| blob | ed37a4e5b969b318fbf4e54b02ea8eb201d37cf2 |
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
3 * Copyright 2012-2013 Ecole Normale Superieure
4 * Copyright 2014-2015 INRIA Rocquencourt
5 * Copyright 2016 Sven Verdoolaege
6 *
7 * Use of this software is governed by the MIT license
8 *
9 * Written by Sven Verdoolaege, K.U.Leuven, Departement
10 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
11 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
12 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
13 * B.P. 105 - 78153 Le Chesnay, France
14 */
16 #include <isl_ctx_private.h>
17 #include <isl_map_private.h>
19 #include <isl/map.h>
20 #include <isl_seq.h>
22 #include <isl_space_private.h>
23 #include <isl_mat_private.h>
24 #include <isl_vec_private.h>
26 #include <bset_to_bmap.c>
27 #include <bset_from_bmap.c>
28 #include <set_to_map.c>
29 #include <set_from_map.c>
32 {
36 }
39 {
44 }
45 }
49 {
51 isl_int gcd;
64 }
67 }
75 }
77 }
85 }
88 }
95 }
99 }
103 {
106 }
108 /* Reduce the coefficient of the variable at position "pos"
109 * in integer division "div", such that it lies in the half-open
110 * interval (1/2,1/2], extracting any excess value from this integer division.
111 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
112 * corresponds to the constant term.
113 *
114 * That is, the integer division is of the form
115 *
116 * floor((... + (c * d + r) * x_pos + ...)/d)
117 *
118 * with -d < 2 * r <= d.
119 * Replace it by
120 *
121 * floor((... + r * x_pos + ...)/d) + c * x_pos
122 *
123 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
124 * Otherwise, c = floor((c * d + r)/d) + 1.
125 *
126 * This is the same normalization that is performed by isl_aff_floor.
127 */
130 {
131 isl_int shift;
146 }
148 /* Does the coefficient of the variable at position "pos"
149 * in integer division "div" need to be reduced?
150 * That is, does it lie outside the half-open interval (1/2,1/2]?
151 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
152 * 2 * c != d.
153 */
156 {
157 isl_bool r;
169 }
171 /* Reduce the coefficients (including the constant term) of
172 * integer division "div", if needed.
173 * In particular, make sure all coefficients lie in
174 * the half-open interval (1/2,1/2].
175 */
178 {
183 isl_bool reduce;
193 }
196 }
198 /* Reduce the coefficients (including the constant term) of
199 * the known integer divisions, if needed
200 * In particular, make sure all coefficients lie in
201 * the half-open interval (1/2,1/2].
202 */
205 {
219 }
222 }
224 /* Remove any common factor in numerator and denominator of the div expression,
225 * not taking into account the constant term.
226 * That is, if the div is of the form
227 *
228 * floor((a + m f(x))/(m d))
229 *
230 * then replace it by
231 *
232 * floor((floor(a/m) + f(x))/d)
233 *
234 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
235 * and can therefore not influence the result of the floor.
236 */
238 {
254 }
256 /* Remove any common factor in numerator and denominator of a div expression,
257 * not taking into account the constant term.
258 * That is, look for any div of the form
259 *
260 * floor((a + m f(x))/(m d))
261 *
262 * and replace it by
263 *
264 * floor((floor(a/m) + f(x))/d)
265 *
266 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
267 * and can therefore not influence the result of the floor.
268 */
271 {
283 }
285 /* Assumes divs have been ordered if keep_divs is set.
286 */
289 {
307 }
317 }
326 /* We need to be careful about circular definitions,
327 * so for now we just remove the definition of div k
328 * if the equality contains any divs.
329 * If keep_divs is set, then the divs have been ordered
330 * and we can keep the definition as long as the result
331 * is still ordered.
332 */
340 }
341 }
343 /* Assumes divs have been ordered if keep_divs is set.
344 */
347 {
355 }
357 /* Check if elimination of div "div" using equality "eq" would not
358 * result in a div depending on a later div.
