| blob | 4cec74fb24a924ac274c19f8a047e366862f621d |
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 {
183 isl_bool reduce;
193 }
196 }
198 /* Reduce the coefficients (including the constant term) of
199 * the known integer divisions, if needed
200 * In particular, make sure all coefficients lie in
201 * the half-open interval (1/2,1/2].
202 */
205 {
219 }
222 }
224 /* Remove any common factor in numerator and denominator of the div expression,
225 * not taking into account the constant term.
226 * That is, if the div is of the form
227 *
228 * floor((a + m f(x))/(m d))
229 *
230 * then replace it by
231 *
232 * floor((floor(a/m) + f(x))/d)
233 *
234 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
235 * and can therefore not influence the result of the floor.
236 */
238 {
254 }
256 /* Remove any common factor in numerator and denominator of a div expression,
257 * not taking into account the constant term.
258 * That is, look for any div of the form
259 *
260 * floor((a + m f(x))/(m d))
261 *
262 * and replace it by
263 *
264 * floor((floor(a/m) + f(x))/d)
265 *
266 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
267 * and can therefore not influence the result of the floor.
268 */
271 {
283 }
285 /* Assumes divs have been ordered if keep_divs is set.
286 */
289 {
307 }
318 }
327 /* We need to be careful about circular definitions,
328 * so for now we just remove the definition of div k
329 * if the equality contains any divs.
330 * If keep_divs is set, then the divs have been ordered
331 * and we can keep the definition as long as the result
332 * is still ordered.
333 */
340 }
341 }
343 /* Assumes divs have been ordered if keep_divs is set.
344 */
347 {
355 }
357 /* Check if elimination of div "div" using equality "eq" would not
358 * result in a div depending on a later div.
359 */
362 {
377 }
380 }
382 /* Eliminate divs based on equalities
383 */
386 {
401 isl_bool ok;
417 }
418 }
422 }
424 /* Eliminate divs based on inequalities
425 */
428 {
458 }
460 }
462 /* Does the equality constraint at position "eq" in "bmap" involve
463 * any local variables in the range [first, first + n)
464 * that are not marked as having an explicit representation?
465 */
468 {
477 isl_bool unknown;
486 }
489 }
491 /* The last local variable involved in the equality constraint
492 * at position "eq" in "bmap" is the local variable at position "div".
493 * It can therefore be used to extract an explicit representation
494 * for that variable.
495 * Do so unless the local variable already has an explicit representation or
496 * the explicit representation would involve any other local variables
497 * that in turn do not have an explicit representation.
498 * An equality constraint involving local variables without an explicit
499 * representation can be used in isl_basic_map_drop_redundant_divs
500 * to separate out an independent local variable. Introducing
501 * an explicit representation here would block this transformation,
502 * while the partial explicit representation in itself is not very useful.
503 * Set *progress if anything is changed.
504 *
505 * The equality constraint is of the form
506 *
507 * f(x) + n e >= 0
508 *
509 * with n a positive number. The explicit representation derived from
510 * this constraint is
511 *
512 * floor((-f(x))/n)
513 */
516 {
518 isl_bool involves;
541 }
545 {
568 }
577 progress);
584 }
591 }
594 }
598 {
600 progress));
601 }
605 {
611 }
613 }
615 /* Hash table of inequalities in a basic map.
616 * "index" is an array of addresses of inequalities in the basic map, some
617 * of which are NULL. The inequalities are hashed on the coefficients
618 * except the constant term.
619 * "size" is the number of elements in the array and is always a power of two
620 * "bits" is the number of bits need to represent an index into the array.
621 * "total" is the total dimension of the basic map.
622 */
628 };
630 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
631 */
634 {
651 }
653 /* Free the memory allocated by create_constraint_index.
654 */
656 {
658 }
660 /* Return the position in ci->index that contains the address of
661 * an inequality that is equal to *ineq up to the constant term,
662 * provided this address is not identical to "ineq".
663 * If there is no such inequality, then return the position where
664 * such an inequality should be inserted.
665 */
667 {
675 }
677 /* Return the position in ci->index that contains the address of
678 * an inequality that is equal to the k'th inequality of "bmap"
679 * up to the constant term, provided it does not point to the very
680 * same inequality.
681 * If there is no such inequality, then return the position where
682 * such an inequality should be inserted.
683 */
686 {
688 }
692 {
694 }
696 /* Fill in the "ci" data structure with the inequalities of "bset".
697 */
700 {
709 }
712 }
714 /* Is the inequality ineq (obviously) redundant with respect
715 * to the constraints in "ci"?
716 *
717 * Look for an inequality in "ci" with the same coefficients and then
718 * check if the contant term of "ineq" is greater than or equal
719 * to the constant term of that inequality. If so, "ineq" is clearly
720 * redundant.
721 *
722 * Note that hash_index_ineq ignores a stored constraint if it has
723 * the same address as the passed inequality. It is ok to pass
724 * the address of a local variable here since it will never be
725 * the same as the address of a constraint in "ci".
726 */
729 {
736 }
738 /* If we can eliminate more than one div, then we need to make
739 * sure we do it from last div to first div, in order not to
740 * change the position of the other divs that still need to
741 * be removed.
742 */
745 {
799 }
801 }
813 }
816 out:
820 }
823 {
835 }
837 }
839 /* Normalize divs that appear in equalities.
840 *
841 * In particular, we assume that bmap contains some equalities
842 * of the form
843 *
844 * a x = m * e_i
845 *
846 * and we want to replace the set of e_i by a minimal set and
847 * such that the new e_i have a canonical representation in terms
848 * of the vector x.
849 * If any of the equalities involves more than one divs, then
850 * we currently simply bail out.
851 *
852 * Let us first additionally assume that all equalities involve
853 * a div. The equalities then express modulo constraints on the
854 * remaining variables and we can use "parameter compression"
855 * to find a minimal set of constraints. The result is a transformation
856 *
857 * x = T(x') = x_0 + G x'
858 *
859 * with G a lower-triangular matrix with all elements below the diagonal
860 * non-negative and smaller than the diagonal element on the same row.
861 * We first normalize x_0 by making the same property hold in the affine
862 * T matrix.
863 * The rows i of G with a 1 on the diagonal do not impose any modulo
864 * constraint and simply express x_i = x'_i.
865 * For each of the remaining rows i, we introduce a div and a corresponding
866 * equality. In particular
867 *
868 * g_ii e_j = x_i - g_i(x')
869 *
870 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
871 * corresponding div (if g_kk != 1).
872 *
873 * If there are any equalities not involving any div, then we
874 * first apply a variable compression on the variables x:
875 *
876 * x = C x'' x'' = C_2 x
877 *
878 * and perform the above parameter compression on A C instead of on A.
