| blob | 50bab5529b5f88167c8c7d440ef9692102f15911 |
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 */
239 {
257 }
259 /* Remove any common factor in numerator and denominator of a div expression,
260 * not taking into account the constant term.
261 * That is, look for any div of the form
262 *
263 * floor((a + m f(x))/(m d))
264 *
265 * and replace it by
266 *
267 * floor((floor(a/m) + f(x))/d)
268 *
269 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
270 * and can therefore not influence the result of the floor.
271 */
274 {
286 }
288 /* Assumes divs have been ordered if keep_divs is set.
289 */
293 {
311 }
322 }
331 /* We need to be careful about circular definitions,
332 * so for now we just remove the definition of div k
333 * if the equality contains any divs.
334 * If keep_divs is set, then the divs have been ordered
335 * and we can keep the definition as long as the result
336 * is still ordered.
337 */
346 }
349 }
351 /* Assumes divs have been ordered if keep_divs is set.
352 */
355 {
363 }
365 /* Check if elimination of div "div" using equality "eq" would not
366 * result in a div depending on a later div.
367 */
370 {
385 }
388 }
390 /* Eliminate divs based on equalities
391 */
394 {
409 isl_bool ok;
425 }
426 }
430 }
432 /* Eliminate divs based on inequalities
433 */
436 {
466 }
468 }
470 /* Does the equality constraint at position "eq" in "bmap" involve
471 * any local variables in the range [first, first + n)
472 * that are not marked as having an explicit representation?
473 */
476 {
485 isl_bool unknown;
494 }
497 }
499 /* The last local variable involved in the equality constraint
500 * at position "eq" in "bmap" is the local variable at position "div".
501 * It can therefore be used to extract an explicit representation
502 * for that variable.
503 * Do so unless the local variable already has an explicit representation or
504 * the explicit representation would involve any other local variables
505 * that in turn do not have an explicit representation.
506 * An equality constraint involving local variables without an explicit
507 * representation can be used in isl_basic_map_drop_redundant_divs
508 * to separate out an independent local variable. Introducing
509 * an explicit representation here would block this transformation,
510 * while the partial explicit representation in itself is not very useful.
511 * Set *progress if anything is changed.
512 *
513 * The equality constraint is of the form
514 *
515 * f(x) + n e >= 0
516 *
517 * with n a positive number. The explicit representation derived from
518 * this constraint is
519 *
520 * floor((-f(x))/n)
521 */
524 {
526 isl_bool involves;
549 }
553 {
576 }
592 }
599 }
602 }
606 {
608 progress));
609 }
613 {
619 }
621 }
623 /* Hash table of inequalities in a basic map.
624 * "index" is an array of addresses of inequalities in the basic map, some
625 * of which are NULL. The inequalities are hashed on the coefficients
626 * except the constant term.
627 * "size" is the number of elements in the array and is always a power of two
628 * "bits" is the number of bits need to represent an index into the array.
629 * "total" is the total dimension of the basic map.
630 */
636 };
638 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
639 */
642 {
659 }
661 /* Free the memory allocated by create_constraint_index.
662 */
664 {
666 }
668 /* Return the position in ci->index that contains the address of
669 * an inequality that is equal to *ineq up to the constant term,
670 * provided this address is not identical to "ineq".
671 * If there is no such inequality, then return the position where
672 * such an inequality should be inserted.
673 */
675 {
683 }
685 /* Return the position in ci->index that contains the address of
686 * an inequality that is equal to the k'th inequality of "bmap"
687 * up to the constant term, provided it does not point to the very
688 * same inequality.
689 * If there is no such inequality, then return the position where
690 * such an inequality should be inserted.
691 */
694 {
696 }
700 {
702 }
704 /* Fill in the "ci" data structure with the inequalities of "bset".
705 */
708 {
717 }
720 }
722 /* Is the inequality ineq (obviously) redundant with respect
723 * to the constraints in "ci"?
724 *
725 * Look for an inequality in "ci" with the same coefficients and then
726 * check if the contant term of "ineq" is greater than or equal
727 * to the constant term of that inequality. If so, "ineq" is clearly
728 * redundant.
729 *
730 * Note that hash_index_ineq ignores a stored constraint if it has
731 * the same address as the passed inequality. It is ok to pass
732 * the address of a local variable here since it will never be
733 * the same as the address of a constraint in "ci".
734 */
737 {
744 }
746 /* If we can eliminate more than one div, then we need to make
747 * sure we do it from last div to first div, in order not to
748 * change the position of the other divs that still need to
749 * be removed.
750 */
753 {
807 }
809 }
821 }
824 out:
828 }
831 {
843 }
845 }
847 /* Normalize divs that appear in equalities.
848 *
849 * In particular, we assume that bmap contains some equalities
850 * of the form
851 *
852 * a x = m * e_i
853 *
854 * and we want to replace the set of e_i by a minimal set and
855 * such that the new e_i have a canonical representation in terms
856 * of the vector x.
857 * If any of the equalities involves more than one divs, then
858 * we currently simply bail out.
859 *
860 * Let us first additionally assume that all equalities involve
861 * a div. The equalities then express modulo constraints on the
862 * remaining variables and we can use "parameter compression"
863 * to find a minimal set of constraints. The result is a transformation
864 *
865 * x = T(x') = x_0 + G x'
866 *
867 * with G a lower-triangular matrix with all elements below the diagonal
868 * non-negative and smaller than the diagonal element on the same row.
869 * We first normalize x_0 by making the same property hold in the affine
870 * T matrix.
871 * The rows i of G with a 1 on the diagonal do not impose any modulo
872 * constraint and simply express x_i = x'_i.
873 * For each of the remaining rows i, we introduce a div and a corresponding
874 * equality. In particular
875 *
876 * g_ii e_j = x_i - g_i(x')
877 *
878 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
879 * corresponding div (if g_kk != 1).
880 *
881 * If there are any equalities not involving any div, then we
882 * first apply a variable compression on the variables x:
883 *
884 * x = C x'' x'' = C_2 x
885 *
886 * and perform the above parameter compression on A C instead of on A.
887 * The resulting compression is then of the form
888 *
889 * x'' = T(x') = x_0 + G x'
890 *
891 * and in constructing the new divs and the corresponding equalities,
892 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
893 * by the corresponding row from C_2.
894 */
897 {
906 isl_int v;
938 }
939 }
948 }
954 }
964 }
971 }
976 /* We have to be careful because dropping equalities may reorder them */
986 }
987 }
993 else
994 needed++;
995 }
1001 }
1011 else
1019 else
1022 }
1026 }
1033 done:
1037 error:
1043 }
1047 {
1057 }
1059 /* Check whether it is ok to define a div based on an inequality.
