| blob | ff73325b13122123154decd952bc856cd4e8dcde |
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 }
1341 }
1342 }
1345 }
1348 {
1353 isl_bool empty;
1369 /* requires equalities in normal form */
1375 }
1377 }
1380 {
1382 }
1387 {
1419 }
1423 {
1425 }
1428 /* If the only constraints a div d=floor(f/m)
1429 * appears in are its two defining constraints
1430 *
1431 * f - m d >=0
1432 * -(f - (m - 1)) + m d >= 0
1433 *
1434 * then it can safely be removed.
1435 */
1437 {
1446 isl_bool red;
1453 }
1460 }
1463 }
1465 /*
1466 * Remove divs that don't occur in any of the constraints or other divs.
1467 * These can arise when dropping constraints from a basic map or
1468 * when the divs of a basic map have been temporarily aligned
1469 * with the divs of another basic map.
1470 */
1473 {
1482 isl_bool redundant;
1492 }
1494 }
1496 /* Mark "bmap" as final, without checking for obviously redundant
1497 * integer divisions. This function should be used when "bmap"
1498 * is known not to involve any such integer divisions.
1499 */
1502 {
1507 }
1509 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1510 */
1512 {
1516 }
1519 {
1521 }
1523 /* Remove definition of any div that is defined in terms of the given variable.
1524 * The div itself is not removed. Functions such as
1525 * eliminate_divs_ineq depend on the other divs remaining in place.
1526 */
1529 {
1543 }
1545 }
1547 /* Eliminate the specified variables from the constraints using
1548 * Fourier-Motzkin. The variables themselves are not removed.
1549 */
1552 {
1586 }
1593 n_lower++;
1595 n_upper++;
1596 }
1620 }
1623 }
1635 }
1636 }
1640 error:
1643 }
1647 {
1650 }
1652 /* Eliminate the specified n dimensions starting at first from the
1653 * constraints, without removing the dimensions from the space.
1654 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1655 * Otherwise, they are projected out and the original space is restored.
1656 */
1660 {
1675 }
1682 }
1687 {
1689 }
1691 /* Remove all constraints from "bmap" that reference any unknown local
1692 * variables (directly or indirectly).
1693 *
1694 * Dropping all constraints on a local variable will make it redundant,
1695 * so it will get removed implicitly by
1696 * isl_basic_map_drop_constraints_involving_dims. Some other local
1697 * variables may also end up becoming redundant if they only appear
1698 * in constraints together with the unknown local variable.
1699 * Therefore, start over after calling
1700 * isl_basic_map_drop_constraints_involving_dims.
1701 */
1704 {
1705 isl_bool known;
1730 }
1733 }
1735 /* Remove all constraints from "map" that reference any unknown local
1736 * variables (directly or indirectly).
1737 *
1738 * Since constraints may get dropped from the basic maps,
1739 * they may no longer be disjoint from each other.
1740 */
1743 {
1745 isl_bool known;
1763 }
1769 }
1771 /* Don't assume equalities are in order, because align_divs
1772 * may have changed the order of the divs.
1773 */
1775 {
1788 }
1789 }
1790 }
1794 {
1796 }
1800 {
1814 }
1816 }
1818 }
1822 {
1825 }
1829 {
1839 }
1860 error:
1864 }
1866 /* For each inequality in "ineq" that is a shifted (more relaxed)
1867 * copy of an inequality in "context", mark the corresponding entry
1868 * in "row" with -1.
1869 * If an inequality only has a non-negative constant term, then
1870 * mark it as well.
1871 */
1874 {
1891 isl_bool redundant;
1897 }
1904 }
1907 error:
1910 }
1914 {
1927 isl_bool redundant;
1939 }
1942 error:
1945 }
1947 /* Remove constraints from "bmap" that are identical to constraints
1948 * in "context" or that are more relaxed (greater constant term).
1949 *
1950 * We perform the test for shifted copies on the pure constraints
1951 * in remove_shifted_constraints.
1952 */
1955 {
1964 }
1977 error:
1981 }
1983 /* Does the (linear part of a) constraint "c" involve any of the "len"
1984 * "relevant" dimensions?
1985 */
1987 {
1995 }
1998 }
2000 /* Drop constraints from "bmap" that do not involve any of
2001 * the dimensions marked "relevant".
2002 */
2005 {
2020 }
2027 }
2030 }
2032 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2033 *
2034 * In particular, for any variable involved in the constraint,
2035 * find the actual group id from before and replace the group
2036 * of the corresponding variable by the minimal group of all
2037 * the variables involved in the constraint considered so far
2038 * (if this minimum is smaller) or replace the minimum by this group
2039 * (if the minimum is larger).
2040 *
2041 * At the end, all the variables in "c" will (indirectly) point
2042 * to the minimal of the groups that they referred to originally.
2043 */
2045 {
2062 }
2063 }
2065 /* Allocate an array of groups of variables, one for each variable
2066 * in "context", initialized to zero.
2067 */
2069 {
2076 }
2078 /* Drop constraints from "bmap" that only involve variables that are
2079 * not related to any of the variables marked with a "-1" in "group".
2080 *
2081 * We construct groups of variables that collect variables that
2082 * (indirectly) appear in some common constraint of "bmap".
2083 * Each group is identified by the first variable in the group,
2084 * except for the special group of variables that was already identified
2085 * in the input as -1 (or are related to those variables).
