| blob | 9759c0df9facd99a5046cdcffd9e4c59af829fe8 |
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 }
68 }
76 }
78 }
86 }
90 }
97 }
101 error:
105 }
109 {
112 }
114 /* Reduce the coefficient of the variable at position "pos"
115 * in integer division "div", such that it lies in the half-open
116 * interval (1/2,1/2], extracting any excess value from this integer division.
117 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
118 * corresponds to the constant term.
119 *
120 * That is, the integer division is of the form
121 *
122 * floor((... + (c * d + r) * x_pos + ...)/d)
123 *
124 * with -d < 2 * r <= d.
125 * Replace it by
126 *
127 * floor((... + r * x_pos + ...)/d) + c * x_pos
128 *
129 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
130 * Otherwise, c = floor((c * d + r)/d) + 1.
131 *
132 * This is the same normalization that is performed by isl_aff_floor.
133 */
136 {
137 isl_int shift;
152 }
154 /* Does the coefficient of the variable at position "pos"
155 * in integer division "div" need to be reduced?
156 * That is, does it lie outside the half-open interval (1/2,1/2]?
157 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
158 * 2 * c != d.
159 */
162 {
163 isl_bool r;
175 }
177 /* Reduce the coefficients (including the constant term) of
178 * integer division "div", if needed.
179 * In particular, make sure all coefficients lie in
180 * the half-open interval (1/2,1/2].
181 */
184 {
186 isl_size total;
192 isl_bool reduce;
202 }
205 }
207 /* Reduce the coefficients (including the constant term) of
208 * the known integer divisions, if needed
209 * In particular, make sure all coefficients lie in
210 * the half-open interval (1/2,1/2].
211 */
214 {
228 }
231 }
233 /* Remove any common factor in numerator and denominator of the div expression,
234 * not taking into account the constant term.
235 * That is, if the div is of the form
236 *
237 * floor((a + m f(x))/(m d))
238 *
239 * then replace it by
240 *
241 * floor((floor(a/m) + f(x))/d)
242 *
243 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
244 * and can therefore not influence the result of the floor.
245 */
248 {
268 }
270 /* Remove any common factor in numerator and denominator of a div expression,
271 * not taking into account the constant term.
272 * That is, look for any div of the form
273 *
274 * floor((a + m f(x))/(m d))
275 *
276 * and replace it by
277 *
278 * floor((floor(a/m) + f(x))/d)
279 *
280 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
281 * and can therefore not influence the result of the floor.
282 */
285 {
297 }
299 /* Assumes divs have been ordered if keep_divs is set.
300 */
304 {
305 isl_size total;
306 isl_size v_div;
324 }
335 }
344 /* We need to be careful about circular definitions,
345 * so for now we just remove the definition of div k
346 * if the equality contains any divs.
347 * If keep_divs is set, then the divs have been ordered
348 * and we can keep the definition as long as the result
349 * is still ordered.
350 */
359 }
362 }
364 /* Assumes divs have been ordered if keep_divs is set.
365 */
368 {
369 isl_size v_div;
381 }
383 /* Check if elimination of div "div" using equality "eq" would not
384 * result in a div depending on a later div.
385 */
388 {
391 isl_size v_div;
408 }
411 }
413 /* Eliminate divs based on equalities
414 */
417 {
432 isl_bool ok;
448 }
449 }
453 }
455 /* Eliminate divs based on inequalities
456 */
459 {
489 }
491 }
493 /* Does the equality constraint at position "eq" in "bmap" involve
494 * any local variables in the range [first, first + n)
495 * that are not marked as having an explicit representation?
496 */
499 {
508 isl_bool unknown;
517 }
520 }
522 /* The last local variable involved in the equality constraint
523 * at position "eq" in "bmap" is the local variable at position "div".
524 * It can therefore be used to extract an explicit representation
525 * for that variable.
526 * Do so unless the local variable already has an explicit representation or
527 * the explicit representation would involve any other local variables
528 * that in turn do not have an explicit representation.
529 * An equality constraint involving local variables without an explicit
530 * representation can be used in isl_basic_map_drop_redundant_divs
531 * to separate out an independent local variable. Introducing
532 * an explicit representation here would block this transformation,
533 * while the partial explicit representation in itself is not very useful.
534 * Set *progress if anything is changed.
535 *
536 * The equality constraint is of the form
537 *
538 * f(x) + n e >= 0
539 *
540 * with n a positive number. The explicit representation derived from
541 * this constraint is
542 *
543 * floor((-f(x))/n)
544 */
547 {
548 isl_size total;
550 isl_bool involves;
575 }
577 /* Perform fangcheng (Gaussian elimination) on the equality
578 * constraints of "bmap".
579 * That is, put them into row-echelon form, starting from the last column
580 * backward and use them to eliminate the corresponding coefficients
581 * from all constraints.
582 *
583 * If "progress" is not NULL, then it gets set if the elimination
584 * results in any changes.
585 * The elimination process may result in some equality constraints
586 * getting interchanged or removed.
587 * If "swap" or "drop" are not NULL, then they get called when
588 * two equality constraints get interchanged or
589 * when a number of final equality constraints get removed.
590 * As a special case, if the input turns out to be empty,
591 * then drop gets called with the number of removed equality
592 * constraints set to the total number of equality constraints.
593 * If "swap" or "drop" are not NULL, then the local variables (if any)
594 * are assumed to be in a valid order.
595 */
600 {
605 isl_size total;
625 }
632 }
644 }
653 }
659 }
663 {
665 }
669 {
671 progress));
672 }
676 {
682 }
684 }
686 /* Hash table of inequalities in a basic map.
687 * "index" is an array of addresses of inequalities in the basic map, some
688 * of which are NULL. The inequalities are hashed on the coefficients
689 * except the constant term.
690 * "size" is the number of elements in the array and is always a power of two
691 * "bits" is the number of bits need to represent an index into the array.
692 * "total" is the total dimension of the basic map.
693 */
698 isl_size total;
699 };
701 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
702 */
705 {
724 }
726 /* Free the memory allocated by create_constraint_index.
727 */
729 {
731 }
733 /* Return the position in ci->index that contains the address of
734 * an inequality that is equal to *ineq up to the constant term,
735 * provided this address is not identical to "ineq".
736 * If there is no such inequality, then return the position where
737 * such an inequality should be inserted.
738 */
740 {
748 }
750 /* Return the position in ci->index that contains the address of
751 * an inequality that is equal to the k'th inequality of "bmap"
752 * up to the constant term, provided it does not point to the very
753 * same inequality.
754 * If there is no such inequality, then return the position where
755 * such an inequality should be inserted.
756 */
759 {
761 }
765 {
767 }
769 /* Fill in the "ci" data structure with the inequalities of "bset".
770 */
773 {
782 }
785 }
787 /* Is the inequality ineq (obviously) redundant with respect
788 * to the constraints in "ci"?
789 *
790 * Look for an inequality in "ci" with the same coefficients and then
791 * check if the contant term of "ineq" is greater than or equal
792 * to the constant term of that inequality. If so, "ineq" is clearly
793 * redundant.
794 *
795 * Note that hash_index_ineq ignores a stored constraint if it has
796 * the same address as the passed inequality. It is ok to pass
797 * the address of a local variable here since it will never be
798 * the same as the address of a constraint in "ci".