1060 * To avoid the introduction of circular definitions of divs, we
1061 * do not allow such a definition if the resulting expression would refer to
1062 * any other undefined divs or if any known div is defined in
1063 * terms of the unknown div.
1064 */
1067 {
1071 /* Not defined in terms of unknown divs */
1079 }
1081 /* No other div defined in terms of this one => avoid loops */
1089 }
1092 }
1094 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1095 * be a better expression than the current one?
1096 *
1097 * If we do not have any expression yet, then any expression would be better.
1098 * Otherwise we check if the last variable involved in the inequality
1099 * (disregarding the div that it would define) is in an earlier position
1100 * than the last variable involved in the current div expression.
1101 */
1104 {
1121 }
1123 /* Given two constraints "k" and "l" that are opposite to each other,
1124 * except for the constant term, check if we can use them
1125 * to obtain an expression for one of the hitherto unknown divs or
1126 * a "better" expression for a div for which we already have an expression.
1127 * "sum" is the sum of the constant terms of the constraints.
1128 * If this sum is strictly smaller than the coefficient of one
1129 * of the divs, then this pair can be used define the div.
1130 * To avoid the introduction of circular definitions of divs, we
1131 * do not use the pair if the resulting expression would refer to
1132 * any other undefined divs or if any known div is defined in
1133 * terms of the unknown div.
1134 */
1138 {
1143 isl_bool set_div;
1158 else
1163 }
1165 }
1169 {
1173 isl_int sum;
1188 }
1196 }
1211 }
1213 /* We need to break out of the loop after these
1214 * changes since the contents of the hash
1215 * will no longer be valid.
1216 * Plus, we probably we want to regauss first.
1217 */
1225 }
1230 }
1232 /* Detect all pairs of inequalities that form an equality.
1233 *
1234 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1235 * Call it repeatedly while it is making progress.
1236 */
1239 {
1251 }
1253 /* Eliminate knowns divs from constraints where they appear with
1254 * a (positive or negative) unit coefficient.
1255 *
1256 * That is, replace
1257 *
1258 * floor(e/m) + f >= 0
1259 *
1260 * by
1261 *
1262 * e + m f >= 0
1263 *
1264 * and
1265 *
1266 * -floor(e/m) + f >= 0
1267 *
1268 * by
1269 *
1270 * -e + m f + m - 1 >= 0
1271 *
1272 * The first conversion is valid because floor(e/m) >= -f is equivalent
1273 * to e/m >= -f because -f is an integral expression.
1274 * The second conversion follows from the fact that
1275 *
1276 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1277 *
1278 *
1279 * Note that one of the div constraints may have been eliminated
1280 * due to being redundant with respect to the constraint that is
1281 * being modified by this function. The modified constraint may
1282 * no longer imply this div constraint, so we add it back to make
1283 * sure we do not lose any information.
1284 *
1285 * We skip integral divs, i.e., those with denominator 1, as we would
1286 * risk eliminating the div from the div constraints. We do not need
1287 * to handle those divs here anyway since the div constraints will turn
1288 * out to form an equality and this equality can then be used to eliminate
1289 * the div from all constraints.
1290 */
1293 {
1325 else
1335 }
1340 }
1341 }
1344 }
1347 {
1352 isl_bool empty;
1368 /* requires equalities in normal form */
1374 }
1376 }
1379 {
1381 }
1386 {
1418 }
1422 {
1424 }
1427 /* If the only constraints a div d=floor(f/m)
1428 * appears in are its two defining constraints
1429 *
1430 * f - m d >=0
1431 * -(f - (m - 1)) + m d >= 0
1432 *
1433 * then it can safely be removed.
1434 */
1436 {
1445 isl_bool red;
1452 }
1459 }
1462 }
1464 /*
1465 * Remove divs that don't occur in any of the constraints or other divs.
1466 * These can arise when dropping constraints from a basic map or
1467 * when the divs of a basic map have been temporarily aligned
1468 * with the divs of another basic map.
1469 */
1472 {
1481 isl_bool redundant;
1491 }
1493 }
1495 /* Mark "bmap" as final, without checking for obviously redundant
1496 * integer divisions. This function should be used when "bmap"
1497 * is known not to involve any such integer divisions.
1498 */
1501 {
1506 }
1508 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1509 */
1511 {
1515 }
1518 {
1520 }
1522 /* Remove definition of any div that is defined in terms of the given variable.
1523 * The div itself is not removed. Functions such as
1524 * eliminate_divs_ineq depend on the other divs remaining in place.
1525 */
1528 {
1542 }
1544 }
1546 /* Eliminate the specified variables from the constraints using
1547 * Fourier-Motzkin. The variables themselves are not removed.
1548 */
1551 {
1585 }
1592 n_lower++;
1594 n_upper++;
1595 }
1619 }
1622 }
1634 }
1635 }
1639 error:
1642 }
1646 {
1649 }
1651 /* Eliminate the specified n dimensions starting at first from the
1652 * constraints, without removing the dimensions from the space.
1653 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1654 * Otherwise, they are projected out and the original space is restored.
1655 */
1659 {
1674 }
1681 }
1686 {
1688 }
1690 /* Remove all constraints from "bmap" that reference any unknown local
1691 * variables (directly or indirectly).
1692 *
1693 * Dropping all constraints on a local variable will make it redundant,
1694 * so it will get removed implicitly by
1695 * isl_basic_map_drop_constraints_involving_dims. Some other local
1696 * variables may also end up becoming redundant if they only appear
1697 * in constraints together with the unknown local variable.
1698 * Therefore, start over after calling
1699 * isl_basic_map_drop_constraints_involving_dims.
1700 */
1703 {
1704 isl_bool known;
1729 }
1732 }
1734 /* Remove all constraints from "map" that reference any unknown local
1735 * variables (directly or indirectly).
1736 *
1737 * Since constraints may get dropped from the basic maps,
1738 * they may no longer be disjoint from each other.
1739 */
1742 {
1744 isl_bool known;
1762 }
1768 }
1770 /* Don't assume equalities are in order, because align_divs
1771 * may have changed the order of the divs.
1772 */
1774 {
1787 }
1788 }
1789 }
1793 {
1795 }
1799 {
1813 }
1815 }
1817 }
1821 {
1824 }
1828 {
1838 }
1859 error:
1863 }
1865 /* For each inequality in "ineq" that is a shifted (more relaxed)
1866 * copy of an inequality in "context", mark the corresponding entry
1867 * in "row" with -1.