2086 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2087 * otherwise the group of i is the group of group[i].
2088 *
2089 * We first initialize groups for the remaining variables.
2090 * Then we iterate over the constraints of "bmap" and update the
2091 * group of the variables in the constraint by the smallest group.
2092 * Finally, we resolve indirect references to groups by running over
2093 * the variables.
2094 *
2095 * After computing the groups, we drop constraints that do not involve
2096 * any variables in the -1 group.
2097 */
2100 {
2117 }
2135 }
2137 /* Drop constraints from "context" that are irrelevant for computing
2138 * the gist of "bset".
2139 *
2140 * In particular, drop constraints in variables that are not related
2141 * to any of the variables involved in the constraints of "bset"
2142 * in the sense that there is no sequence of constraints that connects them.
2143 *
2144 * We first mark all variables that appear in "bset" as belonging
2145 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2146 */
2149 {
2170 }
2176 }
2179 }
2181 /* Drop constraints from "context" that are irrelevant for computing
2182 * the gist of the inequalities "ineq".
2183 * Inequalities in "ineq" for which the corresponding element of row
2184 * is set to -1 have already been marked for removal and should be ignored.
2185 *
2186 * In particular, drop constraints in variables that are not related
2187 * to any of the variables involved in "ineq"
2188 * in the sense that there is no sequence of constraints that connects them.
2189 *
2190 * We first mark all variables that appear in "bset" as belonging
2191 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2192 */
2195 {
2216 }
2219 }
2222 }
2224 /* Do all "n" entries of "row" contain a negative value?
2225 */
2227 {
2235 }
2237 /* Update the inequalities in "bset" based on the information in "row"
2238 * and "tab".
2239 *
2240 * In particular, the array "row" contains either -1, meaning that
2241 * the corresponding inequality of "bset" is redundant, or the index
2242 * of an inequality in "tab".
2243 *
2244 * If the row entry is -1, then drop the inequality.
2245 * Otherwise, if the constraint is marked redundant in the tableau,
2246 * then drop the inequality. Similarly, if it is marked as an equality
2247 * in the tableau, then turn the inequality into an equality and
2248 * perform Gaussian elimination.
2249 */
2252 {
2269 }
2279 }
2280 }
2286 }
2288 /* Update the inequalities in "bset" based on the information in "row"
2289 * and "tab" and free all arguments (other than "bset").
2290 */
2295 {
2304 }
2306 /* Remove all information from bset that is redundant in the context
2307 * of context.
2308 * "ineq" contains the (possibly transformed) inequalities of "bset",
2309 * in the same order.
2310 * The (explicit) equalities of "bset" are assumed to have been taken
2311 * into account by the transformation such that only the inequalities
2312 * are relevant.
2313 * "context" is assumed not to be empty.
2314 *
2315 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2316 * A value of -1 means that the inequality is obviously redundant and may
2317 * not even appear in "tab".
2318 *
2319 * We first mark the inequalities of "bset"
2320 * that are obviously redundant with respect to some inequality in "context".
2321 * Then we remove those constraints from "context" that have become
2322 * irrelevant for computing the gist of "bset".
2323 * Note that this removal of constraints cannot be replaced by
2324 * a factorization because factors in "bset" may still be connected
2325 * to each other through constraints in "context".
2326 *
2327 * If there are any inequalities left, we construct a tableau for
2328 * the context and then add the inequalities of "bset".
2329 * Before adding these inequalities, we freeze all constraints such that
2330 * they won't be considered redundant in terms of the constraints of "bset".
2331 * Then we detect all redundant constraints (among the
2332 * constraints that weren't frozen), first by checking for redundancy in the
2333 * the tableau and then by checking if replacing a constraint by its negation
2334 * would lead to an empty set. This last step is fairly expensive
2335 * and could be optimized by more reuse of the tableau.
2336 * Finally, we update bset according to the results.
2337 */
2340 {
2355 }
2391 }
2414 }
2419 }
2423 error:
2431 }
2433 /* Extract the inequalities of "bset" as an isl_mat.
2434 */
2436 {
2450 }
2452 /* Remove all information from "bset" that is redundant in the context
2453 * of "context", for the case where both "bset" and "context" are
2454 * full-dimensional.
2455 */
2458 {
2463 }
2465 /* Remove all information from "bset" that is redundant in the context
2466 * of "context", for the case where the combined equalities of
2467 * "bset" and "context" allow for a compression that can be obtained
2468 * by preapplication of "T".
2469 *
2470 * "bset" itself is not transformed by "T". Instead, the inequalities
2471 * are extracted from "bset" and those are transformed by "T".
2472 * uset_gist_full then determines which of the transformed inequalities
2473 * are redundant with respect to the transformed "context" and removes
2474 * the corresponding inequalities from "bset".
2475 *
2476 * After preapplying "T" to the inequalities, any common factor is
2477 * removed from the coefficients. If this results in a tightening
2478 * of the constant term, then the same tightening is applied to
2479 * the corresponding untransformed inequality in "bset".
2480 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2481 *
2482 * g f'(x) + r >= 0
2483 *
2484 * with 0 <= r < g, then it is equivalent to
2485 *
2486 * f'(x) >= 0
2487 *
2488 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2489 * subspace compressed by T since the latter would be transformed to
2490 *
2491 * g f'(x) >= 0
2492 */
2496 {
2500 isl_int rem;
2512 }
2535 }
2539 error:
2544 }
2546 /* Project "bset" onto the variables that are involved in "template".
