
2286 */
2289 {
2306 }
2316 }
2317 }
2323 }
2325 /* Update the inequalities in "bset" based on the information in "row"
2326 * and "tab" and free all arguments (other than "bset").
2327 */
2332 {
2341 }
2343 /* Remove all information from bset that is redundant in the context
2344 * of context.
2345 * "ineq" contains the (possibly transformed) inequalities of "bset",
2346 * in the same order.
2347 * The (explicit) equalities of "bset" are assumed to have been taken
2348 * into account by the transformation such that only the inequalities
2349 * are relevant.
2350 * "context" is assumed not to be empty.
2351 *
2352 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2353 * A value of -1 means that the inequality is obviously redundant and may
2354 * not even appear in "tab".
2355 *
2356 * We first mark the inequalities of "bset"
2357 * that are obviously redundant with respect to some inequality in "context".
2358 * Then we remove those constraints from "context" that have become
2359 * irrelevant for computing the gist of "bset".
2360 * Note that this removal of constraints cannot be replaced by
2361 * a factorization because factors in "bset" may still be connected
2362 * to each other through constraints in "context".
2363 *
2364 * If there are any inequalities left, we construct a tableau for
2365 * the context and then add the inequalities of "bset".
2366 * Before adding these inequalities, we freeze all constraints such that
2367 * they won't be considered redundant in terms of the constraints of "bset".
2368 * Then we detect all redundant constraints (among the
2369 * constraints that weren't frozen), first by checking for redundancy in the
2370 * the tableau and then by checking if replacing a constraint by its negation
2371 * would lead to an empty set. This last step is fairly expensive
2372 * and could be optimized by more reuse of the tableau.
2373 * Finally, we update bset according to the results.
2374 */
2377 {
2392 }
2428 }
2451 }
2456 }
2460 error:
2468 }
2470 /* Extract the inequalities of "bset" as an isl_mat.
2471 */
2473 {
2474 isl_size total;
2487 }
2489 /* Remove all information from "bset" that is redundant in the context
2490 * of "context", for the case where both "bset" and "context" are
2491 * full-dimensional.
2492 */
2495 {
2500 }
2502 /* Remove all information from "bset" that is redundant in the context
2503 * of "context", for the case where the combined equalities of
2504 * "bset" and "context" allow for a compression that can be obtained
2505 * by preapplication of "T".
2506 *
2507 * "bset" itself is not transformed by "T". Instead, the inequalities
2508 * are extracted from "bset" and those are transformed by "T".
2509 * uset_gist_full then determines which of the transformed inequalities
2510 * are redundant with respect to the transformed "context" and removes
2511 * the corresponding inequalities from "bset".
2512 *
2513 * After preapplying "T" to the inequalities, any common factor is
2514 * removed from the coefficients. If this results in a tightening
2515 * of the constant term, then the same tightening is applied to
2516 * the corresponding untransformed inequality in "bset".
2517 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2518 *
2519 * g f'(x) + r >= 0
2520 *
2521 * with 0 <= r < g, then it is equivalent to
2522 *
2523 * f'(x) >= 0
2524 *
2525 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2526 * subspace compressed by T since the latter would be transformed to
2527 *
2528 * g f'(x) >= 0
2529 */
2533 {
2537 isl_int rem;
2549 }
2572 }
2576 error:
2581 }
2583 /* Project "bset" onto the variables that are involved in "template".
2584 */
2587 {
2589 isl_size n;
2596 isl_bool involved;
2605 }
2608 }
2610 /* Remove all information from bset that is redundant in the context
2611 * of context. In particular, equalities that are linear combinations
2612 * of those in context are removed. Then the inequalities that are
2613 * redundant in the context of the equalities and inequalities of
2614 * context are removed.
2615 *
2616 * First of all, we drop those constraints from "context"
2617 * that are irrelevant for computing the gist of "bset".
2618 * Alternatively, we could factorize the intersection of "context" and "bset".
2619 *
2620 * We first compute the intersection of the integer affine hulls
2621 * of "bset" and "context",
2622 * compute the gist inside this intersection and then reduce
2623 * the constraints with respect to the equalities of the context
2624 * that only involve variables already involved in the input.
2625 *
2626 * If two constraints are mutually redundant, then uset_gist_full
2627 * will remove the second of those constraints. We therefore first
2628 * sort the constraints so that constraints not involving existentially
2629 * quantified variables are given precedence over those that do.
2630 * We have to perform this sorting before the variable compression,
2631 * because that may effect the order of the variables.