1868 * If an inequality only has a non-negative constant term, then
1869 * mark it as well.
1870 */
1873 {
1890 isl_bool redundant;
1896 }
1903 }
1906 error:
1909 }
1913 {
1926 isl_bool redundant;
1938 }
1941 error:
1944 }
1946 /* Remove constraints from "bmap" that are identical to constraints
1947 * in "context" or that are more relaxed (greater constant term).
1948 *
1949 * We perform the test for shifted copies on the pure constraints
1950 * in remove_shifted_constraints.
1951 */
1954 {
1963 }
1976 error:
1980 }
1982 /* Does the (linear part of a) constraint "c" involve any of the "len"
1983 * "relevant" dimensions?
1984 */
1986 {
1994 }
1997 }
1999 /* Drop constraints from "bmap" that do not involve any of
2000 * the dimensions marked "relevant".
2001 */
2004 {
2019 }
2026 }
2029 }
2031 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2032 *
2033 * In particular, for any variable involved in the constraint,
2034 * find the actual group id from before and replace the group
2035 * of the corresponding variable by the minimal group of all
2036 * the variables involved in the constraint considered so far
2037 * (if this minimum is smaller) or replace the minimum by this group
2038 * (if the minimum is larger).
2039 *
2040 * At the end, all the variables in "c" will (indirectly) point
2041 * to the minimal of the groups that they referred to originally.
2042 */
2044 {
2061 }
2062 }
2064 /* Allocate an array of groups of variables, one for each variable
2065 * in "context", initialized to zero.
2066 */
2068 {
2075 }
2077 /* Drop constraints from "bmap" that only involve variables that are
2078 * not related to any of the variables marked with a "-1" in "group".
2079 *
2080 * We construct groups of variables that collect variables that
2081 * (indirectly) appear in some common constraint of "bmap".
2082 * Each group is identified by the first variable in the group,
2083 * except for the special group of variables that was already identified
2084 * in the input as -1 (or are related to those variables).
2085 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2086 * otherwise the group of i is the group of group[i].
2087 *
2088 * We first initialize groups for the remaining variables.
2089 * Then we iterate over the constraints of "bmap" and update the
2090 * group of the variables in the constraint by the smallest group.
2091 * Finally, we resolve indirect references to groups by running over
2092 * the variables.
2093 *
2094 * After computing the groups, we drop constraints that do not involve
2095 * any variables in the -1 group.
2096 */
2099 {
2116 }
2134 }
2136 /* Drop constraints from "context" that are irrelevant for computing
2137 * the gist of "bset".
2138 *
2139 * In particular, drop constraints in variables that are not related
2140 * to any of the variables involved in the constraints of "bset"
2141 * in the sense that there is no sequence of constraints that connects them.
2142 *
2143 * We first mark all variables that appear in "bset" as belonging
2144 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2145 */
2148 {
2169 }
2175 }
2178 }
2180 /* Drop constraints from "context" that are irrelevant for computing
2181 * the gist of the inequalities "ineq".
2182 * Inequalities in "ineq" for which the corresponding element of row
2183 * is set to -1 have already been marked for removal and should be ignored.
2184 *
2185 * In particular, drop constraints in variables that are not related
2186 * to any of the variables involved in "ineq"
2187 * in the sense that there is no sequence of constraints that connects them.
2188 *
2189 * We first mark all variables that appear in "bset" as belonging
2190 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2191 */
2194 {
2215 }
2218 }
2221 }
2223 /* Do all "n" entries of "row" contain a negative value?
2224 */
2226 {
2234 }
2236 /* Update the inequalities in "bset" based on the information in "row"
2237 * and "tab".
2238 *
2239 * In particular, the array "row" contains either -1, meaning that
2240 * the corresponding inequality of "bset" is redundant, or the index
2241 * of an inequality in "tab".
2242 *
2243 * If the row entry is -1, then drop the inequality.
2244 * Otherwise, if the constraint is marked redundant in the tableau,
2245 * then drop the inequality. Similarly, if it is marked as an equality
2246 * in the tableau, then turn the inequality into an equality and
2247 * perform Gaussian elimination.
2248 */
2251 {
2268 }
2278 }
2279 }
2285 }
2287 /* Update the inequalities in "bset" based on the information in "row"
2288 * and "tab" and free all arguments (other than "bset").
2289 */
2294 {
2303 }
2305 /* Remove all information from bset that is redundant in the context
2306 * of context.
2307 * "ineq" contains the (possibly transformed) inequalities of "bset",
2308 * in the same order.
2309 * The (explicit) equalities of "bset" are assumed to have been taken
2310 * into account by the transformation such that only the inequalities
2311 * are relevant.
2312 * "context" is assumed not to be empty.
2313 *
2314 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2315 * A value of -1 means that the inequality is obviously redundant and may
2316 * not even appear in "tab".
2317 *
2318 * We first mark the inequalities of "bset"
2319 * that are obviously redundant with respect to some inequality in "context".
2320 * Then we remove those constraints from "context" that have become
2321 * irrelevant for computing the gist of "bset".
2322 * Note that this removal of constraints cannot be replaced by
2323 * a factorization because factors in "bset" may still be connected
2324 * to each other through constraints in "context".
2325 *
2326 * If there are any inequalities left, we construct a tableau for
2327 * the context and then add the inequalities of "bset".
2328 * Before adding these inequalities, we freeze all constraints such that
2329 * they won't be considered redundant in terms of the constraints of "bset".
2330 * Then we detect all redundant constraints (among the
2331 * constraints that weren't frozen), first by checking for redundancy in the
2332 * the tableau and then by checking if replacing a constraint by its negation
2333 * would lead to an empty set. This last step is fairly expensive
2334 * and could be optimized by more reuse of the tableau.
2335 * Finally, we update bset according to the results.
2336 */
2339 {
2354 }
2390 }
2413 }
2418 }
2422 error:
2430 }
2432 /* Extract the inequalities of "bset" as an isl_mat.
2433 */
2435 {
2449 }
2451 /* Remove all information from "bset" that is redundant in the context
2452 * of "context", for the case where both "bset" and "context" are
2453 * full-dimensional.
2454 */
2457 {
2462 }
2464 /* Remove all information from "bset" that is redundant in the context
2465 * of "context", for the case where the combined equalities of
2466 * "bset" and "context" allow for a compression that can be obtained
2467 * by preapplication of "T".
2468 *
2469 * "bset" itself is not transformed by "T". Instead, the inequalities
2470 * are extracted from "bset" and those are transformed by "T".