2547 */
2550 {
2559 isl_bool involved;
2568 }
2571 }
2573 /* Remove all information from bset that is redundant in the context
2574 * of context. In particular, equalities that are linear combinations
2575 * of those in context are removed. Then the inequalities that are
2576 * redundant in the context of the equalities and inequalities of
2577 * context are removed.
2578 *
2579 * First of all, we drop those constraints from "context"
2580 * that are irrelevant for computing the gist of "bset".
2581 * Alternatively, we could factorize the intersection of "context" and "bset".
2582 *
2583 * We first compute the intersection of the integer affine hulls
2584 * of "bset" and "context",
2585 * compute the gist inside this intersection and then reduce
2586 * the constraints with respect to the equalities of the context
2587 * that only involve variables already involved in the input.
2588 *
2589 * If two constraints are mutually redundant, then uset_gist_full
2590 * will remove the second of those constraints. We therefore first
2591 * sort the constraints so that constraints not involving existentially
2592 * quantified variables are given precedence over those that do.
2593 * We have to perform this sorting before the variable compression,
2594 * because that may effect the order of the variables.
2595 */
2598 {
2623 }
2628 }
2638 }
2649 }
2652 error:
2656 }
2658 /* Return the number of equality constraints in "bmap" that involve
2659 * local variables. This function assumes that Gaussian elimination
2660 * has been applied to the equality constraints.
2661 */
2663 {
2683 }
2685 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2686 * The constraints are assumed not to involve any local variables.
2687 */
2690 {
2708 }
2713 error:
2718 }
2720 /* Construct and return a variable compression based on the equality
2721 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2722 * "n1" is the number of (initial) equality constraints in "bmap1"
2723 * that do involve local variables.
2724 * "n2" is the number of (initial) equality constraints in "bmap2"
2725 * that do involve local variables.
2726 * "total" is the total number of other variables.
2727 * This function assumes that Gaussian elimination
2728 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2729 * such that the equality constraints not involving local variables
2730 * are those that start at "n1" or "n2".
2731 *
2732 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2733 * then simply compute the compression based on the equality constraints
2734 * in the other basic map.
2735 * Otherwise, combine the equality constraints from both into a new
2736 * basic map such that Gaussian elimination can be applied to this combination
2737 * and then construct a variable compression from the resulting
2738 * equality constraints.
2739 */
2743 {
2753 }
2758 }
2773 }
2775 /* Extract the stride constraints from "bmap", compressed
2776 * with respect to both the stride constraints in "context" and
2777 * the remaining equality constraints in both "bmap" and "context".
2778 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2779 * "context_n_eq" is the number of (initial) stride constraints in "context".
2780 *
2781 * Let x be all variables in "bmap" (and "context") other than the local
2782 * variables. First compute a variable compression
2783 *
2784 * x = V x'
2785 *
2786 * based on the non-stride equality constraints in "bmap" and "context".
2787 * Consider the stride constraints of "context",
2788 *
2789 * A(x) + B(y) = 0
2790 *
2791 * with y the local variables and plug in the variable compression,
2792 * resulting in
2793 *
2794 * A(V x') + B(y) = 0
2795 *
2796 * Use these constraints to compute a parameter compression on x'
2797 *
2798 * x' = T x''
2799 *
2800 * Now consider the stride constraints of "bmap"
2801 *
2802 * C(x) + D(y) = 0
2803 *
2804 * and plug in x = V*T x''.
2805 * That is, return A = [C*V*T D].
2806 */
2810 {
2839 }
2841 /* Remove the prime factors from *g that have an exponent that
2842 * is strictly smaller than the exponent in "c".
2843 * All exponents in *g are known to be smaller than or equal
2844 * to those in "c".
2845 *
2846 * That is, if *g is equal to
2847 *
2848 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2849 *
2850 * and "c" is equal to
2851 *
2852 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2853 *
2854 * then update *g to
2855 *
2856 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2857 * p_n^{e_n * (e_n = f_n)}
2858 *
2859 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2860 * neither does the gcd of *g and c / *g.
2861 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2862 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2863 * Dividing *g by this gcd therefore strictly reduces the exponent
2864 * of the prime factors that need to be removed, while leaving the
2865 * other prime factors untouched.
2866 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2867 * removes all undesired factors, without removing any others.
2868 */
2870 {
2871 isl_int t;
2880 }
2882 }
2884 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2885 * of the same stride constraints in a compressed space that exploits
2886 * all equalities in the context and the other equalities in "bmap".
2887 *
2888 * If the stride constraints of "bmap" are of the form
2889 *
2890 * C(x) + D(y) = 0
2891 *
2892 * then A is of the form
2893 *
2894 * B(x') + D(y) = 0
2895 *
2896 * If any of these constraints involves only a single local variable y,
2897 * then the constraint appears as
2898 *
2899 * f(x) + m y_i = 0
2900 *
2901 * in "bmap" and as
2902 *
2903 * h(x') + m y_i = 0
2904 *
2905 * in "A".
2906 *
2907 * Let g be the gcd of m and the coefficients of h.
2908 * Then, in particular, g is a divisor of the coefficients of h and
2909 *
2910 * f(x) = h(x')
2911 *
2912 * is known to be a multiple of g.
2913 * If some prime factor in m appears with the same exponent in g,
2914 * then it can be removed from m because f(x) is already known
2915 * to be a multiple of g and therefore in particular of this power
2916 * of the prime factors.