2632 */
2635 {
2640 isl_size total;
2661 }
2666 }
2675 }
2686 }
2689 error:
2693 }
2695 /* Return the number of equality constraints in "bmap" that involve
2696 * local variables. This function assumes that Gaussian elimination
2697 * has been applied to the equality constraints.
2698 */
2700 {
2722 }
2724 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2725 * The constraints are assumed not to involve any local variables.
2726 */
2729 {
2731 isl_size total;
2749 }
2754 error:
2759 }
2761 /* Construct and return a variable compression based on the equality
2762 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2763 * "n1" is the number of (initial) equality constraints in "bmap1"
2764 * that do involve local variables.
2765 * "n2" is the number of (initial) equality constraints in "bmap2"
2766 * that do involve local variables.
2767 * "total" is the total number of other variables.
2768 * This function assumes that Gaussian elimination
2769 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2770 * such that the equality constraints not involving local variables
2771 * are those that start at "n1" or "n2".
2772 *
2773 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2774 * then simply compute the compression based on the equality constraints
2775 * in the other basic map.
2776 * Otherwise, combine the equality constraints from both into a new
2777 * basic map such that Gaussian elimination can be applied to this combination
2778 * and then construct a variable compression from the resulting
2779 * equality constraints.
2780 */
2784 {
2794 }
2799 }
2814 }
2816 /* Extract the stride constraints from "bmap", compressed
2817 * with respect to both the stride constraints in "context" and
2818 * the remaining equality constraints in both "bmap" and "context".
2819 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2820 * "context_n_eq" is the number of (initial) stride constraints in "context".
2821 *
2822 * Let x be all variables in "bmap" (and "context") other than the local
2823 * variables. First compute a variable compression
2824 *
2825 * x = V x'
2826 *
2827 * based on the non-stride equality constraints in "bmap" and "context".
2828 * Consider the stride constraints of "context",
2829 *
2830 * A(x) + B(y) = 0
2831 *
2832 * with y the local variables and plug in the variable compression,
2833 * resulting in
2834 *
2835 * A(V x') + B(y) = 0
2836 *
2837 * Use these constraints to compute a parameter compression on x'
2838 *
2839 * x' = T x''
2840 *
2841 * Now consider the stride constraints of "bmap"
2842 *
2843 * C(x) + D(y) = 0
2844 *
2845 * and plug in x = V*T x''.
2846 * That is, return A = [C*V*T D].
2847 */
2851 {
2877 else
2885 }
2887 /* Remove the prime factors from *g that have an exponent that
2888 * is strictly smaller than the exponent in "c".
2889 * All exponents in *g are known to be smaller than or equal
2890 * to those in "c".
2891 *
2892 * That is, if *g is equal to
2893 *
2894 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2895 *
2896 * and "c" is equal to
2897 *
2898 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2899 *
2900 * then update *g to
2901 *
2902 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2903 * p_n^{e_n * (e_n = f_n)}
2904 *
2905 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2906 * neither does the gcd of *g and c / *g.
2907 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2908 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2909 * Dividing *g by this gcd therefore strictly reduces the exponent
2910 * of the prime factors that need to be removed, while leaving the
2911 * other prime factors untouched.
2912 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2913 * removes all undesired factors, without removing any others.
2914 */
2916 {
2917 isl_int t;
2926 }
2928 }
2930 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2931 * of the same stride constraints in a compressed space that exploits
2932 * all equalities in the context and the other equalities in "bmap".
2933 *
2934 * If the stride constraints of "bmap" are of the form
2935 *
2936 * C(x) + D(y) = 0
2937 *
2938 * then A is of the form
2939 *
2940 * B(x') + D(y) = 0
2941 *
2942 * If any of these constraints involves only a single local variable y,
2943 * then the constraint appears as
2944 *
2945 * f(x) + m y_i = 0
2946 *
2947 * in "bmap" and as
2948 *
2949 * h(x') + m y_i = 0
2950 *
2951 * in "A".
2952 *
2953 * Let g be the gcd of m and the coefficients of h.
2954 * Then, in particular, g is a divisor of the coefficients of h and
2955 *
2956 * f(x) = h(x')
2957 *
2958 * is known to be a multiple of g.
2959 * If some prime factor in m appears with the same exponent in g,
2960 * then it can be removed from m because f(x) is already known
2961 * to be a multiple of g and therefore in particular of this power
2962 * of the prime factors.