359 */
362 {
377 }
380 }
382 /* Eliminate divs based on equalities
383 */
386 {
401 isl_bool ok;
417 }
418 }
422 }
424 /* Eliminate divs based on inequalities
425 */
428 {
458 }
460 }
462 /* Does the equality constraint at position "eq" in "bmap" involve
463 * any local variables in the range [first, first + n)
464 * that are not marked as having an explicit representation?
465 */
468 {
477 isl_bool unknown;
486 }
489 }
491 /* The last local variable involved in the equality constraint
492 * at position "eq" in "bmap" is the local variable at position "div".
493 * It can therefore be used to extract an explicit representation
494 * for that variable.
495 * Do so unless the local variable already has an explicit representation or
496 * the explicit representation would involve any other local variables
497 * that in turn do not have an explicit representation.
498 * An equality constraint involving local variables without an explicit
499 * representation can be used in isl_basic_map_drop_redundant_divs
500 * to separate out an independent local variable. Introducing
501 * an explicit representation here would block this transformation,
502 * while the partial explicit representation in itself is not very useful.
503 * Set *progress if anything is changed.
504 *
505 * The equality constraint is of the form
506 *
507 * f(x) + n e >= 0
508 *
509 * with n a positive number. The explicit representation derived from
510 * this constraint is
511 *
512 * floor((-f(x))/n)
513 */
516 {
518 isl_bool involves;
542 }
546 {
569 }
578 progress);
585 }
592 }
595 }
599 {
601 progress));
602 }
606 {
612 }
614 }
616 /* Hash table of inequalities in a basic map.
617 * "index" is an array of addresses of inequalities in the basic map, some
618 * of which are NULL. The inequalities are hashed on the coefficients
619 * except the constant term.
620 * "size" is the number of elements in the array and is always a power of two
621 * "bits" is the number of bits need to represent an index into the array.
622 * "total" is the total dimension of the basic map.
623 */
629 };
631 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
632 */
635 {
652 }
654 /* Free the memory allocated by create_constraint_index.
655 */
657 {
659 }
661 /* Return the position in ci->index that contains the address of
662 * an inequality that is equal to *ineq up to the constant term,
663 * provided this address is not identical to "ineq".
664 * If there is no such inequality, then return the position where
665 * such an inequality should be inserted.
666 */
668 {
676 }
678 /* Return the position in ci->index that contains the address of
679 * an inequality that is equal to the k'th inequality of "bmap"
680 * up to the constant term, provided it does not point to the very
681 * same inequality.
682 * If there is no such inequality, then return the position where
683 * such an inequality should be inserted.
684 */
687 {
689 }
693 {
695 }
697 /* Fill in the "ci" data structure with the inequalities of "bset".
698 */
701 {
710 }
713 }
715 /* Is the inequality ineq (obviously) redundant with respect
716 * to the constraints in "ci"?
717 *
718 * Look for an inequality in "ci" with the same coefficients and then
719 * check if the contant term of "ineq" is greater than or equal
720 * to the constant term of that inequality. If so, "ineq" is clearly
721 * redundant.
722 *
723 * Note that hash_index_ineq ignores a stored constraint if it has
724 * the same address as the passed inequality. It is ok to pass
725 * the address of a local variable here since it will never be
726 * the same as the address of a constraint in "ci".
727 */
730 {
737 }
739 /* If we can eliminate more than one div, then we need to make
740 * sure we do it from last div to first div, in order not to
741 * change the position of the other divs that still need to
742 * be removed.
743 */
746 {
800 }
802 }
814 }
817 out:
821 }
824 {
836 }
838 }
840 /* Normalize divs that appear in equalities.
841 *
842 * In particular, we assume that bmap contains some equalities
843 * of the form
844 *
845 * a x = m * e_i
846 *
847 * and we want to replace the set of e_i by a minimal set and
848 * such that the new e_i have a canonical representation in terms
849 * of the vector x.
850 * If any of the equalities involves more than one divs, then
851 * we currently simply bail out.
852 *
853 * Let us first additionally assume that all equalities involve
854 * a div. The equalities then express modulo constraints on the
855 * remaining variables and we can use "parameter compression"
856 * to find a minimal set of constraints. The result is a transformation
857 *
858 * x = T(x') = x_0 + G x'
859 *
860 * with G a lower-triangular matrix with all elements below the diagonal
861 * non-negative and smaller than the diagonal element on the same row.