2917 * Prime factors that appear with a smaller exponent in g cannot
2918 * be removed from m.
2919 * Let g' be the divisor of g containing all prime factors that
2920 * appear with the same exponent in m and g, then
2921 *
2922 * f(x) + m y_i = 0
2923 *
2924 * can be replaced by
2925 *
2926 * f(x) + m/g' y_i' = 0
2927 *
2928 * Note that (if g' != 1) this changes the explicit representation
2929 * of y_i to that of y_i', so the integer division at position i
2930 * is marked unknown and later recomputed by a call to
2931 * isl_basic_map_gauss.
2932 */
2935 {
2939 isl_int gcd;
2973 }
2980 error:
2984 }
2986 /* Simplify the stride constraints in "bmap" based on
2987 * the remaining equality constraints in "bmap" and all equality
2988 * constraints in "context".
2989 * Only do this if both "bmap" and "context" have stride constraints.
2990 *
2991 * First extract a copy of the stride constraints in "bmap" in a compressed
2992 * space exploiting all the other equality constraints and then
2993 * use this compressed copy to simplify the original stride constraints.
2994 */
2997 {
3019 }
3021 /* Return a basic map that has the same intersection with "context" as "bmap"
3022 * and that is as "simple" as possible.
3023 *
3024 * The core computation is performed on the pure constraints.
3025 * When we add back the meaning of the integer divisions, we need
3026 * to (re)introduce the div constraints. If we happen to have
3027 * discovered that some of these integer divisions are equal to
3028 * some affine combination of other variables, then these div
3029 * constraints may end up getting simplified in terms of the equalities,
3030 * resulting in extra inequalities on the other variables that
3031 * may have been removed already or that may not even have been
3032 * part of the input. We try and remove those constraints of
3033 * this form that are most obviously redundant with respect to
3034 * the context. We also remove those div constraints that are
3035 * redundant with respect to the other constraints in the result.
3036 *
3037 * The stride constraints among the equality constraints in "bmap" are
3038 * also simplified with respecting to the other equality constraints
3039 * in "bmap" and with respect to all equality constraints in "context".
3040 */
3043 {
3054 }
3060 }
3064 }
3086 }
3105 error:
3109 }
3111 /*
3112 * Assumes context has no implicit divs.
3113 */
3116 {
3127 }
3147 }
3148 }
3152 error:
3156 }
3158 /* Drop all inequalities from "bmap" that also appear in "context".
3159 * "context" is assumed to have only known local variables and
3160 * the initial local variables of "bmap" are assumed to be the same
3161 * as those of "context".
3162 * The constraints of both "bmap" and "context" are assumed
3163 * to have been sorted using isl_basic_map_sort_constraints.
3164 *
3165 * Run through the inequality constraints of "bmap" and "context"
3166 * in sorted order.
3167 * If a constraint of "bmap" involves variables not in "context",
3168 * then it cannot appear in "context".
3169 * If a matching constraint is found, it is removed from "bmap".
3170 */
3173 {
3192 }
3198 }
3202 }
3207 }
3210 }
3213 }
3215 /* Drop all equalities from "bmap" that also appear in "context".
3216 * "context" is assumed to have only known local variables and
3217 * the initial local variables of "bmap" are assumed to be the same
3218 * as those of "context".
3219 *
3220 * Run through the equality constraints of "bmap" and "context"
3221 * in sorted order.
3222 * If a constraint of "bmap" involves variables not in "context",
3223 * then it cannot appear in "context".
3224 * If a matching constraint is found, it is removed from "bmap".
3225 */
3228 {
3252 }
3256 }
3261 }
3264 }
3267 }
3269 /* Remove the constraints in "context" from "bmap".
3270 * "context" is assumed to have explicit representations
3271 * for all local variables.
3272 *
3273 * First align the divs of "bmap" to those of "context" and
3274 * sort the constraints. Then drop all constraints from "bmap"
3275 * that appear in "context".
3276 */
3279 {
3294 }
3313 error:
3317 }
3319 /* Replace "map" by the disjunct at position "pos" and free "context".
3320 */
3323 {
3330 }
3332 /* Remove the constraints in "context" from "map".
3333 * If any of the disjuncts in the result turns out to be the universe,
3334 * then return this universe.
3335 * "context" is assumed to have explicit representations
3336 * for all local variables.
3337 */
3340 {
3350 }
3369 }
3376 error:
3380 }
3382 /* Remove the constraints in "context" from "set".
3383 * If any of the disjuncts in the result turns out to be the universe,
3384 * then return this universe.
3385 * "context" is assumed to have explicit representations
3386 * for all local variables.
3387 */
3390 {
3393 }
3395 /* Remove the constraints in "context" from "map".
3396 * If any of the disjuncts in the result turns out to be the universe,
3397 * then return this universe.
3398 * "context" is assumed to consist of a single disjunct and
3399 * to have explicit representations for all local variables.
3400 */
3403 {
3408 }
3410 /* Replace "map" by a universe map in the same space and free "drop".
3411 */
3414 {
3421 }
3423 /* Return a map that has the same intersection with "context" as "map"
3424 * and that is as "simple" as possible.
3425 *
3426 * If "map" is already the universe, then we cannot make it any simpler.
3427 * Similarly, if "context" is the universe, then we cannot exploit it
3428 * to simplify "map"
3429 * If "map" and "context" are identical to each other, then we can
3430 * return the corresponding universe.