2963 * Prime factors that appear with a smaller exponent in g cannot
2964 * be removed from m.
2965 * Let g' be the divisor of g containing all prime factors that
2966 * appear with the same exponent in m and g, then
2967 *
2968 * f(x) + m y_i = 0
2969 *
2970 * can be replaced by
2971 *
2972 * f(x) + m/g' y_i' = 0
2973 *
2974 * Note that (if g' != 1) this changes the explicit representation
2975 * of y_i to that of y_i', so the integer division at position i
2976 * is marked unknown and later recomputed by a call to
2977 * isl_basic_map_gauss.
2978 */
2981 {
2985 isl_int gcd;
3018 }
3025 error:
3029 }
3031 /* Simplify the stride constraints in "bmap" based on
3032 * the remaining equality constraints in "bmap" and all equality
3033 * constraints in "context".
3034 * Only do this if both "bmap" and "context" have stride constraints.
3035 *
3036 * First extract a copy of the stride constraints in "bmap" in a compressed
3037 * space exploiting all the other equality constraints and then
3038 * use this compressed copy to simplify the original stride constraints.
3039 */
3042 {
3064 }
3066 /* Return a basic map that has the same intersection with "context" as "bmap"
3067 * and that is as "simple" as possible.
3068 *
3069 * The core computation is performed on the pure constraints.
3070 * When we add back the meaning of the integer divisions, we need
3071 * to (re)introduce the div constraints. If we happen to have
3072 * discovered that some of these integer divisions are equal to
3073 * some affine combination of other variables, then these div
3074 * constraints may end up getting simplified in terms of the equalities,
3075 * resulting in extra inequalities on the other variables that
3076 * may have been removed already or that may not even have been
3077 * part of the input. We try and remove those constraints of
3078 * this form that are most obviously redundant with respect to
3079 * the context. We also remove those div constraints that are
3080 * redundant with respect to the other constraints in the result.
3081 *
3082 * The stride constraints among the equality constraints in "bmap" are
3083 * also simplified with respecting to the other equality constraints
3084 * in "bmap" and with respect to all equality constraints in "context".
3085 */
3088 {
3100 }
3106 }
3110 }
3133 }
3152 error:
3156 }
3158 /*
3159 * Assumes context has no implicit divs.
3160 */
3163 {
3174 }
3194 }
3195 }
3199 error:
3203 }
3205 /* Drop all inequalities from "bmap" that also appear in "context".
3206 * "context" is assumed to have only known local variables and
3207 * the initial local variables of "bmap" are assumed to be the same
3208 * as those of "context".
3209 * The constraints of both "bmap" and "context" are assumed
3210 * to have been sorted using isl_basic_map_sort_constraints.
3211 *
3212 * Run through the inequality constraints of "bmap" and "context"
3213 * in sorted order.
3214 * If a constraint of "bmap" involves variables not in "context",
3215 * then it cannot appear in "context".
3216 * If a matching constraint is found, it is removed from "bmap".
3217 */
3220 {
3241 }
3247 }
3251 }
3256 }
3259 }
3262 }
3264 /* Drop all equalities from "bmap" that also appear in "context".
3265 * "context" is assumed to have only known local variables and
3266 * the initial local variables of "bmap" are assumed to be the same
3267 * as those of "context".
3268 *
3269 * Run through the equality constraints of "bmap" and "context"
3270 * in sorted order.
3271 * If a constraint of "bmap" involves variables not in "context",
3272 * then it cannot appear in "context".
3273 * If a matching constraint is found, it is removed from "bmap".
3274 */
3277 {
3303 }
3307 }
3312 }
3315 }
3318 }
3320 /* Remove the constraints in "context" from "bmap".
3321 * "context" is assumed to have explicit representations
3322 * for all local variables.
3323 *
3324 * First align the divs of "bmap" to those of "context" and
3325 * sort the constraints. Then drop all constraints from "bmap"
3326 * that appear in "context".
3327 */
3330 {
3345 }
3364 error:
3368 }
3370 /* Replace "map" by the disjunct at position "pos" and free "context".
3371 */
3374 {
3381 }
3383 /* Remove the constraints in "context" from "map".
3384 * If any of the disjuncts in the result turns out to be the universe,
3385 * then return this universe.
3386 * "context" is assumed to have explicit representations
3387 * for all local variables.
3388 */
3391 {
3401 }
3420 }
3427 error:
3431 }
3433 /* Remove the constraints in "context" from "set".