799 */
802 {
809 }
811 /* If we can eliminate more than one div, then we need to make
812 * sure we do it from last div to first div, in order not to
813 * change the position of the other divs that still need to
814 * be removed.
815 */
818 {
825 isl_size v_div;
874 }
876 }
888 }
891 out:
895 }
898 {
900 isl_size v_div;
912 }
914 }
916 /* Normalize divs that appear in equalities.
917 *
918 * In particular, we assume that bmap contains some equalities
919 * of the form
920 *
921 * a x = m * e_i
922 *
923 * and we want to replace the set of e_i by a minimal set and
924 * such that the new e_i have a canonical representation in terms
925 * of the vector x.
926 * If any of the equalities involves more than one divs, then
927 * we currently simply bail out.
928 *
929 * Let us first additionally assume that all equalities involve
930 * a div. The equalities then express modulo constraints on the
931 * remaining variables and we can use "parameter compression"
932 * to find a minimal set of constraints. The result is a transformation
933 *
934 * x = T(x') = x_0 + G x'
935 *
936 * with G a lower-triangular matrix with all elements below the diagonal
937 * non-negative and smaller than the diagonal element on the same row.
938 * We first normalize x_0 by making the same property hold in the affine
939 * T matrix.
940 * The rows i of G with a 1 on the diagonal do not impose any modulo
941 * constraint and simply express x_i = x'_i.
942 * For each of the remaining rows i, we introduce a div and a corresponding
943 * equality. In particular
944 *
945 * g_ii e_j = x_i - g_i(x')
946 *
947 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
948 * corresponding div (if g_kk != 1).
949 *
950 * If there are any equalities not involving any div, then we
951 * first apply a variable compression on the variables x:
952 *
953 * x = C x'' x'' = C_2 x
954 *
955 * and perform the above parameter compression on A C instead of on A.
956 * The resulting compression is then of the form
957 *
958 * x'' = T(x') = x_0 + G x'
959 *
960 * and in constructing the new divs and the corresponding equalities,
961 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
962 * by the corresponding row from C_2.
963 */
966 {
968 isl_size v_div;
975 isl_int v;
1009 }
1010 }
1019 }
1025 }
1035 }
1042 }
1047 /* We have to be careful because dropping equalities may reorder them */
1058 }
1059 }
1065 else
1066 needed++;
1067 }
1072 }
1082 else
1090 else
1093 }
1097 }
1104 done:
1108 error:
1115 }
1119 {
1129 }
1131 /* Check whether it is ok to define a div based on an inequality.
1132 * To avoid the introduction of circular definitions of divs, we
1133 * do not allow such a definition if the resulting expression would refer to
1134 * any other undefined divs or if any known div is defined in
1135 * terms of the unknown div.
1136 */
1139 {
1143 /* Not defined in terms of unknown divs */
1151 }
1153 /* No other div defined in terms of this one => avoid loops */
1161 }
1164 }
1166 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1167 * be a better expression than the current one?
1168 *
1169 * If we do not have any expression yet, then any expression would be better.
1170 * Otherwise we check if the last variable involved in the inequality
1171 * (disregarding the div that it would define) is in an earlier position
1172 * than the last variable involved in the current div expression.
1173 */
1176 {
1193 }
1195 /* Given two constraints "k" and "l" that are opposite to each other,
1196 * except for the constant term, check if we can use them
1197 * to obtain an expression for one of the hitherto unknown divs or
1198 * a "better" expression for a div for which we already have an expression.
1199 * "sum" is the sum of the constant terms of the constraints.
1200 * If this sum is strictly smaller than the coefficient of one
1201 * of the divs, then this pair can be used define the div.
1202 * To avoid the introduction of circular definitions of divs, we
1203 * do not use the pair if the resulting expression would refer to
1204 * any other undefined divs or if any known div is defined in
1205 * terms of the unknown div.
1206 */
1210 {
1215 isl_bool set_div;
1230 else
1235 }
1237 }
1241 {
1245 isl_int sum;
1260 }
1268 }
1283 }
1285 /* We need to break out of the loop after these
1286 * changes since the contents of the hash
1287 * will no longer be valid.
1288 * Plus, we probably we want to regauss first.
1289 */
1297 }
1302 }
1304 /* Detect all pairs of inequalities that form an equality.
1305 *
1306 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1307 * Call it repeatedly while it is making progress.
1308 */
1311 {
1323 }
1325 /* Given a known integer division "div" that is not integral
1326 * (with denominator 1), eliminate it from the constraints in "bmap"
1327 * where it appears with a (positive or negative) unit coefficient.
1328 * If "progress" is not NULL, then it gets set if the elimination
1329 * results in any changes.
1330 *
1331 * That is, replace
1332 *
1333 * floor(e/m) + f >= 0
1334 *
1335 * by
1336 *
1337 * e + m f >= 0
1338 *
1339 * and
1340 *
1341 * -floor(e/m) + f >= 0
1342 *
1343 * by
1344 *
1345 * -e + m f + m - 1 >= 0
1346 *
1347 * The first conversion is valid because floor(e/m) >= -f is equivalent
1348 * to e/m >= -f because -f is an integral expression.
1349 * The second conversion follows from the fact that
1350 *
1351 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1352 *
1353 *
1354 * Note that one of the div constraints may have been eliminated
1355 * due to being redundant with respect to the constraint that is
1356 * being modified by this function. The modified constraint may
1357 * no longer imply this div constraint, so we add it back to make
1358 * sure we do not lose any information.
1359 */
1362 {
1390 else
1399 }
1405 }
1408 }
1410 /* Eliminate selected known divs from constraints where they appear with
1411 * a (positive or negative) unit coefficient.
1412 * In particular, only handle those for which "select" returns isl_bool_true.
1413 * If "progress" is not NULL, then it gets set if the elimination
1414 * results in any changes.
1415 *
1416 * We skip integral divs, i.e., those with denominator 1, as we would
1417 * risk eliminating the div from the div constraints. We do not need
1418 * to handle those divs here anyway since the div constraints will turn
1419 * out to form an equality and this equality can then be used to eliminate
1420 * the div from all constraints.
1421 */
1426 {
1433 isl_bool selected;
1447 }
1450 }
1452 /* eliminate_selected_unit_divs callback that selects every
1453 * integer division.
1454 */
1456 {
1458 }
1460 /* Eliminate known divs from constraints where they appear with
1461 * a (positive or negative) unit coefficient.
1462 * If "progress" is not NULL, then it gets set if the elimination
1463 * results in any changes.
1464 */
1467 {
1469 }
1471 /* eliminate_selected_unit_divs callback that selects
1472 * integer divisions that only appear with
1473 * a (positive or negative) unit coefficient
1474 * (outside their div constraints).
1475 */
1477 {
1487 isl_bool skip;
1500 }
1503 }
1505 /* Eliminate known divs from constraints where they appear with
1506 * a (positive or negative) unit coefficient,
1507 * but only if they do not appear in any other constraints
1508 * (other than the div constraints).
1509 */
1512 {
1514 }
1517 {
1522 isl_bool empty;
1538 /* requires equalities in normal form */
1544 }
1546 }
1550 {
1552 }
1557 {
1589 }
1591 /* If the only constraints a div d=floor(f/m)
1592 * appears in are its two defining constraints
1593 *
1594 * f - m d >=0
1595 * -(f - (m - 1)) + m d >= 0
1596 *
1597 * then it can safely be removed.