3431 *
3432 * If either "map" or "context" consists of multiple disjuncts,
3433 * then check if "context" happens to be a subset of "map",
3434 * in which case all constraints can be removed.
3435 * In case of multiple disjuncts, the standard procedure
3436 * may not be able to detect that all constraints can be removed.
3437 *
3438 * If none of these cases apply, we have to work a bit harder.
3439 * During this computation, we make use of a single disjunct context,
3440 * so if the original context consists of more than one disjunct
3441 * then we need to approximate the context by a single disjunct set.
3442 * Simply taking the simple hull may drop constraints that are
3443 * only implicitly available in each disjunct. We therefore also
3444 * look for constraints among those defining "map" that are valid
3445 * for the context. These can then be used to simplify away
3446 * the corresponding constraints in "map".
3447 */
3450 {
3454 isl_bool subset;
3465 }
3481 }
3497 list);
3498 }
3500 error:
3504 }
3508 {
3510 }
3514 {
3517 }
3521 {
3524 }
3528 {
3533 }
3537 {
3539 }
3541 /* Compute the gist of "bmap" with respect to the constraints "context"
3542 * on the domain.
3543 */
3546 {
3552 }
3556 {
3560 }
3564 {
3568 }
3572 {
3576 }
3580 {
3582 }
3584 /* Quick check to see if two basic maps are disjoint.
3585 * In particular, we reduce the equalities and inequalities of
3586 * one basic map in the context of the equalities of the other
3587 * basic map and check if we get a contradiction.
3588 */
3591 {
3623 }
3631 }
3640 }
3644 disjoint:
3648 error:
3652 }
3656 {
3659 }
3661 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3662 */
3666 {
3677 }
3678 }
3681 }
3683 /* Are "map1" and "map2" obviously disjoint, based on information
3684 * that can be derived without looking at the individual basic maps?
3685 *
3686 * In particular, if one of them is empty or if they live in different spaces
3687 * (ignoring parameters), then they are clearly disjoint.
3688 */
3691 {
3692 isl_bool disjoint;
3693 isl_bool match;
3717 }
3719 /* Are "map1" and "map2" obviously disjoint?
3720 *
3721 * If one of them is empty or if they live in different spaces (ignoring
3722 * parameters), then they are clearly disjoint.
3723 * This is checked by isl_map_plain_is_disjoint_global.
3724 *
3725 * If they have different parameters, then we skip any further tests.
3726 *
3727 * If they are obviously equal, but not obviously empty, then we will
3728 * not be able to detect if they are disjoint.
3729 *
3730 * Otherwise we check if each basic map in "map1" is obviously disjoint
3731 * from each basic map in "map2".
3732 */
3735 {
3736 isl_bool disjoint;
3737 isl_bool intersect;
3738 isl_bool match;
3753 }
3755 /* Are "map1" and "map2" disjoint?
3756 * The parameters are assumed to have been aligned.
3757 *
3758 * In particular, check whether all pairs of basic maps are disjoint.
3759 */
3762 {
3764 }
3766 /* Are "map1" and "map2" disjoint?
3767 *
3768 * They are disjoint if they are "obviously disjoint" or if one of them
3769 * is empty. Otherwise, they are not disjoint if one of them is universal.
3770 * If the two inputs are (obviously) equal and not empty, then they are
3771 * not disjoint.
3772 * If none of these cases apply, then check if all pairs of basic maps
3773 * are disjoint after aligning the parameters.
3774 */
3776 {
3777 isl_bool disjoint;
3778 isl_bool intersect;
3806 }
3808 /* Are "bmap1" and "bmap2" disjoint?
3809 *
3810 * They are disjoint if they are "obviously disjoint" or if one of them
3811 * is empty. Otherwise, they are not disjoint if one of them is universal.
3812 * If none of these cases apply, we compute the intersection and see if
3813 * the result is empty.
3814 */
3817 {
3818 isl_bool disjoint;
3819 isl_bool intersect;
3848 }
3850 /* Are "bset1" and "bset2" disjoint?
3851 */
3854 {
3856 }
3860 {
3862 }
3864 /* Are "set1" and "set2" disjoint?
3865 */
3867 {
3869 }
3871 /* Is "v" equal to 0, 1 or -1?
3872 */
3874 {
3876 }
3878 /* Check if we can combine a given div with lower bound l and upper
3879 * bound u with some other div and if so return that other div.
3880 * Otherwise return -1.
3881 *
3882 * We first check that
3883 * - the bounds are opposites of each other (except for the constant
3884 * term)
3885 * - the bounds do not reference any other div
3886 * - no div is defined in terms of this div
3887 *
3888 * Let m be the size of the range allowed on the div by the bounds.
3889 * That is, the bounds are of the form
3890 *
3891 * e <= a <= e + m - 1
3892 *
3893 * with e some expression in the other variables.
3894 * We look for another div b such that no third div is defined in terms
3895 * of this second div b and such that in any constraint that contains
3896 * a (except for the given lower and upper bound), also contains b
3897 * with a coefficient that is m times that of b.
3898 * That is, all constraints (except for the lower and upper bound)
3899 * are of the form
3900 *
3901 * e + f (a + m b) >= 0
3902 *
3903 * Furthermore, in the constraints that only contain b, the coefficient
3904 * of b should be equal to 1 or -1.