879 * The resulting compression is then of the form
880 *
881 * x'' = T(x') = x_0 + G x'
882 *
883 * and in constructing the new divs and the corresponding equalities,
884 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
885 * by the corresponding row from C_2.
886 */
889 {
898 isl_int v;
930 }
931 }
940 }
946 }
956 }
963 }
968 /* We have to be careful because dropping equalities may reorder them */
978 }
979 }
985 else
986 needed++;
987 }
993 }
1003 else
1011 else
1014 }
1018 }
1025 done:
1029 error:
1035 }
1039 {
1049 }
1051 /* Check whether it is ok to define a div based on an inequality.
1052 * To avoid the introduction of circular definitions of divs, we
1053 * do not allow such a definition if the resulting expression would refer to
1054 * any other undefined divs or if any known div is defined in
1055 * terms of the unknown div.
1056 */
1059 {
1063 /* Not defined in terms of unknown divs */
1071 }
1073 /* No other div defined in terms of this one => avoid loops */
1081 }
1084 }
1086 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1087 * be a better expression than the current one?
1088 *
1089 * If we do not have any expression yet, then any expression would be better.
1090 * Otherwise we check if the last variable involved in the inequality
1091 * (disregarding the div that it would define) is in an earlier position
1092 * than the last variable involved in the current div expression.
1093 */
1096 {
1113 }
1115 /* Given two constraints "k" and "l" that are opposite to each other,
1116 * except for the constant term, check if we can use them
1117 * to obtain an expression for one of the hitherto unknown divs or
1118 * a "better" expression for a div for which we already have an expression.
1119 * "sum" is the sum of the constant terms of the constraints.
1120 * If this sum is strictly smaller than the coefficient of one
1121 * of the divs, then this pair can be used define the div.
1122 * To avoid the introduction of circular definitions of divs, we
1123 * do not use the pair if the resulting expression would refer to
1124 * any other undefined divs or if any known div is defined in
1125 * terms of the unknown div.
1126 */
1130 {
1135 isl_bool set_div;
1150 else
1155 }
1157 }
1161 {
1165 isl_int sum;
1180 }
1188 }
1203 }
1205 /* We need to break out of the loop after these
1206 * changes since the contents of the hash
1207 * will no longer be valid.
1208 * Plus, we probably we want to regauss first.
1209 */
1217 }
1222 }
1224 /* Detect all pairs of inequalities that form an equality.
1225 *
1226 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1227 * Call it repeatedly while it is making progress.
1228 */
1231 {
1243 }
1245 /* Eliminate knowns divs from constraints where they appear with
1246 * a (positive or negative) unit coefficient.
1247 *
1248 * That is, replace
1249 *
1250 * floor(e/m) + f >= 0
1251 *
1252 * by
1253 *
1254 * e + m f >= 0
1255 *
1256 * and
1257 *
1258 * -floor(e/m) + f >= 0
1259 *
1260 * by
1261 *
1262 * -e + m f + m - 1 >= 0
1263 *
1264 * The first conversion is valid because floor(e/m) >= -f is equivalent
1265 * to e/m >= -f because -f is an integral expression.
1266 * The second conversion follows from the fact that
1267 *
1268 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1269 *
1270 *
1271 * Note that one of the div constraints may have been eliminated
1272 * due to being redundant with respect to the constraint that is
1273 * being modified by this function. The modified constraint may
1274 * no longer imply this div constraint, so we add it back to make
1275 * sure we do not lose any information.
1276 *
1277 * We skip integral divs, i.e., those with denominator 1, as we would
1278 * risk eliminating the div from the div constraints. We do not need
1279 * to handle those divs here anyway since the div constraints will turn
1280 * out to form an equality and this equality can then be used to eliminate
1281 * the div from all constraints.
1282 */
1285 {
1317 else
1327 }
1332 }
1333 }
1336 }
1339 {
1344 isl_bool empty;
1360 /* requires equalities in normal form */
1366 }
1368 }
1371 {
1373 }
1378 {
1410 }
1414 {
1416 }
1419 /* If the only constraints a div d=floor(f/m)
1420 * appears in are its two defining constraints
1421 *
1422 * f - m d >=0
1423 * -(f - (m - 1)) + m d >= 0
1424 *
1425 * then it can safely be removed.
1426 */
1428 {
1437 isl_bool red;
1444 }
1451 }
1454 }
1456 /*
1457 * Remove divs that don't occur in any of the constraints or other divs.
1458 * These can arise when dropping constraints from a basic map or
1459 * when the divs of a basic map have been temporarily aligned
1460 * with the divs of another basic map.
1461 */
1464 {
1473 isl_bool redundant;
1483 }
1485 }
1487 /* Mark "bmap" as final, without checking for obviously redundant
1488 * integer divisions. This function should be used when "bmap"
1489 * is known not to involve any such integer divisions.
1490 */
1493 {
1498 }
1500 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1501 */
1503 {
1507 }
1510 {
1512 }
1514 /* Remove definition of any div that is defined in terms of the given variable.
1515 * The div itself is not removed. Functions such as
1516 * eliminate_divs_ineq depend on the other divs remaining in place.
1517 */
1520 {
1534 }
1536 }
1538 /* Eliminate the specified variables from the constraints using
1539 * Fourier-Motzkin. The variables themselves are not removed.
1540 */
1543 {
1575 }
1582 n_lower++;
1584 n_upper++;
1585 }
1609 }
1612 }
1624 }
1625 }
1629 error:
1632 }
1636 {
1639 }
1641 /* Eliminate the specified n dimensions starting at first from the
1642 * constraints, without removing the dimensions from the space.
1643 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1644 * Otherwise, they are projected out and the original space is restored.
1645 */
1649 {
1664 }
1671 }
1676 {
1678 }
1680 /* Remove all constraints from "bmap" that reference any unknown local
1681 * variables (directly or indirectly).
1682 *
1683 * Dropping all constraints on a local variable will make it redundant,
1684 * so it will get removed implicitly by
1685 * isl_basic_map_drop_constraints_involving_dims. Some other local
1686 * variables may also end up becoming redundant if they only appear
1687 * in constraints together with the unknown local variable.
1688 * Therefore, start over after calling
1689 * isl_basic_map_drop_constraints_involving_dims.
1690 */
1693 {
1694 isl_bool known;
1719 }
1722 }
1724 /* Remove all constraints from "map" that reference any unknown local
1725 * variables (directly or indirectly).
1726 *
1727 * Since constraints may get dropped from the basic maps,
1728 * they may no longer be disjoint from each other.