1598 */
1600 {
1613 isl_bool red;
1620 }
1627 }
1630 }
1632 /*
1633 * Remove divs that don't occur in any of the constraints or other divs.
1634 * These can arise when dropping constraints from a basic map or
1635 * when the divs of a basic map have been temporarily aligned
1636 * with the divs of another basic map.
1637 */
1640 {
1642 isl_size v_div;
1649 isl_bool redundant;
1659 }
1661 }
1663 /* Mark "bmap" as final, without checking for obviously redundant
1664 * integer divisions. This function should be used when "bmap"
1665 * is known not to involve any such integer divisions.
1666 */
1669 {
1674 }
1676 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1677 */
1679 {
1683 }
1687 {
1689 }
1691 /* Remove definition of any div that is defined in terms of the given variable.
1692 * The div itself is not removed. Functions such as
1693 * eliminate_divs_ineq depend on the other divs remaining in place.
1694 */
1697 {
1711 }
1713 }
1715 /* Eliminate the specified variables from the constraints using
1716 * Fourier-Motzkin. The variables themselves are not removed.
1717 */
1720 {
1723 isl_size total;
1754 }
1761 n_lower++;
1763 n_upper++;
1764 }
1788 }
1791 }
1803 }
1804 }
1808 error:
1811 }
1815 {
1818 }
1820 /* Eliminate the specified n dimensions starting at first from the
1821 * constraints, without removing the dimensions from the space.
1822 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1823 * Otherwise, they are projected out and the original space is restored.
1824 */
1828 {
1843 }
1850 }
1855 {
1857 }
1859 /* Remove all constraints from "bmap" that reference any unknown local
1860 * variables (directly or indirectly).
1861 *
1862 * Dropping all constraints on a local variable will make it redundant,
1863 * so it will get removed implicitly by
1864 * isl_basic_map_drop_constraints_involving_dims. Some other local
1865 * variables may also end up becoming redundant if they only appear
1866 * in constraints together with the unknown local variable.
1867 * Therefore, start over after calling
1868 * isl_basic_map_drop_constraints_involving_dims.
1869 */
1872 {
1873 isl_bool known;
1874 isl_size n_div;
1901 }
1904 }
1906 /* Remove all constraints from "bset" that reference any unknown local
1907 * variables (directly or indirectly).
1908 */
1911 {
1917 }
1919 /* Remove all constraints from "map" that reference any unknown local
1920 * variables (directly or indirectly).
1921 *
1922 * Since constraints may get dropped from the basic maps,
1923 * they may no longer be disjoint from each other.
1924 */
1927 {
1929 isl_bool known;
1947 }
1953 }
1955 /* Don't assume equalities are in order, because align_divs
1956 * may have changed the order of the divs.
1957 */
1960 {
1971 }
1972 }
1973 }
1977 {
1979 }
1983 {
1995 }
1997 }
1999 }
2003 {
2006 }
2010 {
2013 isl_size dim;
2021 }
2043 error:
2047 }
2049 /* For each inequality in "ineq" that is a shifted (more relaxed)
2050 * copy of an inequality in "context", mark the corresponding entry
2051 * in "row" with -1.
2052 * If an inequality only has a non-negative constant term, then
2053 * mark it as well.
2054 */
2057 {
2077 isl_bool redundant;
2083 }
2090 }
2093 error:
2096 }
2100 {
2113 isl_bool redundant;
2125 }
2128 error:
2131 }
2133 /* Remove constraints from "bmap" that are identical to constraints
2134 * in "context" or that are more relaxed (greater constant term).
2135 *
2136 * We perform the test for shifted copies on the pure constraints
2137 * in remove_shifted_constraints.
2138 */
2141 {
2150 }
2164 error:
2168 }
2170 /* Does the (linear part of a) constraint "c" involve any of the "len"
2171 * "relevant" dimensions?
2172 */
2174 {
2182 }
2185 }
2187 /* Drop constraints from "bmap" that do not involve any of
2188 * the dimensions marked "relevant".
2189 */
2192 {
2194 isl_size dim;
2210 }
2217 }
2220 }
2222 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2223 *
2224 * In particular, for any variable involved in the constraint,
2225 * find the actual group id from before and replace the group
2226 * of the corresponding variable by the minimal group of all
2227 * the variables involved in the constraint considered so far
2228 * (if this minimum is smaller) or replace the minimum by this group
2229 * (if the minimum is larger).
2230 *
2231 * At the end, all the variables in "c" will (indirectly) point
2232 * to the minimal of the groups that they referred to originally.
2233 */
2235 {
2252 }
2253 }
2255 /* Allocate an array of groups of variables, one for each variable
2256 * in "context", initialized to zero.
2257 */
2259 {
2261 isl_size dim;
2268 }
2270 /* Drop constraints from "bmap" that only involve variables that are
2271 * not related to any of the variables marked with a "-1" in "group".
2272 *
2273 * We construct groups of variables that collect variables that
2274 * (indirectly) appear in some common constraint of "bmap".
2275 * Each group is identified by the first variable in the group,
2276 * except for the special group of variables that was already identified
2277 * in the input as -1 (or are related to those variables).
2278 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2279 * otherwise the group of i is the group of group[i].
2280 *
2281 * We first initialize groups for the remaining variables.
2282 * Then we iterate over the constraints of "bmap" and update the
2283 * group of the variables in the constraint by the smallest group.
2284 * Finally, we resolve indirect references to groups by running over
2285 * the variables.
2286 *
2287 * After computing the groups, we drop constraints that do not involve
2288 * any variables in the -1 group.
2289 */
2292 {
2293 isl_size dim;
2308 }
2326 }
2328 /* Drop constraints from "context" that are irrelevant for computing
2329 * the gist of "bset".
2330 *
2331 * In particular, drop constraints in variables that are not related
2332 * to any of the variables involved in the constraints of "bset"
2333 * in the sense that there is no sequence of constraints that connects them.
2334 *
2335 * We first mark all variables that appear in "bset" as belonging
2336 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2337 */
2340 {
2342 isl_size dim;
2361 }
2367 }
2370 }
2372 /* Drop constraints from "context" that are irrelevant for computing
2373 * the gist of the inequalities "ineq".
2374 * Inequalities in "ineq" for which the corresponding element of row
2375 * is set to -1 have already been marked for removal and should be ignored.
2376 *
2377 * In particular, drop constraints in variables that are not related
2378 * to any of the variables involved in "ineq"
2379 * in the sense that there is no sequence of constraints that connects them.
2380 *
2381 * We first mark all variables that appear in "bset" as belonging
2382 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2383 */
2386 {
2388 isl_size dim;
2390 isl_size n;
2408 }
2411 }
2414 }
2416 /* Do all "n" entries of "row" contain a negative value?
2417 */
2419 {
2427 }
2429 /* Update the inequalities in "bset" based on the information in "row"
2430 * and "tab".
2431 *
2432 * In particular, the array "row" contains either -1, meaning that
2433 * the corresponding inequality of "bset" is redundant, or the index
2434 * of an inequality in "tab".
2435 *
2436 * If the row entry is -1, then drop the inequality.
2437 * Otherwise, if the constraint is marked redundant in the tableau,
2438 * then drop the inequality. Similarly, if it is marked as an equality
2439 * in the tableau, then turn the inequality into an equality and
2440 * perform Gaussian elimination.
2441 */
2444 {
2461 }
2471 }
2472 }
2478 }
2480 /* Update the inequalities in "bset" based on the information in "row"
2481 * and "tab" and free all arguments (other than "bset").