2471 * uset_gist_full then determines which of the transformed inequalities
2472 * are redundant with respect to the transformed "context" and removes
2473 * the corresponding inequalities from "bset".
2474 *
2475 * After preapplying "T" to the inequalities, any common factor is
2476 * removed from the coefficients. If this results in a tightening
2477 * of the constant term, then the same tightening is applied to
2478 * the corresponding untransformed inequality in "bset".
2479 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2480 *
2481 * g f'(x) + r >= 0
2482 *
2483 * with 0 <= r < g, then it is equivalent to
2484 *
2485 * f'(x) >= 0
2486 *
2487 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2488 * subspace compressed by T since the latter would be transformed to
2489 *
2490 * g f'(x) >= 0
2491 */
2495 {
2499 isl_int rem;
2511 }
2534 }
2538 error:
2543 }
2545 /* Project "bset" onto the variables that are involved in "template".
2546 */
2549 {
2558 isl_bool involved;
2567 }
2570 }
2572 /* Remove all information from bset that is redundant in the context
2573 * of context. In particular, equalities that are linear combinations
2574 * of those in context are removed. Then the inequalities that are
2575 * redundant in the context of the equalities and inequalities of
2576 * context are removed.
2577 *
2578 * First of all, we drop those constraints from "context"
2579 * that are irrelevant for computing the gist of "bset".
2580 * Alternatively, we could factorize the intersection of "context" and "bset".
2581 *
2582 * We first compute the intersection of the integer affine hulls
2583 * of "bset" and "context",
2584 * compute the gist inside this intersection and then reduce
2585 * the constraints with respect to the equalities of the context
2586 * that only involve variables already involved in the input.
2587 *
2588 * If two constraints are mutually redundant, then uset_gist_full
2589 * will remove the second of those constraints. We therefore first
2590 * sort the constraints so that constraints not involving existentially
2591 * quantified variables are given precedence over those that do.
2592 * We have to perform this sorting before the variable compression,
2593 * because that may effect the order of the variables.
2594 */
2597 {
2622 }
2627 }
2637 }
2648 }
2651 error:
2655 }
2657 /* Return the number of equality constraints in "bmap" that involve
2658 * local variables. This function assumes that Gaussian elimination
2659 * has been applied to the equality constraints.
2660 */
2662 {
2682 }
2684 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2685 * The constraints are assumed not to involve any local variables.
2686 */
2689 {
2707 }
2712 error:
2717 }
2719 /* Construct and return a variable compression based on the equality
2720 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2721 * "n1" is the number of (initial) equality constraints in "bmap1"
2722 * that do involve local variables.
2723 * "n2" is the number of (initial) equality constraints in "bmap2"
2724 * that do involve local variables.
2725 * "total" is the total number of other variables.
2726 * This function assumes that Gaussian elimination
2727 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2728 * such that the equality constraints not involving local variables
2729 * are those that start at "n1" or "n2".
2730 *
2731 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2732 * then simply compute the compression based on the equality constraints
2733 * in the other basic map.
2734 * Otherwise, combine the equality constraints from both into a new
2735 * basic map such that Gaussian elimination can be applied to this combination
2736 * and then construct a variable compression from the resulting
2737 * equality constraints.
2738 */
2742 {
2752 }
2757 }
2772 }
2774 /* Extract the stride constraints from "bmap", compressed
2775 * with respect to both the stride constraints in "context" and
2776 * the remaining equality constraints in both "bmap" and "context".
2777 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2778 * "context_n_eq" is the number of (initial) stride constraints in "context".
2779 *
2780 * Let x be all variables in "bmap" (and "context") other than the local
2781 * variables. First compute a variable compression
2782 *
2783 * x = V x'
2784 *
2785 * based on the non-stride equality constraints in "bmap" and "context".
2786 * Consider the stride constraints of "context",
2787 *
2788 * A(x) + B(y) = 0
2789 *
2790 * with y the local variables and plug in the variable compression,
2791 * resulting in
2792 *
2793 * A(V x') + B(y) = 0
2794 *
2795 * Use these constraints to compute a parameter compression on x'
2796 *
2797 * x' = T x''
2798 *
2799 * Now consider the stride constraints of "bmap"
2800 *
2801 * C(x) + D(y) = 0
2802 *
2803 * and plug in x = V*T x''.
2804 * That is, return A = [C*V*T D].
2805 */
2809 {
2838 }
2840 /* Remove the prime factors from *g that have an exponent that
2841 * is strictly smaller than the exponent in "c".
2842 * All exponents in *g are known to be smaller than or equal
2843 * to those in "c".
2844 *
2845 * That is, if *g is equal to
2846 *
2847 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2848 *
2849 * and "c" is equal to
2850 *
2851 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2852 *
2853 * then update *g to
2854 *
2855 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2856 * p_n^{e_n * (e_n = f_n)}
2857 *
2858 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2859 * neither does the gcd of *g and c / *g.
2860 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2861 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2862 * Dividing *g by this gcd therefore strictly reduces the exponent
2863 * of the prime factors that need to be removed, while leaving the
2864 * other prime factors untouched.
2865 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2866 * removes all undesired factors, without removing any others.
2867 */
2869 {
2870 isl_int t;
2879 }
2881 }
2883 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2884 * of the same stride constraints in a compressed space that exploits
2885 * all equalities in the context and the other equalities in "bmap".
2886 *
2887 * If the stride constraints of "bmap" are of the form
2888 *
2889 * C(x) + D(y) = 0
2890 *
2891 * then A is of the form
2892 *
2893 * B(x') + D(y) = 0
2894 *
2895 * If any of these constraints involves only a single local variable y,
2896 * then the constraint appears as
2897 *
2898 * f(x) + m y_i = 0
2899 *
2900 * in "bmap" and as
2901 *
2902 * h(x') + m y_i = 0
2903 *
2904 * in "A".
2905 *
2906 * Let g be the gcd of m and the coefficients of h.
2907 * Then, in particular, g is a divisor of the coefficients of h and
2908 *
2909 * f(x) = h(x')
2910 *
2911 * is known to be a multiple of g.
2912 * If some prime factor in m appears with the same exponent in g,
2913 * then it can be removed from m because f(x) is already known
2914 * to be a multiple of g and therefore in particular of this power
2915 * of the prime factors.
2916 * Prime factors that appear with a smaller exponent in g cannot
2917 * be removed from m.
2918 * Let g' be the divisor of g containing all prime factors that
2919 * appear with the same exponent in m and g, then
2920 *
2921 * f(x) + m y_i = 0
2922 *
2923 * can be replaced by
2924 *
2925 * f(x) + m/g' y_i' = 0
2926 *
2927 * Note that (if g' != 1) this changes the explicit representation
2928 * of y_i to that of y_i', so the integer division at position i
2929 * is marked unknown and later recomputed by a call to
2930 * isl_basic_map_gauss.