1729 */
1732 {
1734 isl_bool known;
1752 }
1758 }
1760 /* Don't assume equalities are in order, because align_divs
1761 * may have changed the order of the divs.
1762 */
1764 {
1777 }
1778 }
1779 }
1783 {
1785 }
1789 {
1803 }
1805 }
1807 }
1811 {
1814 }
1818 {
1828 }
1849 error:
1853 }
1855 /* For each inequality in "ineq" that is a shifted (more relaxed)
1856 * copy of an inequality in "context", mark the corresponding entry
1857 * in "row" with -1.
1858 * If an inequality only has a non-negative constant term, then
1859 * mark it as well.
1860 */
1863 {
1880 isl_bool redundant;
1886 }
1893 }
1896 error:
1899 }
1903 {
1916 isl_bool redundant;
1928 }
1931 error:
1934 }
1936 /* Remove constraints from "bmap" that are identical to constraints
1937 * in "context" or that are more relaxed (greater constant term).
1938 *
1939 * We perform the test for shifted copies on the pure constraints
1940 * in remove_shifted_constraints.
1941 */
1944 {
1953 }
1966 error:
1970 }
1972 /* Does the (linear part of a) constraint "c" involve any of the "len"
1973 * "relevant" dimensions?
1974 */
1976 {
1984 }
1987 }
1989 /* Drop constraints from "bmap" that do not involve any of
1990 * the dimensions marked "relevant".
1991 */
1994 {
2009 }
2016 }
2019 }
2021 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2022 *
2023 * In particular, for any variable involved in the constraint,
2024 * find the actual group id from before and replace the group
2025 * of the corresponding variable by the minimal group of all
2026 * the variables involved in the constraint considered so far
2027 * (if this minimum is smaller) or replace the minimum by this group
2028 * (if the minimum is larger).
2029 *
2030 * At the end, all the variables in "c" will (indirectly) point
2031 * to the minimal of the groups that they referred to originally.
2032 */
2034 {
2051 }
2052 }
2054 /* Allocate an array of groups of variables, one for each variable
2055 * in "context", initialized to zero.
2056 */
2058 {
2065 }
2067 /* Drop constraints from "bmap" that only involve variables that are
2068 * not related to any of the variables marked with a "-1" in "group".
2069 *
2070 * We construct groups of variables that collect variables that
2071 * (indirectly) appear in some common constraint of "bmap".
2072 * Each group is identified by the first variable in the group,
2073 * except for the special group of variables that was already identified
2074 * in the input as -1 (or are related to those variables).
2075 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2076 * otherwise the group of i is the group of group[i].
2077 *
2078 * We first initialize groups for the remaining variables.
2079 * Then we iterate over the constraints of "bmap" and update the
2080 * group of the variables in the constraint by the smallest group.
2081 * Finally, we resolve indirect references to groups by running over
2082 * the variables.
2083 *
2084 * After computing the groups, we drop constraints that do not involve
2085 * any variables in the -1 group.
2086 */
2089 {
2106 }
2124 }
2126 /* Drop constraints from "context" that are irrelevant for computing
2127 * the gist of "bset".
2128 *
2129 * In particular, drop constraints in variables that are not related
2130 * to any of the variables involved in the constraints of "bset"
2131 * in the sense that there is no sequence of constraints that connects them.
2132 *
2133 * We first mark all variables that appear in "bset" as belonging
2134 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2135 */
2138 {
2159 }
2165 }
2168 }
2170 /* Drop constraints from "context" that are irrelevant for computing
2171 * the gist of the inequalities "ineq".
2172 * Inequalities in "ineq" for which the corresponding element of row
2173 * is set to -1 have already been marked for removal and should be ignored.
2174 *
2175 * In particular, drop constraints in variables that are not related
2176 * to any of the variables involved in "ineq"
2177 * in the sense that there is no sequence of constraints that connects them.
2178 *
2179 * We first mark all variables that appear in "bset" as belonging
2180 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2181 */
2184 {
2205 }
2208 }
2211 }
2213 /* Do all "n" entries of "row" contain a negative value?
2214 */
2216 {
2224 }
2226 /* Update the inequalities in "bset" based on the information in "row"
2227 * and "tab".
2228 *
2229 * In particular, the array "row" contains either -1, meaning that
2230 * the corresponding inequality of "bset" is redundant, or the index
2231 * of an inequality in "tab".
2232 *
2233 * If the row entry is -1, then drop the inequality.
2234 * Otherwise, if the constraint is marked redundant in the tableau,
2235 * then drop the inequality. Similarly, if it is marked as an equality
2236 * in the tableau, then turn the inequality into an equality and
2237 * perform Gaussian elimination.
2238 */
2241 {
2258 }
2268 }
2269 }
2275 }
2277 /* Update the inequalities in "bset" based on the information in "row"
2278 * and "tab" and free all arguments (other than "bset").
2279 */
2284 {
2293 }
2295 /* Remove all information from bset that is redundant in the context
2296 * of context.
2297 * "ineq" contains the (possibly transformed) inequalities of "bset",
2298 * in the same order.
2299 * The (explicit) equalities of "bset" are assumed to have been taken
2300 * into account by the transformation such that only the inequalities
2301 * are relevant.
2302 * "context" is assumed not to be empty.
2303 *
2304 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2305 * A value of -1 means that the inequality is obviously redundant and may
2306 * not even appear in "tab".
2307 *
2308 * We first mark the inequalities of "bset"
2309 * that are obviously redundant with respect to some inequality in "context".
2310 * Then we remove those constraints from "context" that have become
2311 * irrelevant for computing the gist of "bset".
2312 * Note that this removal of constraints cannot be replaced by
2313 * a factorization because factors in "bset" may still be connected
2314 * to each other through constraints in "context".
2315 *
2316 * If there are any inequalities left, we construct a tableau for
2317 * the context and then add the inequalities of "bset".
2318 * Before adding these inequalities, we freeze all constraints such that
2319 * they won't be considered redundant in terms of the constraints of "bset".
2320 * Then we detect all redundant constraints (among the
2321 * constraints that weren't frozen), first by checking for redundancy in the
2322 * the tableau and then by checking if replacing a constraint by its negation
2323 * would lead to an empty set. This last step is fairly expensive
2324 * and could be optimized by more reuse of the tableau.
2325 * Finally, we update bset according to the results.
2326 */
2329 {
2344 }
2380 }
2403 }
2408 }
2412 error:
2420 }
2422 /* Extract the inequalities of "bset" as an isl_mat.
2423 */
2425 {
2439 }
2441 /* Remove all information from "bset" that is redundant in the context
2442 * of "context", for the case where both "bset" and "context" are
2443 * full-dimensional.