3434 * If any of the disjuncts in the result turns out to be the universe,
3435 * then return this universe.
3436 * "context" is assumed to have explicit representations
3437 * for all local variables.
3438 */
3441 {
3444 }
3446 /* Remove the constraints in "context" from "map".
3447 * If any of the disjuncts in the result turns out to be the universe,
3448 * then return this universe.
3449 * "context" is assumed to consist of a single disjunct and
3450 * to have explicit representations for all local variables.
3451 */
3454 {
3459 }
3461 /* Replace "map" by a universe map in the same space and free "drop".
3462 */
3465 {
3472 }
3474 /* Return a map that has the same intersection with "context" as "map"
3475 * and that is as "simple" as possible.
3476 *
3477 * If "map" is already the universe, then we cannot make it any simpler.
3478 * Similarly, if "context" is the universe, then we cannot exploit it
3479 * to simplify "map"
3480 * If "map" and "context" are identical to each other, then we can
3481 * return the corresponding universe.
3482 *
3483 * If either "map" or "context" consists of multiple disjuncts,
3484 * then check if "context" happens to be a subset of "map",
3485 * in which case all constraints can be removed.
3486 * In case of multiple disjuncts, the standard procedure
3487 * may not be able to detect that all constraints can be removed.
3488 *
3489 * If none of these cases apply, we have to work a bit harder.
3490 * During this computation, we make use of a single disjunct context,
3491 * so if the original context consists of more than one disjunct
3492 * then we need to approximate the context by a single disjunct set.
3493 * Simply taking the simple hull may drop constraints that are
3494 * only implicitly available in each disjunct. We therefore also
3495 * look for constraints among those defining "map" that are valid
3496 * for the context. These can then be used to simplify away
3497 * the corresponding constraints in "map".
3498 */
3501 {
3505 isl_bool subset;
3516 }
3532 }
3548 list);
3549 }
3551 error:
3555 }
3559 {
3561 }
3565 {
3568 }
3572 {
3575 }
3579 {
3584 }
3588 {
3590 }
3592 /* Compute the gist of "bmap" with respect to the constraints "context"
3593 * on the domain.
3594 */
3597 {
3603 }
3607 {
3611 }
3615 {
3619 }
3623 {
3627 }
3631 {
3633 }
3635 /* Quick check to see if two basic maps are disjoint.
3636 * In particular, we reduce the equalities and inequalities of
3637 * one basic map in the context of the equalities of the other
3638 * basic map and check if we get a contradiction.
3639 */
3642 {
3645 isl_size total;
3676 }
3684 }
3693 }
3697 disjoint:
3701 error:
3705 }
3709 {
3712 }
3714 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3715 */
3719 {
3730 }
3731 }
3734 }
3736 /* Are "map1" and "map2" obviously disjoint, based on information
3737 * that can be derived without looking at the individual basic maps?
3738 *
3739 * In particular, if one of them is empty or if they live in different spaces
3740 * (ignoring parameters), then they are clearly disjoint.
3741 */
3744 {
3745 isl_bool disjoint;
3746 isl_bool match;
3770 }
3772 /* Are "map1" and "map2" obviously disjoint?
3773 *
3774 * If one of them is empty or if they live in different spaces (ignoring
3775 * parameters), then they are clearly disjoint.
3776 * This is checked by isl_map_plain_is_disjoint_global.
3777 *
3778 * If they have different parameters, then we skip any further tests.
3779 *
3780 * If they are obviously equal, but not obviously empty, then we will
3781 * not be able to detect if they are disjoint.
3782 *
3783 * Otherwise we check if each basic map in "map1" is obviously disjoint
3784 * from each basic map in "map2".
3785 */
3788 {
3789 isl_bool disjoint;
3790 isl_bool intersect;
3791 isl_bool match;
3806 }
3808 /* Are "map1" and "map2" disjoint?
3809 * The parameters are assumed to have been aligned.
3810 *
3811 * In particular, check whether all pairs of basic maps are disjoint.
3812 */
3815 {
3817 }
3819 /* Are "map1" and "map2" disjoint?
3820 *
3821 * They are disjoint if they are "obviously disjoint" or if one of them
3822 * is empty. Otherwise, they are not disjoint if one of them is universal.
3823 * If the two inputs are (obviously) equal and not empty, then they are
3824 * not disjoint.
3825 * If none of these cases apply, then check if all pairs of basic maps
3826 * are disjoint after aligning the parameters.