3905 * If so, we return b so that "a + m b" can be replaced by
3906 * a single div "c = a + m b".
3907 */
3910 {
3932 }
3942 }
3954 }
3965 }
3978 }
3983 }
3987 }
3989 /* Internal data structure used during the construction and/or evaluation of
3990 * an inequality that ensures that a pair of bounds always allows
3991 * for an integer value.
3992 *
3993 * "tab" is the tableau in which the inequality is evaluated. It may
3994 * be NULL until it is actually needed.
3995 * "v" contains the inequality coefficients.
3996 * "g", "fl" and "fu" are temporary scalars used during the construction and
3997 * evaluation.
3998 */
4002 isl_int g;
4003 isl_int fl;
4004 isl_int fu;
4005 };
4007 /* Free all the memory allocated by the fields of "data".
4008 */
4010 {
4016 }
4018 /* Is the inequality stored in data->v satisfied by "bmap"?
4019 * That is, does it only attain non-negative values?
4020 * data->tab is a tableau corresponding to "bmap".
4021 */
4024 {
4035 }
4037 /* Given a lower and an upper bound on div i, do they always allow
4038 * for an integer value of the given div?
4039 * Determine this property by constructing an inequality
4040 * such that the property is guaranteed when the inequality is nonnegative.
4041 * The lower bound is inequality l, while the upper bound is inequality u.
4042 * The constructed inequality is stored in data->v.
4043 *
4044 * Let the upper bound be
4045 *
4046 * -n_u a + e_u >= 0
4047 *
4048 * and the lower bound
4049 *
4050 * n_l a + e_l >= 0
4051 *
4052 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4053 * We have
4054 *
4055 * - f_u e_l <= f_u f_l g a <= f_l e_u
4056 *
4057 * Since all variables are integer valued, this is equivalent to
4058 *
4059 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4060 *
4061 * If this interval is at least f_u f_l g, then it contains at least
4062 * one integer value for a.
4063 * That is, the test constraint is
4064 *
4065 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4066 *
4067 * or
4068 *
4069 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4070 *
4071 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4072 * then the constraint can be scaled down by a factor g',
4073 * with the constant term replaced by
4074 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4075 * Note that the result of applying Fourier-Motzkin to this pair
4076 * of constraints is
4077 *
4078 * f_l e_u + f_u e_l >= 0
4079 *
4080 * If the constant term of the scaled down version of this constraint,
4081 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4082 * term of the scaled down test constraint, then the test constraint
4083 * is known to hold and no explicit evaluation is required.
4084 * This is essentially the Omega test.
4085 *
4086 * If the test constraint consists of only a constant term, then
4087 * it is sufficient to look at the sign of this constant term.
4088 */
4091 {
4116 }
4126 }
4128 /* Remove more kinds of divs that are not strictly needed.
4129 * In particular, if all pairs of lower and upper bounds on a div
4130 * are such that they allow at least one integer value of the div,
4131 * then we can eliminate the div using Fourier-Motzkin without
4132 * introducing any spurious solutions.
4133 *
4134 * If at least one of the two constraints has a unit coefficient for the div,
4135 * then the presence of such a value is guaranteed so there is no need to check.
4136 * In particular, the value attained by the bound with unit coefficient
4137 * can serve as this intermediate value.
4138 */
4141 {
4164 isl_bool has_int;
4172 }
4193 }
4196 }
4200 }
4204 }
4207 }
4218 error:
4223 }
4225 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4226 * and the upper bound u, div1 always occurs together with div2 in the form
4227 * (div1 + m div2), where m is the constant range on the variable div1
4228 * allowed by l and u, replace the pair div1 and div2 by a single
4229 * div that is equal to div1 + m div2.
4230 *
4231 * The new div will appear in the location that contains div2.
4232 * We need to modify all constraints that contain
4233 * div2 = (div - div1) / m
4234 * The coefficient of div2 is known to be equal to 1 or -1.
4235 * (If a constraint does not contain div2, it will also not contain div1.)
4236 * If the constraint also contains div1, then we know they appear
4237 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4238 * i.e., the coefficient of div is f.
4239 *
4240 * Otherwise, we first need to introduce div1 into the constraint.
4241 * Let l be
4242 *
4243 * div1 + f >=0
4244 *
4245 * and u
4246 *
4247 * -div1 + f' >= 0
4248 *
4249 * A lower bound on div2
4250 *
4251 * div2 + t >= 0
4252 *
4253 * can be replaced by
4254 *
4255 * m div2 + div1 + m t + f >= 0
4256 *
4257 * An upper bound
4258 *
4259 * -div2 + t >= 0
4260 *
4261 * can be replaced by
4262 *
4263 * -(m div2 + div1) + m t + f' >= 0
4264 *
4265 * These constraint are those that we would obtain from eliminating
4266 * div1 using Fourier-Motzkin.
4267 *
4268 * After all constraints have been modified, we drop the lower and upper
4269 * bound and then drop div1.
4270 * Since the new div is only placed in the same location that used
4271 * to store div2, but otherwise has a different meaning, any possible
4272 * explicit representation of the original div2 is removed.
4273 */
4276 {
4278 isl_int m;
4300 else
4303 }
4307 }
4316 }
4320 }
4322 /* First check if we can coalesce any pair of divs and
4323 * then continue with dropping more redundant divs.