2482 */
2487 {
2496 }
2498 /* Remove all information from bset that is redundant in the context
2499 * of context.
2500 * "ineq" contains the (possibly transformed) inequalities of "bset",
2501 * in the same order.
2502 * The (explicit) equalities of "bset" are assumed to have been taken
2503 * into account by the transformation such that only the inequalities
2504 * are relevant.
2505 * "context" is assumed not to be empty.
2506 *
2507 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2508 * A value of -1 means that the inequality is obviously redundant and may
2509 * not even appear in "tab".
2510 *
2511 * We first mark the inequalities of "bset"
2512 * that are obviously redundant with respect to some inequality in "context".
2513 * Then we remove those constraints from "context" that have become
2514 * irrelevant for computing the gist of "bset".
2515 * Note that this removal of constraints cannot be replaced by
2516 * a factorization because factors in "bset" may still be connected
2517 * to each other through constraints in "context".
2518 *
2519 * If there are any inequalities left, we construct a tableau for
2520 * the context and then add the inequalities of "bset".
2521 * Before adding these inequalities, we freeze all constraints such that
2522 * they won't be considered redundant in terms of the constraints of "bset".
2523 * Then we detect all redundant constraints (among the
2524 * constraints that weren't frozen), first by checking for redundancy in the
2525 * the tableau and then by checking if replacing a constraint by its negation
2526 * would lead to an empty set. This last step is fairly expensive
2527 * and could be optimized by more reuse of the tableau.
2528 * Finally, we update bset according to the results.
2529 */
2532 {
2547 }
2583 }
2606 }
2611 }
2615 error:
2623 }
2625 /* Extract the inequalities of "bset" as an isl_mat.
2626 */
2628 {
2629 isl_size total;
2642 }
2644 /* Remove all information from "bset" that is redundant in the context
2645 * of "context", for the case where both "bset" and "context" are
2646 * full-dimensional.
2647 */
2650 {
2655 }
2657 /* Replace "bset" by an empty basic set in the same space.
2658 */
2661 {
2667 }
2669 /* Remove all information from "bset" that is redundant in the context
2670 * of "context", for the case where the combined equalities of
2671 * "bset" and "context" allow for a compression that can be obtained
2672 * by preapplication of "T".
2673 * If the compression of "context" is empty, meaning that "bset" and
2674 * "context" do not intersect, then return the empty set.
2675 *
2676 * "bset" itself is not transformed by "T". Instead, the inequalities
2677 * are extracted from "bset" and those are transformed by "T".
2678 * uset_gist_full then determines which of the transformed inequalities
2679 * are redundant with respect to the transformed "context" and removes
2680 * the corresponding inequalities from "bset".
2681 *
2682 * After preapplying "T" to the inequalities, any common factor is
2683 * removed from the coefficients. If this results in a tightening
2684 * of the constant term, then the same tightening is applied to
2685 * the corresponding untransformed inequality in "bset".
2686 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2687 *
2688 * g f'(x) + r >= 0
2689 *
2690 * with 0 <= r < g, then it is equivalent to
2691 *
2692 * f'(x) >= 0
2693 *
2694 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2695 * subspace compressed by T since the latter would be transformed to
2696 *
2697 * g f'(x) >= 0
2698 */
2702 {
2707 isl_int rem;
2719 }
2744 }
2748 error:
2753 }
2755 /* Project "bset" onto the variables that are involved in "template".
2756 */
2759 {
2761 isl_size n;
2768 isl_bool involved;
2777 }
2780 }
2782 /* Remove all information from bset that is redundant in the context
2783 * of context. In particular, equalities that are linear combinations
2784 * of those in context are removed. Then the inequalities that are
2785 * redundant in the context of the equalities and inequalities of
2786 * context are removed.
2787 *
2788 * First of all, we drop those constraints from "context"
2789 * that are irrelevant for computing the gist of "bset".
2790 * Alternatively, we could factorize the intersection of "context" and "bset".
2791 *
2792 * We first compute the intersection of the integer affine hulls
2793 * of "bset" and "context",
2794 * compute the gist inside this intersection and then reduce
2795 * the constraints with respect to the equalities of the context
2796 * that only involve variables already involved in the input.
2797 * If the intersection of the affine hulls turns out to be empty,
2798 * then return the empty set.
2799 *
2800 * If two constraints are mutually redundant, then uset_gist_full
2801 * will remove the second of those constraints. We therefore first
2802 * sort the constraints so that constraints not involving existentially
2803 * quantified variables are given precedence over those that do.
2804 * We have to perform this sorting before the variable compression,
2805 * because that may effect the order of the variables.
2806 */
2809 {
2814 isl_size total;
2835 }
2840 }
2849 }
2860 }
2863 error:
2867 }
2869 /* Return the number of equality constraints in "bmap" that involve
2870 * local variables. This function assumes that Gaussian elimination
2871 * has been applied to the equality constraints.
2872 */
2874 {
2896 }
2898 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2899 * The constraints are assumed not to involve any local variables.
2900 */
2903 {
2905 isl_size total;
2923 }
2928 error:
2933 }
2935 /* Construct and return a variable compression based on the equality
2936 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2937 * "n1" is the number of (initial) equality constraints in "bmap1"
2938 * that do involve local variables.
2939 * "n2" is the number of (initial) equality constraints in "bmap2"
2940 * that do involve local variables.
2941 * "total" is the total number of other variables.
2942 * This function assumes that Gaussian elimination
2943 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2944 * such that the equality constraints not involving local variables
2945 * are those that start at "n1" or "n2".
2946 *
2947 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2948 * then simply compute the compression based on the equality constraints
2949 * in the other basic map.
2950 * Otherwise, combine the equality constraints from both into a new
2951 * basic map such that Gaussian elimination can be applied to this combination
2952 * and then construct a variable compression from the resulting
2953 * equality constraints.
2954 */
2958 {
2968 }
2973 }
2988 }
2990 /* Extract the stride constraints from "bmap", compressed
2991 * with respect to both the stride constraints in "context" and
2992 * the remaining equality constraints in both "bmap" and "context".
2993 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2994 * "context_n_eq" is the number of (initial) stride constraints in "context".
2995 *
2996 * Let x be all variables in "bmap" (and "context") other than the local
2997 * variables. First compute a variable compression
2998 *
2999 * x = V x'
3000 *
3001 * based on the non-stride equality constraints in "bmap" and "context".
3002 * Consider the stride constraints of "context",
3003 *
3004 * A(x) + B(y) = 0
3005 *
3006 * with y the local variables and plug in the variable compression,
3007 * resulting in
3008 *
3009 * A(V x') + B(y) = 0
3010 *
3011 * Use these constraints to compute a parameter compression on x'
3012 *
3013 * x' = T x''
3014 *
3015 * Now consider the stride constraints of "bmap"
3016 *
3017 * C(x) + D(y) = 0
3018 *
3019 * and plug in x = V*T x''.
3020 * That is, return A = [C*V*T D].
3021 */
3025 {
3051 else
3059 }
3061 /* Remove the prime factors from *g that have an exponent that
3062 * is strictly smaller than the exponent in "c".
3063 * All exponents in *g are known to be smaller than or equal
3064 * to those in "c".