3827 */
3829 {
3830 isl_bool disjoint;
3831 isl_bool intersect;
3859 }
3861 /* Are "bmap1" and "bmap2" disjoint?
3862 *
3863 * They are disjoint if they are "obviously disjoint" or if one of them
3864 * is empty. Otherwise, they are not disjoint if one of them is universal.
3865 * If none of these cases apply, we compute the intersection and see if
3866 * the result is empty.
3867 */
3870 {
3871 isl_bool disjoint;
3872 isl_bool intersect;
3901 }
3903 /* Are "bset1" and "bset2" disjoint?
3904 */
3907 {
3909 }
3913 {
3915 }
3917 /* Are "set1" and "set2" disjoint?
3918 */
3920 {
3922 }
3924 /* Is "v" equal to 0, 1 or -1?
3925 */
3927 {
3929 }
3931 /* Are the "n" coefficients starting at "first" of inequality constraints
3932 * "i" and "j" of "bmap" opposite to each other?
3933 */
3936 {
3938 }
3940 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
3941 * apart from the constant term?
3942 */
3944 {
3945 isl_size total;
3951 }
3953 /* Check if we can combine a given div with lower bound l and upper
3954 * bound u with some other div and if so return that other div.
3955 * Otherwise, return a position beyond the integer divisions.
3956 * Return -1 on error.
3957 *
3958 * We first check that
3959 * - the bounds are opposites of each other (except for the constant
3960 * term)
3961 * - the bounds do not reference any other div
3962 * - no div is defined in terms of this div
3963 *
3964 * Let m be the size of the range allowed on the div by the bounds.
3965 * That is, the bounds are of the form
3966 *
3967 * e <= a <= e + m - 1
3968 *
3969 * with e some expression in the other variables.
3970 * We look for another div b such that no third div is defined in terms
3971 * of this second div b and such that in any constraint that contains
3972 * a (except for the given lower and upper bound), also contains b
3973 * with a coefficient that is m times that of b.
3974 * That is, all constraints (except for the lower and upper bound)
3975 * are of the form
3976 *
3977 * e + f (a + m b) >= 0
3978 *
3979 * Furthermore, in the constraints that only contain b, the coefficient
3980 * of b should be equal to 1 or -1.
3981 * If so, we return b so that "a + m b" can be replaced by
3982 * a single div "c = a + m b".
3983 */
3986 {
3991 isl_bool opp;
4013 }
4023 }
4036 }
4047 }
4060 }
4065 }
4069 }
4071 /* Internal data structure used during the construction and/or evaluation of
4072 * an inequality that ensures that a pair of bounds always allows
4073 * for an integer value.
4074 *
4075 * "tab" is the tableau in which the inequality is evaluated. It may
4076 * be NULL until it is actually needed.
4077 * "v" contains the inequality coefficients.
4078 * "g", "fl" and "fu" are temporary scalars used during the construction and
4079 * evaluation.
4080 */
4084 isl_int g;
4085 isl_int fl;
4086 isl_int fu;
4087 };
4089 /* Free all the memory allocated by the fields of "data".
4090 */
4092 {
4098 }
4100 /* Is the inequality stored in data->v satisfied by "bmap"?
4101 * That is, does it only attain non-negative values?
4102 * data->tab is a tableau corresponding to "bmap".
4103 */
4106 {
4117 }
4119 /* Given a lower and an upper bound on div i, do they always allow
4120 * for an integer value of the given div?
4121 * Determine this property by constructing an inequality
4122 * such that the property is guaranteed when the inequality is nonnegative.
4123 * The lower bound is inequality l, while the upper bound is inequality u.
4124 * The constructed inequality is stored in data->v.
4125 *
4126 * Let the upper bound be
4127 *
4128 * -n_u a + e_u >= 0
4129 *
4130 * and the lower bound
4131 *
4132 * n_l a + e_l >= 0
4133 *
4134 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4135 * We have
4136 *
4137 * - f_u e_l <= f_u f_l g a <= f_l e_u
4138 *
4139 * Since all variables are integer valued, this is equivalent to
4140 *
4141 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4142 *
4143 * If this interval is at least f_u f_l g, then it contains at least
4144 * one integer value for a.
4145 * That is, the test constraint is
4146 *
4147 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4148 *
4149 * or
4150 *
4151 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4152 *
4153 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4154 * then the constraint can be scaled down by a factor g',
4155 * with the constant term replaced by
4156 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4157 * Note that the result of applying Fourier-Motzkin to this pair
4158 * of constraints is
4159 *
4160 * f_l e_u + f_u e_l >= 0
4161 *
4162 * If the constant term of the scaled down version of this constraint,
4163 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4164 * term of the scaled down test constraint, then the test constraint
4165 * is known to hold and no explicit evaluation is required.