2444 */
2447 {
2452 }
2454 /* Remove all information from "bset" that is redundant in the context
2455 * of "context", for the case where the combined equalities of
2456 * "bset" and "context" allow for a compression that can be obtained
2457 * by preapplication of "T".
2458 *
2459 * "bset" itself is not transformed by "T". Instead, the inequalities
2460 * are extracted from "bset" and those are transformed by "T".
2461 * uset_gist_full then determines which of the transformed inequalities
2462 * are redundant with respect to the transformed "context" and removes
2463 * the corresponding inequalities from "bset".
2464 *
2465 * After preapplying "T" to the inequalities, any common factor is
2466 * removed from the coefficients. If this results in a tightening
2467 * of the constant term, then the same tightening is applied to
2468 * the corresponding untransformed inequality in "bset".
2469 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2470 *
2471 * g f'(x) + r >= 0
2472 *
2473 * with 0 <= r < g, then it is equivalent to
2474 *
2475 * f'(x) >= 0
2476 *
2477 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2478 * subspace compressed by T since the latter would be transformed to
2479 *
2480 * g f'(x) >= 0
2481 */
2485 {
2489 isl_int rem;
2501 }
2524 }
2528 error:
2533 }
2535 /* Project "bset" onto the variables that are involved in "template".
2536 */
2539 {
2548 isl_bool involved;
2557 }
2560 }
2562 /* Remove all information from bset that is redundant in the context
2563 * of context. In particular, equalities that are linear combinations
2564 * of those in context are removed. Then the inequalities that are
2565 * redundant in the context of the equalities and inequalities of
2566 * context are removed.
2567 *
2568 * First of all, we drop those constraints from "context"
2569 * that are irrelevant for computing the gist of "bset".
2570 * Alternatively, we could factorize the intersection of "context" and "bset".
2571 *
2572 * We first compute the intersection of the integer affine hulls
2573 * of "bset" and "context",
2574 * compute the gist inside this intersection and then reduce
2575 * the constraints with respect to the equalities of the context
2576 * that only involve variables already involved in the input.
2577 *
2578 * If two constraints are mutually redundant, then uset_gist_full
2579 * will remove the second of those constraints. We therefore first
2580 * sort the constraints so that constraints not involving existentially
2581 * quantified variables are given precedence over those that do.
2582 * We have to perform this sorting before the variable compression,
2583 * because that may effect the order of the variables.
2584 */
2587 {
2612 }
2617 }
2627 }
2638 }
2641 error:
2645 }
2647 /* Return the number of equality constraints in "bmap" that involve
2648 * local variables. This function assumes that Gaussian elimination
2649 * has been applied to the equality constraints.
2650 */
2652 {
2672 }
2674 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2675 * The constraints are assumed not to involve any local variables.
2676 */
2679 {
2697 }
2702 error:
2707 }
2709 /* Construct and return a variable compression based on the equality
2710 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2711 * "n1" is the number of (initial) equality constraints in "bmap1"
2712 * that do involve local variables.
2713 * "n2" is the number of (initial) equality constraints in "bmap2"
2714 * that do involve local variables.
2715 * "total" is the total number of other variables.
2716 * This function assumes that Gaussian elimination
2717 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2718 * such that the equality constraints not involving local variables
2719 * are those that start at "n1" or "n2".
2720 *
2721 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2722 * then simply compute the compression based on the equality constraints
2723 * in the other basic map.
2724 * Otherwise, combine the equality constraints from both into a new
2725 * basic map such that Gaussian elimination can be applied to this combination
2726 * and then construct a variable compression from the resulting
2727 * equality constraints.
2728 */
2732 {
2742 }
2747 }
2762 }
2764 /* Extract the stride constraints from "bmap", compressed
2765 * with respect to both the stride constraints in "context" and
2766 * the remaining equality constraints in both "bmap" and "context".
2767 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2768 * "context_n_eq" is the number of (initial) stride constraints in "context".
2769 *
2770 * Let x be all variables in "bmap" (and "context") other than the local
2771 * variables. First compute a variable compression
2772 *
2773 * x = V x'
2774 *
2775 * based on the non-stride equality constraints in "bmap" and "context".
2776 * Consider the stride constraints of "context",
2777 *
2778 * A(x) + B(y) = 0
2779 *
2780 * with y the local variables and plug in the variable compression,
2781 * resulting in
2782 *
2783 * A(V x') + B(y) = 0
2784 *
2785 * Use these constraints to compute a parameter compression on x'
2786 *
2787 * x' = T x''
2788 *
2789 * Now consider the stride constraints of "bmap"
2790 *
2791 * C(x) + D(y) = 0
2792 *
2793 * and plug in x = V*T x''.
2794 * That is, return A = [C*V*T D].
2795 */
2799 {
2828 }
2830 /* Remove the prime factors from *g that have an exponent that
2831 * is strictly smaller than the exponent in "c".
2832 * All exponents in *g are known to be smaller than or equal
2833 * to those in "c".
2834 *
2835 * That is, if *g is equal to
2836 *
2837 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2838 *
2839 * and "c" is equal to
2840 *
2841 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2842 *
2843 * then update *g to
2844 *
2845 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2846 * p_n^{e_n * (e_n = f_n)}
2847 *
2848 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2849 * neither does the gcd of *g and c / *g.
2850 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2851 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2852 * Dividing *g by this gcd therefore strictly reduces the exponent
2853 * of the prime factors that need to be removed, while leaving the
2854 * other prime factors untouched.
2855 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2856 * removes all undesired factors, without removing any others.
2857 */
2859 {
2860 isl_int t;
2869 }
2871 }
2873 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2874 * of the same stride constraints in a compressed space that exploits
2875 * all equalities in the context and the other equalities in "bmap".
2876 *
2877 * If the stride constraints of "bmap" are of the form
2878 *
2879 * C(x) + D(y) = 0
2880 *
2881 * then A is of the form
2882 *
2883 * B(x') + D(y) = 0
2884 *
2885 * If any of these constraints involves only a single local variable y,
2886 * then the constraint appears as
2887 *
2888 * f(x) + m y_i = 0
2889 *
2890 * in "bmap" and as
2891 *
2892 * h(x') + m y_i = 0
2893 *
2894 * in "A".
2895 *
2896 * Let g be the gcd of m and the coefficients of h.
2897 * Then, in particular, g is a divisor of the coefficients of h and
2898 *
2899 * f(x) = h(x')
2900 *
2901 * is known to be a multiple of g.
2902 * If some prime factor in m appears with the same exponent in g,
2903 * then it can be removed from m because f(x) is already known
2904 * to be a multiple of g and therefore in particular of this power
2905 * of the prime factors.
2906 * Prime factors that appear with a smaller exponent in g cannot
2907 * be removed from m.
