| blob | e73e81abe4c954f62a891985dc34666225af39c0 |
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 * result 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 /* Eliminate knowns divs from constraints where they appear with
1326 * a (positive or negative) unit coefficient.
1327 *
1328 * That is, replace
1329 *
1330 * floor(e/m) + f >= 0
1331 *
1332 * by
1333 *
1334 * e + m f >= 0
1335 *
1336 * and
1337 *
1338 * -floor(e/m) + f >= 0
1339 *
1340 * by
1341 *
1342 * -e + m f + m - 1 >= 0
1343 *
1344 * The first conversion is valid because floor(e/m) >= -f is equivalent
1345 * to e/m >= -f because -f is an integral expression.
1346 * The second conversion follows from the fact that
1347 *
1348 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1349 *
1350 *
1351 * Note that one of the div constraints may have been eliminated
1352 * due to being redundant with respect to the constraint that is
1353 * being modified by this function. The modified constraint may
1354 * no longer imply this div constraint, so we add it back to make
1355 * sure we do not lose any information.
1356 *
1357 * We skip integral divs, i.e., those with denominator 1, as we would
1358 * risk eliminating the div from the div constraints. We do not need
1359 * to handle those divs here anyway since the div constraints will turn
1360 * out to form an equality and this equality can then be used to eliminate
1361 * the div from all constraints.
1362 */
1365 {
1397 else
1407 }
1413 }
1414 }
1417 }
1420 {
1425 isl_bool empty;
1441 /* requires equalities in normal form */
1447 }
1449 }
1453 {
1455 }
1460 {
1492 }
1494 /* If the only constraints a div d=floor(f/m)
1495 * appears in are its two defining constraints
1496 *
1497 * f - m d >=0
1498 * -(f - (m - 1)) + m d >= 0
1499 *
1500 * then it can safely be removed.
1501 */
1503 {
1516 isl_bool red;
1523 }
1530 }
1533 }
1535 /*
1536 * Remove divs that don't occur in any of the constraints or other divs.
1537 * These can arise when dropping constraints from a basic map or
1538 * when the divs of a basic map have been temporarily aligned
1539 * with the divs of another basic map.
1540 */
1543 {
1545 isl_size v_div;
1552 isl_bool redundant;
1562 }
1564 }
1566 /* Mark "bmap" as final, without checking for obviously redundant
1567 * integer divisions. This function should be used when "bmap"
1568 * is known not to involve any such integer divisions.
1569 */
1572 {
1577 }
1579 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1580 */
1582 {
1586 }
1590 {
1592 }
1594 /* Remove definition of any div that is defined in terms of the given variable.
1595 * The div itself is not removed. Functions such as
1596 * eliminate_divs_ineq depend on the other divs remaining in place.
1597 */
1600 {
1614 }
1616 }
1618 /* Eliminate the specified variables from the constraints using
1619 * Fourier-Motzkin. The variables themselves are not removed.
1620 */
1623 {
1626 isl_size total;
1657 }
1664 n_lower++;
1666 n_upper++;
1667 }
1691 }
1694 }
1706 }
1707 }
1711 error:
1714 }
1718 {
1721 }
1723 /* Eliminate the specified n dimensions starting at first from the
1724 * constraints, without removing the dimensions from the space.
1725 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1726 * Otherwise, they are projected out and the original space is restored.
1727 */
1731 {
1746 }
1753 }
1758 {
1760 }
1762 /* Remove all constraints from "bmap" that reference any unknown local
1763 * variables (directly or indirectly).
1764 *
1765 * Dropping all constraints on a local variable will make it redundant,
1766 * so it will get removed implicitly by
1767 * isl_basic_map_drop_constraints_involving_dims. Some other local
1768 * variables may also end up becoming redundant if they only appear
1769 * in constraints together with the unknown local variable.
1770 * Therefore, start over after calling
1771 * isl_basic_map_drop_constraints_involving_dims.
1772 */
1775 {
1776 isl_bool known;
1777 isl_size n_div;
1804 }
1807 }
1809 /* Remove all constraints from "bset" that reference any unknown local
1810 * variables (directly or indirectly).
1811 */
1814 {
1820 }
1822 /* Remove all constraints from "map" that reference any unknown local
1823 * variables (directly or indirectly).
1824 *
1825 * Since constraints may get dropped from the basic maps,
1826 * they may no longer be disjoint from each other.
1827 */
1830 {
1832 isl_bool known;
1850 }
1856 }
1858 /* Don't assume equalities are in order, because align_divs
1859 * may have changed the order of the divs.
1860 */
1862 {
1875 }
1876 }
1877 }
1881 {
1883 }
1887 {
1901 }
1903 }
1905 }
1909 {
1912 }
1916 {
1919 isl_size dim;
1927 }
1949 error:
1953 }
1955 /* For each inequality in "ineq" that is a shifted (more relaxed)
1956 * copy of an inequality in "context", mark the corresponding entry
1957 * in "row" with -1.
1958 * If an inequality only has a non-negative constant term, then
1959 * mark it as well.
1960 */
1963 {
1983 isl_bool redundant;
1989 }
1996 }
1999 error:
2002 }
2006 {
2019 isl_bool redundant;
2031 }
2034 error:
2037 }
2039 /* Remove constraints from "bmap" that are identical to constraints
2040 * in "context" or that are more relaxed (greater constant term).
2041 *
2042 * We perform the test for shifted copies on the pure constraints
2043 * in remove_shifted_constraints.
2044 */
2047 {
2056 }
2069 error:
2073 }
2075 /* Does the (linear part of a) constraint "c" involve any of the "len"
2076 * "relevant" dimensions?
2077 */
2079 {
2087 }
2090 }
2092 /* Drop constraints from "bmap" that do not involve any of
2093 * the dimensions marked "relevant".