3065 *
3066 * That is, if *g is equal to
3067 *
3068 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
3069 *
3070 * and "c" is equal to
3071 *
3072 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
3073 *
3074 * then update *g to
3075 *
3076 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
3077 * p_n^{e_n * (e_n = f_n)}
3078 *
3079 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
3080 * neither does the gcd of *g and c / *g.
3081 * If e_i < f_i, then the gcd of *g and c / *g has a positive
3082 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
3083 * Dividing *g by this gcd therefore strictly reduces the exponent
3084 * of the prime factors that need to be removed, while leaving the
3085 * other prime factors untouched.
3086 * Repeating this process until gcd(*g, c / *g) = 1 therefore
3087 * removes all undesired factors, without removing any others.
3088 */
3090 {
3091 isl_int t;
3100 }
3102 }
3104 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
3105 * of the same stride constraints in a compressed space that exploits
3106 * all equalities in the context and the other equalities in "bmap".
3107 *
3108 * If the stride constraints of "bmap" are of the form
3109 *
3110 * C(x) + D(y) = 0
3111 *
3112 * then A is of the form
3113 *
3114 * B(x') + D(y) = 0
3115 *
3116 * If any of these constraints involves only a single local variable y,
3117 * then the constraint appears as
3118 *
3119 * f(x) + m y_i = 0
3120 *
3121 * in "bmap" and as
3122 *
3123 * h(x') + m y_i = 0
3124 *
3125 * in "A".
3126 *
3127 * Let g be the gcd of m and the coefficients of h.
3128 * Then, in particular, g is a divisor of the coefficients of h and
3129 *
3130 * f(x) = h(x')
3131 *
3132 * is known to be a multiple of g.
3133 * If some prime factor in m appears with the same exponent in g,
3134 * then it can be removed from m because f(x) is already known
3135 * to be a multiple of g and therefore in particular of this power
3136 * of the prime factors.
3137 * Prime factors that appear with a smaller exponent in g cannot
3138 * be removed from m.
3139 * Let g' be the divisor of g containing all prime factors that
3140 * appear with the same exponent in m and g, then
3141 *
3142 * f(x) + m y_i = 0
3143 *
3144 * can be replaced by
3145 *
3146 * f(x) + m/g' y_i' = 0
3147 *
3148 * Note that (if g' != 1) this changes the explicit representation
3149 * of y_i to that of y_i', so the integer division at position i
3150 * is marked unknown and later recomputed by a call to
3151 * isl_basic_map_gauss.
3152 */
3155 {
3159 isl_int gcd;
3192 }
3199 error:
3203 }
3205 /* Simplify the stride constraints in "bmap" based on
3206 * the remaining equality constraints in "bmap" and all equality
3207 * constraints in "context".
3208 * Only do this if both "bmap" and "context" have stride constraints.
3209 *
3210 * First extract a copy of the stride constraints in "bmap" in a compressed
3211 * space exploiting all the other equality constraints and then
3212 * use this compressed copy to simplify the original stride constraints.
3213 */
3216 {
3238 }
3240 /* Return a basic map that has the same intersection with "context" as "bmap"
3241 * and that is as "simple" as possible.
3242 *
3243 * The core computation is performed on the pure constraints.
3244 * When we add back the meaning of the integer divisions, we need
3245 * to (re)introduce the div constraints. If we happen to have
3246 * discovered that some of these integer divisions are equal to
3247 * some affine combination of other variables, then these div
3248 * constraints may end up getting simplified in terms of the equalities,
3249 * resulting in extra inequalities on the other variables that
3250 * may have been removed already or that may not even have been
3251 * part of the input. We try and remove those constraints of
3252 * this form that are most obviously redundant with respect to
3253 * the context. We also remove those div constraints that are
3254 * redundant with respect to the other constraints in the result.
3255 *
3256 * The stride constraints among the equality constraints in "bmap" are
3257 * also simplified with respecting to the other equality constraints
3258 * in "bmap" and with respect to all equality constraints in "context".
3259 */
3262 {
3274 }
3280 }
3284 }
3308 }
3325 error:
3329 }
3331 /*
3332 * Assumes context has no implicit divs.
3333 */
3336 {
3347 }
3366 }
3367 }
3371 error:
3375 }
3377 /* Drop all inequalities from "bmap" that also appear in "context".
3378 * "context" is assumed to have only known local variables and
3379 * the initial local variables of "bmap" are assumed to be the same
3380 * as those of "context".
3381 * The constraints of both "bmap" and "context" are assumed
3382 * to have been sorted using isl_basic_map_sort_constraints.
3383 *
3384 * Run through the inequality constraints of "bmap" and "context"
3385 * in sorted order.
3386 * If a constraint of "bmap" involves variables not in "context",
3387 * then it cannot appear in "context".
3388 * If a matching constraint is found, it is removed from "bmap".
3389 */
3392 {
3413 }
3419 }
3423 }
3428 }
3431 }
3434 }
3436 /* Drop all equalities from "bmap" that also appear in "context".
3437 * "context" is assumed to have only known local variables and
3438 * the initial local variables of "bmap" are assumed to be the same
3439 * as those of "context".
3440 *
3441 * Run through the equality constraints of "bmap" and "context"
3442 * in sorted order.
3443 * If a constraint of "bmap" involves variables not in "context",
3444 * then it cannot appear in "context".
3445 * If a matching constraint is found, it is removed from "bmap".
3446 */
3449 {
3475 }
3479 }
3484 }
3487 }
3490 }
3492 /* Remove the constraints in "context" from "bmap".
3493 * "context" is assumed to have explicit representations
3494 * for all local variables.
3495 *
3496 * First align the divs of "bmap" to those of "context" and
3497 * sort the constraints. Then drop all constraints from "bmap"
3498 * that appear in "context".
3499 */
3502 {
3517 }
3537 error:
3541 }
3543 /* Replace "map" by the disjunct at position "pos" and free "context".
3544 */
3547 {
3554 }
3556 /* Remove the constraints in "context" from "map".
3557 * If any of the disjuncts in the result turns out to be the universe,
3558 * then return this universe.
3559 * "context" is assumed to have explicit representations
3560 * for all local variables.
3561 */
3564 {
3574 }
3593 }
3600 error:
3604 }
3606 /* Remove the constraints in "context" from "set".
3607 * If any of the disjuncts in the result turns out to be the universe,
3608 * then return this universe.
3609 * "context" is assumed to have explicit representations
3610 * for all local variables.
3611 */
3614 {
3617 }
3619 /* Remove the constraints in "context" from "map".
3620 * If any of the disjuncts in the result turns out to be the universe,
3621 * then return this universe.
3622 * "context" is assumed to consist of a single disjunct and
3623 * to have explicit representations for all local variables.
3624 */
3627 {
3632 }
3634 /* Replace "map" by a universe map in the same space and free "drop".
3635 */
3638 {
3645 }
3647 /* Return a map that has the same intersection with "context" as "map"
3648 * and that is as "simple" as possible.
3649 *
3650 * If "map" is already the universe, then we cannot make it any simpler.
3651 * Similarly, if "context" is the universe, then we cannot exploit it
3652 * to simplify "map"
3653 * If "map" and "context" are identical to each other, then we can
3654 * return the corresponding universe.
3655 *
3656 * If either "map" or "context" consists of multiple disjuncts,
3657 * then check if "context" happens to be a subset of "map",
3658 * in which case all constraints can be removed.
3659 * In case of multiple disjuncts, the standard procedure
3660 * may not be able to detect that all constraints can be removed.