862 * We first normalize x_0 by making the same property hold in the affine
863 * T matrix.
864 * The rows i of G with a 1 on the diagonal do not impose any modulo
865 * constraint and simply express x_i = x'_i.
866 * For each of the remaining rows i, we introduce a div and a corresponding
867 * equality. In particular
868 *
869 * g_ii e_j = x_i - g_i(x')
870 *
871 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
872 * corresponding div (if g_kk != 1).
873 *
874 * If there are any equalities not involving any div, then we
875 * first apply a variable compression on the variables x:
876 *
877 * x = C x'' x'' = C_2 x
878 *
879 * and perform the above parameter compression on A C instead of on A.
880 * The resulting compression is then of the form
881 *
882 * x'' = T(x') = x_0 + G x'
883 *
884 * and in constructing the new divs and the corresponding equalities,
885 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
886 * by the corresponding row from C_2.
887 */
890 {
899 isl_int v;
931 }
932 }
941 }
947 }
957 }
964 }
969 /* We have to be careful because dropping equalities may reorder them */
979 }
980 }
986 else
987 needed++;
988 }
994 }
1004 else
1012 else
1015 }
1019 }
1026 done:
1030 error:
1036 }
1040 {
1050 }
1052 /* Check whether it is ok to define a div based on an inequality.
1053 * To avoid the introduction of circular definitions of divs, we
1054 * do not allow such a definition if the resulting expression would refer to
1055 * any other undefined divs or if any known div is defined in
1056 * terms of the unknown div.
1057 */
1060 {
1064 /* Not defined in terms of unknown divs */
1072 }
1074 /* No other div defined in terms of this one => avoid loops */
1082 }
1085 }
1087 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1088 * be a better expression than the current one?
1089 *
1090 * If we do not have any expression yet, then any expression would be better.
1091 * Otherwise we check if the last variable involved in the inequality
1092 * (disregarding the div that it would define) is in an earlier position
1093 * than the last variable involved in the current div expression.
1094 */
1097 {
1114 }
1116 /* Given two constraints "k" and "l" that are opposite to each other,
1117 * except for the constant term, check if we can use them
1118 * to obtain an expression for one of the hitherto unknown divs or
1119 * a "better" expression for a div for which we already have an expression.
1120 * "sum" is the sum of the constant terms of the constraints.
1121 * If this sum is strictly smaller than the coefficient of one
1122 * of the divs, then this pair can be used define the div.
1123 * To avoid the introduction of circular definitions of divs, we
1124 * do not use the pair if the resulting expression would refer to
1125 * any other undefined divs or if any known div is defined in
1126 * terms of the unknown div.
1127 */
1131 {
1136 isl_bool set_div;
1151 else
1156 }
1158 }
1162 {
1166 isl_int sum;
1181 }
1189 }
1204 }
1206 /* We need to break out of the loop after these
1207 * changes since the contents of the hash
1208 * will no longer be valid.
1209 * Plus, we probably we want to regauss first.
1210 */
1218 }
1223 }
1225 /* Detect all pairs of inequalities that form an equality.
1226 *
1227 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1228 * Call it repeatedly while it is making progress.
1229 */
1232 {
1244 }
1246 /* Eliminate knowns divs from constraints where they appear with
1247 * a (positive or negative) unit coefficient.
1248 *
1249 * That is, replace
1250 *
1251 * floor(e/m) + f >= 0
1252 *
1253 * by
1254 *
1255 * e + m f >= 0
1256 *
1257 * and
1258 *
1259 * -floor(e/m) + f >= 0
1260 *
1261 * by
1262 *
1263 * -e + m f + m - 1 >= 0
1264 *
1265 * The first conversion is valid because floor(e/m) >= -f is equivalent
1266 * to e/m >= -f because -f is an integral expression.
1267 * The second conversion follows from the fact that
1268 *
1269 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1270 *
1271 *
1272 * Note that one of the div constraints may have been eliminated
1273 * due to being redundant with respect to the constraint that is
1274 * being modified by this function. The modified constraint may
1275 * no longer imply this div constraint, so we add it back to make
1276 * sure we do not lose any information.