2094 */
2097 {
2099 isl_size dim;
2115 }
2122 }
2125 }
2127 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2128 *
2129 * In particular, for any variable involved in the constraint,
2130 * find the actual group id from before and replace the group
2131 * of the corresponding variable by the minimal group of all
2132 * the variables involved in the constraint considered so far
2133 * (if this minimum is smaller) or replace the minimum by this group
2134 * (if the minimum is larger).
2135 *
2136 * At the end, all the variables in "c" will (indirectly) point
2137 * to the minimal of the groups that they referred to originally.
2138 */
2140 {
2157 }
2158 }
2160 /* Allocate an array of groups of variables, one for each variable
2161 * in "context", initialized to zero.
2162 */
2164 {
2166 isl_size dim;
2173 }
2175 /* Drop constraints from "bmap" that only involve variables that are
2176 * not related to any of the variables marked with a "-1" in "group".
2177 *
2178 * We construct groups of variables that collect variables that
2179 * (indirectly) appear in some common constraint of "bmap".
2180 * Each group is identified by the first variable in the group,
2181 * except for the special group of variables that was already identified
2182 * in the input as -1 (or are related to those variables).
2183 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2184 * otherwise the group of i is the group of group[i].
2185 *
2186 * We first initialize groups for the remaining variables.
2187 * Then we iterate over the constraints of "bmap" and update the
2188 * group of the variables in the constraint by the smallest group.
2189 * Finally, we resolve indirect references to groups by running over
2190 * the variables.
2191 *
2192 * After computing the groups, we drop constraints that do not involve
2193 * any variables in the -1 group.
2194 */
2197 {
2198 isl_size dim;
2213 }
2231 }
2233 /* Drop constraints from "context" that are irrelevant for computing
2234 * the gist of "bset".
2235 *
2236 * In particular, drop constraints in variables that are not related
2237 * to any of the variables involved in the constraints of "bset"
2238 * in the sense that there is no sequence of constraints that connects them.
2239 *
2240 * We first mark all variables that appear in "bset" as belonging
2241 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2242 */
2245 {
2247 isl_size dim;
2266 }
2272 }
2275 }
2277 /* Drop constraints from "context" that are irrelevant for computing
2278 * the gist of the inequalities "ineq".
2279 * Inequalities in "ineq" for which the corresponding element of row
2280 * is set to -1 have already been marked for removal and should be ignored.
2281 *
2282 * In particular, drop constraints in variables that are not related
2283 * to any of the variables involved in "ineq"
2284 * in the sense that there is no sequence of constraints that connects them.
2285 *
2286 * We first mark all variables that appear in "bset" as belonging
2287 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2288 */
2291 {
2293 isl_size dim;
2295 isl_size n;
2313 }
2316 }
2319 }
2321 /* Do all "n" entries of "row" contain a negative value?
2322 */
2324 {
2332 }
2334 /* Update the inequalities in "bset" based on the information in "row"
2335 * and "tab".
2336 *
2337 * In particular, the array "row" contains either -1, meaning that
2338 * the corresponding inequality of "bset" is redundant, or the index
2339 * of an inequality in "tab".
2340 *
2341 * If the row entry is -1, then drop the inequality.
2342 * Otherwise, if the constraint is marked redundant in the tableau,
2343 * then drop the inequality. Similarly, if it is marked as an equality
2344 * in the tableau, then turn the inequality into an equality and
2345 * perform Gaussian elimination.
2346 */
2349 {
2366 }
2376 }
2377 }
2383 }
2385 /* Update the inequalities in "bset" based on the information in "row"
2386 * and "tab" and free all arguments (other than "bset").
2387 */
2392 {
2401 }
2403 /* Remove all information from bset that is redundant in the context
2404 * of context.
2405 * "ineq" contains the (possibly transformed) inequalities of "bset",
2406 * in the same order.
2407 * The (explicit) equalities of "bset" are assumed to have been taken
2408 * into account by the transformation such that only the inequalities
2409 * are relevant.
2410 * "context" is assumed not to be empty.
2411 *
2412 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2413 * A value of -1 means that the inequality is obviously redundant and may
2414 * not even appear in "tab".
2415 *
2416 * We first mark the inequalities of "bset"
2417 * that are obviously redundant with respect to some inequality in "context".
2418 * Then we remove those constraints from "context" that have become
2419 * irrelevant for computing the gist of "bset".
2420 * Note that this removal of constraints cannot be replaced by
2421 * a factorization because factors in "bset" may still be connected
2422 * to each other through constraints in "context".
2423 *
2424 * If there are any inequalities left, we construct a tableau for
2425 * the context and then add the inequalities of "bset".
2426 * Before adding these inequalities, we freeze all constraints such that
2427 * they won't be considered redundant in terms of the constraints of "bset".
2428 * Then we detect all redundant constraints (among the
2429 * constraints that weren't frozen), first by checking for redundancy in the
2430 * the tableau and then by checking if replacing a constraint by its negation
2431 * would lead to an empty set. This last step is fairly expensive
2432 * and could be optimized by more reuse of the tableau.
2433 * Finally, we update bset according to the results.
2434 */
2437 {
2452 }
2488 }
2511 }
2516 }
2520 error:
2528 }
2530 /* Extract the inequalities of "bset" as an isl_mat.
2531 */
2533 {
2534 isl_size total;
2547 }
2549 /* Remove all information from "bset" that is redundant in the context
2550 * of "context", for the case where both "bset" and "context" are
2551 * full-dimensional.
2552 */
2555 {
2560 }
2562 /* Replace "bset" by an empty basic set in the same space.
2563 */
2566 {
2572 }
2574 /* Remove all information from "bset" that is redundant in the context
2575 * of "context", for the case where the combined equalities of
2576 * "bset" and "context" allow for a compression that can be obtained
2577 * by preapplication of "T".