3661 *
3662 * If none of these cases apply, we have to work a bit harder.
3663 * During this computation, we make use of a single disjunct context,
3664 * so if the original context consists of more than one disjunct
3665 * then we need to approximate the context by a single disjunct set.
3666 * Simply taking the simple hull may drop constraints that are
3667 * only implicitly available in each disjunct. We therefore also
3668 * look for constraints among those defining "map" that are valid
3669 * for the context. These can then be used to simplify away
3670 * the corresponding constraints in "map".
3671 */
3674 {
3678 isl_bool subset;
3689 }
3708 }
3724 list);
3725 }
3727 error:
3731 }
3735 {
3738 }
3742 {
3745 }
3749 {
3754 }
3758 {
3760 }
3762 /* Compute the gist of "bmap" with respect to the constraints "context"
3763 * on the domain.
3764 */
3767 {
3773 }
3777 {
3781 }
3785 {
3789 }
3793 {
3797 }
3801 {
3803 }
3805 /* Quick check to see if two basic maps are disjoint.
3806 * In particular, we reduce the equalities and inequalities of
3807 * one basic map in the context of the equalities of the other
3808 * basic map and check if we get a contradiction.
3809 */
3812 {
3815 isl_size total;
3844 }
3852 }
3861 }
3865 disjoint:
3869 error:
3873 }
3877 {
3880 }
3882 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3883 */
3887 {
3898 }
3899 }
3902 }
3904 /* Are "map1" and "map2" obviously disjoint, based on information
3905 * that can be derived without looking at the individual basic maps?
3906 *
3907 * In particular, if one of them is empty or if they live in different spaces
3908 * (ignoring parameters), then they are clearly disjoint.
3909 */
3912 {
3913 isl_bool disjoint;
3914 isl_bool match;
3936 }
3938 /* Are "map1" and "map2" obviously disjoint?
3939 *
3940 * If one of them is empty or if they live in different spaces (ignoring
3941 * parameters), then they are clearly disjoint.
3942 * This is checked by isl_map_plain_is_disjoint_global.
3943 *
3944 * If they have different parameters, then we skip any further tests.
3945 *
3946 * If they are obviously equal, but not obviously empty, then we will
3947 * not be able to detect if they are disjoint.
3948 *
3949 * Otherwise we check if each basic map in "map1" is obviously disjoint
3950 * from each basic map in "map2".
3951 */
3954 {
3955 isl_bool disjoint;
3956 isl_bool intersect;
3957 isl_bool match;
3972 }
3974 /* Are "map1" and "map2" disjoint?
3975 * The parameters are assumed to have been aligned.
3976 *
3977 * In particular, check whether all pairs of basic maps are disjoint.
3978 */
3981 {
3983 }
3985 /* Are "map1" and "map2" disjoint?
3986 *
3987 * They are disjoint if they are "obviously disjoint" or if one of them
3988 * is empty. Otherwise, they are not disjoint if one of them is universal.
3989 * If the two inputs are (obviously) equal and not empty, then they are
3990 * not disjoint.
3991 * If none of these cases apply, then check if all pairs of basic maps
3992 * are disjoint after aligning the parameters.
3993 */
3995 {
3996 isl_bool disjoint;
3997 isl_bool intersect;
4025 }
4027 /* Are "bmap1" and "bmap2" disjoint?
4028 *
4029 * They are disjoint if they are "obviously disjoint" or if one of them
4030 * is empty. Otherwise, they are not disjoint if one of them is universal.
4031 * If none of these cases apply, we compute the intersection and see if
4032 * the result is empty.
4033 */
4036 {
4037 isl_bool disjoint;
4038 isl_bool intersect;
4067 }
4069 /* Are "bset1" and "bset2" disjoint?
4070 */
4073 {
4075 }
4079 {
4081 }
4083 /* Are "set1" and "set2" disjoint?
4084 */
4086 {
4088 }
4090 /* Is "v" equal to 0, 1 or -1?
4091 */
4093 {
4095 }
4097 /* Are the "n" coefficients starting at "first" of inequality constraints
4098 * "i" and "j" of "bmap" opposite to each other?
4099 */
4102 {
4104 }
4106 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4107 * apart from the constant term?
4108 */
4110 {
4111 isl_size total;
4117 }
4119 /* Check if we can combine a given div with lower bound l and upper
4120 * bound u with some other div and if so return that other div.
4121 * Otherwise, return a position beyond the integer divisions.
4122 * Return -1 on error.
4123 *
4124 * We first check that
4125 * - the bounds are opposites of each other (except for the constant
4126 * term)
4127 * - the bounds do not reference any other div
4128 * - no div is defined in terms of this div
4129 *
4130 * Let m be the size of the range allowed on the div by the bounds.
4131 * That is, the bounds are of the form
4132 *
4133 * e <= a <= e + m - 1
4134 *
4135 * with e some expression in the other variables.
4136 * We look for another div b such that no third div is defined in terms
4137 * of this second div b and such that in any constraint that contains
4138 * a (except for the given lower and upper bound), also contains b
4139 * with a coefficient that is m times that of b.
4140 * That is, all constraints (except for the lower and upper bound)
4141 * are of the form
4142 *
4143 * e + f (a + m b) >= 0
4144 *
4145 * Furthermore, in the constraints that only contain b, the coefficient
4146 * of b should be equal to 1 or -1.
4147 * If so, we return b so that "a + m b" can be replaced by
4148 * a single div "c = a + m b".
4149 */
4152 {
4155 isl_size v_div;
4157 isl_bool opp;
4179 }
4189 }
4202 }
4213 }
4226 }
4231 }
4235 }
4237 /* Internal data structure used during the construction and/or evaluation of
4238 * an inequality that ensures that a pair of bounds always allows
4239 * for an integer value.
4240 *
4241 * "tab" is the tableau in which the inequality is evaluated. It may
4242 * be NULL until it is actually needed.
4243 * "v" contains the inequality coefficients.
4244 * "g", "fl" and "fu" are temporary scalars used during the construction and
4245 * evaluation.
4246 */
4250 isl_int g;
4251 isl_int fl;
4252 isl_int fu;
4253 };
4255 /* Free all the memory allocated by the fields of "data".
4256 */
4258 {
4264 }
4266 /* Is the inequality stored in data->v satisfied by "bmap"?
4267 * That is, does it only attain non-negative values?
4268 * data->tab is a tableau corresponding to "bmap".
4269 */
4272 {
4283 }
4285 /* Given a lower and an upper bound on div i, do they always allow
4286 * for an integer value of the given div?
4287 * Determine this property by constructing an inequality
4288 * such that the property is guaranteed when the inequality is nonnegative.
4289 * The lower bound is inequality l, while the upper bound is inequality u.
4290 * The constructed inequality is stored in data->v.
4291 *
4292 * Let the upper bound be
4293 *
4294 * -n_u a + e_u >= 0
4295 *
4296 * and the lower bound
4297 *
4298 * n_l a + e_l >= 0
4299 *
4300 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4301 * We have
4302 *
4303 * - f_u e_l <= f_u f_l g a <= f_l e_u
4304 *
4305 * Since all variables are integer valued, this is equivalent to
4306 *
4307 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4308 *
4309 * If this interval is at least f_u f_l g, then it contains at least
4310 * one integer value for a.