1277 *
1278 * We skip integral divs, i.e., those with denominator 1, as we would
1279 * risk eliminating the div from the div constraints. We do not need
1280 * to handle those divs here anyway since the div constraints will turn
1281 * out to form an equality and this equality can then be used to eliminate
1282 * the div from all constraints.
1283 */
1286 {
1318 else
1328 }
1333 }
1334 }
1337 }
1340 {
1345 isl_bool empty;
1361 /* requires equalities in normal form */
1367 }
1369 }
1372 {
1374 }
1379 {
1411 }
1415 {
1417 }
1420 /* If the only constraints a div d=floor(f/m)
1421 * appears in are its two defining constraints
1422 *
1423 * f - m d >=0
1424 * -(f - (m - 1)) + m d >= 0
1425 *
1426 * then it can safely be removed.
1427 */
1429 {
1438 isl_bool red;
1445 }
1452 }
1455 }
1457 /*
1458 * Remove divs that don't occur in any of the constraints or other divs.
1459 * These can arise when dropping constraints from a basic map or
1460 * when the divs of a basic map have been temporarily aligned
1461 * with the divs of another basic map.
1462 */
1464 {
1471 isl_bool redundant;
1479 }
1481 }
1483 /* Mark "bmap" as final, without checking for obviously redundant
1484 * integer divisions. This function should be used when "bmap"
1485 * is known not to involve any such integer divisions.
1486 */
1489 {
1494 }
1496 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1497 */
1499 {
1503 }
1506 {
1508 }
1510 /* Remove definition of any div that is defined in terms of the given variable.
1511 * The div itself is not removed. Functions such as
1512 * eliminate_divs_ineq depend on the other divs remaining in place.
1513 */
1516 {
1530 }
1532 }
1534 /* Eliminate the specified variables from the constraints using
1535 * Fourier-Motzkin. The variables themselves are not removed.
1536 */
1539 {
1571 }
1578 n_lower++;
1580 n_upper++;
1581 }
1605 }
1608 }
1620 }
1621 }
1626 error:
1629 }
1633 {
1636 }
1638 /* Eliminate the specified n dimensions starting at first from the
1639 * constraints, without removing the dimensions from the space.
1640 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1641 * Otherwise, they are projected out and the original space is restored.
1642 */
1646 {
1662 }
1669 error:
1672 }
1677 {
1679 }
1681 /* Remove all constraints from "bmap" that reference any unknown local
1682 * variables (directly or indirectly).
1683 *
1684 * Dropping all constraints on a local variable will make it redundant,
1685 * so it will get removed implicitly by
1686 * isl_basic_map_drop_constraints_involving_dims. Some other local
1687 * variables may also end up becoming redundant if they only appear
1688 * in constraints together with the unknown local variable.
1689 * Therefore, start over after calling
1690 * isl_basic_map_drop_constraints_involving_dims.
1691 */
1694 {
1695 isl_bool known;
1720 }
1723 }
1725 /* Remove all constraints from "map" that reference any unknown local
1726 * variables (directly or indirectly).
1727 *
1728 * Since constraints may get dropped from the basic maps,
1729 * they may no longer be disjoint from each other.
1730 */
1733 {
1735 isl_bool known;
1753 }
1759 }
1761 /* Don't assume equalities are in order, because align_divs
1762 * may have changed the order of the divs.
1763 */
1765 {
1778 }
1779 }
1780 }
1784 {
1786 }
1790 {
1804 }
1806 }
1808 }
1812 {
1815 }
1819 {
1829 }
1850 error:
1854 }
1856 /* For each inequality in "ineq" that is a shifted (more relaxed)
1857 * copy of an inequality in "context", mark the corresponding entry
1858 * in "row" with -1.
1859 * If an inequality only has a non-negative constant term, then
1860 * mark it as well.
1861 */
1864 {
1881 isl_bool redundant;
1887 }
1894 }
1897 error:
1900 }
1904 {
1917 isl_bool redundant;
1929 }
1932 error:
1935 }
1937 /* Remove constraints from "bmap" that are identical to constraints
1938 * in "context" or that are more relaxed (greater constant term).
