| blob | 3d05c51675fd421e2326362d54f3058964fd8563 |
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
3 * Copyright 2012-2013 Ecole Normale Superieure
4 * Copyright 2014-2015 INRIA Rocquencourt
5 * Copyright 2016 Sven Verdoolaege
6 *
7 * Use of this software is governed by the MIT license
8 *
9 * Written by Sven Verdoolaege, K.U.Leuven, Departement
10 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
11 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
12 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
13 * B.P. 105 - 78153 Le Chesnay, France
14 */
16 #include <isl_ctx_private.h>
17 #include <isl_map_private.h>
19 #include <isl/map.h>
20 #include <isl_seq.h>
22 #include <isl_space_private.h>
23 #include <isl_mat_private.h>
24 #include <isl_vec_private.h>
26 #include <bset_to_bmap.c>
27 #include <bset_from_bmap.c>
28 #include <set_to_map.c>
29 #include <set_from_map.c>
32 {
36 }
39 {
44 }
45 }
49 {
51 isl_int gcd;
64 }
67 }
75 }
77 }
85 }
88 }
95 }
99 }
103 {
106 }
108 /* Reduce the coefficient of the variable at position "pos"
109 * in integer division "div", such that it lies in the half-open
110 * interval (1/2,1/2], extracting any excess value from this integer division.
111 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
112 * corresponds to the constant term.
113 *
114 * That is, the integer division is of the form
115 *
116 * floor((... + (c * d + r) * x_pos + ...)/d)
117 *
118 * with -d < 2 * r <= d.
119 * Replace it by
120 *
121 * floor((... + r * x_pos + ...)/d) + c * x_pos
122 *
123 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
124 * Otherwise, c = floor((c * d + r)/d) + 1.
125 *
126 * This is the same normalization that is performed by isl_aff_floor.
127 */
130 {
131 isl_int shift;
146 }
148 /* Does the coefficient of the variable at position "pos"
149 * in integer division "div" need to be reduced?
150 * That is, does it lie outside the half-open interval (1/2,1/2]?
151 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
152 * 2 * c != d.
153 */
156 {
157 isl_bool r;
169 }
171 /* Reduce the coefficients (including the constant term) of
172 * integer division "div", if needed.
173 * In particular, make sure all coefficients lie in
174 * the half-open interval (1/2,1/2].
175 */
178 {
180 isl_size total;
186 isl_bool reduce;
196 }
199 }
201 /* Reduce the coefficients (including the constant term) of
202 * the known integer divisions, if needed
203 * In particular, make sure all coefficients lie in
204 * the half-open interval (1/2,1/2].
205 */
208 {
222 }
225 }
227 /* Remove any common factor in numerator and denominator of the div expression,
228 * not taking into account the constant term.
229 * That is, if the div is of the form
230 *
231 * floor((a + m f(x))/(m d))
232 *
233 * then replace it by
234 *
235 * floor((floor(a/m) + f(x))/d)
236 *
237 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
238 * and can therefore not influence the result of the floor.
239 */
242 {
262 }
264 /* Remove any common factor in numerator and denominator of a div expression,
265 * not taking into account the constant term.
266 * That is, look for any div of the form
267 *
268 * floor((a + m f(x))/(m d))
269 *
270 * and replace it by
271 *
272 * floor((floor(a/m) + f(x))/d)
273 *
274 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
275 * and can therefore not influence the result of the floor.
276 */
279 {
291 }
293 /* Assumes divs have been ordered if keep_divs is set.
294 */
298 {
299 isl_size total;
300 isl_size v_div;
318 }
329 }
338 /* We need to be careful about circular definitions,
339 * so for now we just remove the definition of div k
340 * if the equality contains any divs.
341 * If keep_divs is set, then the divs have been ordered
342 * and we can keep the definition as long as the result
343 * is still ordered.
344 */
353 }
356 }
358 /* Assumes divs have been ordered if keep_divs is set.
359 */
362 {
363 isl_size v_div;
375 }
377 /* Check if elimination of div "div" using equality "eq" would not
378 * result in a div depending on a later div.
379 */
382 {
385 isl_size v_div;
402 }
405 }
407 /* Eliminate divs based on equalities
408 */
411 {
426 isl_bool ok;
442 }
443 }
447 }
449 /* Eliminate divs based on inequalities
450 */
453 {
483 }
485 }
487 /* Does the equality constraint at position "eq" in "bmap" involve
488 * any local variables in the range [first, first + n)
489 * that are not marked as having an explicit representation?
490 */
493 {
502 isl_bool unknown;
511 }
514 }
516 /* The last local variable involved in the equality constraint
517 * at position "eq" in "bmap" is the local variable at position "div".
518 * It can therefore be used to extract an explicit representation
519 * for that variable.
520 * Do so unless the local variable already has an explicit representation or
521 * the explicit representation would involve any other local variables
522 * that in turn do not have an explicit representation.
523 * An equality constraint involving local variables without an explicit
524 * representation can be used in isl_basic_map_drop_redundant_divs
525 * to separate out an independent local variable. Introducing
526 * an explicit representation here would block this transformation,
527 * while the partial explicit representation in itself is not very useful.
528 * Set *progress if anything is changed.
529 *
530 * The equality constraint is of the form
531 *
532 * f(x) + n e >= 0
533 *
534 * with n a positive number. The explicit representation derived from
535 * this constraint is
536 *
537 * floor((-f(x))/n)
538 */
541 {
542 isl_size total;
544 isl_bool involves;
569 }
571 /* Perform fangcheng (Gaussian elimination) on the equality
572 * constraints of "bmap".
573 * That is, put them into row-echelon form, starting from the last column
574 * backward and use them to eliminate the corresponding coefficients
575 * from all constraints.
576 *
577 * If "progress" is not NULL, then it gets set if the elimination
578 * result in any changes.
579 * The elimination process may result in some equality constraints
580 * getting interchanged or removed.
581 * If "swap" or "drop" are not NULL, then they get called when
582 * two equality constraints get interchanged or
583 * when a number of final equality constraints get removed.
584 * As a special case, if the input turns out to be empty,
585 * then drop gets called with the number of removed equality
586 * constraints set to the total number of equality constraints.
587 * If "swap" or "drop" are not NULL, then the local variables (if any)
588 * are assumed to be in a valid order.
589 */
594 {
599 isl_size total;
619 }
626 }
638 }
647 }
653 }
657 {
659 }
663 {
665 progress));
666 }
670 {
676 }
678 }
680 /* Hash table of inequalities in a basic map.
681 * "index" is an array of addresses of inequalities in the basic map, some
682 * of which are NULL. The inequalities are hashed on the coefficients
683 * except the constant term.
684 * "size" is the number of elements in the array and is always a power of two
685 * "bits" is the number of bits need to represent an index into the array.
686 * "total" is the total dimension of the basic map.
687 */
692 isl_size total;
693 };
695 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
696 */
699 {
718 }
720 /* Free the memory allocated by create_constraint_index.
721 */
723 {
725 }
727 /* Return the position in ci->index that contains the address of
728 * an inequality that is equal to *ineq up to the constant term,
729 * provided this address is not identical to "ineq".
730 * If there is no such inequality, then return the position where
731 * such an inequality should be inserted.