2578 * If the compression of "context" is empty, meaning that "bset" and
2579 * "context" do not intersect, then return the empty set.
2580 *
2581 * "bset" itself is not transformed by "T". Instead, the inequalities
2582 * are extracted from "bset" and those are transformed by "T".
2583 * uset_gist_full then determines which of the transformed inequalities
2584 * are redundant with respect to the transformed "context" and removes
2585 * the corresponding inequalities from "bset".
2586 *
2587 * After preapplying "T" to the inequalities, any common factor is
2588 * removed from the coefficients. If this results in a tightening
2589 * of the constant term, then the same tightening is applied to
2590 * the corresponding untransformed inequality in "bset".
2591 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2592 *
2593 * g f'(x) + r >= 0
2594 *
2595 * with 0 <= r < g, then it is equivalent to
2596 *
2597 * f'(x) >= 0
2598 *
2599 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2600 * subspace compressed by T since the latter would be transformed to
2601 *
2602 * g f'(x) >= 0
2603 */
2607 {
2612 isl_int rem;
2624 }
2649 }
2653 error:
2658 }
2660 /* Project "bset" onto the variables that are involved in "template".
2661 */
2664 {
2666 isl_size n;
2673 isl_bool involved;
2682 }
2685 }
2687 /* Remove all information from bset that is redundant in the context
2688 * of context. In particular, equalities that are linear combinations
2689 * of those in context are removed. Then the inequalities that are
2690 * redundant in the context of the equalities and inequalities of
2691 * context are removed.
2692 *
2693 * First of all, we drop those constraints from "context"
2694 * that are irrelevant for computing the gist of "bset".
2695 * Alternatively, we could factorize the intersection of "context" and "bset".
2696 *
2697 * We first compute the intersection of the integer affine hulls
2698 * of "bset" and "context",
2699 * compute the gist inside this intersection and then reduce
2700 * the constraints with respect to the equalities of the context
2701 * that only involve variables already involved in the input.
2702 * If the intersection of the affine hulls turns out to be empty,
2703 * then return the empty set.
2704 *
2705 * If two constraints are mutually redundant, then uset_gist_full
2706 * will remove the second of those constraints. We therefore first
2707 * sort the constraints so that constraints not involving existentially
2708 * quantified variables are given precedence over those that do.
2709 * We have to perform this sorting before the variable compression,
2710 * because that may effect the order of the variables.
2711 */
2714 {
2719 isl_size total;
2740 }
2745 }
2754 }
2765 }
2768 error:
2772 }
2774 /* Return the number of equality constraints in "bmap" that involve
2775 * local variables. This function assumes that Gaussian elimination
2776 * has been applied to the equality constraints.
2777 */
2779 {
2801 }
2803 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2804 * The constraints are assumed not to involve any local variables.
2805 */
2808 {
2810 isl_size total;
2828 }
2833 error:
2838 }
2840 /* Construct and return a variable compression based on the equality
2841 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2842 * "n1" is the number of (initial) equality constraints in "bmap1"
2843 * that do involve local variables.
2844 * "n2" is the number of (initial) equality constraints in "bmap2"
2845 * that do involve local variables.
2846 * "total" is the total number of other variables.
2847 * This function assumes that Gaussian elimination
2848 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2849 * such that the equality constraints not involving local variables
2850 * are those that start at "n1" or "n2".
2851 *
2852 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2853 * then simply compute the compression based on the equality constraints
2854 * in the other basic map.
2855 * Otherwise, combine the equality constraints from both into a new
2856 * basic map such that Gaussian elimination can be applied to this combination
2857 * and then construct a variable compression from the resulting
2858 * equality constraints.
2859 */
2863 {
2873 }
2878 }
2893 }
2895 /* Extract the stride constraints from "bmap", compressed
2896 * with respect to both the stride constraints in "context" and
2897 * the remaining equality constraints in both "bmap" and "context".
2898 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2899 * "context_n_eq" is the number of (initial) stride constraints in "context".
2900 *
2901 * Let x be all variables in "bmap" (and "context") other than the local
2902 * variables. First compute a variable compression
2903 *
2904 * x = V x'
2905 *
2906 * based on the non-stride equality constraints in "bmap" and "context".
2907 * Consider the stride constraints of "context",
2908 *
2909 * A(x) + B(y) = 0
2910 *
2911 * with y the local variables and plug in the variable compression,
2912 * resulting in
2913 *
2914 * A(V x') + B(y) = 0
2915 *
2916 * Use these constraints to compute a parameter compression on x'
2917 *
2918 * x' = T x''
2919 *
2920 * Now consider the stride constraints of "bmap"
2921 *
2922 * C(x) + D(y) = 0
2923 *
2924 * and plug in x = V*T x''.
2925 * That is, return A = [C*V*T D].
2926 */
2930 {
2956 else
2964 }
2966 /* Remove the prime factors from *g that have an exponent that
2967 * is strictly smaller than the exponent in "c".
2968 * All exponents in *g are known to be smaller than or equal
2969 * to those in "c".
2970 *
2971 * That is, if *g is equal to
2972 *
2973 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2974 *
2975 * and "c" is equal to
2976 *
2977 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2978 *
2979 * then update *g to
2980 *
2981 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2982 * p_n^{e_n * (e_n = f_n)}
2983 *
2984 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2985 * neither does the gcd of *g and c / *g.
2986 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2987 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2988 * Dividing *g by this gcd therefore strictly reduces the exponent
2989 * of the prime factors that need to be removed, while leaving the
2990 * other prime factors untouched.
2991 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2992 * removes all undesired factors, without removing any others.