2908 * Let g' be the divisor of g containing all prime factors that
2909 * appear with the same exponent in m and g, then
2910 *
2911 * f(x) + m y_i = 0
2912 *
2913 * can be replaced by
2914 *
2915 * f(x) + m/g' y_i' = 0
2916 *
2917 * Note that (if g' != 1) this changes the explicit representation
2918 * of y_i to that of y_i', so the integer division at position i
2919 * is marked unknown and later recomputed by a call to
2920 * isl_basic_map_gauss.
2921 */
2924 {
2928 isl_int gcd;
2962 }
2969 error:
2973 }
2975 /* Simplify the stride constraints in "bmap" based on
2976 * the remaining equality constraints in "bmap" and all equality
2977 * constraints in "context".
2978 * Only do this if both "bmap" and "context" have stride constraints.
2979 *
2980 * First extract a copy of the stride constraints in "bmap" in a compressed
2981 * space exploiting all the other equality constraints and then
2982 * use this compressed copy to simplify the original stride constraints.
2983 */
2986 {
3008 }
3010 /* Return a basic map that has the same intersection with "context" as "bmap"
3011 * and that is as "simple" as possible.
3012 *
3013 * The core computation is performed on the pure constraints.
3014 * When we add back the meaning of the integer divisions, we need
3015 * to (re)introduce the div constraints. If we happen to have
3016 * discovered that some of these integer divisions are equal to
3017 * some affine combination of other variables, then these div
3018 * constraints may end up getting simplified in terms of the equalities,
3019 * resulting in extra inequalities on the other variables that
3020 * may have been removed already or that may not even have been
3021 * part of the input. We try and remove those constraints of
3022 * this form that are most obviously redundant with respect to
3023 * the context. We also remove those div constraints that are
3024 * redundant with respect to the other constraints in the result.
3025 *
3026 * The stride constraints among the equality constraints in "bmap" are
3027 * also simplified with respecting to the other equality constraints
3028 * in "bmap" and with respect to all equality constraints in "context".
3029 */
3032 {
3043 }
3049 }
3053 }
3075 }
3094 error:
3098 }
3100 /*
3101 * Assumes context has no implicit divs.
3102 */
3105 {
3116 }
3136 }
3137 }
3141 error:
3145 }
3147 /* Drop all inequalities from "bmap" that also appear in "context".
3148 * "context" is assumed to have only known local variables and
3149 * the initial local variables of "bmap" are assumed to be the same
3150 * as those of "context".
3151 * The constraints of both "bmap" and "context" are assumed
3152 * to have been sorted using isl_basic_map_sort_constraints.
3153 *
3154 * Run through the inequality constraints of "bmap" and "context"
3155 * in sorted order.
3156 * If a constraint of "bmap" involves variables not in "context",
3157 * then it cannot appear in "context".
3158 * If a matching constraint is found, it is removed from "bmap".
3159 */
3162 {
3181 }
3187 }
3191 }
3196 }
3199 }
3202 }
3204 /* Drop all equalities from "bmap" that also appear in "context".
3205 * "context" is assumed to have only known local variables and
3206 * the initial local variables of "bmap" are assumed to be the same
3207 * as those of "context".
3208 *
3209 * Run through the equality constraints of "bmap" and "context"
3210 * in sorted order.
3211 * If a constraint of "bmap" involves variables not in "context",
3212 * then it cannot appear in "context".
3213 * If a matching constraint is found, it is removed from "bmap".
3214 */
3217 {
3241 }
3245 }
3250 }
3253 }
3256 }
3258 /* Remove the constraints in "context" from "bmap".
3259 * "context" is assumed to have explicit representations
3260 * for all local variables.
3261 *
3262 * First align the divs of "bmap" to those of "context" and
3263 * sort the constraints. Then drop all constraints from "bmap"
3264 * that appear in "context".
3265 */
3268 {
3283 }
3302 error:
3306 }
3308 /* Replace "map" by the disjunct at position "pos" and free "context".
3309 */
3312 {
3319 }
3321 /* Remove the constraints in "context" from "map".
3322 * If any of the disjuncts in the result turns out to be the universe,
3323 * then return this universe.
3324 * "context" is assumed to have explicit representations
3325 * for all local variables.
3326 */
3329 {
3339 }
3358 }
3365 error:
3369 }
3371 /* Remove the constraints in "context" from "set".
3372 * If any of the disjuncts in the result turns out to be the universe,
3373 * then return this universe.
3374 * "context" is assumed to have explicit representations
3375 * for all local variables.
3376 */
3379 {
3382 }
3384 /* Remove the constraints in "context" from "map".
3385 * If any of the disjuncts in the result turns out to be the universe,
3386 * then return this universe.
3387 * "context" is assumed to consist of a single disjunct and
3388 * to have explicit representations for all local variables.
3389 */
3392 {
3397 }
3399 /* Replace "map" by a universe map in the same space and free "drop".
3400 */
3403 {
3410 }
3412 /* Return a map that has the same intersection with "context" as "map"
3413 * and that is as "simple" as possible.
3414 *
3415 * If "map" is already the universe, then we cannot make it any simpler.
3416 * Similarly, if "context" is the universe, then we cannot exploit it
3417 * to simplify "map"
3418 * If "map" and "context" are identical to each other, then we can
3419 * return the corresponding universe.
3420 *
3421 * If either "map" or "context" consists of multiple disjuncts,
3422 * then check if "context" happens to be a subset of "map",
3423 * in which case all constraints can be removed.
3424 * In case of multiple disjuncts, the standard procedure
3425 * may not be able to detect that all constraints can be removed.
3426 *
3427 * If none of these cases apply, we have to work a bit harder.
3428 * During this computation, we make use of a single disjunct context,
3429 * so if the original context consists of more than one disjunct
3430 * then we need to approximate the context by a single disjunct set.
3431 * Simply taking the simple hull may drop constraints that are
3432 * only implicitly available in each disjunct. We therefore also
3433 * look for constraints among those defining "map" that are valid
3434 * for the context. These can then be used to simplify away
3435 * the corresponding constraints in "map".
3436 */
3439 {
3443 isl_bool subset;
3454 }
3470 }
3486 list);
3487 }
3489 error:
3493 }
3497 {
3499 }
3503 {
3506 }
3510 {
3513 }
3517 {
3522 }
3526 {
3528 }
3530 /* Compute the gist of "bmap" with respect to the constraints "context"
3531 * on the domain.
3532 */
3535 {
3541 }
3545 {
3549 }
3553 {
3557 }
3561 {
3565 }
3569 {
3571 }
3573 /* Quick check to see if two basic maps are disjoint.