732 */
734 {
742 }
744 /* Return the position in ci->index that contains the address of
745 * an inequality that is equal to the k'th inequality of "bmap"
746 * up to the constant term, provided it does not point to the very
747 * same inequality.
748 * If there is no such inequality, then return the position where
749 * such an inequality should be inserted.
750 */
753 {
755 }
759 {
761 }
763 /* Fill in the "ci" data structure with the inequalities of "bset".
764 */
767 {
776 }
779 }
781 /* Is the inequality ineq (obviously) redundant with respect
782 * to the constraints in "ci"?
783 *
784 * Look for an inequality in "ci" with the same coefficients and then
785 * check if the contant term of "ineq" is greater than or equal
786 * to the constant term of that inequality. If so, "ineq" is clearly
787 * redundant.
788 *
789 * Note that hash_index_ineq ignores a stored constraint if it has
790 * the same address as the passed inequality. It is ok to pass
791 * the address of a local variable here since it will never be
792 * the same as the address of a constraint in "ci".
793 */
796 {
803 }
805 /* If we can eliminate more than one div, then we need to make
806 * sure we do it from last div to first div, in order not to
807 * change the position of the other divs that still need to
808 * be removed.
809 */
812 {
819 isl_size v_div;
868 }
870 }
882 }
885 out:
889 }
892 {
894 isl_size v_div;
906 }
908 }
910 /* Normalize divs that appear in equalities.
911 *
912 * In particular, we assume that bmap contains some equalities
913 * of the form
914 *
915 * a x = m * e_i
916 *
917 * and we want to replace the set of e_i by a minimal set and
918 * such that the new e_i have a canonical representation in terms
919 * of the vector x.
920 * If any of the equalities involves more than one divs, then
921 * we currently simply bail out.
922 *
923 * Let us first additionally assume that all equalities involve
924 * a div. The equalities then express modulo constraints on the
925 * remaining variables and we can use "parameter compression"
926 * to find a minimal set of constraints. The result is a transformation
927 *
928 * x = T(x') = x_0 + G x'
929 *
930 * with G a lower-triangular matrix with all elements below the diagonal
931 * non-negative and smaller than the diagonal element on the same row.
932 * We first normalize x_0 by making the same property hold in the affine
933 * T matrix.
934 * The rows i of G with a 1 on the diagonal do not impose any modulo
935 * constraint and simply express x_i = x'_i.
936 * For each of the remaining rows i, we introduce a div and a corresponding
937 * equality. In particular
938 *
939 * g_ii e_j = x_i - g_i(x')
940 *
941 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
942 * corresponding div (if g_kk != 1).
943 *
944 * If there are any equalities not involving any div, then we
945 * first apply a variable compression on the variables x:
946 *
947 * x = C x'' x'' = C_2 x
948 *
949 * and perform the above parameter compression on A C instead of on A.
950 * The resulting compression is then of the form
951 *
952 * x'' = T(x') = x_0 + G x'
953 *
954 * and in constructing the new divs and the corresponding equalities,
955 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
956 * by the corresponding row from C_2.
957 */
960 {
962 isl_size v_div;
969 isl_int v;
1003 }
1004 }
1013 }
1019 }
1029 }
1036 }
1041 /* We have to be careful because dropping equalities may reorder them */
1051 }
1052 }
1058 else
1059 needed++;
1060 }
1066 }
1076 else
1084 else
1087 }
1091 }
1098 done:
1102 error:
1108 }
1112 {
1122 }
1124 /* Check whether it is ok to define a div based on an inequality.
1125 * To avoid the introduction of circular definitions of divs, we
1126 * do not allow such a definition if the resulting expression would refer to
1127 * any other undefined divs or if any known div is defined in
1128 * terms of the unknown div.
1129 */
1132 {
1136 /* Not defined in terms of unknown divs */
1144 }
1146 /* No other div defined in terms of this one => avoid loops */
1154 }
1157 }
1159 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1160 * be a better expression than the current one?
1161 *
1162 * If we do not have any expression yet, then any expression would be better.
1163 * Otherwise we check if the last variable involved in the inequality
1164 * (disregarding the div that it would define) is in an earlier position
1165 * than the last variable involved in the current div expression.
1166 */
1169 {
1186 }
1188 /* Given two constraints "k" and "l" that are opposite to each other,
1189 * except for the constant term, check if we can use them
1190 * to obtain an expression for one of the hitherto unknown divs or
1191 * a "better" expression for a div for which we already have an expression.
1192 * "sum" is the sum of the constant terms of the constraints.
1193 * If this sum is strictly smaller than the coefficient of one
1194 * of the divs, then this pair can be used define the div.
1195 * To avoid the introduction of circular definitions of divs, we
1196 * do not use the pair if the resulting expression would refer to
1197 * any other undefined divs or if any known div is defined in
1198 * terms of the unknown div.
1199 */
1203 {
1208 isl_bool set_div;
1223 else
1228 }
1230 }
1234 {
1238 isl_int sum;
1253 }
1261 }
1276 }
1278 /* We need to break out of the loop after these
1279 * changes since the contents of the hash
1280 * will no longer be valid.
1281 * Plus, we probably we want to regauss first.
1282 */
1290 }
1295 }
1297 /* Detect all pairs of inequalities that form an equality.
1298 *
1299 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1300 * Call it repeatedly while it is making progress.
1301 */
1304 {
1316 }
1318 /* Eliminate knowns divs from constraints where they appear with
1319 * a (positive or negative) unit coefficient.
1320 *
1321 * That is, replace
1322 *
1323 * floor(e/m) + f >= 0
1324 *
1325 * by
1326 *
1327 * e + m f >= 0
1328 *
1329 * and
1330 *
1331 * -floor(e/m) + f >= 0
1332 *
1333 * by
1334 *
1335 * -e + m f + m - 1 >= 0
1336 *
1337 * The first conversion is valid because floor(e/m) >= -f is equivalent
1338 * to e/m >= -f because -f is an integral expression.
1339 * The second conversion follows from the fact that
1340 *
1341 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1342 *
1343 *
1344 * Note that one of the div constraints may have been eliminated
1345 * due to being redundant with respect to the constraint that is
1346 * being modified by this function. The modified constraint may
1347 * no longer imply this div constraint, so we add it back to make
1348 * sure we do not lose any information.
1349 *
1350 * We skip integral divs, i.e., those with denominator 1, as we would
1351 * risk eliminating the div from the div constraints. We do not need
1352 * to handle those divs here anyway since the div constraints will turn
1353 * out to form an equality and this equality can then be used to eliminate
1354 * the div from all constraints.
1355 */
1358 {
1390 else
1400 }
1406 }
1407 }
1410 }
1413 {
1418 isl_bool empty;
1434 /* requires equalities in normal form */
1440 }
1442 }
1445 {
1447 }
1452 {
1484 }
1488 {
1490 }
1493 /* If the only constraints a div d=floor(f/m)
1494 * appears in are its two defining constraints
1495 *
1496 * f - m d >=0
1497 * -(f - (m - 1)) + m d >= 0
1498 *
1499 * then it can safely be removed.
1500 */
1502 {
1511 isl_bool red;
1518 }
1525 }
1528 }
1530 /*
1531 * Remove divs that don't occur in any of the constraints or other divs.