2993 */
2995 {
2996 isl_int t;
3005 }
3007 }
3009 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
3010 * of the same stride constraints in a compressed space that exploits
3011 * all equalities in the context and the other equalities in "bmap".
3012 *
3013 * If the stride constraints of "bmap" are of the form
3014 *
3015 * C(x) + D(y) = 0
3016 *
3017 * then A is of the form
3018 *
3019 * B(x') + D(y) = 0
3020 *
3021 * If any of these constraints involves only a single local variable y,
3022 * then the constraint appears as
3023 *
3024 * f(x) + m y_i = 0
3025 *
3026 * in "bmap" and as
3027 *
3028 * h(x') + m y_i = 0
3029 *
3030 * in "A".
3031 *
3032 * Let g be the gcd of m and the coefficients of h.
3033 * Then, in particular, g is a divisor of the coefficients of h and
3034 *
3035 * f(x) = h(x')
3036 *
3037 * is known to be a multiple of g.
3038 * If some prime factor in m appears with the same exponent in g,
3039 * then it can be removed from m because f(x) is already known
3040 * to be a multiple of g and therefore in particular of this power
3041 * of the prime factors.
3042 * Prime factors that appear with a smaller exponent in g cannot
3043 * be removed from m.
3044 * Let g' be the divisor of g containing all prime factors that
3045 * appear with the same exponent in m and g, then
3046 *
3047 * f(x) + m y_i = 0
3048 *
3049 * can be replaced by
3050 *
3051 * f(x) + m/g' y_i' = 0
3052 *
3053 * Note that (if g' != 1) this changes the explicit representation
3054 * of y_i to that of y_i', so the integer division at position i
3055 * is marked unknown and later recomputed by a call to
3056 * isl_basic_map_gauss.
3057 */
3060 {
3064 isl_int gcd;
3097 }
3104 error:
3108 }
3110 /* Simplify the stride constraints in "bmap" based on
3111 * the remaining equality constraints in "bmap" and all equality
3112 * constraints in "context".
3113 * Only do this if both "bmap" and "context" have stride constraints.
3114 *
3115 * First extract a copy of the stride constraints in "bmap" in a compressed
3116 * space exploiting all the other equality constraints and then
3117 * use this compressed copy to simplify the original stride constraints.
3118 */
3121 {
3143 }
3145 /* Return a basic map that has the same intersection with "context" as "bmap"
3146 * and that is as "simple" as possible.
3147 *
3148 * The core computation is performed on the pure constraints.
3149 * When we add back the meaning of the integer divisions, we need
3150 * to (re)introduce the div constraints. If we happen to have
3151 * discovered that some of these integer divisions are equal to
3152 * some affine combination of other variables, then these div
3153 * constraints may end up getting simplified in terms of the equalities,
3154 * resulting in extra inequalities on the other variables that
3155 * may have been removed already or that may not even have been
3156 * part of the input. We try and remove those constraints of
3157 * this form that are most obviously redundant with respect to
3158 * the context. We also remove those div constraints that are
3159 * redundant with respect to the other constraints in the result.
3160 *
3161 * The stride constraints among the equality constraints in "bmap" are
3162 * also simplified with respecting to the other equality constraints
3163 * in "bmap" and with respect to all equality constraints in "context".
3164 */
3167 {
3179 }
3185 }
3189 }
3212 }
3229 error:
3233 }
3235 /*
3236 * Assumes context has no implicit divs.
3237 */
3240 {
3251 }
3270 }
3271 }
3275 error:
3279 }
3281 /* Drop all inequalities from "bmap" that also appear in "context".
3282 * "context" is assumed to have only known local variables and
3283 * the initial local variables of "bmap" are assumed to be the same
3284 * as those of "context".
3285 * The constraints of both "bmap" and "context" are assumed
3286 * to have been sorted using isl_basic_map_sort_constraints.
3287 *
3288 * Run through the inequality constraints of "bmap" and "context"
3289 * in sorted order.
3290 * If a constraint of "bmap" involves variables not in "context",
3291 * then it cannot appear in "context".
3292 * If a matching constraint is found, it is removed from "bmap".
3293 */
3296 {
3317 }
3323 }
3327 }
3332 }
3335 }
3338 }
3340 /* Drop all equalities from "bmap" that also appear in "context".
3341 * "context" is assumed to have only known local variables and
3342 * the initial local variables of "bmap" are assumed to be the same
3343 * as those of "context".
3344 *
3345 * Run through the equality constraints of "bmap" and "context"
3346 * in sorted order.
3347 * If a constraint of "bmap" involves variables not in "context",
3348 * then it cannot appear in "context".
3349 * If a matching constraint is found, it is removed from "bmap".
3350 */
3353 {
3379 }
3383 }
3388 }
3391 }
3394 }
3396 /* Remove the constraints in "context" from "bmap".
3397 * "context" is assumed to have explicit representations
3398 * for all local variables.
3399 *
3400 * First align the divs of "bmap" to those of "context" and
3401 * sort the constraints. Then drop all constraints from "bmap"
3402 * that appear in "context".
3403 */
3406 {
3421 }
3440 error:
3444 }
3446 /* Replace "map" by the disjunct at position "pos" and free "context".
3447 */
3450 {
3457 }
3459 /* Remove the constraints in "context" from "map".
3460 * If any of the disjuncts in the result turns out to be the universe,
3461 * then return this universe.
3462 * "context" is assumed to have explicit representations
3463 * for all local variables.
3464 */
3467 {
3477 }
3496 }
3503 error:
3507 }
3509 /* Remove the constraints in "context" from "set".
3510 * If any of the disjuncts in the result turns out to be the universe,
3511 * then return this universe.
3512 * "context" is assumed to have explicit representations
3513 * for all local variables.
3514 */
3517 {
3520 }
3522 /* Remove the constraints in "context" from "map".