1939 *
1940 * We perform the test for shifted copies on the pure constraints
1941 * in remove_shifted_constraints.
1942 */
1945 {
1954 }
1967 error:
1971 }
1973 /* Does the (linear part of a) constraint "c" involve any of the "len"
1974 * "relevant" dimensions?
1975 */
1977 {
1985 }
1988 }
1990 /* Drop constraints from "bmap" that do not involve any of
1991 * the dimensions marked "relevant".
1992 */
1995 {
2010 }
2017 }
2020 }
2022 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2023 *
2024 * In particular, for any variable involved in the constraint,
2025 * find the actual group id from before and replace the group
2026 * of the corresponding variable by the minimal group of all
2027 * the variables involved in the constraint considered so far
2028 * (if this minimum is smaller) or replace the minimum by this group
2029 * (if the minimum is larger).
2030 *
2031 * At the end, all the variables in "c" will (indirectly) point
2032 * to the minimal of the groups that they referred to originally.
2033 */
2035 {
2052 }
2053 }
2055 /* Allocate an array of groups of variables, one for each variable
2056 * in "context", initialized to zero.
2057 */
2059 {
2066 }
2068 /* Drop constraints from "bmap" that only involve variables that are
2069 * not related to any of the variables marked with a "-1" in "group".
2070 *
2071 * We construct groups of variables that collect variables that
2072 * (indirectly) appear in some common constraint of "bmap".
2073 * Each group is identified by the first variable in the group,
2074 * except for the special group of variables that was already identified
2075 * in the input as -1 (or are related to those variables).
2076 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2077 * otherwise the group of i is the group of group[i].
2078 *
2079 * We first initialize groups for the remaining variables.
2080 * Then we iterate over the constraints of "bmap" and update the
2081 * group of the variables in the constraint by the smallest group.
2082 * Finally, we resolve indirect references to groups by running over
2083 * the variables.
2084 *
2085 * After computing the groups, we drop constraints that do not involve
2086 * any variables in the -1 group.
2087 */
2090 {
2107 }
2125 }
2127 /* Drop constraints from "context" that are irrelevant for computing
2128 * the gist of "bset".
2129 *
2130 * In particular, drop constraints in variables that are not related
2131 * to any of the variables involved in the constraints of "bset"
2132 * in the sense that there is no sequence of constraints that connects them.
2133 *
2134 * We first mark all variables that appear in "bset" as belonging
2135 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2136 */
2139 {
2160 }
2166 }
2169 }
2171 /* Drop constraints from "context" that are irrelevant for computing
2172 * the gist of the inequalities "ineq".
2173 * Inequalities in "ineq" for which the corresponding element of row
2174 * is set to -1 have already been marked for removal and should be ignored.
2175 *
2176 * In particular, drop constraints in variables that are not related
2177 * to any of the variables involved in "ineq"
2178 * in the sense that there is no sequence of constraints that connects them.
2179 *
2180 * We first mark all variables that appear in "bset" as belonging
2181 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2182 */
2185 {
2206 }
2209 }
2212 }
2214 /* Do all "n" entries of "row" contain a negative value?
2215 */
2217 {
2225 }
2227 /* Update the inequalities in "bset" based on the information in "row"
2228 * and "tab".
2229 *
2230 * In particular, the array "row" contains either -1, meaning that
2231 * the corresponding inequality of "bset" is redundant, or the index
2232 * of an inequality in "tab".
2233 *
2234 * If the row entry is -1, then drop the inequality.
2235 * Otherwise, if the constraint is marked redundant in the tableau,
2236 * then drop the inequality. Similarly, if it is marked as an equality
2237 * in the tableau, then turn the inequality into an equality and
2238 * perform Gaussian elimination.
2239 */
2242 {
2259 }
2269 }
2270 }
2276 }
2278 /* Update the inequalities in "bset" based on the information in "row"
2279 * and "tab" and free all arguments (other than "bset").
2280 */
2285 {
2294 }
2296 /* Remove all information from bset that is redundant in the context
2297 * of context.
2298 * "ineq" contains the (possibly transformed) inequalities of "bset",
2299 * in the same order.