2931 */
2934 {
2938 isl_int gcd;
2972 }
2979 error:
2983 }
2985 /* Simplify the stride constraints in "bmap" based on
2986 * the remaining equality constraints in "bmap" and all equality
2987 * constraints in "context".
2988 * Only do this if both "bmap" and "context" have stride constraints.
2989 *
2990 * First extract a copy of the stride constraints in "bmap" in a compressed
2991 * space exploiting all the other equality constraints and then
2992 * use this compressed copy to simplify the original stride constraints.
2993 */
2996 {
3018 }
3020 /* Return a basic map that has the same intersection with "context" as "bmap"
3021 * and that is as "simple" as possible.
3022 *
3023 * The core computation is performed on the pure constraints.
3024 * When we add back the meaning of the integer divisions, we need
3025 * to (re)introduce the div constraints. If we happen to have
3026 * discovered that some of these integer divisions are equal to
3027 * some affine combination of other variables, then these div
3028 * constraints may end up getting simplified in terms of the equalities,
3029 * resulting in extra inequalities on the other variables that
3030 * may have been removed already or that may not even have been
3031 * part of the input. We try and remove those constraints of
3032 * this form that are most obviously redundant with respect to
3033 * the context. We also remove those div constraints that are
3034 * redundant with respect to the other constraints in the result.
3035 *
3036 * The stride constraints among the equality constraints in "bmap" are
3037 * also simplified with respecting to the other equality constraints
3038 * in "bmap" and with respect to all equality constraints in "context".
3039 */
3042 {
3053 }
3059 }
3063 }
3085 }
3104 error:
3108 }
3110 /*
3111 * Assumes context has no implicit divs.
3112 */
3115 {
3126 }
3146 }
3147 }
3151 error:
3155 }
3157 /* Drop all inequalities from "bmap" that also appear in "context".
3158 * "context" is assumed to have only known local variables and
3159 * the initial local variables of "bmap" are assumed to be the same
3160 * as those of "context".
3161 * The constraints of both "bmap" and "context" are assumed
3162 * to have been sorted using isl_basic_map_sort_constraints.
3163 *
3164 * Run through the inequality constraints of "bmap" and "context"
3165 * in sorted order.
3166 * If a constraint of "bmap" involves variables not in "context",
3167 * then it cannot appear in "context".
3168 * If a matching constraint is found, it is removed from "bmap".
3169 */
3172 {
3191 }
3197 }
3201 }
3206 }
3209 }
3212 }
3214 /* Drop all equalities from "bmap" that also appear in "context".
3215 * "context" is assumed to have only known local variables and
3216 * the initial local variables of "bmap" are assumed to be the same
3217 * as those of "context".
3218 *
3219 * Run through the equality constraints of "bmap" and "context"
3220 * in sorted order.
3221 * If a constraint of "bmap" involves variables not in "context",
3222 * then it cannot appear in "context".
3223 * If a matching constraint is found, it is removed from "bmap".
3224 */
3227 {
3251 }
3255 }
3260 }
3263 }
3266 }
3268 /* Remove the constraints in "context" from "bmap".
3269 * "context" is assumed to have explicit representations
3270 * for all local variables.
3271 *
3272 * First align the divs of "bmap" to those of "context" and
3273 * sort the constraints. Then drop all constraints from "bmap"
3274 * that appear in "context".
3275 */
3278 {
3293 }
3312 error:
3316 }
3318 /* Replace "map" by the disjunct at position "pos" and free "context".
3319 */
3322 {
3329 }
3331 /* Remove the constraints in "context" from "map".
3332 * If any of the disjuncts in the result turns out to be the universe,
3333 * then return this universe.
3334 * "context" is assumed to have explicit representations
3335 * for all local variables.
3336 */
3339 {
3349 }
3368 }
3375 error:
3379 }
3381 /* Remove the constraints in "context" from "set".
3382 * If any of the disjuncts in the result turns out to be the universe,
3383 * then return this universe.
3384 * "context" is assumed to have explicit representations
3385 * for all local variables.
3386 */
3389 {
3392 }
3394 /* Remove the constraints in "context" from "map".
3395 * If any of the disjuncts in the result turns out to be the universe,
3396 * then return this universe.
3397 * "context" is assumed to consist of a single disjunct and
3398 * to have explicit representations for all local variables.
3399 */
3402 {
3407 }
3409 /* Replace "map" by a universe map in the same space and free "drop".
3410 */
3413 {
3420 }
3422 /* Return a map that has the same intersection with "context" as "map"
3423 * and that is as "simple" as possible.
3424 *
3425 * If "map" is already the universe, then we cannot make it any simpler.
3426 * Similarly, if "context" is the universe, then we cannot exploit it
3427 * to simplify "map"
3428 * If "map" and "context" are identical to each other, then we can
3429 * return the corresponding universe.
3430 *
3431 * If either "map" or "context" consists of multiple disjuncts,
3432 * then check if "context" happens to be a subset of "map",
3433 * in which case all constraints can be removed.
3434 * In case of multiple disjuncts, the standard procedure
3435 * may not be able to detect that all constraints can be removed.
3436 *
3437 * If none of these cases apply, we have to work a bit harder.
3438 * During this computation, we make use of a single disjunct context,
3439 * so if the original context consists of more than one disjunct
3440 * then we need to approximate the context by a single disjunct set.
3441 * Simply taking the simple hull may drop constraints that are
3442 * only implicitly available in each disjunct. We therefore also
3443 * look for constraints among those defining "map" that are valid
3444 * for the context. These can then be used to simplify away
3445 * the corresponding constraints in "map".
3446 */
3449 {
3453 isl_bool subset;
3464 }
3480 }
3496 list);
3497 }
3499 error:
3503 }
3507 {
3509 }
3513 {
3516 }
3520 {
3523 }
3527 {
3532 }
3536 {
3538 }
3540 /* Compute the gist of "bmap" with respect to the constraints "context"
3541 * on the domain.
3542 */
3545 {
3551 }
3555 {
3559 }
3563 {
3567 }
3571 {
3575 }
3579 {
3581 }
3583 /* Quick check to see if two basic maps are disjoint.
3584 * In particular, we reduce the equalities and inequalities of
3585 * one basic map in the context of the equalities of the other
3586 * basic map and check if we get a contradiction.