3523 * If any of the disjuncts in the result turns out to be the universe,
3524 * then return this universe.
3525 * "context" is assumed to consist of a single disjunct and
3526 * to have explicit representations for all local variables.
3527 */
3530 {
3535 }
3537 /* Replace "map" by a universe map in the same space and free "drop".
3538 */
3541 {
3548 }
3550 /* Return a map that has the same intersection with "context" as "map"
3551 * and that is as "simple" as possible.
3552 *
3553 * If "map" is already the universe, then we cannot make it any simpler.
3554 * Similarly, if "context" is the universe, then we cannot exploit it
3555 * to simplify "map"
3556 * If "map" and "context" are identical to each other, then we can
3557 * return the corresponding universe.
3558 *
3559 * If either "map" or "context" consists of multiple disjuncts,
3560 * then check if "context" happens to be a subset of "map",
3561 * in which case all constraints can be removed.
3562 * In case of multiple disjuncts, the standard procedure
3563 * may not be able to detect that all constraints can be removed.
3564 *
3565 * If none of these cases apply, we have to work a bit harder.
3566 * During this computation, we make use of a single disjunct context,
3567 * so if the original context consists of more than one disjunct
3568 * then we need to approximate the context by a single disjunct set.
3569 * Simply taking the simple hull may drop constraints that are
3570 * only implicitly available in each disjunct. We therefore also
3571 * look for constraints among those defining "map" that are valid
3572 * for the context. These can then be used to simplify away
3573 * the corresponding constraints in "map".
3574 */
3577 {
3581 isl_bool subset;
3592 }
3611 }
3627 list);
3628 }
3630 error:
3634 }
3638 {
3641 }
3645 {
3648 }
3652 {
3657 }
3661 {
3663 }
3665 /* Compute the gist of "bmap" with respect to the constraints "context"
3666 * on the domain.
3667 */
3670 {
3676 }
3680 {
3684 }
3688 {
3692 }
3696 {
3700 }
3704 {
3706 }
3708 /* Quick check to see if two basic maps are disjoint.
3709 * In particular, we reduce the equalities and inequalities of
3710 * one basic map in the context of the equalities of the other
3711 * basic map and check if we get a contradiction.
3712 */
3715 {
3718 isl_size total;
3747 }
3755 }
3764 }
3768 disjoint:
3772 error:
3776 }
3780 {
3783 }
3785 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3786 */
3790 {
3801 }
3802 }
3805 }
3807 /* Are "map1" and "map2" obviously disjoint, based on information
3808 * that can be derived without looking at the individual basic maps?
3809 *
3810 * In particular, if one of them is empty or if they live in different spaces
3811 * (ignoring parameters), then they are clearly disjoint.
3812 */
3815 {
3816 isl_bool disjoint;
3817 isl_bool match;
3841 }
3843 /* Are "map1" and "map2" obviously disjoint?
3844 *
3845 * If one of them is empty or if they live in different spaces (ignoring
3846 * parameters), then they are clearly disjoint.
3847 * This is checked by isl_map_plain_is_disjoint_global.
3848 *
3849 * If they have different parameters, then we skip any further tests.
3850 *
3851 * If they are obviously equal, but not obviously empty, then we will
3852 * not be able to detect if they are disjoint.
3853 *
3854 * Otherwise we check if each basic map in "map1" is obviously disjoint
3855 * from each basic map in "map2".
3856 */
3859 {
3860 isl_bool disjoint;
3861 isl_bool intersect;
3862 isl_bool match;
3877 }
3879 /* Are "map1" and "map2" disjoint?
3880 * The parameters are assumed to have been aligned.
3881 *
3882 * In particular, check whether all pairs of basic maps are disjoint.
3883 */
3886 {
3888 }
3890 /* Are "map1" and "map2" disjoint?
3891 *
3892 * They are disjoint if they are "obviously disjoint" or if one of them
3893 * is empty. Otherwise, they are not disjoint if one of them is universal.
3894 * If the two inputs are (obviously) equal and not empty, then they are
3895 * not disjoint.
3896 * If none of these cases apply, then check if all pairs of basic maps
3897 * are disjoint after aligning the parameters.
3898 */
3900 {
3901 isl_bool disjoint;
3902 isl_bool intersect;
3930 }
3932 /* Are "bmap1" and "bmap2" disjoint?
3933 *
3934 * They are disjoint if they are "obviously disjoint" or if one of them
3935 * is empty. Otherwise, they are not disjoint if one of them is universal.
3936 * If none of these cases apply, we compute the intersection and see if
3937 * the result is empty.
3938 */
3941 {
3942 isl_bool disjoint;
3943 isl_bool intersect;
3972 }
3974 /* Are "bset1" and "bset2" disjoint?
3975 */
3978 {
3980 }
3984 {
3986 }
3988 /* Are "set1" and "set2" disjoint?
3989 */
3991 {
3993 }
3995 /* Is "v" equal to 0, 1 or -1?
3996 */
3998 {
4000 }
4002 /* Are the "n" coefficients starting at "first" of inequality constraints
4003 * "i" and "j" of "bmap" opposite to each other?
4004 */
4007 {
4009 }
4011 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4012 * apart from the constant term?
4013 */
4015 {
4016 isl_size total;
4022 }
4024 /* Check if we can combine a given div with lower bound l and upper
4025 * bound u with some other div and if so return that other div.
4026 * Otherwise, return a position beyond the integer divisions.
4027 * Return -1 on error.
4028 *
4029 * We first check that
4030 * - the bounds are opposites of each other (except for the constant
4031 * term)
4032 * - the bounds do not reference any other div
4033 * - no div is defined in terms of this div
4034 *
4035 * Let m be the size of the range allowed on the div by the bounds.