4166 * This is essentially the Omega test.
4167 *
4168 * If the test constraint consists of only a constant term, then
4169 * it is sufficient to look at the sign of this constant term.
4170 */
4173 {
4175 isl_size n_div;
4202 }
4212 }
4214 /* Remove more kinds of divs that are not strictly needed.
4215 * In particular, if all pairs of lower and upper bounds on a div
4216 * are such that they allow at least one integer value of the div,
4217 * then we can eliminate the div using Fourier-Motzkin without
4218 * introducing any spurious solutions.
4219 *
4220 * If at least one of the two constraints has a unit coefficient for the div,
4221 * then the presence of such a value is guaranteed so there is no need to check.
4222 * In particular, the value attained by the bound with unit coefficient
4223 * can serve as this intermediate value.
4224 */
4227 {
4231 isl_size n_div;
4251 isl_bool has_int;
4259 }
4280 }
4283 }
4287 }
4291 }
4294 }
4305 error:
4310 }
4312 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4313 * and the upper bound u, div1 always occurs together with div2 in the form
4314 * (div1 + m div2), where m is the constant range on the variable div1
4315 * allowed by l and u, replace the pair div1 and div2 by a single
4316 * div that is equal to div1 + m div2.
4317 *
4318 * The new div will appear in the location that contains div2.
4319 * We need to modify all constraints that contain
4320 * div2 = (div - div1) / m
4321 * The coefficient of div2 is known to be equal to 1 or -1.
4322 * (If a constraint does not contain div2, it will also not contain div1.)
4323 * If the constraint also contains div1, then we know they appear
4324 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4325 * i.e., the coefficient of div is f.
4326 *
4327 * Otherwise, we first need to introduce div1 into the constraint.
4328 * Let l be
4329 *
4330 * div1 + f >=0
4331 *
4332 * and u
4333 *
4334 * -div1 + f' >= 0
4335 *
4336 * A lower bound on div2
4337 *
4338 * div2 + t >= 0
4339 *
4340 * can be replaced by
4341 *
4342 * m div2 + div1 + m t + f >= 0
4343 *
4344 * An upper bound
4345 *
4346 * -div2 + t >= 0
4347 *
4348 * can be replaced by
4349 *
4350 * -(m div2 + div1) + m t + f' >= 0
4351 *
4352 * These constraint are those that we would obtain from eliminating
4353 * div1 using Fourier-Motzkin.
4354 *
4355 * After all constraints have been modified, we drop the lower and upper
4356 * bound and then drop div1.
4357 * Since the new div is only placed in the same location that used
4358 * to store div2, but otherwise has a different meaning, any possible
4359 * explicit representation of the original div2 is removed.
4360 */
4363 {
4365 isl_int m;
4390 else
4393 }
4397 }
4406 }
4410 }
4412 /* First check if we can coalesce any pair of divs and
4413 * then continue with dropping more redundant divs.
4414 *
4415 * We loop over all pairs of lower and upper bounds on a div
4416 * with coefficient 1 and -1, respectively, check if there
4417 * is any other div "c" with which we can coalesce the div
4418 * and if so, perform the coalescing.
4419 */
4422 {
4425 isl_size n_div;
4451 }
4452 }
4453 }
4458 }
4461 error:
4465 }
4467 /* Are the "n" coefficients starting at "first" of inequality constraints
4468 * "i" and "j" of "bmap" equal to each other?
4469 */
4472 {
4474 }
4476 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4477 * apart from the constant term and the coefficient at position "pos"?
4478 */
4481 {
4482 isl_size total;
4489 }
4491 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4492 * apart from the constant term and the coefficient at position "pos"?
4493 */
4496 {
4497 isl_size total;
4504 }
4506 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4507 * been modified, simplying it if "simplify" is set.
4508 * Free the temporary data structure "pairs" that was associated
4509 * to the old version of "bmap".
4510 */
4513 {
4518 }
4520 /* Is "div" the single unknown existentially quantified variable
4521 * in inequality constraint "ineq" of "bmap"?
4522 * "div" is known to have a non-zero coefficient in "ineq".
4523 */
4526 {
4528 isl_size n_div;
4530 isl_bool known;
4542 isl_bool known;
4551 }
4554 }
4556 /* Does integer division "div" have coefficient 1 in inequality constraint
4557 * "ineq" of "map"?