3574 * In particular, we reduce the equalities and inequalities of
3575 * one basic map in the context of the equalities of the other
3576 * basic map and check if we get a contradiction.
3577 */
3580 {
3612 }
3620 }
3629 }
3633 disjoint:
3637 error:
3641 }
3645 {
3648 }
3650 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3651 */
3655 {
3666 }
3667 }
3670 }
3672 /* Are "map1" and "map2" obviously disjoint, based on information
3673 * that can be derived without looking at the individual basic maps?
3674 *
3675 * In particular, if one of them is empty or if they live in different spaces
3676 * (ignoring parameters), then they are clearly disjoint.
3677 */
3680 {
3681 isl_bool disjoint;
3682 isl_bool match;
3706 }
3708 /* Are "map1" and "map2" obviously disjoint?
3709 *
3710 * If one of them is empty or if they live in different spaces (ignoring
3711 * parameters), then they are clearly disjoint.
3712 * This is checked by isl_map_plain_is_disjoint_global.
3713 *
3714 * If they have different parameters, then we skip any further tests.
3715 *
3716 * If they are obviously equal, but not obviously empty, then we will
3717 * not be able to detect if they are disjoint.
3718 *
3719 * Otherwise we check if each basic map in "map1" is obviously disjoint
3720 * from each basic map in "map2".
3721 */
3724 {
3725 isl_bool disjoint;
3726 isl_bool intersect;
3727 isl_bool match;
3742 }
3744 /* Are "map1" and "map2" disjoint?
3745 * The parameters are assumed to have been aligned.
3746 *
3747 * In particular, check whether all pairs of basic maps are disjoint.
3748 */
3751 {
3753 }
3755 /* Are "map1" and "map2" disjoint?
3756 *
3757 * They are disjoint if they are "obviously disjoint" or if one of them
3758 * is empty. Otherwise, they are not disjoint if one of them is universal.
3759 * If the two inputs are (obviously) equal and not empty, then they are
3760 * not disjoint.
3761 * If none of these cases apply, then check if all pairs of basic maps
3762 * are disjoint after aligning the parameters.
3763 */
3765 {
3766 isl_bool disjoint;
3767 isl_bool intersect;
3795 }
3797 /* Are "bmap1" and "bmap2" disjoint?
3798 *
3799 * They are disjoint if they are "obviously disjoint" or if one of them
3800 * is empty. Otherwise, they are not disjoint if one of them is universal.
3801 * If none of these cases apply, we compute the intersection and see if
3802 * the result is empty.
3803 */
3806 {
3807 isl_bool disjoint;
3808 isl_bool intersect;
3837 }
3839 /* Are "bset1" and "bset2" disjoint?
3840 */
3843 {
3845 }
3849 {
3851 }
3853 /* Are "set1" and "set2" disjoint?
3854 */
3856 {
3858 }
3860 /* Is "v" equal to 0, 1 or -1?
3861 */
3863 {
3865 }
3867 /* Check if we can combine a given div with lower bound l and upper
3868 * bound u with some other div and if so return that other div.
3869 * Otherwise return -1.
3870 *
3871 * We first check that
3872 * - the bounds are opposites of each other (except for the constant
3873 * term)
3874 * - the bounds do not reference any other div
3875 * - no div is defined in terms of this div
3876 *
3877 * Let m be the size of the range allowed on the div by the bounds.
3878 * That is, the bounds are of the form
3879 *
3880 * e <= a <= e + m - 1
3881 *
3882 * with e some expression in the other variables.
3883 * We look for another div b such that no third div is defined in terms
3884 * of this second div b and such that in any constraint that contains
3885 * a (except for the given lower and upper bound), also contains b
3886 * with a coefficient that is m times that of b.
3887 * That is, all constraints (except for the lower and upper bound)
3888 * are of the form
3889 *
3890 * e + f (a + m b) >= 0
3891 *
3892 * Furthermore, in the constraints that only contain b, the coefficient
3893 * of b should be equal to 1 or -1.
3894 * If so, we return b so that "a + m b" can be replaced by
3895 * a single div "c = a + m b".
3896 */
3899 {
3921 }
3931 }
3943 }
3954 }
3967 }
3972 }
3976 }
3978 /* Internal data structure used during the construction and/or evaluation of
3979 * an inequality that ensures that a pair of bounds always allows
3980 * for an integer value.
3981 *
3982 * "tab" is the tableau in which the inequality is evaluated. It may
3983 * be NULL until it is actually needed.
3984 * "v" contains the inequality coefficients.
3985 * "g", "fl" and "fu" are temporary scalars used during the construction and
3986 * evaluation.
3987 */
3991 isl_int g;
3992 isl_int fl;
3993 isl_int fu;
3994 };
3996 /* Free all the memory allocated by the fields of "data".
3997 */
3999 {
4005 }
4007 /* Is the inequality stored in data->v satisfied by "bmap"?
4008 * That is, does it only attain non-negative values?
4009 * data->tab is a tableau corresponding to "bmap".
4010 */
4013 {
4024 }
4026 /* Given a lower and an upper bound on div i, do they always allow
4027 * for an integer value of the given div?
4028 * Determine this property by constructing an inequality
4029 * such that the property is guaranteed when the inequality is nonnegative.
4030 * The lower bound is inequality l, while the upper bound is inequality u.
4031 * The constructed inequality is stored in data->v.
4032 *
4033 * Let the upper bound be
4034 *
4035 * -n_u a + e_u >= 0
4036 *
4037 * and the lower bound
4038 *
4039 * n_l a + e_l >= 0
4040 *
4041 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4042 * We have
4043 *
4044 * - f_u e_l <= f_u f_l g a <= f_l e_u
4045 *
4046 * Since all variables are integer valued, this is equivalent to
4047 *
4048 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4049 *
4050 * If this interval is at least f_u f_l g, then it contains at least
4051 * one integer value for a.
4052 * That is, the test constraint is
4053 *
4054 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4055 *
4056 * or
4057 *
4058 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4059 *
4060 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4061 * then the constraint can be scaled down by a factor g',
4062 * with the constant term replaced by
4063 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4064 * Note that the result of applying Fourier-Motzkin to this pair
4065 * of constraints is
4066 *
4067 * f_l e_u + f_u e_l >= 0
4068 *
4069 * If the constant term of the scaled down version of this constraint,
4070 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4071 * term of the scaled down test constraint, then the test constraint
4072 * is known to hold and no explicit evaluation is required.
4073 * This is essentially the Omega test.
4074 *
4075 * If the test constraint consists of only a constant term, then
4076 * it is sufficient to look at the sign of this constant term.
4077 */
4080 {
4105 }
4115 }
4117 /* Remove more kinds of divs that are not strictly needed.