4036 * That is, the bounds are of the form
4037 *
4038 * e <= a <= e + m - 1
4039 *
4040 * with e some expression in the other variables.
4041 * We look for another div b such that no third div is defined in terms
4042 * of this second div b and such that in any constraint that contains
4043 * a (except for the given lower and upper bound), also contains b
4044 * with a coefficient that is m times that of b.
4045 * That is, all constraints (except for the lower and upper bound)
4046 * are of the form
4047 *
4048 * e + f (a + m b) >= 0
4049 *
4050 * Furthermore, in the constraints that only contain b, the coefficient
4051 * of b should be equal to 1 or -1.
4052 * If so, we return b so that "a + m b" can be replaced by
4053 * a single div "c = a + m b".
4054 */
4057 {
4060 isl_size v_div;
4062 isl_bool opp;
4084 }
4094 }
4107 }
4118 }
4131 }
4136 }
4140 }
4142 /* Internal data structure used during the construction and/or evaluation of
4143 * an inequality that ensures that a pair of bounds always allows
4144 * for an integer value.
4145 *
4146 * "tab" is the tableau in which the inequality is evaluated. It may
4147 * be NULL until it is actually needed.
4148 * "v" contains the inequality coefficients.
4149 * "g", "fl" and "fu" are temporary scalars used during the construction and
4150 * evaluation.
4151 */
4155 isl_int g;
4156 isl_int fl;
4157 isl_int fu;
4158 };
4160 /* Free all the memory allocated by the fields of "data".
4161 */
4163 {
4169 }
4171 /* Is the inequality stored in data->v satisfied by "bmap"?
4172 * That is, does it only attain non-negative values?
4173 * data->tab is a tableau corresponding to "bmap".
4174 */
4177 {
4188 }
4190 /* Given a lower and an upper bound on div i, do they always allow
4191 * for an integer value of the given div?
4192 * Determine this property by constructing an inequality
4193 * such that the property is guaranteed when the inequality is nonnegative.
4194 * The lower bound is inequality l, while the upper bound is inequality u.
4195 * The constructed inequality is stored in data->v.
4196 *
4197 * Let the upper bound be
4198 *
4199 * -n_u a + e_u >= 0
4200 *
4201 * and the lower bound
4202 *
4203 * n_l a + e_l >= 0
4204 *
4205 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4206 * We have
4207 *
4208 * - f_u e_l <= f_u f_l g a <= f_l e_u
4209 *
4210 * Since all variables are integer valued, this is equivalent to
4211 *
4212 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4213 *
4214 * If this interval is at least f_u f_l g, then it contains at least
4215 * one integer value for a.
4216 * That is, the test constraint is
4217 *
4218 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4219 *
4220 * or
4221 *
4222 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4223 *
4224 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4225 * then the constraint can be scaled down by a factor g',
4226 * with the constant term replaced by
4227 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4228 * Note that the result of applying Fourier-Motzkin to this pair
4229 * of constraints is
4230 *
4231 * f_l e_u + f_u e_l >= 0
4232 *
4233 * If the constant term of the scaled down version of this constraint,
4234 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4235 * term of the scaled down test constraint, then the test constraint
4236 * is known to hold and no explicit evaluation is required.
4237 * This is essentially the Omega test.
4238 *
4239 * If the test constraint consists of only a constant term, then
4240 * it is sufficient to look at the sign of this constant term.
4241 */
4244 {
4246 isl_size n_div;
4273 }
4283 }
4285 /* Remove more kinds of divs that are not strictly needed.
4286 * In particular, if all pairs of lower and upper bounds on a div
4287 * are such that they allow at least one integer value of the div,
4288 * then we can eliminate the div using Fourier-Motzkin without
4289 * introducing any spurious solutions.
4290 *
4291 * If at least one of the two constraints has a unit coefficient for the div,
4292 * then the presence of such a value is guaranteed so there is no need to check.
4293 * In particular, the value attained by the bound with unit coefficient
4294 * can serve as this intermediate value.
4295 */
4298 {
4302 isl_size n_div;
4322 isl_bool has_int;
4330 }
4351 }
4354 }
4358 }
4362 }
4365 }
4376 error:
4381 }
4383 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4384 * and the upper bound u, div1 always occurs together with div2 in the form
4385 * (div1 + m div2), where m is the constant range on the variable div1
4386 * allowed by l and u, replace the pair div1 and div2 by a single
4387 * div that is equal to div1 + m div2.
4388 *
4389 * The new div will appear in the location that contains div2.
4390 * We need to modify all constraints that contain
4391 * div2 = (div - div1) / m
4392 * The coefficient of div2 is known to be equal to 1 or -1.
4393 * (If a constraint does not contain div2, it will also not contain div1.)
4394 * If the constraint also contains div1, then we know they appear
4395 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4396 * i.e., the coefficient of div is f.
4397 *
4398 * Otherwise, we first need to introduce div1 into the constraint.
4399 * Let l be
4400 *
4401 * div1 + f >=0
4402 *
4403 * and u
4404 *
4405 * -div1 + f' >= 0
4406 *
4407 * A lower bound on div2
4408 *
4409 * div2 + t >= 0
4410 *
4411 * can be replaced by
4412 *
4413 * m div2 + div1 + m t + f >= 0
4414 *
4415 * An upper bound
4416 *
4417 * -div2 + t >= 0
4418 *
4419 * can be replaced by
4420 *
4421 * -(m div2 + div1) + m t + f' >= 0
4422 *
4423 * These constraint are those that we would obtain from eliminating
4424 * div1 using Fourier-Motzkin.
4425 *
4426 * After all constraints have been modified, we drop the lower and upper
4427 * bound and then drop div1.
4428 * Since the new div is only placed in the same location that used
4429 * to store div2, but otherwise has a different meaning, any possible
4430 * explicit representation of the original div2 is removed.