4558 */
4560 {
4568 }
4570 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4571 * then try and drop redundant divs again,
4572 * freeing the temporary data structure "pairs" that was associated
4573 * to the old version of "bmap".
4574 */
4577 {
4581 }
4583 /* Drop the integer division at position "div", along with the two
4584 * inequality constraints "ineq1" and "ineq2" in which it appears
4585 * from "bmap" and then try and drop redundant divs again,
4586 * freeing the temporary data structure "pairs" that was associated
4587 * to the old version of "bmap".
4588 */
4592 {
4599 }
4602 }
4604 /* Given two inequality constraints
4605 *
4606 * f(x) + n d + c >= 0, (ineq)
4607 *
4608 * with d the variable at position "pos", and
4609 *
4610 * f(x) + c0 >= 0, (lower)
4611 *
4612 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4613 * determined by the first constraint.
4614 * That is, store
4615 *
4616 * ceil((c0 - c)/n)
4617 *
4618 * in *l.
4619 */
4622 {
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, (upper)
4635 *
4636 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4637 * determined by the first constraint.
4638 * That is, store
4639 *
4640 * ceil((-c1 - c)/n)
4641 *
4642 * in *u.
4643 */
4646 {
4650 }
4652 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4653 * does the corresponding lower bound have a fixed value in "bmap"?
4654 *
4655 * In particular, "ineq" is of the form
4656 *
4657 * f(x) + n d + c >= 0
4658 *
4659 * with n > 0, c the constant term and
4660 * d the existentially quantified variable "div".
4661 * That is, the lower bound is
4662 *
4663 * ceil((-f(x) - c)/n)
4664 *
4665 * Look for a pair of constraints
4666 *
4667 * f(x) + c0 >= 0
4668 * -f(x) + c1 >= 0
4669 *
4670 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4671 * That is, check that
4672 *
4673 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4674 *
4675 * If so, return the index of inequality f(x) + c0 >= 0.
4676 * Otherwise, return bmap->n_ineq.
4677 * Return -1 on error.
4678 */
4680 {
4703 }
4711 }
4728 }
4730 /* Given a lower bound constraint "ineq" on the existentially quantified
4731 * variable "div", such that the corresponding lower bound has
4732 * a fixed value in "bmap", assign this fixed value to the variable and
4733 * then try and drop redundant divs again,
4734 * freeing the temporary data structure "pairs" that was associated
4735 * to the old version of "bmap".
4736 * "lower" determines the constant value for the lower bound.
4737 *
4738 * In particular, "ineq" is of the form
4739 *
4740 * f(x) + n d + c >= 0,
4741 *
4742 * while "lower" is of the form
4743 *
4744 * f(x) + c0 >= 0
4745 *
4746 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4747 * is ceil((c0 - c)/n).
4748 */
4751 {
4752 isl_int c;
4765 }
4767 /* Remove divs that are not strictly needed based on the inequality
4768 * constraints.
4769 * In particular, if a div only occurs positively (or negatively)
4770 * in constraints, then it can simply be dropped.
4771 * Also, if a div occurs in only two constraints and if moreover
4772 * those two constraints are opposite to each other, except for the constant
4773 * term and if the sum of the constant terms is such that for any value
4774 * of the other values, there is always at least one integer value of the
4775 * div, i.e., if one plus this sum is greater than or equal to
4776 * the (absolute value) of the coefficient of the div in the constraints,
4777 * then we can also simply drop the div.
4778 *
4779 * If an existentially quantified variable does not have an explicit
4780 * representation, appears in only a single lower bound that does not
4781 * involve any other such existentially quantified variables and appears
4782 * in this lower bound with coefficient 1,
4783 * then fix the variable to the value of the lower bound. That is,
4784 * turn the inequality into an equality.
4785 * If for any value of the other variables, there is any value
4786 * for the existentially quantified variable satisfying the constraints,
4787 * then this lower bound also satisfies the constraints.
4788 * It is therefore safe to pick this lower bound.
4789 *
4790 * The same reasoning holds even if the coefficient is not one.
4791 * However, fixing the variable to the value of the lower bound may
4792 * in general introduce an extra integer division, in which case
4793 * it may be better to pick another value.
4794 * If this integer division has a known constant value, then plugging
4795 * in this constant value removes the existentially quantified variable
4796 * completely. In particular, if the lower bound is of the form
4797 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4798 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4799 * then the existentially quantified variable can be assigned this
4800 * shared value.