3587 */
3590 {
3622 }
3630 }
3639 }
3643 disjoint:
3647 error:
3651 }
3655 {
3658 }
3660 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3661 */
3665 {
3676 }
3677 }
3680 }
3682 /* Are "map1" and "map2" obviously disjoint, based on information
3683 * that can be derived without looking at the individual basic maps?
3684 *
3685 * In particular, if one of them is empty or if they live in different spaces
3686 * (ignoring parameters), then they are clearly disjoint.
3687 */
3690 {
3691 isl_bool disjoint;
3692 isl_bool match;
3716 }
3718 /* Are "map1" and "map2" obviously disjoint?
3719 *
3720 * If one of them is empty or if they live in different spaces (ignoring
3721 * parameters), then they are clearly disjoint.
3722 * This is checked by isl_map_plain_is_disjoint_global.
3723 *
3724 * If they have different parameters, then we skip any further tests.
3725 *
3726 * If they are obviously equal, but not obviously empty, then we will
3727 * not be able to detect if they are disjoint.
3728 *
3729 * Otherwise we check if each basic map in "map1" is obviously disjoint
3730 * from each basic map in "map2".
3731 */
3734 {
3735 isl_bool disjoint;
3736 isl_bool intersect;
3737 isl_bool match;
3752 }
3754 /* Are "map1" and "map2" disjoint?
3755 * The parameters are assumed to have been aligned.
3756 *
3757 * In particular, check whether all pairs of basic maps are disjoint.
3758 */
3761 {
3763 }
3765 /* Are "map1" and "map2" disjoint?
3766 *
3767 * They are disjoint if they are "obviously disjoint" or if one of them
3768 * is empty. Otherwise, they are not disjoint if one of them is universal.
3769 * If the two inputs are (obviously) equal and not empty, then they are
3770 * not disjoint.
3771 * If none of these cases apply, then check if all pairs of basic maps
3772 * are disjoint after aligning the parameters.
3773 */
3775 {
3776 isl_bool disjoint;
3777 isl_bool intersect;
3805 }
3807 /* Are "bmap1" and "bmap2" disjoint?
3808 *
3809 * They are disjoint if they are "obviously disjoint" or if one of them
3810 * is empty. Otherwise, they are not disjoint if one of them is universal.
3811 * If none of these cases apply, we compute the intersection and see if
3812 * the result is empty.
3813 */
3816 {
3817 isl_bool disjoint;
3818 isl_bool intersect;
3847 }
3849 /* Are "bset1" and "bset2" disjoint?
3850 */
3853 {
3855 }
3859 {
3861 }
3863 /* Are "set1" and "set2" disjoint?
3864 */
3866 {
3868 }
3870 /* Is "v" equal to 0, 1 or -1?
3871 */
3873 {
3875 }
3877 /* Check if we can combine a given div with lower bound l and upper
3878 * bound u with some other div and if so return that other div.
3879 * Otherwise return -1.
3880 *
3881 * We first check that
3882 * - the bounds are opposites of each other (except for the constant
3883 * term)
3884 * - the bounds do not reference any other div
3885 * - no div is defined in terms of this div
3886 *
3887 * Let m be the size of the range allowed on the div by the bounds.
3888 * That is, the bounds are of the form
3889 *
3890 * e <= a <= e + m - 1
3891 *
3892 * with e some expression in the other variables.
3893 * We look for another div b such that no third div is defined in terms
3894 * of this second div b and such that in any constraint that contains
3895 * a (except for the given lower and upper bound), also contains b
3896 * with a coefficient that is m times that of b.
3897 * That is, all constraints (except for the lower and upper bound)
3898 * are of the form
3899 *
3900 * e + f (a + m b) >= 0
3901 *
3902 * Furthermore, in the constraints that only contain b, the coefficient
3903 * of b should be equal to 1 or -1.
3904 * If so, we return b so that "a + m b" can be replaced by
3905 * a single div "c = a + m b".
3906 */
3909 {
3931 }
3941 }
3953 }
3964 }
3977 }
3982 }
3986 }
3988 /* Internal data structure used during the construction and/or evaluation of
3989 * an inequality that ensures that a pair of bounds always allows
3990 * for an integer value.
3991 *
3992 * "tab" is the tableau in which the inequality is evaluated. It may
3993 * be NULL until it is actually needed.
3994 * "v" contains the inequality coefficients.
3995 * "g", "fl" and "fu" are temporary scalars used during the construction and
3996 * evaluation.
3997 */
4001 isl_int g;
4002 isl_int fl;
4003 isl_int fu;
4004 };
4006 /* Free all the memory allocated by the fields of "data".
4007 */
4009 {
4015 }
4017 /* Is the inequality stored in data->v satisfied by "bmap"?
4018 * That is, does it only attain non-negative values?
4019 * data->tab is a tableau corresponding to "bmap".
4020 */
4023 {
4034 }
4036 /* Given a lower and an upper bound on div i, do they always allow
4037 * for an integer value of the given div?
4038 * Determine this property by constructing an inequality
4039 * such that the property is guaranteed when the inequality is nonnegative.
4040 * The lower bound is inequality l, while the upper bound is inequality u.
4041 * The constructed inequality is stored in data->v.
4042 *
4043 * Let the upper bound be
4044 *
4045 * -n_u a + e_u >= 0
4046 *
4047 * and the lower bound
4048 *
4049 * n_l a + e_l >= 0
4050 *
4051 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4052 * We have
4053 *
4054 * - f_u e_l <= f_u f_l g a <= f_l e_u
4055 *
4056 * Since all variables are integer valued, this is equivalent to
4057 *
4058 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4059 *
4060 * If this interval is at least f_u f_l g, then it contains at least
4061 * one integer value for a.
4062 * That is, the test constraint is
4063 *
4064 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4065 *
4066 * or
4067 *
4068 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4069 *
4070 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4071 * then the constraint can be scaled down by a factor g',
4072 * with the constant term replaced by
4073 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4074 * Note that the result of applying Fourier-Motzkin to this pair
4075 * of constraints is
4076 *
4077 * f_l e_u + f_u e_l >= 0
4078 *
4079 * If the constant term of the scaled down version of this constraint,
4080 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4081 * term of the scaled down test constraint, then the test constraint
4082 * is known to hold and no explicit evaluation is required.
4083 * This is essentially the Omega test.
4084 *
4085 * If the test constraint consists of only a constant term, then
4086 * it is sufficient to look at the sign of this constant term.
4087 */
4090 {
4115 }
4125 }
4127 /* Remove more kinds of divs that are not strictly needed.
4128 * In particular, if all pairs of lower and upper bounds on a div
4129 * are such that they allow at least one integer value of the div,
4130 * then we can eliminate the div using Fourier-Motzkin without
4131 * introducing any spurious solutions.