2300 * The (explicit) equalities of "bset" are assumed to have been taken
2301 * into account by the transformation such that only the inequalities
2302 * are relevant.
2303 * "context" is assumed not to be empty.
2304 *
2305 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2306 * A value of -1 means that the inequality is obviously redundant and may
2307 * not even appear in "tab".
2308 *
2309 * We first mark the inequalities of "bset"
2310 * that are obviously redundant with respect to some inequality in "context".
2311 * Then we remove those constraints from "context" that have become
2312 * irrelevant for computing the gist of "bset".
2313 * Note that this removal of constraints cannot be replaced by
2314 * a factorization because factors in "bset" may still be connected
2315 * to each other through constraints in "context".
2316 *
2317 * If there are any inequalities left, we construct a tableau for
2318 * the context and then add the inequalities of "bset".
2319 * Before adding these inequalities, we freeze all constraints such that
2320 * they won't be considered redundant in terms of the constraints of "bset".
2321 * Then we detect all redundant constraints (among the
2322 * constraints that weren't frozen), first by checking for redundancy in the
2323 * the tableau and then by checking if replacing a constraint by its negation
2324 * would lead to an empty set. This last step is fairly expensive
2325 * and could be optimized by more reuse of the tableau.
2326 * Finally, we update bset according to the results.
2327 */
2330 {
2345 }
2381 }
2405 }
2410 }
2414 error:
2422 }
2424 /* Extract the inequalities of "bset" as an isl_mat.
2425 */
2427 {
2441 }
2443 /* Remove all information from "bset" that is redundant in the context
2444 * of "context", for the case where both "bset" and "context" are
2445 * full-dimensional.
2446 */
2449 {
2454 }
2456 /* Remove all information from "bset" that is redundant in the context
2457 * of "context", for the case where the combined equalities of
2458 * "bset" and "context" allow for a compression that can be obtained
2459 * by preapplication of "T".
2460 *
2461 * "bset" itself is not transformed by "T". Instead, the inequalities
2462 * are extracted from "bset" and those are transformed by "T".
2463 * uset_gist_full then determines which of the transformed inequalities
2464 * are redundant with respect to the transformed "context" and removes
2465 * the corresponding inequalities from "bset".
2466 *
2467 * After preapplying "T" to the inequalities, any common factor is
2468 * removed from the coefficients. If this results in a tightening
2469 * of the constant term, then the same tightening is applied to
2470 * the corresponding untransformed inequality in "bset".
2471 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2472 *
2473 * g f'(x) + r >= 0
2474 *
2475 * with 0 <= r < g, then it is equivalent to
2476 *
2477 * f'(x) >= 0
2478 *
2479 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2480 * subspace compressed by T since the latter would be transformed to
2481 *
2482 * g f'(x) >= 0
2483 */
2487 {
2491 isl_int rem;
2503 }
2526 }
2530 error:
2535 }
2537 /* Project "bset" onto the variables that are involved in "template".
2538 */
2541 {
2550 isl_bool involved;
2559 }
2562 }
2564 /* Remove all information from bset that is redundant in the context
2565 * of context. In particular, equalities that are linear combinations
2566 * of those in context are removed. Then the inequalities that are
2567 * redundant in the context of the equalities and inequalities of
2568 * context are removed.
2569 *
2570 * First of all, we drop those constraints from "context"
2571 * that are irrelevant for computing the gist of "bset".
2572 * Alternatively, we could factorize the intersection of "context" and "bset".
2573 *
2574 * We first compute the intersection of the integer affine hulls
2575 * of "bset" and "context",
2576 * compute the gist inside this intersection and then reduce
2577 * the constraints with respect to the equalities of the context
2578 * that only involve variables already involved in the input.
2579 *
2580 * If two constraints are mutually redundant, then uset_gist_full
2581 * will remove the second of those constraints. We therefore first
2582 * sort the constraints so that constraints not involving existentially
2583 * quantified variables are given precedence over those that do.
2584 * We have to perform this sorting before the variable compression,
2585 * because that may effect the order of the variables.