4801 *
4802 * We skip divs that appear in equalities or in the definition of other divs.
4803 * Divs that appear in the definition of other divs usually occur in at least
4804 * 4 constraints, but the constraints may have been simplified.
4805 *
4806 * If any divs are left after these simple checks then we move on
4807 * to more complicated cases in drop_more_redundant_divs.
4808 */
4811 {
4855 }
4859 }
4860 }
4868 }
4871 else
4891 pairs);
4897 pairs);
4899 }
4916 else
4923 }
4926 }
4933 error:
4937 }
4939 /* Consider the coefficients at "c" as a row vector and replace
4940 * them with their product with "T". "T" is assumed to be a square matrix.
4941 */
4943 {
4965 }
4967 /* Plug in T for the variables in "bmap" starting at "pos".
4968 * T is a linear unimodular matrix, i.e., without constant term.
4969 */
4972 {
4999 }
5003 error:
5007 }
5009 /* Remove divs that are not strictly needed.
5010 *
5011 * First look for an equality constraint involving two or more
5012 * existentially quantified variables without an explicit
5013 * representation. Replace the combination that appears
5014 * in the equality constraint by a single existentially quantified
5015 * variable such that the equality can be used to derive
5016 * an explicit representation for the variable.
5017 * If there are no more such equality constraints, then continue
5018 * with isl_basic_map_drop_redundant_divs_ineq.
5019 *
5020 * In particular, if the equality constraint is of the form
5021 *
5022 * f(x) + \sum_i c_i a_i = 0
5023 *
5024 * with a_i existentially quantified variable without explicit
5025 * representation, then apply a transformation on the existentially
5026 * quantified variables to turn the constraint into
5027 *
5028 * f(x) + g a_1' = 0
5029 *
5030 * with g the gcd of the c_i.
5031 * In order to easily identify which existentially quantified variables
5032 * have a complete explicit representation, i.e., without being defined
5033 * in terms of other existentially quantified variables without
5034 * an explicit representation, the existentially quantified variables
5035 * are first sorted.
5036 *
5037 * The variable transformation is computed by extending the row
5038 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5039 *
5040 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5041 * [a_2'] [ a_2 ]
5042 * ... = U ....
5043 * [a_n'] [ a_n ]
5044 *
5045 * with [c_1/g ... c_n/g] representing the first row of U.
5046 * The inverse of U is then plugged into the original constraints.
5047 * The call to isl_basic_map_simplify makes sure the explicit
5048 * representation for a_1' is extracted from the equality constraint.
5049 */
5052 {
5056 isl_size n_div;
5090 }
5109 }
5111 /* Does "bmap" satisfy any equality that involves more than 2 variables
5112 * and/or has coefficients different from -1 and 1?
5113 */
5115 {
5117 isl_size total;
5146 }
5149 }
5151 /* Remove any common factor g from the constraint coefficients in "v".
5152 * The constant term is stored in the first position and is replaced
5153 * by floor(c/g). If any common factor is removed and if this results
5154 * in a tightening of the constraint, then set *tightened.
5155 */
5158 {
5178 }
5180 /* If "bmap" is an integer set that satisfies any equality involving
5181 * more than 2 variables and/or has coefficients different from -1 and 1,
5182 * then use variable compression to reduce the coefficients by removing
5183 * any (hidden) common factor.
5184 * In particular, apply the variable compression to each constraint,
5185 * factor out any common factor in the non-constant coefficients and
5186 * then apply the inverse of the compression.
5187 * At the end, we mark the basic map as having reduced constants.
5188 * If this flag is still set on the next invocation of this function,
5189 * then we skip the computation.
5190 *
5191 * Removing a common factor may result in a tightening of some of
5192 * the constraints. If this happens, then we may end up with two
5193 * opposite inequalities that can be replaced by an equality.
5194 * We therefore call isl_basic_map_detect_inequality_pairs,
5195 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5196 * and isl_basic_map_gauss if such a pair was found.
5197 *
5198 * Note that this function may leave the result in an inconsistent state.
5199 * In particular, the constraints may not be gaussed.
5200 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5201 * for some of the test cases to pass successfully.
5202 * Any potential modification of the representation is therefore only
5203 * performed on a single copy of the basic map.
5204 */
5207 {
5208 isl_size total;
5209 isl_bool multi;
5247 }
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 {
5322 }
5328 }
5336 }
5339 }