1532 * These can arise when dropping constraints from a basic map or
1533 * when the divs of a basic map have been temporarily aligned
1534 * with the divs of another basic map.
1535 */
1538 {
1540 isl_size v_div;
1547 isl_bool redundant;
1557 }
1559 }
1561 /* Mark "bmap" as final, without checking for obviously redundant
1562 * integer divisions. This function should be used when "bmap"
1563 * is known not to involve any such integer divisions.
1564 */
1567 {
1572 }
1574 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1575 */
1577 {
1581 }
1584 {
1586 }
1588 /* Remove definition of any div that is defined in terms of the given variable.
1589 * The div itself is not removed. Functions such as
1590 * eliminate_divs_ineq depend on the other divs remaining in place.
1591 */
1594 {
1608 }
1610 }
1612 /* Eliminate the specified variables from the constraints using
1613 * Fourier-Motzkin. The variables themselves are not removed.
1614 */
1617 {
1620 isl_size total;
1651 }
1658 n_lower++;
1660 n_upper++;
1661 }
1685 }
1688 }
1700 }
1701 }
1705 error:
1708 }
1712 {
1715 }
1717 /* Eliminate the specified n dimensions starting at first from the
1718 * constraints, without removing the dimensions from the space.
1719 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1720 * Otherwise, they are projected out and the original space is restored.
1721 */
1725 {
1740 }
1747 }
1752 {
1754 }
1756 /* Remove all constraints from "bmap" that reference any unknown local
1757 * variables (directly or indirectly).
1758 *
1759 * Dropping all constraints on a local variable will make it redundant,
1760 * so it will get removed implicitly by
1761 * isl_basic_map_drop_constraints_involving_dims. Some other local
1762 * variables may also end up becoming redundant if they only appear
1763 * in constraints together with the unknown local variable.
1764 * Therefore, start over after calling
1765 * isl_basic_map_drop_constraints_involving_dims.
1766 */
1769 {
1770 isl_bool known;
1771 isl_size n_div;
1798 }
1801 }
1803 /* Remove all constraints from "map" that reference any unknown local
1804 * variables (directly or indirectly).
1805 *
1806 * Since constraints may get dropped from the basic maps,
1807 * they may no longer be disjoint from each other.
1808 */
1811 {
1813 isl_bool known;
1831 }
1837 }
1839 /* Don't assume equalities are in order, because align_divs
1840 * may have changed the order of the divs.
1841 */
1843 {
1856 }
1857 }
1858 }
1862 {
1864 }
1868 {
1882 }
1884 }
1886 }
1890 {
1893 }
1897 {
1900 isl_size dim;
1908 }
1930 error:
1934 }
1936 /* For each inequality in "ineq" that is a shifted (more relaxed)
1937 * copy of an inequality in "context", mark the corresponding entry
1938 * in "row" with -1.
1939 * If an inequality only has a non-negative constant term, then
1940 * mark it as well.
1941 */
1944 {
1964 isl_bool redundant;
1970 }
1977 }
1980 error:
1983 }
1987 {
2000 isl_bool redundant;
2012 }
2015 error:
2018 }
2020 /* Remove constraints from "bmap" that are identical to constraints
2021 * in "context" or that are more relaxed (greater constant term).
2022 *
2023 * We perform the test for shifted copies on the pure constraints
2024 * in remove_shifted_constraints.
2025 */
2028 {
2037 }
2050 error:
2054 }
2056 /* Does the (linear part of a) constraint "c" involve any of the "len"
2057 * "relevant" dimensions?
2058 */
2060 {
2068 }
2071 }
2073 /* Drop constraints from "bmap" that do not involve any of
2074 * the dimensions marked "relevant".
2075 */
2078 {
2080 isl_size dim;
2096 }
2103 }
2106 }
2108 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2109 *
2110 * In particular, for any variable involved in the constraint,
2111 * find the actual group id from before and replace the group
2112 * of the corresponding variable by the minimal group of all
2113 * the variables involved in the constraint considered so far
2114 * (if this minimum is smaller) or replace the minimum by this group
2115 * (if the minimum is larger).
2116 *
2117 * At the end, all the variables in "c" will (indirectly) point
2118 * to the minimal of the groups that they referred to originally.
2119 */
2121 {
2138 }
2139 }
2141 /* Allocate an array of groups of variables, one for each variable
2142 * in "context", initialized to zero.
2143 */
2145 {
2147 isl_size dim;
2154 }
2156 /* Drop constraints from "bmap" that only involve variables that are
2157 * not related to any of the variables marked with a "-1" in "group".
2158 *
2159 * We construct groups of variables that collect variables that
2160 * (indirectly) appear in some common constraint of "bmap".
2161 * Each group is identified by the first variable in the group,
2162 * except for the special group of variables that was already identified
2163 * in the input as -1 (or are related to those variables).
2164 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2165 * otherwise the group of i is the group of group[i].
2166 *
2167 * We first initialize groups for the remaining variables.
2168 * Then we iterate over the constraints of "bmap" and update the
2169 * group of the variables in the constraint by the smallest group.
2170 * Finally, we resolve indirect references to groups by running over
2171 * the variables.
2172 *
2173 * After computing the groups, we drop constraints that do not involve
2174 * any variables in the -1 group.
2175 */
2178 {
2179 isl_size dim;
2194 }
2212 }
2214 /* Drop constraints from "context" that are irrelevant for computing
2215 * the gist of "bset".
2216 *
2217 * In particular, drop constraints in variables that are not related
2218 * to any of the variables involved in the constraints of "bset"
2219 * in the sense that there is no sequence of constraints that connects them.
2220 *
2221 * We first mark all variables that appear in "bset" as belonging
2222 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2223 */
2226 {
2228 isl_size dim;
2247 }
2253 }
2256 }
2258 /* Drop constraints from "context" that are irrelevant for computing
2259 * the gist of the inequalities "ineq".
2260 * Inequalities in "ineq" for which the corresponding element of row
2261 * is set to -1 have already been marked for removal and should be ignored.
2262 *
2263 * In particular, drop constraints in variables that are not related
2264 * to any of the variables involved in "ineq"
2265 * in the sense that there is no sequence of constraints that connects them.
2266 *
2267 * We first mark all variables that appear in "bset" as belonging
2268 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2269 */
2272 {
2274 isl_size dim;
2276 isl_size n;
2294 }
2297 }
2300 }
2302 /* Do all "n" entries of "row" contain a negative value?
2303 */
2305 {
2313 }
2315 /* Update the inequalities in "bset" based on the information in "row"
2316 * and "tab".
2317 *
2318 * In particular, the array "row" contains either -1, meaning that
2319 * the corresponding inequality of "bset" is redundant, or the index
2320 * of an inequality in "tab".
2321 *
2322 * If the row entry is -1, then drop the inequality.
2323 * Otherwise, if the constraint is marked redundant in the tableau,
2324 * then drop the inequality. Similarly, if it is marked as an equality
2325 * in the tableau, then turn the inequality into an equality and
2326 * perform Gaussian elimination.
2327 */
2330 {
2347 }
2357 }
2358 }
2364 }
2366 /* Update the inequalities in "bset" based on the information in "row"
2367 * and "tab" and free all arguments (other than "bset").