4311 * That is, the test constraint is
4312 *
4313 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4314 *
4315 * or
4316 *
4317 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4318 *
4319 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4320 * then the constraint can be scaled down by a factor g',
4321 * with the constant term replaced by
4322 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4323 * Note that the result of applying Fourier-Motzkin to this pair
4324 * of constraints is
4325 *
4326 * f_l e_u + f_u e_l >= 0
4327 *
4328 * If the constant term of the scaled down version of this constraint,
4329 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4330 * term of the scaled down test constraint, then the test constraint
4331 * is known to hold and no explicit evaluation is required.
4332 * This is essentially the Omega test.
4333 *
4334 * If the test constraint consists of only a constant term, then
4335 * it is sufficient to look at the sign of this constant term.
4336 */
4339 {
4341 isl_size n_div;
4368 }
4378 }
4380 /* Remove more kinds of divs that are not strictly needed.
4381 * In particular, if all pairs of lower and upper bounds on a div
4382 * are such that they allow at least one integer value of the div,
4383 * then we can eliminate the div using Fourier-Motzkin without
4384 * introducing any spurious solutions.
4385 *
4386 * If at least one of the two constraints has a unit coefficient for the div,
4387 * then the presence of such a value is guaranteed so there is no need to check.
4388 * In particular, the value attained by the bound with unit coefficient
4389 * can serve as this intermediate value.
4390 */
4393 {
4397 isl_size n_div;
4417 isl_bool has_int;
4425 }
4446 }
4449 }
4453 }
4457 }
4460 }
4471 error:
4476 }
4478 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4479 * and the upper bound u, div1 always occurs together with div2 in the form
4480 * (div1 + m div2), where m is the constant range on the variable div1
4481 * allowed by l and u, replace the pair div1 and div2 by a single
4482 * div that is equal to div1 + m div2.
4483 *
4484 * The new div will appear in the location that contains div2.
4485 * We need to modify all constraints that contain
4486 * div2 = (div - div1) / m
4487 * The coefficient of div2 is known to be equal to 1 or -1.
4488 * (If a constraint does not contain div2, it will also not contain div1.)
4489 * If the constraint also contains div1, then we know they appear
4490 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4491 * i.e., the coefficient of div is f.
4492 *
4493 * Otherwise, we first need to introduce div1 into the constraint.
4494 * Let l be
4495 *
4496 * div1 + f >=0
4497 *
4498 * and u
4499 *
4500 * -div1 + f' >= 0
4501 *
4502 * A lower bound on div2
4503 *
4504 * div2 + t >= 0
4505 *
4506 * can be replaced by
4507 *
4508 * m div2 + div1 + m t + f >= 0
4509 *
4510 * An upper bound
4511 *
4512 * -div2 + t >= 0
4513 *
4514 * can be replaced by
4515 *
4516 * -(m div2 + div1) + m t + f' >= 0
4517 *
4518 * These constraint are those that we would obtain from eliminating
4519 * div1 using Fourier-Motzkin.
4520 *
4521 * After all constraints have been modified, we drop the lower and upper
4522 * bound and then drop div1.
4523 * Since the new div is only placed in the same location that used
4524 * to store div2, but otherwise has a different meaning, any possible
4525 * explicit representation of the original div2 is removed.
4526 */
4529 {
4531 isl_int m;
4532 isl_size v_div;
4556 else
4559 }
4563 }
4572 }
4576 }
4578 /* First check if we can coalesce any pair of divs and
4579 * then continue with dropping more redundant divs.
4580 *
4581 * We loop over all pairs of lower and upper bounds on a div
4582 * with coefficient 1 and -1, respectively, check if there
4583 * is any other div "c" with which we can coalesce the div
4584 * and if so, perform the coalescing.
4585 */
4588 {
4590 isl_size v_div;
4591 isl_size n_div;
4617 }
4618 }
4619 }
4624 }
4627 error:
4631 }
4633 /* Are the "n" coefficients starting at "first" of inequality constraints
4634 * "i" and "j" of "bmap" equal to each other?
4635 */
4638 {
4640 }
4642 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4643 * apart from the constant term and the coefficient at position "pos"?
4644 */
4647 {
4648 isl_size total;
4655 }
4657 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4658 * apart from the constant term and the coefficient at position "pos"?
4659 */
4662 {
4663 isl_size total;
4670 }
4672 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4673 * been modified, simplying it if "simplify" is set.
4674 * Free the temporary data structure "pairs" that was associated
4675 * to the old version of "bmap".
4676 */
4679 {
4684 }
4686 /* Is "div" the single unknown existentially quantified variable
4687 * in inequality constraint "ineq" of "bmap"?
4688 * "div" is known to have a non-zero coefficient in "ineq".
4689 */
4692 {
4694 isl_size n_div;
4696 isl_bool known;
4708 isl_bool known;
4717 }
4720 }
4722 /* Does integer division "div" have coefficient 1 in inequality constraint
4723 * "ineq" of "map"?
4724 */
4726 {
4734 }
4736 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4737 * then try and drop redundant divs again,
4738 * freeing the temporary data structure "pairs" that was associated
4739 * to the old version of "bmap".
4740 */
4743 {
4747 }
4749 /* Drop the integer division at position "div", along with the two
4750 * inequality constraints "ineq1" and "ineq2" in which it appears
4751 * from "bmap" and then try and drop redundant divs again,
4752 * freeing the temporary data structure "pairs" that was associated
4753 * to the old version of "bmap".
4754 */
4758 {
4765 }
4768 }
4770 /* Given two inequality constraints
4771 *
4772 * f(x) + n d + c >= 0, (ineq)
4773 *
4774 * with d the variable at position "pos", and
4775 *
4776 * f(x) + c0 >= 0, (lower)
4777 *
4778 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4779 * determined by the first constraint.
4780 * That is, store
4781 *
4782 * ceil((c0 - c)/n)
4783 *
4784 * in *l.
4785 */
4788 {
4792 }
4794 /* Given two inequality constraints
4795 *
4796 * f(x) + n d + c >= 0, (ineq)
4797 *
4798 * with d the variable at position "pos", and
4799 *
4800 * -f(x) - c0 >= 0, (upper)
4801 *
4802 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4803 * determined by the first constraint.
4804 * That is, store
4805 *
4806 * ceil((-c1 - c)/n)
4807 *
4808 * in *u.
4809 */
4812 {
4816 }
4818 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4819 * does the corresponding lower bound have a fixed value in "bmap"?
4820 *
4821 * In particular, "ineq" is of the form
4822 *
4823 * f(x) + n d + c >= 0
4824 *
4825 * with n > 0, c the constant term and
4826 * d the existentially quantified variable "div".
4827 * That is, the lower bound is
4828 *
4829 * ceil((-f(x) - c)/n)
4830 *
4831 * Look for a pair of constraints
4832 *
4833 * f(x) + c0 >= 0
4834 * -f(x) + c1 >= 0
4835 *
4836 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4837 * That is, check that
4838 *
4839 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4840 *
4841 * If so, return the index of inequality f(x) + c0 >= 0.
4842 * Otherwise, return bmap->n_ineq.
4843 * Return -1 on error.
4844 */
4846 {
4869 }
4877 }
4894 }
4896 /* Given a lower bound constraint "ineq" on the existentially quantified
4897 * variable "div", such that the corresponding lower bound has
4898 * a fixed value in "bmap", assign this fixed value to the variable and
4899 * then try and drop redundant divs again,
4900 * freeing the temporary data structure "pairs" that was associated
4901 * to the old version of "bmap".