4132 *
4133 * If at least one of the two constraints has a unit coefficient for the div,
4134 * then the presence of such a value is guaranteed so there is no need to check.
4135 * In particular, the value attained by the bound with unit coefficient
4136 * can serve as this intermediate value.
4137 */
4140 {
4163 isl_bool has_int;
4171 }
4192 }
4195 }
4199 }
4203 }
4206 }
4217 error:
4222 }
4224 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4225 * and the upper bound u, div1 always occurs together with div2 in the form
4226 * (div1 + m div2), where m is the constant range on the variable div1
4227 * allowed by l and u, replace the pair div1 and div2 by a single
4228 * div that is equal to div1 + m div2.
4229 *
4230 * The new div will appear in the location that contains div2.
4231 * We need to modify all constraints that contain
4232 * div2 = (div - div1) / m
4233 * The coefficient of div2 is known to be equal to 1 or -1.
4234 * (If a constraint does not contain div2, it will also not contain div1.)
4235 * If the constraint also contains div1, then we know they appear
4236 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4237 * i.e., the coefficient of div is f.
4238 *
4239 * Otherwise, we first need to introduce div1 into the constraint.
4240 * Let l be
4241 *
4242 * div1 + f >=0
4243 *
4244 * and u
4245 *
4246 * -div1 + f' >= 0
4247 *
4248 * A lower bound on div2
4249 *
4250 * div2 + t >= 0
4251 *
4252 * can be replaced by
4253 *
4254 * m div2 + div1 + m t + f >= 0
4255 *
4256 * An upper bound
4257 *
4258 * -div2 + t >= 0
4259 *
4260 * can be replaced by
4261 *
4262 * -(m div2 + div1) + m t + f' >= 0
4263 *
4264 * These constraint are those that we would obtain from eliminating
4265 * div1 using Fourier-Motzkin.
4266 *
4267 * After all constraints have been modified, we drop the lower and upper
4268 * bound and then drop div1.
4269 * Since the new div is only placed in the same location that used
4270 * to store div2, but otherwise has a different meaning, any possible
4271 * explicit representation of the original div2 is removed.
4272 */
4275 {
4277 isl_int m;
4299 else
4302 }
4306 }
4315 }
4319 }
4321 /* First check if we can coalesce any pair of divs and
4322 * then continue with dropping more redundant divs.
4323 *
4324 * We loop over all pairs of lower and upper bounds on a div
4325 * with coefficient 1 and -1, respectively, check if there
4326 * is any other div "c" with which we can coalesce the div
4327 * and if so, perform the coalescing.
4328 */
4331 {
4354 }
4355 }
4356 }
4361 }
4364 }
4366 /* Are the "n" coefficients starting at "first" of inequality constraints
4367 * "i" and "j" of "bmap" equal to each other?
4368 */
4371 {
4373 }
4375 /* Are the "n" coefficients starting at "first" of inequality constraints
4376 * "i" and "j" of "bmap" opposite to each other?
4377 */
4380 {
4382 }
4384 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4385 * apart from the constant term?
4386 */
4388 {
4393 }
4395 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4396 * apart from the constant term and the coefficient at position "pos"?
4397 */
4400 {
4406 }
4408 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4409 * apart from the constant term and the coefficient at position "pos"?
4410 */
4413 {
4419 }
4421 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4422 * been modified, simplying it if "simplify" is set.
4423 * Free the temporary data structure "pairs" that was associated
4424 * to the old version of "bmap".
4425 */
4428 {
4433 }
4435 /* Is "div" the single unknown existentially quantified variable
4436 * in inequality constraint "ineq" of "bmap"?
4437 * "div" is known to have a non-zero coefficient in "ineq".
4438 */
4441 {
4444 isl_bool known;
4454 isl_bool known;
4463 }
4466 }
4468 /* Does integer division "div" have coefficient 1 in inequality constraint
4469 * "ineq" of "map"?
4470 */
4472 {
4480 }
4482 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4483 * then try and drop redundant divs again,
4484 * freeing the temporary data structure "pairs" that was associated
4485 * to the old version of "bmap".
4486 */
4489 {
4493 }
4495 /* Drop the integer division at position "div", along with the two
4496 * inequality constraints "ineq1" and "ineq2" in which it appears
4497 * from "bmap" and then try and drop redundant divs again,
4498 * freeing the temporary data structure "pairs" that was associated
4499 * to the old version of "bmap".
4500 */
4504 {
4511 }
4514 }
4516 /* Given two inequality constraints
4517 *
4518 * f(x) + n d + c >= 0, (ineq)
4519 *
4520 * with d the variable at position "pos", and
4521 *
4522 * f(x) + c0 >= 0, (lower)
4523 *
4524 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4525 * determined by the first constraint.
4526 * That is, store
4527 *
4528 * ceil((c0 - c)/n)
4529 *
4530 * in *l.
4531 */
4534 {
4538 }
4540 /* Given two inequality constraints
4541 *
4542 * f(x) + n d + c >= 0, (ineq)
4543 *
4544 * with d the variable at position "pos", and
4545 *
4546 * -f(x) - c0 >= 0, (upper)
4547 *
4548 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4549 * determined by the first constraint.
4550 * That is, store
4551 *
4552 * ceil((-c1 - c)/n)
4553 *
4554 * in *u.
4555 */
4558 {
4562 }
4564 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4565 * does the corresponding lower bound have a fixed value in "bmap"?
4566 *
4567 * In particular, "ineq" is of the form
4568 *
4569 * f(x) + n d + c >= 0
4570 *
4571 * with n > 0, c the constant term and
4572 * d the existentially quantified variable "div".
4573 * That is, the lower bound is
4574 *
4575 * ceil((-f(x) - c)/n)
4576 *
4577 * Look for a pair of constraints
4578 *
4579 * f(x) + c0 >= 0
4580 * -f(x) + c1 >= 0
4581 *
4582 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4583 * That is, check that
4584 *
4585 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4586 *
4587 * If so, return the index of inequality f(x) + c0 >= 0.
4588 * Otherwise, return -1.
4589 */
4591 {
4608 }
4612 }
4613 }
4630 }
4632 /* Given a lower bound constraint "ineq" on the existentially quantified
4633 * variable "div", such that the corresponding lower bound has
4634 * a fixed value in "bmap", assign this fixed value to the variable and
4635 * then try and drop redundant divs again,
4636 * freeing the temporary data structure "pairs" that was associated
4637 * to the old version of "bmap".
4638 * "lower" determines the constant value for the lower bound.