2368 */
2373 {
2382 }
2384 /* Remove all information from bset that is redundant in the context
2385 * of context.
2386 * "ineq" contains the (possibly transformed) inequalities of "bset",
2387 * in the same order.
2388 * The (explicit) equalities of "bset" are assumed to have been taken
2389 * into account by the transformation such that only the inequalities
2390 * are relevant.
2391 * "context" is assumed not to be empty.
2392 *
2393 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2394 * A value of -1 means that the inequality is obviously redundant and may
2395 * not even appear in "tab".
2396 *
2397 * We first mark the inequalities of "bset"
2398 * that are obviously redundant with respect to some inequality in "context".
2399 * Then we remove those constraints from "context" that have become
2400 * irrelevant for computing the gist of "bset".
2401 * Note that this removal of constraints cannot be replaced by
2402 * a factorization because factors in "bset" may still be connected
2403 * to each other through constraints in "context".
2404 *
2405 * If there are any inequalities left, we construct a tableau for
2406 * the context and then add the inequalities of "bset".
2407 * Before adding these inequalities, we freeze all constraints such that
2408 * they won't be considered redundant in terms of the constraints of "bset".
2409 * Then we detect all redundant constraints (among the
2410 * constraints that weren't frozen), first by checking for redundancy in the
2411 * the tableau and then by checking if replacing a constraint by its negation
2412 * would lead to an empty set. This last step is fairly expensive
2413 * and could be optimized by more reuse of the tableau.
2414 * Finally, we update bset according to the results.
2415 */
2418 {
2433 }
2469 }
2492 }
2497 }
2501 error:
2509 }
2511 /* Extract the inequalities of "bset" as an isl_mat.
2512 */
2514 {
2515 isl_size total;
2528 }
2530 /* Remove all information from "bset" that is redundant in the context
2531 * of "context", for the case where both "bset" and "context" are
2532 * full-dimensional.
2533 */
2536 {
2541 }
2543 /* Remove all information from "bset" that is redundant in the context
2544 * of "context", for the case where the combined equalities of
2545 * "bset" and "context" allow for a compression that can be obtained
2546 * by preapplication of "T".
2547 *
2548 * "bset" itself is not transformed by "T". Instead, the inequalities
2549 * are extracted from "bset" and those are transformed by "T".
2550 * uset_gist_full then determines which of the transformed inequalities
2551 * are redundant with respect to the transformed "context" and removes
2552 * the corresponding inequalities from "bset".
2553 *
2554 * After preapplying "T" to the inequalities, any common factor is
2555 * removed from the coefficients. If this results in a tightening
2556 * of the constant term, then the same tightening is applied to
2557 * the corresponding untransformed inequality in "bset".
2558 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2559 *
2560 * g f'(x) + r >= 0
2561 *
2562 * with 0 <= r < g, then it is equivalent to
2563 *
2564 * f'(x) >= 0
2565 *
2566 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2567 * subspace compressed by T since the latter would be transformed to
2568 *
2569 * g f'(x) >= 0
2570 */
2574 {
2579 isl_int rem;
2591 }
2616 }
2620 error:
2625 }
2627 /* Project "bset" onto the variables that are involved in "template".
2628 */
2631 {
2633 isl_size n;
2640 isl_bool involved;
2649 }
2652 }
2654 /* Remove all information from bset that is redundant in the context
2655 * of context. In particular, equalities that are linear combinations
2656 * of those in context are removed. Then the inequalities that are
2657 * redundant in the context of the equalities and inequalities of
2658 * context are removed.
2659 *
2660 * First of all, we drop those constraints from "context"
2661 * that are irrelevant for computing the gist of "bset".
2662 * Alternatively, we could factorize the intersection of "context" and "bset".
2663 *
2664 * We first compute the intersection of the integer affine hulls
2665 * of "bset" and "context",
2666 * compute the gist inside this intersection and then reduce
2667 * the constraints with respect to the equalities of the context
2668 * that only involve variables already involved in the input.
2669 *
2670 * If two constraints are mutually redundant, then uset_gist_full
2671 * will remove the second of those constraints. We therefore first
2672 * sort the constraints so that constraints not involving existentially
2673 * quantified variables are given precedence over those that do.
2674 * We have to perform this sorting before the variable compression,
2675 * because that may effect the order of the variables.
2676 */
2679 {
2684 isl_size total;
2705 }
2710 }
2719 }
2730 }
2733 error:
2737 }
2739 /* Return the number of equality constraints in "bmap" that involve
2740 * local variables. This function assumes that Gaussian elimination
2741 * has been applied to the equality constraints.
2742 */
2744 {
2766 }
2768 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2769 * The constraints are assumed not to involve any local variables.
2770 */
2773 {
2775 isl_size total;
2793 }
2798 error:
2803 }
2805 /* Construct and return a variable compression based on the equality
2806 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2807 * "n1" is the number of (initial) equality constraints in "bmap1"
2808 * that do involve local variables.
2809 * "n2" is the number of (initial) equality constraints in "bmap2"
2810 * that do involve local variables.
2811 * "total" is the total number of other variables.
2812 * This function assumes that Gaussian elimination
2813 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2814 * such that the equality constraints not involving local variables
2815 * are those that start at "n1" or "n2".
2816 *
2817 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2818 * then simply compute the compression based on the equality constraints
2819 * in the other basic map.
2820 * Otherwise, combine the equality constraints from both into a new
2821 * basic map such that Gaussian elimination can be applied to this combination
2822 * and then construct a variable compression from the resulting
2823 * equality constraints.
2824 */
2828 {
2838 }
2843 }
2858 }
2860 /* Extract the stride constraints from "bmap", compressed
2861 * with respect to both the stride constraints in "context" and
2862 * the remaining equality constraints in both "bmap" and "context".
2863 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2864 * "context_n_eq" is the number of (initial) stride constraints in "context".
2865 *
2866 * Let x be all variables in "bmap" (and "context") other than the local
2867 * variables. First compute a variable compression
2868 *
2869 * x = V x'
2870 *
2871 * based on the non-stride equality constraints in "bmap" and "context".
2872 * Consider the stride constraints of "context",
2873 *
2874 * A(x) + B(y) = 0
2875 *
2876 * with y the local variables and plug in the variable compression,
2877 * resulting in
2878 *
2879 * A(V x') + B(y) = 0
2880 *
2881 * Use these constraints to compute a parameter compression on x'
2882 *
2883 * x' = T x''
2884 *
2885 * Now consider the stride constraints of "bmap"
2886 *
2887 * C(x) + D(y) = 0
2888 *
2889 * and plug in x = V*T x''.
2890 * That is, return A = [C*V*T D].
2891 */
2895 {
2921 else
2929 }
2931 /* Remove the prime factors from *g that have an exponent that
2932 * is strictly smaller than the exponent in "c".
2933 * All exponents in *g are known to be smaller than or equal
2934 * to those in "c".
2935 *
2936 * That is, if *g is equal to
2937 *
2938 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2939 *
2940 * and "c" is equal to
2941 *
2942 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2943 *
2944 * then update *g to
2945 *
2946 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2947 * p_n^{e_n * (e_n = f_n)}
2948 *
2949 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2950 * neither does the gcd of *g and c / *g.