4324 *
4325 * We loop over all pairs of lower and upper bounds on a div
4326 * with coefficient 1 and -1, respectively, check if there
4327 * is any other div "c" with which we can coalesce the div
4328 * and if so, perform the coalescing.
4329 */
4332 {
4355 }
4356 }
4357 }
4362 }
4365 }
4367 /* Are the "n" coefficients starting at "first" of inequality constraints
4368 * "i" and "j" of "bmap" equal to each other?
4369 */
4372 {
4374 }
4376 /* Are the "n" coefficients starting at "first" of inequality constraints
4377 * "i" and "j" of "bmap" opposite to each other?
4378 */
4381 {
4383 }
4385 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4386 * apart from the constant term?
4387 */
4389 {
4394 }
4396 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4397 * apart from the constant term and the coefficient at position "pos"?
4398 */
4401 {
4407 }
4409 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4410 * apart from the constant term and the coefficient at position "pos"?
4411 */
4414 {
4420 }
4422 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4423 * been modified, simplying it if "simplify" is set.
4424 * Free the temporary data structure "pairs" that was associated
4425 * to the old version of "bmap".
4426 */
4429 {
4434 }
4436 /* Is "div" the single unknown existentially quantified variable
4437 * in inequality constraint "ineq" of "bmap"?
4438 * "div" is known to have a non-zero coefficient in "ineq".
4439 */
4442 {
4445 isl_bool known;
4455 isl_bool known;
4464 }
4467 }
4469 /* Does integer division "div" have coefficient 1 in inequality constraint
4470 * "ineq" of "map"?
4471 */
4473 {
4481 }
4483 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4484 * then try and drop redundant divs again,
4485 * freeing the temporary data structure "pairs" that was associated
4486 * to the old version of "bmap".
4487 */
4490 {
4494 }
4496 /* Drop the integer division at position "div", along with the two
4497 * inequality constraints "ineq1" and "ineq2" in which it appears
4498 * from "bmap" and then try and drop redundant divs again,
4499 * freeing the temporary data structure "pairs" that was associated
4500 * to the old version of "bmap".
4501 */
4505 {
4512 }
4515 }
4517 /* Given two inequality constraints
4518 *
4519 * f(x) + n d + c >= 0, (ineq)
4520 *
4521 * with d the variable at position "pos", and
4522 *
4523 * f(x) + c0 >= 0, (lower)
4524 *
4525 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4526 * determined by the first constraint.
4527 * That is, store
4528 *
4529 * ceil((c0 - c)/n)
4530 *
4531 * in *l.
4532 */
4535 {
4539 }
4541 /* Given two inequality constraints
4542 *
4543 * f(x) + n d + c >= 0, (ineq)
4544 *
4545 * with d the variable at position "pos", and
4546 *
4547 * -f(x) - c0 >= 0, (upper)
4548 *
4549 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4550 * determined by the first constraint.
4551 * That is, store
4552 *
4553 * ceil((-c1 - c)/n)
4554 *
4555 * in *u.
4556 */
4559 {
4563 }
4565 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4566 * does the corresponding lower bound have a fixed value in "bmap"?
4567 *
4568 * In particular, "ineq" is of the form
4569 *
4570 * f(x) + n d + c >= 0
4571 *
4572 * with n > 0, c the constant term and
4573 * d the existentially quantified variable "div".
4574 * That is, the lower bound is
4575 *
4576 * ceil((-f(x) - c)/n)
4577 *
4578 * Look for a pair of constraints
4579 *
4580 * f(x) + c0 >= 0
4581 * -f(x) + c1 >= 0
4582 *
4583 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4584 * That is, check that
4585 *
4586 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4587 *
4588 * If so, return the index of inequality f(x) + c0 >= 0.
4589 * Otherwise, return bmap->n_ineq.
4590 * Return -1 on error.
4591 */
4593 {
4616 }
4624 }
4641 }
4643 /* Given a lower bound constraint "ineq" on the existentially quantified
4644 * variable "div", such that the corresponding lower bound has
4645 * a fixed value in "bmap", assign this fixed value to the variable and
4646 * then try and drop redundant divs again,
4647 * freeing the temporary data structure "pairs" that was associated
4648 * to the old version of "bmap".
4649 * "lower" determines the constant value for the lower bound.
4650 *
4651 * In particular, "ineq" is of the form
4652 *
4653 * f(x) + n d + c >= 0,
4654 *
4655 * while "lower" is of the form
4656 *
4657 * f(x) + c0 >= 0
4658 *
4659 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4660 * is ceil((c0 - c)/n).
4661 */
4664 {
4665 isl_int c;
4678 }
4680 /* Remove divs that are not strictly needed based on the inequality
4681 * constraints.
4682 * In particular, if a div only occurs positively (or negatively)
4683 * in constraints, then it can simply be dropped.
4684 * Also, if a div occurs in only two constraints and if moreover
4685 * those two constraints are opposite to each other, except for the constant
4686 * term and if the sum of the constant terms is such that for any value
4687 * of the other values, there is always at least one integer value of the
4688 * div, i.e., if one plus this sum is greater than or equal to
4689 * the (absolute value) of the coefficient of the div in the constraints,
4690 * then we can also simply drop the div.
4691 *
4692 * If an existentially quantified variable does not have an explicit
4693 * representation, appears in only a single lower bound that does not
4694 * involve any other such existentially quantified variables and appears
4695 * in this lower bound with coefficient 1,
4696 * then fix the variable to the value of the lower bound. That is,
4697 * turn the inequality into an equality.