4902 * "lower" determines the constant value for the lower bound.
4903 *
4904 * In particular, "ineq" is of the form
4905 *
4906 * f(x) + n d + c >= 0,
4907 *
4908 * while "lower" is of the form
4909 *
4910 * f(x) + c0 >= 0
4911 *
4912 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4913 * is ceil((c0 - c)/n).
4914 */
4917 {
4918 isl_int c;
4931 }
4933 /* Do any of the integer divisions of "bmap" involve integer division "div"?
4934 *
4935 * The integer division "div" could only ever appear in any later
4936 * integer division (with an explicit representation).
4937 */
4939 {
4949 isl_bool unknown;
4958 }
4961 }
4963 /* Remove divs that are not strictly needed based on the inequality
4964 * constraints.
4965 * In particular, if a div only occurs positively (or negatively)
4966 * in constraints, then it can simply be dropped.
4967 * Also, if a div occurs in only two constraints and if moreover
4968 * those two constraints are opposite to each other, except for the constant
4969 * term and if the sum of the constant terms is such that for any value
4970 * of the other values, there is always at least one integer value of the
4971 * div, i.e., if one plus this sum is greater than or equal to
4972 * the (absolute value) of the coefficient of the div in the constraints,
4973 * then we can also simply drop the div.
4974 *
4975 * If an existentially quantified variable does not have an explicit
4976 * representation, appears in only a single lower bound that does not
4977 * involve any other such existentially quantified variables and appears
4978 * in this lower bound with coefficient 1,
4979 * then fix the variable to the value of the lower bound. That is,
4980 * turn the inequality into an equality.
4981 * If for any value of the other variables, there is any value
4982 * for the existentially quantified variable satisfying the constraints,
4983 * then this lower bound also satisfies the constraints.
4984 * It is therefore safe to pick this lower bound.
4985 *
4986 * The same reasoning holds even if the coefficient is not one.
4987 * However, fixing the variable to the value of the lower bound may
4988 * in general introduce an extra integer division, in which case
4989 * it may be better to pick another value.
4990 * If this integer division has a known constant value, then plugging
4991 * in this constant value removes the existentially quantified variable
4992 * completely. In particular, if the lower bound is of the form
4993 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4994 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4995 * then the existentially quantified variable can be assigned this
4996 * shared value.
4997 *
4998 * We skip divs that appear in equalities or in the definition of other divs.
4999 * Divs that appear in the definition of other divs usually occur in at least
5000 * 4 constraints, but the constraints may have been simplified.
5001 *
5002 * If any divs are left after these simple checks then we move on
5003 * to more complicated cases in drop_more_redundant_divs.
5004 */
5007 {
5009 isl_size off;
5012 isl_size n_ineq;
5053 }
5057 }
5058 }
5066 }
5069 else
5089 pairs);
5095 pairs);
5097 }
5114 else
5121 }
5124 }
5131 error:
5135 }
5137 /* Consider the coefficients at "c" as a row vector and replace
5138 * them with their product with "T". "T" is assumed to be a square matrix.
5139 */
5141 {
5142 isl_size n;
5163 }
5165 /* Plug in T for the variables in "bmap" starting at "pos".
5166 * T is a linear unimodular matrix, i.e., without constant term.
5167 */
5170 {
5198 }
5202 error:
5206 }
5208 /* Remove divs that are not strictly needed.
5209 *
5210 * First look for an equality constraint involving two or more
5211 * existentially quantified variables without an explicit
5212 * representation. Replace the combination that appears
5213 * in the equality constraint by a single existentially quantified
5214 * variable such that the equality can be used to derive
5215 * an explicit representation for the variable.
5216 * If there are no more such equality constraints, then continue
5217 * with isl_basic_map_drop_redundant_divs_ineq.
5218 *
5219 * In particular, if the equality constraint is of the form
5220 *
5221 * f(x) + \sum_i c_i a_i = 0
5222 *
5223 * with a_i existentially quantified variable without explicit
5224 * representation, then apply a transformation on the existentially
5225 * quantified variables to turn the constraint into
5226 *
5227 * f(x) + g a_1' = 0
5228 *
5229 * with g the gcd of the c_i.
5230 * In order to easily identify which existentially quantified variables
5231 * have a complete explicit representation, i.e., without being defined
5232 * in terms of other existentially quantified variables without
5233 * an explicit representation, the existentially quantified variables
5234 * are first sorted.
5235 *
5236 * The variable transformation is computed by extending the row
5237 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5238 *
5239 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5240 * [a_2'] [ a_2 ]
5241 * ... = U ....
5242 * [a_n'] [ a_n ]
5243 *
5244 * with [c_1/g ... c_n/g] representing the first row of U.
5245 * The inverse of U is then plugged into the original constraints.
5246 * The call to isl_basic_map_simplify makes sure the explicit
5247 * representation for a_1' is extracted from the equality constraint.
5248 */
5251 {
5255 isl_size n_div;
5289 }
5308 }
5310 /* Does "bmap" satisfy any equality that involves more than 2 variables
5311 * and/or has coefficients different from -1 and 1?
5312 */
5314 {
5316 isl_size total;
5345 }
5348 }
5350 /* Remove any common factor g from the constraint coefficients in "v".
5351 * The constant term is stored in the first position and is replaced
5352 * by floor(c/g). If any common factor is removed and if this results
5353 * in a tightening of the constraint, then set *tightened.
5354 */
5357 {
5377 }
5379 /* If "bmap" is an integer set that satisfies any equality involving
5380 * more than 2 variables and/or has coefficients different from -1 and 1,
5381 * then use variable compression to reduce the coefficients by removing
5382 * any (hidden) common factor.
5383 * In particular, apply the variable compression to each constraint,
5384 * factor out any common factor in the non-constant coefficients and
5385 * then apply the inverse of the compression.
5386 * At the end, we mark the basic map as having reduced constants.
5387 * If this flag is still set on the next invocation of this function,
5388 * then we skip the computation.
5389 *
5390 * Removing a common factor may result in a tightening of some of
5391 * the constraints. If this happens, then we may end up with two
5392 * opposite inequalities that can be replaced by an equality.
5393 * We therefore call isl_basic_map_detect_inequality_pairs,
5394 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5395 * and isl_basic_map_gauss if such a pair was found.
5396 *
5397 * Tightening may also result in some other constraints becoming
5398 * (rationally) redundant with respect to the tightened constraint
5399 * (in combination with other constraints). The basic map may
5400 * therefore no longer be assumed to have no redundant constraints.
5401 *
5402 * Note that this function may leave the result in an inconsistent state.
5403 * In particular, the constraints may not be gaussed.
5404 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5405 * for some of the test cases to pass successfully.
5406 * Any potential modification of the representation is therefore only
5407 * performed on a single copy of the basic map.
5408 */
5411 {
5412 isl_size total;
5413 isl_bool multi;
5451 }
5466 }
5482 }
5483 }
5486 error:
5491 }
5493 /* Shift the integer division at position "div" of "bmap"
5494 * by "shift" times the variable at position "pos".
5495 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5496 * corresponds to the constant term.
5497 *
5498 * That is, if the integer division has the form
5499 *
5500 * floor(f(x)/d)
5501 *
5502 * then replace it by
5503 *
5504 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5505 */
5508 {
5527 }
5533 }
5541 }
5544 }