2951 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2952 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2953 * Dividing *g by this gcd therefore strictly reduces the exponent
2954 * of the prime factors that need to be removed, while leaving the
2955 * other prime factors untouched.
2956 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2957 * removes all undesired factors, without removing any others.
2958 */
2960 {
2961 isl_int t;
2970 }
2972 }
2974 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2975 * of the same stride constraints in a compressed space that exploits
2976 * all equalities in the context and the other equalities in "bmap".
2977 *
2978 * If the stride constraints of "bmap" are of the form
2979 *
2980 * C(x) + D(y) = 0
2981 *
2982 * then A is of the form
2983 *
2984 * B(x') + D(y) = 0
2985 *
2986 * If any of these constraints involves only a single local variable y,
2987 * then the constraint appears as
2988 *
2989 * f(x) + m y_i = 0
2990 *
2991 * in "bmap" and as
2992 *
2993 * h(x') + m y_i = 0
2994 *
2995 * in "A".
2996 *
2997 * Let g be the gcd of m and the coefficients of h.
2998 * Then, in particular, g is a divisor of the coefficients of h and
2999 *
3000 * f(x) = h(x')
3001 *
3002 * is known to be a multiple of g.
3003 * If some prime factor in m appears with the same exponent in g,
3004 * then it can be removed from m because f(x) is already known
3005 * to be a multiple of g and therefore in particular of this power
3006 * of the prime factors.
3007 * Prime factors that appear with a smaller exponent in g cannot
3008 * be removed from m.
3009 * Let g' be the divisor of g containing all prime factors that
3010 * appear with the same exponent in m and g, then
3011 *
3012 * f(x) + m y_i = 0
3013 *
3014 * can be replaced by
3015 *
3016 * f(x) + m/g' y_i' = 0
3017 *
3018 * Note that (if g' != 1) this changes the explicit representation
3019 * of y_i to that of y_i', so the integer division at position i
3020 * is marked unknown and later recomputed by a call to
3021 * isl_basic_map_gauss.
3022 */
3025 {
3029 isl_int gcd;
3062 }
3069 error:
3073 }
3075 /* Simplify the stride constraints in "bmap" based on
3076 * the remaining equality constraints in "bmap" and all equality
3077 * constraints in "context".
3078 * Only do this if both "bmap" and "context" have stride constraints.
3079 *
3080 * First extract a copy of the stride constraints in "bmap" in a compressed
3081 * space exploiting all the other equality constraints and then
3082 * use this compressed copy to simplify the original stride constraints.
3083 */
3086 {
3108 }
3110 /* Return a basic map that has the same intersection with "context" as "bmap"
3111 * and that is as "simple" as possible.
3112 *
3113 * The core computation is performed on the pure constraints.
3114 * When we add back the meaning of the integer divisions, we need
3115 * to (re)introduce the div constraints. If we happen to have
3116 * discovered that some of these integer divisions are equal to
3117 * some affine combination of other variables, then these div
3118 * constraints may end up getting simplified in terms of the equalities,
3119 * resulting in extra inequalities on the other variables that
3120 * may have been removed already or that may not even have been
3121 * part of the input. We try and remove those constraints of
3122 * this form that are most obviously redundant with respect to
3123 * the context. We also remove those div constraints that are
3124 * redundant with respect to the other constraints in the result.
3125 *
3126 * The stride constraints among the equality constraints in "bmap" are
3127 * also simplified with respecting to the other equality constraints
3128 * in "bmap" and with respect to all equality constraints in "context".
3129 */
3132 {
3144 }
3150 }
3154 }
3177 }
3194 error:
3198 }
3200 /*
3201 * Assumes context has no implicit divs.
3202 */
3205 {
3216 }
3236 }
3237 }
3241 error:
3245 }
3247 /* Drop all inequalities from "bmap" that also appear in "context".
3248 * "context" is assumed to have only known local variables and
3249 * the initial local variables of "bmap" are assumed to be the same
3250 * as those of "context".
3251 * The constraints of both "bmap" and "context" are assumed
3252 * to have been sorted using isl_basic_map_sort_constraints.
3253 *
3254 * Run through the inequality constraints of "bmap" and "context"
3255 * in sorted order.
3256 * If a constraint of "bmap" involves variables not in "context",
3257 * then it cannot appear in "context".
3258 * If a matching constraint is found, it is removed from "bmap".
3259 */
3262 {
3283 }
3289 }
3293 }
3298 }
3301 }
3304 }
3306 /* Drop all equalities from "bmap" that also appear in "context".
3307 * "context" is assumed to have only known local variables and
3308 * the initial local variables of "bmap" are assumed to be the same
3309 * as those of "context".
3310 *
3311 * Run through the equality constraints of "bmap" and "context"
3312 * in sorted order.
3313 * If a constraint of "bmap" involves variables not in "context",
3314 * then it cannot appear in "context".
3315 * If a matching constraint is found, it is removed from "bmap".
3316 */
3319 {
3345 }
3349 }
3354 }
3357 }
3360 }
3362 /* Remove the constraints in "context" from "bmap".
3363 * "context" is assumed to have explicit representations
3364 * for all local variables.
3365 *
3366 * First align the divs of "bmap" to those of "context" and
3367 * sort the constraints. Then drop all constraints from "bmap"
3368 * that appear in "context".
3369 */
3372 {
3387 }
3406 error:
3410 }
3412 /* Replace "map" by the disjunct at position "pos" and free "context".
3413 */
3416 {
3423 }
3425 /* Remove the constraints in "context" from "map".
3426 * If any of the disjuncts in the result turns out to be the universe,
3427 * then return this universe.
3428 * "context" is assumed to have explicit representations
3429 * for all local variables.
3430 */
3433 {
3443 }
3462 }
3469 error:
3473 }
3475 /* Remove the constraints in "context" from "set".
3476 * If any of the disjuncts in the result turns out to be the universe,
3477 * then return this universe.
3478 * "context" is assumed to have explicit representations
3479 * for all local variables.
3480 */
3483 {
3486 }
3488 /* Remove the constraints in "context" from "map".
3489 * If any of the disjuncts in the result turns out to be the universe,
3490 * then return this universe.
3491 * "context" is assumed to consist of a single disjunct and
3492 * to have explicit representations for all local variables.
3493 */
3496 {
3501 }
3503 /* Replace "map" by a universe map in the same space and free "drop".
3504 */
3507 {
3514 }
3516 /* Return a map that has the same intersection with "context" as "map"
3517 * and that is as "simple" as possible.
3518 *
3519 * If "map" is already the universe, then we cannot make it any simpler.
3520 * Similarly, if "context" is the universe, then we cannot exploit it
3521 * to simplify "map"
3522 * If "map" and "context" are identical to each other, then we can
3523 * return the corresponding universe.
3524 *
3525 * If either "map" or "context" consists of multiple disjuncts,
3526 * then check if "context" happens to be a subset of "map",
3527 * in which case all constraints can be removed.
3528 * In case of multiple disjuncts, the standard procedure
3529 * may not be able to detect that all constraints can be removed.
3530 *
3531 * If none of these cases apply, we have to work a bit harder.
3532 * During this computation, we make use of a single disjunct context,
3533 * so if the original context consists of more than one disjunct
3534 * then we need to approximate the context by a single disjunct set.