4118 * In particular, if all pairs of lower and upper bounds on a div
4119 * are such that they allow at least one integer value of the div,
4120 * then we can eliminate the div using Fourier-Motzkin without
4121 * introducing any spurious solutions.
4122 *
4123 * If at least one of the two constraints has a unit coefficient for the div,
4124 * then the presence of such a value is guaranteed so there is no need to check.
4125 * In particular, the value attained by the bound with unit coefficient
4126 * can serve as this intermediate value.
4127 */
4130 {
4153 isl_bool has_int;
4161 }
4182 }
4185 }
4189 }
4193 }
4196 }
4207 error:
4212 }
4214 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4215 * and the upper bound u, div1 always occurs together with div2 in the form
4216 * (div1 + m div2), where m is the constant range on the variable div1
4217 * allowed by l and u, replace the pair div1 and div2 by a single
4218 * div that is equal to div1 + m div2.
4219 *
4220 * The new div will appear in the location that contains div2.
4221 * We need to modify all constraints that contain
4222 * div2 = (div - div1) / m
4223 * The coefficient of div2 is known to be equal to 1 or -1.
4224 * (If a constraint does not contain div2, it will also not contain div1.)
4225 * If the constraint also contains div1, then we know they appear
4226 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4227 * i.e., the coefficient of div is f.
4228 *
4229 * Otherwise, we first need to introduce div1 into the constraint.
4230 * Let l be
4231 *
4232 * div1 + f >=0
4233 *
4234 * and u
4235 *
4236 * -div1 + f' >= 0
4237 *
4238 * A lower bound on div2
4239 *
4240 * div2 + t >= 0
4241 *
4242 * can be replaced by
4243 *
4244 * m div2 + div1 + m t + f >= 0
4245 *
4246 * An upper bound
4247 *
4248 * -div2 + t >= 0
4249 *
4250 * can be replaced by
4251 *
4252 * -(m div2 + div1) + m t + f' >= 0
4253 *
4254 * These constraint are those that we would obtain from eliminating
4255 * div1 using Fourier-Motzkin.
4256 *
4257 * After all constraints have been modified, we drop the lower and upper
4258 * bound and then drop div1.
4259 * Since the new div is only placed in the same location that used
4260 * to store div2, but otherwise has a different meaning, any possible
4261 * explicit representation of the original div2 is removed.
4262 */
4265 {
4267 isl_int m;
4289 else
4292 }
4296 }
4305 }
4309 }
4311 /* First check if we can coalesce any pair of divs and
4312 * then continue with dropping more redundant divs.
4313 *
4314 * We loop over all pairs of lower and upper bounds on a div
4315 * with coefficient 1 and -1, respectively, check if there
4316 * is any other div "c" with which we can coalesce the div
4317 * and if so, perform the coalescing.
4318 */
4321 {
4344 }
4345 }
4346 }
4351 }
4354 }
4356 /* Are the "n" coefficients starting at "first" of inequality constraints
4357 * "i" and "j" of "bmap" equal to each other?
4358 */
4361 {
4363 }
4365 /* Are the "n" coefficients starting at "first" of inequality constraints
4366 * "i" and "j" of "bmap" opposite to each other?
4367 */
4370 {
4372 }
4374 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4375 * apart from the constant term?
4376 */
4378 {
4383 }
4385 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4386 * apart from the constant term and the coefficient at position "pos"?
4387 */
4390 {
4396 }
4398 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4399 * apart from the constant term and the coefficient at position "pos"?
4400 */
4403 {
4409 }
4411 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4412 * been modified, simplying it if "simplify" is set.
4413 * Free the temporary data structure "pairs" that was associated
4414 * to the old version of "bmap".
4415 */
4418 {
4423 }
4425 /* Is "div" the single unknown existentially quantified variable
4426 * in inequality constraint "ineq" of "bmap"?
4427 * "div" is known to have a non-zero coefficient in "ineq".
4428 */
4431 {
4434 isl_bool known;
4444 isl_bool known;
4453 }
4456 }
4458 /* Does integer division "div" have coefficient 1 in inequality constraint
4459 * "ineq" of "map"?
4460 */
4462 {
4470 }
4472 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4473 * then try and drop redundant divs again,
4474 * freeing the temporary data structure "pairs" that was associated
4475 * to the old version of "bmap".
4476 */
4479 {
4483 }
4485 /* Drop the integer division at position "div", along with the two
4486 * inequality constraints "ineq1" and "ineq2" in which it appears
4487 * from "bmap" and then try and drop redundant divs again,
4488 * freeing the temporary data structure "pairs" that was associated
4489 * to the old version of "bmap".
4490 */
4494 {
4501 }
4504 }
4506 /* Given two inequality constraints
4507 *
4508 * f(x) + n d + c >= 0, (ineq)
4509 *
4510 * with d the variable at position "pos", and
4511 *
4512 * f(x) + c0 >= 0, (lower)
4513 *
4514 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4515 * determined by the first constraint.
4516 * That is, store
4517 *
4518 * ceil((c0 - c)/n)
4519 *
4520 * in *l.
4521 */
4524 {
4528 }
4530 /* Given two inequality constraints
4531 *
4532 * f(x) + n d + c >= 0, (ineq)
4533 *
4534 * with d the variable at position "pos", and
4535 *
4536 * -f(x) - c0 >= 0, (upper)
4537 *
4538 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4539 * determined by the first constraint.
4540 * That is, store
4541 *
4542 * ceil((-c1 - c)/n)
4543 *
4544 * in *u.
4545 */
4548 {
4552 }
4554 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4555 * does the corresponding lower bound have a fixed value in "bmap"?
4556 *
4557 * In particular, "ineq" is of the form
4558 *
4559 * f(x) + n d + c >= 0
4560 *
4561 * with n > 0, c the constant term and
4562 * d the existentially quantified variable "div".
4563 * That is, the lower bound is
4564 *
4565 * ceil((-f(x) - c)/n)
4566 *
4567 * Look for a pair of constraints
4568 *
4569 * f(x) + c0 >= 0
4570 * -f(x) + c1 >= 0
4571 *
4572 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4573 * That is, check that
4574 *
4575 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4576 *
4577 * If so, return the index of inequality f(x) + c0 >= 0.
4578 * Otherwise, return -1.
4579 */
4581 {
4598 }
4602 }
4603 }
4620 }
4622 /* Given a lower bound constraint "ineq" on the existentially quantified
4623 * variable "div", such that the corresponding lower bound has
4624 * a fixed value in "bmap", assign this fixed value to the variable and
4625 * then try and drop redundant divs again,
4626 * freeing the temporary data structure "pairs" that was associated
4627 * to the old version of "bmap".