4639 *
4640 * In particular, "ineq" is of the form
4641 *
4642 * f(x) + n d + c >= 0,
4643 *
4644 * while "lower" is of the form
4645 *
4646 * f(x) + c0 >= 0
4647 *
4648 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4649 * is ceil((c0 - c)/n).
4650 */
4653 {
4654 isl_int c;
4667 }
4669 /* Remove divs that are not strictly needed based on the inequality
4670 * constraints.
4671 * In particular, if a div only occurs positively (or negatively)
4672 * in constraints, then it can simply be dropped.
4673 * Also, if a div occurs in only two constraints and if moreover
4674 * those two constraints are opposite to each other, except for the constant
4675 * term and if the sum of the constant terms is such that for any value
4676 * of the other values, there is always at least one integer value of the
4677 * div, i.e., if one plus this sum is greater than or equal to
4678 * the (absolute value) of the coefficient of the div in the constraints,
4679 * then we can also simply drop the div.
4680 *
4681 * If an existentially quantified variable does not have an explicit
4682 * representation, appears in only a single lower bound that does not
4683 * involve any other such existentially quantified variables and appears
4684 * in this lower bound with coefficient 1,
4685 * then fix the variable to the value of the lower bound. That is,
4686 * turn the inequality into an equality.
4687 * If for any value of the other variables, there is any value
4688 * for the existentially quantified variable satisfying the constraints,
4689 * then this lower bound also satisfies the constraints.
4690 * It is therefore safe to pick this lower bound.
4691 *
4692 * The same reasoning holds even if the coefficient is not one.
4693 * However, fixing the variable to the value of the lower bound may
4694 * in general introduce an extra integer division, in which case
4695 * it may be better to pick another value.
4696 * If this integer division has a known constant value, then plugging
4697 * in this constant value removes the existentially quantified variable
4698 * completely. In particular, if the lower bound is of the form
4699 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4700 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4701 * then the existentially quantified variable can be assigned this
4702 * shared value.
4703 *
4704 * We skip divs that appear in equalities or in the definition of other divs.
4705 * Divs that appear in the definition of other divs usually occur in at least
4706 * 4 constraints, but the constraints may have been simplified.
4707 *
4708 * If any divs are left after these simple checks then we move on
4709 * to more complicated cases in drop_more_redundant_divs.
4710 */
4713 {
4753 }
4757 }
4758 }
4766 }
4769 else
4789 pairs);
4793 pairs);
4795 }
4812 else
4819 }
4822 }
4829 error:
4833 }
4835 /* Consider the coefficients at "c" as a row vector and replace
4836 * them with their product with "T". "T" is assumed to be a square matrix.
4837 */
4839 {
4861 }
4863 /* Plug in T for the variables in "bmap" starting at "pos".
4864 * T is a linear unimodular matrix, i.e., without constant term.
4865 */
4868 {
4895 }
4899 error:
4903 }
4905 /* Remove divs that are not strictly needed.
4906 *
4907 * First look for an equality constraint involving two or more
4908 * existentially quantified variables without an explicit
4909 * representation. Replace the combination that appears
4910 * in the equality constraint by a single existentially quantified
4911 * variable such that the equality can be used to derive
4912 * an explicit representation for the variable.
4913 * If there are no more such equality constraints, then continue
4914 * with isl_basic_map_drop_redundant_divs_ineq.
4915 *
4916 * In particular, if the equality constraint is of the form
4917 *
4918 * f(x) + \sum_i c_i a_i = 0
4919 *
4920 * with a_i existentially quantified variable without explicit
4921 * representation, then apply a transformation on the existentially
4922 * quantified variables to turn the constraint into
4923 *
4924 * f(x) + g a_1' = 0
4925 *
4926 * with g the gcd of the c_i.
4927 * In order to easily identify which existentially quantified variables
4928 * have a complete explicit representation, i.e., without being defined
4929 * in terms of other existentially quantified variables without
4930 * an explicit representation, the existentially quantified variables
4931 * are first sorted.
4932 *
4933 * The variable transformation is computed by extending the row
4934 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4935 *
4936 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4937 * [a_2'] [ a_2 ]
4938 * ... = U ....
4939 * [a_n'] [ a_n ]
4940 *
4941 * with [c_1/g ... c_n/g] representing the first row of U.
4942 * The inverse of U is then plugged into the original constraints.
4943 * The call to isl_basic_map_simplify makes sure the explicit
4944 * representation for a_1' is extracted from the equality constraint.
4945 */
4948 {
4983 }
5002 }
5004 /* Does "bmap" satisfy any equality that involves more than 2 variables
5005 * and/or has coefficients different from -1 and 1?
5006 */
5008 {
5037 }
5040 }
5042 /* Remove any common factor g from the constraint coefficients in "v".
5043 * The constant term is stored in the first position and is replaced
5044 * by floor(c/g). If any common factor is removed and if this results
5045 * in a tightening of the constraint, then set *tightened.
5046 */
5049 {
5069 }
5071 /* If "bmap" is an integer set that satisfies any equality involving
5072 * more than 2 variables and/or has coefficients different from -1 and 1,
5073 * then use variable compression to reduce the coefficients by removing
5074 * any (hidden) common factor.
5075 * In particular, apply the variable compression to each constraint,
5076 * factor out any common factor in the non-constant coefficients and
5077 * then apply the inverse of the compression.
5078 * At the end, we mark the basic map as having reduced constants.
5079 * If this flag is still set on the next invocation of this function,
5080 * then we skip the computation.
5081 *
5082 * Removing a common factor may result in a tightening of some of
5083 * the constraints. If this happens, then we may end up with two
5084 * opposite inequalities that can be replaced by an equality.
5085 * We therefore call isl_basic_map_detect_inequality_pairs,
5086 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5087 * and isl_basic_map_gauss if such a pair was found.
5088 *
5089 * Note that this function may leave the result in an inconsistent state.
5090 * In particular, the constraints may not be gaussed.
5091 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5092 * for some of the test cases to pass successfully.
5093 * Any potential modification of the representation is therefore only
5094 * performed on a single copy of the basic map.
5095 */
5098 {
5132 }
5147 }
5162 }
5163 }
5166 error:
5171 }
5173 /* Shift the integer division at position "div" of "bmap"
5174 * by "shift" times the variable at position "pos".
5175 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5176 * corresponds to the constant term.
5177 *
5178 * That is, if the integer division has the form
5179 *
5180 * floor(f(x)/d)
5181 *
5182 * then replace it by
5183 *
5184 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5185 */
5188 {
5207 }
5213 }
5221 }
5224 }