3535 * Simply taking the simple hull may drop constraints that are
3536 * only implicitly available in each disjunct. We therefore also
3537 * look for constraints among those defining "map" that are valid
3538 * for the context. These can then be used to simplify away
3539 * the corresponding constraints in "map".
3540 */
3543 {
3547 isl_bool subset;
3558 }
3576 }
3592 list);
3593 }
3595 error:
3599 }
3603 {
3605 }
3609 {
3612 }
3616 {
3619 }
3623 {
3628 }
3632 {
3634 }
3636 /* Compute the gist of "bmap" with respect to the constraints "context"
3637 * on the domain.
3638 */
3641 {
3647 }
3651 {
3655 }
3659 {
3663 }
3667 {
3671 }
3675 {
3677 }
3679 /* Quick check to see if two basic maps are disjoint.
3680 * In particular, we reduce the equalities and inequalities of
3681 * one basic map in the context of the equalities of the other
3682 * basic map and check if we get a contradiction.
3683 */
3686 {
3689 isl_size total;
3720 }
3728 }
3737 }
3741 disjoint:
3745 error:
3749 }
3753 {
3756 }
3758 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3759 */
3763 {
3774 }
3775 }
3778 }
3780 /* Are "map1" and "map2" obviously disjoint, based on information
3781 * that can be derived without looking at the individual basic maps?
3782 *
3783 * In particular, if one of them is empty or if they live in different spaces
3784 * (ignoring parameters), then they are clearly disjoint.
3785 */
3788 {
3789 isl_bool disjoint;
3790 isl_bool match;
3814 }
3816 /* Are "map1" and "map2" obviously disjoint?
3817 *
3818 * If one of them is empty or if they live in different spaces (ignoring
3819 * parameters), then they are clearly disjoint.
3820 * This is checked by isl_map_plain_is_disjoint_global.
3821 *
3822 * If they have different parameters, then we skip any further tests.
3823 *
3824 * If they are obviously equal, but not obviously empty, then we will
3825 * not be able to detect if they are disjoint.
3826 *
3827 * Otherwise we check if each basic map in "map1" is obviously disjoint
3828 * from each basic map in "map2".
3829 */
3832 {
3833 isl_bool disjoint;
3834 isl_bool intersect;
3835 isl_bool match;
3850 }
3852 /* Are "map1" and "map2" disjoint?
3853 * The parameters are assumed to have been aligned.
3854 *
3855 * In particular, check whether all pairs of basic maps are disjoint.
3856 */
3859 {
3861 }
3863 /* Are "map1" and "map2" disjoint?
3864 *
3865 * They are disjoint if they are "obviously disjoint" or if one of them
3866 * is empty. Otherwise, they are not disjoint if one of them is universal.
3867 * If the two inputs are (obviously) equal and not empty, then they are
3868 * not disjoint.
3869 * If none of these cases apply, then check if all pairs of basic maps
3870 * are disjoint after aligning the parameters.
3871 */
3873 {
3874 isl_bool disjoint;
3875 isl_bool intersect;
3903 }
3905 /* Are "bmap1" and "bmap2" disjoint?
3906 *
3907 * They are disjoint if they are "obviously disjoint" or if one of them
3908 * is empty. Otherwise, they are not disjoint if one of them is universal.
3909 * If none of these cases apply, we compute the intersection and see if
3910 * the result is empty.
3911 */
3914 {
3915 isl_bool disjoint;
3916 isl_bool intersect;
3945 }
3947 /* Are "bset1" and "bset2" disjoint?
3948 */
3951 {
3953 }
3957 {
3959 }
3961 /* Are "set1" and "set2" disjoint?
3962 */
3964 {
3966 }
3968 /* Is "v" equal to 0, 1 or -1?
3969 */
3971 {
3973 }
3975 /* Are the "n" coefficients starting at "first" of inequality constraints
3976 * "i" and "j" of "bmap" opposite to each other?
3977 */
3980 {
3982 }
3984 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
3985 * apart from the constant term?
3986 */
3988 {
3989 isl_size total;
3995 }
3997 /* Check if we can combine a given div with lower bound l and upper
3998 * bound u with some other div and if so return that other div.
3999 * Otherwise, return a position beyond the integer divisions.
4000 * Return -1 on error.
4001 *
4002 * We first check that
4003 * - the bounds are opposites of each other (except for the constant
4004 * term)
4005 * - the bounds do not reference any other div
4006 * - no div is defined in terms of this div
4007 *
4008 * Let m be the size of the range allowed on the div by the bounds.
4009 * That is, the bounds are of the form
4010 *
4011 * e <= a <= e + m - 1
4012 *
4013 * with e some expression in the other variables.
4014 * We look for another div b such that no third div is defined in terms
4015 * of this second div b and such that in any constraint that contains
4016 * a (except for the given lower and upper bound), also contains b
4017 * with a coefficient that is m times that of b.
4018 * That is, all constraints (except for the lower and upper bound)
4019 * are of the form
4020 *
4021 * e + f (a + m b) >= 0
4022 *
4023 * Furthermore, in the constraints that only contain b, the coefficient
4024 * of b should be equal to 1 or -1.
4025 * If so, we return b so that "a + m b" can be replaced by
4026 * a single div "c = a + m b".
4027 */
4030 {
4033 isl_size v_div;
4035 isl_bool opp;
4057 }
4067 }
4080 }
4091 }
4104 }
4109 }
4113 }
4115 /* Internal data structure used during the construction and/or evaluation of
4116 * an inequality that ensures that a pair of bounds always allows
4117 * for an integer value.
4118 *
4119 * "tab" is the tableau in which the inequality is evaluated. It may
4120 * be NULL until it is actually needed.
4121 * "v" contains the inequality coefficients.
4122 * "g", "fl" and "fu" are temporary scalars used during the construction and
4123 * evaluation.
4124 */
4128 isl_int g;
4129 isl_int fl;
4130 isl_int fu;
4131 };
4133 /* Free all the memory allocated by the fields of "data".
4134 */
4136 {
4142 }
4144 /* Is the inequality stored in data->v satisfied by "bmap"?
4145 * That is, does it only attain non-negative values?
4146 * data->tab is a tableau corresponding to "bmap".
4147 */
4150 {
4161 }
4163 /* Given a lower and an upper bound on div i, do they always allow
4164 * for an integer value of the given div?
4165 * Determine this property by constructing an inequality
4166 * such that the property is guaranteed when the inequality is nonnegative.
4167 * The lower bound is inequality l, while the upper bound is inequality u.
4168 * The constructed inequality is stored in data->v.
4169 *
4170 * Let the upper bound be
4171 *
4172 * -n_u a + e_u >= 0
4173 *
4174 * and the lower bound
4175 *
4176 * n_l a + e_l >= 0
4177 *
4178 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4179 * We have
4180 *
4181 * - f_u e_l <= f_u f_l g a <= f_l e_u
4182 *
4183 * Since all variables are integer valued, this is equivalent to
4184 *
4185 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4186 *
4187 * If this interval is at least f_u f_l g, then it contains at least
4188 * one integer value for a.