4431 */
4434 {
4436 isl_int m;
4437 isl_size v_div;
4461 else
4464 }
4468 }
4477 }
4481 }
4483 /* First check if we can coalesce any pair of divs and
4484 * then continue with dropping more redundant divs.
4485 *
4486 * We loop over all pairs of lower and upper bounds on a div
4487 * with coefficient 1 and -1, respectively, check if there
4488 * is any other div "c" with which we can coalesce the div
4489 * and if so, perform the coalescing.
4490 */
4493 {
4495 isl_size v_div;
4496 isl_size n_div;
4522 }
4523 }
4524 }
4529 }
4532 error:
4536 }
4538 /* Are the "n" coefficients starting at "first" of inequality constraints
4539 * "i" and "j" of "bmap" equal to each other?
4540 */
4543 {
4545 }
4547 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4548 * apart from the constant term and the coefficient at position "pos"?
4549 */
4552 {
4553 isl_size total;
4560 }
4562 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4563 * apart from the constant term and the coefficient at position "pos"?
4564 */
4567 {
4568 isl_size total;
4575 }
4577 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4578 * been modified, simplying it if "simplify" is set.
4579 * Free the temporary data structure "pairs" that was associated
4580 * to the old version of "bmap".
4581 */
4584 {
4589 }
4591 /* Is "div" the single unknown existentially quantified variable
4592 * in inequality constraint "ineq" of "bmap"?
4593 * "div" is known to have a non-zero coefficient in "ineq".
4594 */
4597 {
4599 isl_size n_div;
4601 isl_bool known;
4613 isl_bool known;
4622 }
4625 }
4627 /* Does integer division "div" have coefficient 1 in inequality constraint
4628 * "ineq" of "map"?
4629 */
4631 {
4639 }
4641 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4642 * then try and drop redundant divs again,
4643 * freeing the temporary data structure "pairs" that was associated
4644 * to the old version of "bmap".
4645 */
4648 {
4652 }
4654 /* Drop the integer division at position "div", along with the two
4655 * inequality constraints "ineq1" and "ineq2" in which it appears
4656 * from "bmap" and then try and drop redundant divs again,
4657 * freeing the temporary data structure "pairs" that was associated
4658 * to the old version of "bmap".
4659 */
4663 {
4670 }
4673 }
4675 /* Given two inequality constraints
4676 *
4677 * f(x) + n d + c >= 0, (ineq)
4678 *
4679 * with d the variable at position "pos", and
4680 *
4681 * f(x) + c0 >= 0, (lower)
4682 *
4683 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4684 * determined by the first constraint.
4685 * That is, store
4686 *
4687 * ceil((c0 - c)/n)
4688 *
4689 * in *l.
4690 */
4693 {
4697 }
4699 /* Given two inequality constraints
4700 *
4701 * f(x) + n d + c >= 0, (ineq)
4702 *
4703 * with d the variable at position "pos", and
4704 *
4705 * -f(x) - c0 >= 0, (upper)
4706 *
4707 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4708 * determined by the first constraint.
4709 * That is, store
4710 *
4711 * ceil((-c1 - c)/n)
4712 *
4713 * in *u.
4714 */
4717 {
4721 }
4723 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4724 * does the corresponding lower bound have a fixed value in "bmap"?
4725 *
4726 * In particular, "ineq" is of the form
4727 *
4728 * f(x) + n d + c >= 0
4729 *
4730 * with n > 0, c the constant term and
4731 * d the existentially quantified variable "div".
4732 * That is, the lower bound is
4733 *
4734 * ceil((-f(x) - c)/n)
4735 *
4736 * Look for a pair of constraints
4737 *
4738 * f(x) + c0 >= 0
4739 * -f(x) + c1 >= 0
4740 *
4741 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4742 * That is, check that
4743 *
4744 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4745 *
4746 * If so, return the index of inequality f(x) + c0 >= 0.
4747 * Otherwise, return bmap->n_ineq.
4748 * Return -1 on error.
4749 */
4751 {
4774 }
4782 }
4799 }
4801 /* Given a lower bound constraint "ineq" on the existentially quantified
4802 * variable "div", such that the corresponding lower bound has
4803 * a fixed value in "bmap", assign this fixed value to the variable and
4804 * then try and drop redundant divs again,
4805 * freeing the temporary data structure "pairs" that was associated
4806 * to the old version of "bmap".
4807 * "lower" determines the constant value for the lower bound.
4808 *
4809 * In particular, "ineq" is of the form
4810 *
4811 * f(x) + n d + c >= 0,
4812 *
4813 * while "lower" is of the form
4814 *
4815 * f(x) + c0 >= 0
4816 *
4817 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4818 * is ceil((c0 - c)/n).
4819 */
4822 {
4823 isl_int c;
4836 }
4838 /* Remove divs that are not strictly needed based on the inequality
4839 * constraints.
4840 * In particular, if a div only occurs positively (or negatively)
4841 * in constraints, then it can simply be dropped.
4842 * Also, if a div occurs in only two constraints and if moreover
4843 * those two constraints are opposite to each other, except for the constant
4844 * term and if the sum of the constant terms is such that for any value
4845 * of the other values, there is always at least one integer value of the
4846 * div, i.e., if one plus this sum is greater than or equal to
4847 * the (absolute value) of the coefficient of the div in the constraints,
4848 * then we can also simply drop the div.
4849 *
4850 * If an existentially quantified variable does not have an explicit
4851 * representation, appears in only a single lower bound that does not
4852 * involve any other such existentially quantified variables and appears
4853 * in this lower bound with coefficient 1,
4854 * then fix the variable to the value of the lower bound. That is,
4855 * turn the inequality into an equality.