4628 * "lower" determines the constant value for the lower bound.
4629 *
4630 * In particular, "ineq" is of the form
4631 *
4632 * f(x) + n d + c >= 0,
4633 *
4634 * while "lower" is of the form
4635 *
4636 * f(x) + c0 >= 0
4637 *
4638 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4639 * is ceil((c0 - c)/n).
4640 */
4643 {
4644 isl_int c;
4657 }
4659 /* Remove divs that are not strictly needed based on the inequality
4660 * constraints.
4661 * In particular, if a div only occurs positively (or negatively)
4662 * in constraints, then it can simply be dropped.
4663 * Also, if a div occurs in only two constraints and if moreover
4664 * those two constraints are opposite to each other, except for the constant
4665 * term and if the sum of the constant terms is such that for any value
4666 * of the other values, there is always at least one integer value of the
4667 * div, i.e., if one plus this sum is greater than or equal to
4668 * the (absolute value) of the coefficient of the div in the constraints,
4669 * then we can also simply drop the div.
4670 *
4671 * If an existentially quantified variable does not have an explicit
4672 * representation, appears in only a single lower bound that does not
4673 * involve any other such existentially quantified variables and appears
4674 * in this lower bound with coefficient 1,
4675 * then fix the variable to the value of the lower bound. That is,
4676 * turn the inequality into an equality.
4677 * If for any value of the other variables, there is any value
4678 * for the existentially quantified variable satisfying the constraints,
4679 * then this lower bound also satisfies the constraints.
4680 * It is therefore safe to pick this lower bound.
4681 *
4682 * The same reasoning holds even if the coefficient is not one.
4683 * However, fixing the variable to the value of the lower bound may
4684 * in general introduce an extra integer division, in which case
4685 * it may be better to pick another value.
4686 * If this integer division has a known constant value, then plugging
4687 * in this constant value removes the existentially quantified variable
4688 * completely. In particular, if the lower bound is of the form
4689 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4690 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4691 * then the existentially quantified variable can be assigned this
4692 * shared value.
4693 *
4694 * We skip divs that appear in equalities or in the definition of other divs.
4695 * Divs that appear in the definition of other divs usually occur in at least
4696 * 4 constraints, but the constraints may have been simplified.
4697 *
4698 * If any divs are left after these simple checks then we move on
4699 * to more complicated cases in drop_more_redundant_divs.
4700 */
4703 {
4743 }
4747 }
4748 }
4756 }
4759 else
4779 pairs);
4783 pairs);
4785 }
4802 else
4809 }
4812 }
4819 error:
4823 }
4825 /* Consider the coefficients at "c" as a row vector and replace
4826 * them with their product with "T". "T" is assumed to be a square matrix.
4827 */
4829 {
4851 }
4853 /* Plug in T for the variables in "bmap" starting at "pos".
4854 * T is a linear unimodular matrix, i.e., without constant term.
4855 */
4858 {
4885 }
4889 error:
4893 }
4895 /* Remove divs that are not strictly needed.
4896 *
4897 * First look for an equality constraint involving two or more
4898 * existentially quantified variables without an explicit
4899 * representation. Replace the combination that appears
4900 * in the equality constraint by a single existentially quantified
4901 * variable such that the equality can be used to derive
4902 * an explicit representation for the variable.
4903 * If there are no more such equality constraints, then continue
4904 * with isl_basic_map_drop_redundant_divs_ineq.
4905 *
4906 * In particular, if the equality constraint is of the form
4907 *
4908 * f(x) + \sum_i c_i a_i = 0
4909 *
4910 * with a_i existentially quantified variable without explicit
4911 * representation, then apply a transformation on the existentially
4912 * quantified variables to turn the constraint into
4913 *
4914 * f(x) + g a_1' = 0
4915 *
4916 * with g the gcd of the c_i.
4917 * In order to easily identify which existentially quantified variables
4918 * have a complete explicit representation, i.e., without being defined
4919 * in terms of other existentially quantified variables without
4920 * an explicit representation, the existentially quantified variables
4921 * are first sorted.
4922 *
4923 * The variable transformation is computed by extending the row
4924 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4925 *
4926 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4927 * [a_2'] [ a_2 ]
4928 * ... = U ....
4929 * [a_n'] [ a_n ]
4930 *
4931 * with [c_1/g ... c_n/g] representing the first row of U.
4932 * The inverse of U is then plugged into the original constraints.
4933 * The call to isl_basic_map_simplify makes sure the explicit
4934 * representation for a_1' is extracted from the equality constraint.
4935 */
4938 {
4973 }
4992 }
4994 /* Does "bmap" satisfy any equality that involves more than 2 variables
4995 * and/or has coefficients different from -1 and 1?
4996 */
4998 {
5027 }
5030 }
5032 /* Remove any common factor g from the constraint coefficients in "v".
5033 * The constant term is stored in the first position and is replaced
5034 * by floor(c/g). If any common factor is removed and if this results
5035 * in a tightening of the constraint, then set *tightened.
5036 */
5039 {
5059 }
5061 /* If "bmap" is an integer set that satisfies any equality involving
5062 * more than 2 variables and/or has coefficients different from -1 and 1,
5063 * then use variable compression to reduce the coefficients by removing
5064 * any (hidden) common factor.
5065 * In particular, apply the variable compression to each constraint,
5066 * factor out any common factor in the non-constant coefficients and
5067 * then apply the inverse of the compression.
5068 * At the end, we mark the basic map as having reduced constants.
5069 * If this flag is still set on the next invocation of this function,
5070 * then we skip the computation.
5071 *
5072 * Removing a common factor may result in a tightening of some of
5073 * the constraints. If this happens, then we may end up with two
5074 * opposite inequalities that can be replaced by an equality.
5075 * We therefore call isl_basic_map_detect_inequality_pairs,
5076 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5077 * and isl_basic_map_gauss if such a pair was found.
5078 *
5079 * Note that this function may leave the result in an inconsistent state.
5080 * In particular, the constraints may not be gaussed.
5081 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5082 * for some of the test cases to pass successfully.
5083 * Any potential modification of the representation is therefore only
5084 * performed on a single copy of the basic map.
5085 */
5088 {
5122 }
5137 }
5152 }
5153 }
5156 error:
5161 }
5163 /* Shift the integer division at position "div" of "bmap"
5164 * by "shift" times the variable at position "pos".
5165 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5166 * corresponds to the constant term.
5167 *
5168 * That is, if the integer division has the form
5169 *
5170 * floor(f(x)/d)
5171 *
5172 * then replace it by
5173 *
5174 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5175 */
5178 {
5197 }
5203 }
5211 }
5214 }