4189 * That is, the test constraint is
4190 *
4191 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4192 *
4193 * or
4194 *
4195 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4196 *
4197 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4198 * then the constraint can be scaled down by a factor g',
4199 * with the constant term replaced by
4200 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4201 * Note that the result of applying Fourier-Motzkin to this pair
4202 * of constraints is
4203 *
4204 * f_l e_u + f_u e_l >= 0
4205 *
4206 * If the constant term of the scaled down version of this constraint,
4207 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4208 * term of the scaled down test constraint, then the test constraint
4209 * is known to hold and no explicit evaluation is required.
4210 * This is essentially the Omega test.
4211 *
4212 * If the test constraint consists of only a constant term, then
4213 * it is sufficient to look at the sign of this constant term.
4214 */
4217 {
4219 isl_size n_div;
4246 }
4256 }
4258 /* Remove more kinds of divs that are not strictly needed.
4259 * In particular, if all pairs of lower and upper bounds on a div
4260 * are such that they allow at least one integer value of the div,
4261 * then we can eliminate the div using Fourier-Motzkin without
4262 * introducing any spurious solutions.
4263 *
4264 * If at least one of the two constraints has a unit coefficient for the div,
4265 * then the presence of such a value is guaranteed so there is no need to check.
4266 * In particular, the value attained by the bound with unit coefficient
4267 * can serve as this intermediate value.
4268 */
4271 {
4275 isl_size n_div;
4295 isl_bool has_int;
4303 }
4324 }
4327 }
4331 }
4335 }
4338 }
4349 error:
4354 }
4356 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4357 * and the upper bound u, div1 always occurs together with div2 in the form
4358 * (div1 + m div2), where m is the constant range on the variable div1
4359 * allowed by l and u, replace the pair div1 and div2 by a single
4360 * div that is equal to div1 + m div2.
4361 *
4362 * The new div will appear in the location that contains div2.
4363 * We need to modify all constraints that contain
4364 * div2 = (div - div1) / m
4365 * The coefficient of div2 is known to be equal to 1 or -1.
4366 * (If a constraint does not contain div2, it will also not contain div1.)
4367 * If the constraint also contains div1, then we know they appear
4368 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4369 * i.e., the coefficient of div is f.
4370 *
4371 * Otherwise, we first need to introduce div1 into the constraint.
4372 * Let l be
4373 *
4374 * div1 + f >=0
4375 *
4376 * and u
4377 *
4378 * -div1 + f' >= 0
4379 *
4380 * A lower bound on div2
4381 *
4382 * div2 + t >= 0
4383 *
4384 * can be replaced by
4385 *
4386 * m div2 + div1 + m t + f >= 0
4387 *
4388 * An upper bound
4389 *
4390 * -div2 + t >= 0
4391 *
4392 * can be replaced by
4393 *
4394 * -(m div2 + div1) + m t + f' >= 0
4395 *
4396 * These constraint are those that we would obtain from eliminating
4397 * div1 using Fourier-Motzkin.
4398 *
4399 * After all constraints have been modified, we drop the lower and upper
4400 * bound and then drop div1.
4401 * Since the new div is only placed in the same location that used
4402 * to store div2, but otherwise has a different meaning, any possible
4403 * explicit representation of the original div2 is removed.
4404 */
4407 {
4409 isl_int m;
4410 isl_size v_div;
4434 else
4437 }
4441 }
4450 }
4454 }
4456 /* First check if we can coalesce any pair of divs and
4457 * then continue with dropping more redundant divs.
4458 *
4459 * We loop over all pairs of lower and upper bounds on a div
4460 * with coefficient 1 and -1, respectively, check if there
4461 * is any other div "c" with which we can coalesce the div
4462 * and if so, perform the coalescing.
4463 */
4466 {
4468 isl_size v_div;
4469 isl_size n_div;
4495 }
4496 }
4497 }
4502 }
4505 error:
4509 }
4511 /* Are the "n" coefficients starting at "first" of inequality constraints
4512 * "i" and "j" of "bmap" equal to each other?
4513 */
4516 {
4518 }
4520 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4521 * apart from the constant term and the coefficient at position "pos"?
4522 */
4525 {
4526 isl_size total;
4533 }
4535 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4536 * apart from the constant term and the coefficient at position "pos"?
4537 */
4540 {
4541 isl_size total;
4548 }
4550 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4551 * been modified, simplying it if "simplify" is set.
4552 * Free the temporary data structure "pairs" that was associated
4553 * to the old version of "bmap".
4554 */
4557 {
4562 }
4564 /* Is "div" the single unknown existentially quantified variable
4565 * in inequality constraint "ineq" of "bmap"?
4566 * "div" is known to have a non-zero coefficient in "ineq".
4567 */
4570 {
4572 isl_size n_div;
4574 isl_bool known;
4586 isl_bool known;
4595 }
4598 }
4600 /* Does integer division "div" have coefficient 1 in inequality constraint
4601 * "ineq" of "map"?
4602 */
4604 {
4612 }
4614 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4615 * then try and drop redundant divs again,
4616 * freeing the temporary data structure "pairs" that was associated
4617 * to the old version of "bmap".
4618 */
4621 {
4625 }
4627 /* Drop the integer division at position "div", along with the two
4628 * inequality constraints "ineq1" and "ineq2" in which it appears
4629 * from "bmap" and then try and drop redundant divs again,
4630 * freeing the temporary data structure "pairs" that was associated
4631 * to the old version of "bmap".
4632 */
4636 {
4643 }
4646 }
4648 /* Given two inequality constraints
4649 *
4650 * f(x) + n d + c >= 0, (ineq)
4651 *
4652 * with d the variable at position "pos", and
4653 *
4654 * f(x) + c0 >= 0, (lower)
4655 *
4656 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4657 * determined by the first constraint.
4658 * That is, store
4659 *
4660 * ceil((c0 - c)/n)
4661 *
4662 * in *l.
4663 */
4666 {
4670 }
4672 /* Given two inequality constraints
4673 *
4674 * f(x) + n d + c >= 0, (ineq)
4675 *
4676 * with d the variable at position "pos", and
4677 *
4678 * -f(x) - c0 >= 0, (upper)
4679 *
4680 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4681 * determined by the first constraint.
4682 * That is, store
4683 *
4684 * ceil((-c1 - c)/n)
4685 *
4686 * in *u.
4687 */
4690 {
4694 }
4696 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4697 * does the corresponding lower bound have a fixed value in "bmap"?
4698 *
4699 * In particular, "ineq" is of the form
4700 *
4701 * f(x) + n d + c >= 0
4702 *
4703 * with n > 0, c the constant term and
4704 * d the existentially quantified variable "div".
4705 * That is, the lower bound is
4706 *
4707 * ceil((-f(x) - c)/n)
4708 *
4709 * Look for a pair of constraints
4710 *
4711 * f(x) + c0 >= 0
4712 * -f(x) + c1 >= 0
4713 *
4714 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4715 * That is, check that
4716 *
4717 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4718 *
4719 * If so, return the index of inequality f(x) + c0 >= 0.
4720 * Otherwise, return bmap->n_ineq.
4721 * Return -1 on error.