4856 * If for any value of the other variables, there is any value
4857 * for the existentially quantified variable satisfying the constraints,
4858 * then this lower bound also satisfies the constraints.
4859 * It is therefore safe to pick this lower bound.
4860 *
4861 * The same reasoning holds even if the coefficient is not one.
4862 * However, fixing the variable to the value of the lower bound may
4863 * in general introduce an extra integer division, in which case
4864 * it may be better to pick another value.
4865 * If this integer division has a known constant value, then plugging
4866 * in this constant value removes the existentially quantified variable
4867 * completely. In particular, if the lower bound is of the form
4868 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4869 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4870 * then the existentially quantified variable can be assigned this
4871 * shared value.
4872 *
4873 * We skip divs that appear in equalities or in the definition of other divs.
4874 * Divs that appear in the definition of other divs usually occur in at least
4875 * 4 constraints, but the constraints may have been simplified.
4876 *
4877 * If any divs are left after these simple checks then we move on
4878 * to more complicated cases in drop_more_redundant_divs.
4879 */
4882 {
4884 isl_size off;
4887 isl_size n_ineq;
4928 }
4932 }
4933 }
4941 }
4944 else
4964 pairs);
4970 pairs);
4972 }
4989 else
4996 }
4999 }
5006 error:
5010 }
5012 /* Consider the coefficients at "c" as a row vector and replace
5013 * them with their product with "T". "T" is assumed to be a square matrix.
5014 */
5016 {
5017 isl_size n;
5038 }
5040 /* Plug in T for the variables in "bmap" starting at "pos".
5041 * T is a linear unimodular matrix, i.e., without constant term.
5042 */
5045 {
5073 }
5077 error:
5081 }
5083 /* Remove divs that are not strictly needed.
5084 *
5085 * First look for an equality constraint involving two or more
5086 * existentially quantified variables without an explicit
5087 * representation. Replace the combination that appears
5088 * in the equality constraint by a single existentially quantified
5089 * variable such that the equality can be used to derive
5090 * an explicit representation for the variable.
5091 * If there are no more such equality constraints, then continue
5092 * with isl_basic_map_drop_redundant_divs_ineq.
5093 *
5094 * In particular, if the equality constraint is of the form
5095 *
5096 * f(x) + \sum_i c_i a_i = 0
5097 *
5098 * with a_i existentially quantified variable without explicit
5099 * representation, then apply a transformation on the existentially
5100 * quantified variables to turn the constraint into
5101 *
5102 * f(x) + g a_1' = 0
5103 *
5104 * with g the gcd of the c_i.
5105 * In order to easily identify which existentially quantified variables
5106 * have a complete explicit representation, i.e., without being defined
5107 * in terms of other existentially quantified variables without
5108 * an explicit representation, the existentially quantified variables
5109 * are first sorted.
5110 *
5111 * The variable transformation is computed by extending the row
5112 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5113 *
5114 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5115 * [a_2'] [ a_2 ]
5116 * ... = U ....
5117 * [a_n'] [ a_n ]
5118 *
5119 * with [c_1/g ... c_n/g] representing the first row of U.
5120 * The inverse of U is then plugged into the original constraints.
5121 * The call to isl_basic_map_simplify makes sure the explicit
5122 * representation for a_1' is extracted from the equality constraint.
5123 */
5126 {
5130 isl_size n_div;
5164 }
5183 }
5185 /* Does "bmap" satisfy any equality that involves more than 2 variables
5186 * and/or has coefficients different from -1 and 1?
5187 */
5189 {
5191 isl_size total;
5220 }
5223 }
5225 /* Remove any common factor g from the constraint coefficients in "v".
5226 * The constant term is stored in the first position and is replaced
5227 * by floor(c/g). If any common factor is removed and if this results
5228 * in a tightening of the constraint, then set *tightened.
5229 */
5232 {
5252 }
5254 /* If "bmap" is an integer set that satisfies any equality involving
5255 * more than 2 variables and/or has coefficients different from -1 and 1,
5256 * then use variable compression to reduce the coefficients by removing
5257 * any (hidden) common factor.
5258 * In particular, apply the variable compression to each constraint,
5259 * factor out any common factor in the non-constant coefficients and
5260 * then apply the inverse of the compression.
5261 * At the end, we mark the basic map as having reduced constants.
5262 * If this flag is still set on the next invocation of this function,
5263 * then we skip the computation.
5264 *
5265 * Removing a common factor may result in a tightening of some of
5266 * the constraints. If this happens, then we may end up with two
5267 * opposite inequalities that can be replaced by an equality.
5268 * We therefore call isl_basic_map_detect_inequality_pairs,
5269 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5270 * and isl_basic_map_gauss if such a pair was found.
5271 *
5272 * Tightening may also result in some other constraints becoming
5273 * (rationally) redundant with respect to the tightened constraint
5274 * (in combination with other constraints). The basic map may
5275 * therefore no longer be assumed to have no redundant constraints.
5276 *
5277 * Note that this function may leave the result in an inconsistent state.
5278 * In particular, the constraints may not be gaussed.
5279 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5280 * for some of the test cases to pass successfully.
5281 * Any potential modification of the representation is therefore only
5282 * performed on a single copy of the basic map.
5283 */
5286 {
5287 isl_size total;
5288 isl_bool multi;
5326 }
5341 }
5357 }
5358 }
5361 error:
5366 }
5368 /* Shift the integer division at position "div" of "bmap"
5369 * by "shift" times the variable at position "pos".
5370 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5371 * corresponds to the constant term.
5372 *
5373 * That is, if the integer division has the form
5374 *
5375 * floor(f(x)/d)
5376 *
5377 * then replace it by
5378 *
5379 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5380 */
5383 {
5402 }
5408 }
5416 }
5419 }