4698 * If for any value of the other variables, there is any value
4699 * for the existentially quantified variable satisfying the constraints,
4700 * then this lower bound also satisfies the constraints.
4701 * It is therefore safe to pick this lower bound.
4702 *
4703 * The same reasoning holds even if the coefficient is not one.
4704 * However, fixing the variable to the value of the lower bound may
4705 * in general introduce an extra integer division, in which case
4706 * it may be better to pick another value.
4707 * If this integer division has a known constant value, then plugging
4708 * in this constant value removes the existentially quantified variable
4709 * completely. In particular, if the lower bound is of the form
4710 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4711 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4712 * then the existentially quantified variable can be assigned this
4713 * shared value.
4714 *
4715 * We skip divs that appear in equalities or in the definition of other divs.
4716 * Divs that appear in the definition of other divs usually occur in at least
4717 * 4 constraints, but the constraints may have been simplified.
4718 *
4719 * If any divs are left after these simple checks then we move on
4720 * to more complicated cases in drop_more_redundant_divs.
4721 */
4724 {
4766 }
4770 }
4771 }
4779 }
4782 else
4802 pairs);
4808 pairs);
4810 }
4827 else
4834 }
4837 }
4844 error:
4848 }
4850 /* Consider the coefficients at "c" as a row vector and replace
4851 * them with their product with "T". "T" is assumed to be a square matrix.
4852 */
4854 {
4876 }
4878 /* Plug in T for the variables in "bmap" starting at "pos".
4879 * T is a linear unimodular matrix, i.e., without constant term.
4880 */
4883 {
4910 }
4914 error:
4918 }
4920 /* Remove divs that are not strictly needed.
4921 *
4922 * First look for an equality constraint involving two or more
4923 * existentially quantified variables without an explicit
4924 * representation. Replace the combination that appears
4925 * in the equality constraint by a single existentially quantified
4926 * variable such that the equality can be used to derive
4927 * an explicit representation for the variable.
4928 * If there are no more such equality constraints, then continue
4929 * with isl_basic_map_drop_redundant_divs_ineq.
4930 *
4931 * In particular, if the equality constraint is of the form
4932 *
4933 * f(x) + \sum_i c_i a_i = 0
4934 *
4935 * with a_i existentially quantified variable without explicit
4936 * representation, then apply a transformation on the existentially
4937 * quantified variables to turn the constraint into
4938 *
4939 * f(x) + g a_1' = 0
4940 *
4941 * with g the gcd of the c_i.
4942 * In order to easily identify which existentially quantified variables
4943 * have a complete explicit representation, i.e., without being defined
4944 * in terms of other existentially quantified variables without
4945 * an explicit representation, the existentially quantified variables
4946 * are first sorted.
4947 *
4948 * The variable transformation is computed by extending the row
4949 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4950 *
4951 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4952 * [a_2'] [ a_2 ]
4953 * ... = U ....
4954 * [a_n'] [ a_n ]
4955 *
4956 * with [c_1/g ... c_n/g] representing the first row of U.
4957 * The inverse of U is then plugged into the original constraints.
4958 * The call to isl_basic_map_simplify makes sure the explicit
4959 * representation for a_1' is extracted from the equality constraint.
4960 */
4963 {
4998 }
5017 }
5019 /* Does "bmap" satisfy any equality that involves more than 2 variables
5020 * and/or has coefficients different from -1 and 1?
5021 */
5023 {
5052 }
5055 }
5057 /* Remove any common factor g from the constraint coefficients in "v".
5058 * The constant term is stored in the first position and is replaced
5059 * by floor(c/g). If any common factor is removed and if this results
5060 * in a tightening of the constraint, then set *tightened.
5061 */
5064 {
5084 }
5086 /* If "bmap" is an integer set that satisfies any equality involving
5087 * more than 2 variables and/or has coefficients different from -1 and 1,
5088 * then use variable compression to reduce the coefficients by removing
5089 * any (hidden) common factor.
5090 * In particular, apply the variable compression to each constraint,
5091 * factor out any common factor in the non-constant coefficients and
5092 * then apply the inverse of the compression.
5093 * At the end, we mark the basic map as having reduced constants.
5094 * If this flag is still set on the next invocation of this function,
5095 * then we skip the computation.
5096 *
5097 * Removing a common factor may result in a tightening of some of
5098 * the constraints. If this happens, then we may end up with two
5099 * opposite inequalities that can be replaced by an equality.
5100 * We therefore call isl_basic_map_detect_inequality_pairs,
5101 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5102 * and isl_basic_map_gauss if such a pair was found.
5103 *
5104 * Note that this function may leave the result in an inconsistent state.
5105 * In particular, the constraints may not be gaussed.
5106 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5107 * for some of the test cases to pass successfully.
5108 * Any potential modification of the representation is therefore only
5109 * performed on a single copy of the basic map.
5110 */
5113 {
5115 isl_bool multi;
5151 }
5166 }
5181 }
5182 }
5185 error:
5190 }
5192 /* Shift the integer division at position "div" of "bmap"
5193 * by "shift" times the variable at position "pos".
5194 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5195 * corresponds to the constant term.
5196 *
5197 * That is, if the integer division has the form
5198 *
5199 * floor(f(x)/d)
5200 *
5201 * then replace it by
5202 *
5203 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5204 */
5207 {
5226 }
5232 }
5240 }
5243 }