4722 */
4724 {
4747 }
4755 }
4772 }
4774 /* Given a lower bound constraint "ineq" on the existentially quantified
4775 * variable "div", such that the corresponding lower bound has
4776 * a fixed value in "bmap", assign this fixed value to the variable and
4777 * then try and drop redundant divs again,
4778 * freeing the temporary data structure "pairs" that was associated
4779 * to the old version of "bmap".
4780 * "lower" determines the constant value for the lower bound.
4781 *
4782 * In particular, "ineq" is of the form
4783 *
4784 * f(x) + n d + c >= 0,
4785 *
4786 * while "lower" is of the form
4787 *
4788 * f(x) + c0 >= 0
4789 *
4790 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4791 * is ceil((c0 - c)/n).
4792 */
4795 {
4796 isl_int c;
4809 }
4811 /* Remove divs that are not strictly needed based on the inequality
4812 * constraints.
4813 * In particular, if a div only occurs positively (or negatively)
4814 * in constraints, then it can simply be dropped.
4815 * Also, if a div occurs in only two constraints and if moreover
4816 * those two constraints are opposite to each other, except for the constant
4817 * term and if the sum of the constant terms is such that for any value
4818 * of the other values, there is always at least one integer value of the
4819 * div, i.e., if one plus this sum is greater than or equal to
4820 * the (absolute value) of the coefficient of the div in the constraints,
4821 * then we can also simply drop the div.
4822 *
4823 * If an existentially quantified variable does not have an explicit
4824 * representation, appears in only a single lower bound that does not
4825 * involve any other such existentially quantified variables and appears
4826 * in this lower bound with coefficient 1,
4827 * then fix the variable to the value of the lower bound. That is,
4828 * turn the inequality into an equality.
4829 * If for any value of the other variables, there is any value
4830 * for the existentially quantified variable satisfying the constraints,
4831 * then this lower bound also satisfies the constraints.
4832 * It is therefore safe to pick this lower bound.
4833 *
4834 * The same reasoning holds even if the coefficient is not one.
4835 * However, fixing the variable to the value of the lower bound may
4836 * in general introduce an extra integer division, in which case
4837 * it may be better to pick another value.
4838 * If this integer division has a known constant value, then plugging
4839 * in this constant value removes the existentially quantified variable
4840 * completely. In particular, if the lower bound is of the form
4841 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4842 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4843 * then the existentially quantified variable can be assigned this
4844 * shared value.
4845 *
4846 * We skip divs that appear in equalities or in the definition of other divs.
4847 * Divs that appear in the definition of other divs usually occur in at least
4848 * 4 constraints, but the constraints may have been simplified.
4849 *
4850 * If any divs are left after these simple checks then we move on
4851 * to more complicated cases in drop_more_redundant_divs.
4852 */
4855 {
4857 isl_size off;
4899 }
4903 }
4904 }
4912 }
4915 else
4935 pairs);
4941 pairs);
4943 }
4960 else
4967 }
4970 }
4977 error:
4981 }
4983 /* Consider the coefficients at "c" as a row vector and replace
4984 * them with their product with "T". "T" is assumed to be a square matrix.
4985 */
4987 {
4988 isl_size n;
5009 }
5011 /* Plug in T for the variables in "bmap" starting at "pos".
5012 * T is a linear unimodular matrix, i.e., without constant term.
5013 */
5016 {
5044 }
5048 error:
5052 }
5054 /* Remove divs that are not strictly needed.
5055 *
5056 * First look for an equality constraint involving two or more
5057 * existentially quantified variables without an explicit
5058 * representation. Replace the combination that appears
5059 * in the equality constraint by a single existentially quantified
5060 * variable such that the equality can be used to derive
5061 * an explicit representation for the variable.
5062 * If there are no more such equality constraints, then continue
5063 * with isl_basic_map_drop_redundant_divs_ineq.
5064 *
5065 * In particular, if the equality constraint is of the form
5066 *
5067 * f(x) + \sum_i c_i a_i = 0
5068 *
5069 * with a_i existentially quantified variable without explicit
5070 * representation, then apply a transformation on the existentially
5071 * quantified variables to turn the constraint into
5072 *
5073 * f(x) + g a_1' = 0
5074 *
5075 * with g the gcd of the c_i.
5076 * In order to easily identify which existentially quantified variables
5077 * have a complete explicit representation, i.e., without being defined
5078 * in terms of other existentially quantified variables without
5079 * an explicit representation, the existentially quantified variables
5080 * are first sorted.
5081 *
5082 * The variable transformation is computed by extending the row
5083 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5084 *
5085 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5086 * [a_2'] [ a_2 ]
5087 * ... = U ....
5088 * [a_n'] [ a_n ]
5089 *
5090 * with [c_1/g ... c_n/g] representing the first row of U.
5091 * The inverse of U is then plugged into the original constraints.
5092 * The call to isl_basic_map_simplify makes sure the explicit
5093 * representation for a_1' is extracted from the equality constraint.
5094 */
5097 {
5101 isl_size n_div;
5135 }
5154 }
5156 /* Does "bmap" satisfy any equality that involves more than 2 variables
5157 * and/or has coefficients different from -1 and 1?
5158 */
5160 {
5162 isl_size total;
5191 }
5194 }
5196 /* Remove any common factor g from the constraint coefficients in "v".
5197 * The constant term is stored in the first position and is replaced
5198 * by floor(c/g). If any common factor is removed and if this results
5199 * in a tightening of the constraint, then set *tightened.
5200 */
5203 {
5223 }
5225 /* If "bmap" is an integer set that satisfies any equality involving
5226 * more than 2 variables and/or has coefficients different from -1 and 1,
5227 * then use variable compression to reduce the coefficients by removing
5228 * any (hidden) common factor.
5229 * In particular, apply the variable compression to each constraint,
5230 * factor out any common factor in the non-constant coefficients and
5231 * then apply the inverse of the compression.
5232 * At the end, we mark the basic map as having reduced constants.
5233 * If this flag is still set on the next invocation of this function,
5234 * then we skip the computation.
5235 *
5236 * Removing a common factor may result in a tightening of some of
5237 * the constraints. If this happens, then we may end up with two
5238 * opposite inequalities that can be replaced by an equality.
5239 * We therefore call isl_basic_map_detect_inequality_pairs,
5240 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5241 * and isl_basic_map_gauss if such a pair was found.
5242 *
5243 * Tightening may also result in some other constraints becoming
5244 * (rationally) redundant with respect to the tightened constraint
5245 * (in combination with other constraints). The basic map may
5246 * therefore no longer be assumed to have no redundant constraints.
5247 *
5248 * Note that this function may leave the result in an inconsistent state.
5249 * In particular, the constraints may not be gaussed.
5250 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5251 * for some of the test cases to pass successfully.
5252 * Any potential modification of the representation is therefore only
5253 * performed on a single copy of the basic map.
5254 */
5257 {
5258 isl_size total;
5259 isl_bool multi;
5297 }
5312 }
5328 }
5329 }
5332 error:
5337 }
5339 /* Shift the integer division at position "div" of "bmap"
5340 * by "shift" times the variable at position "pos".
5341 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5342 * corresponds to the constant term.
5343 *
5344 * That is, if the integer division has the form
5345 *
5346 * floor(f(x)/d)
5347 *
5348 * then replace it by
5349 *
5350 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5351 */
5354 {
5373 }
5379 }
5387 }
5390 }