2586 */
2589 {
2614 }
2619 }
2629 }
2640 }
2643 error:
2647 }
2649 /* Return the number of equality constraints in "bmap" that involve
2650 * local variables. This function assumes that Gaussian elimination
2651 * has been applied to the equality constraints.
2652 */
2654 {
2674 }
2676 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2677 * The constraints are assumed not to involve any local variables.
2678 */
2681 {
2699 }
2704 error:
2709 }
2711 /* Construct and return a variable compression based on the equality
2712 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2713 * "n1" is the number of (initial) equality constraints in "bmap1"
2714 * that do involve local variables.
2715 * "n2" is the number of (initial) equality constraints in "bmap2"
2716 * that do involve local variables.
2717 * "total" is the total number of other variables.
2718 * This function assumes that Gaussian elimination
2719 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2720 * such that the equality constraints not involving local variables
2721 * are those that start at "n1" or "n2".
2722 *
2723 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2724 * then simply compute the compression based on the equality constraints
2725 * in the other basic map.
2726 * Otherwise, combine the equality constraints from both into a new
2727 * basic map such that Gaussian elimination can be applied to this combination
2728 * and then construct a variable compression from the resulting
2729 * equality constraints.
2730 */
2734 {
2744 }
2749 }
2764 }
2766 /* Extract the stride constraints from "bmap", compressed
2767 * with respect to both the stride constraints in "context" and
2768 * the remaining equality constraints in both "bmap" and "context".
2769 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2770 * "context_n_eq" is the number of (initial) stride constraints in "context".
2771 *
2772 * Let x be all variables in "bmap" (and "context") other than the local
2773 * variables. First compute a variable compression
2774 *
2775 * x = V x'
2776 *
2777 * based on the non-stride equality constraints in "bmap" and "context".
2778 * Consider the stride constraints of "context",
2779 *
2780 * A(x) + B(y) = 0
2781 *
2782 * with y the local variables and plug in the variable compression,
2783 * resulting in
2784 *
2785 * A(V x') + B(y) = 0
2786 *
2787 * Use these constraints to compute a parameter compression on x'
2788 *
2789 * x' = T x''
2790 *
2791 * Now consider the stride constraints of "bmap"
2792 *
2793 * C(x) + D(y) = 0
2794 *
2795 * and plug in x = V*T x''.
2796 * That is, return A = [C*V*T D].
2797 */
2801 {
2830 }
2832 /* Remove the prime factors from *g that have an exponent that
2833 * is strictly smaller than the exponent in "c".
2834 * All exponents in *g are known to be smaller than or equal
2835 * to those in "c".
2836 *
2837 * That is, if *g is equal to
2838 *
2839 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2840 *
2841 * and "c" is equal to
2842 *
2843 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2844 *
2845 * then update *g to
2846 *
2847 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2848 * p_n^{e_n * (e_n = f_n)}
2849 *
2850 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2851 * neither does the gcd of *g and c / *g.
2852 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2853 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2854 * Dividing *g by this gcd therefore strictly reduces the exponent
2855 * of the prime factors that need to be removed, while leaving the
2856 * other prime factors untouched.
2857 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2858 * removes all undesired factors, without removing any others.
2859 */
2861 {
2862 isl_int t;
2871 }
2873 }
2875 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2876 * of the same stride constraints in a compressed space that exploits
2877 * all equalities in the context and the other equalities in "bmap".
2878 *
2879 * If the stride constraints of "bmap" are of the form
2880 *
2881 * C(x) + D(y) = 0
2882 *
2883 * then A is of the form
2884 *
2885 * B(x') + D(y) = 0
2886 *
2887 * If any of these constraints involves only a single local variable y,
2888 * then the constraint appears as
2889 *
2890 * f(x) + m y_i = 0
2891 *
2892 * in "bmap" and as
2893 *
2894 * h(x') + m y_i = 0
2895 *
2896 * in "A".
2897 *
2898 * Let g be the gcd of m and the coefficients of h.
2899 * Then, in particular, g is a divisor of the coefficients of h and
2900 *
2901 * f(x) = h(x')
2902 *
2903 * is known to be a multiple of g.
2904 * If some prime factor in m appears with the same exponent in g,
2905 * then it can be removed from m because f(x) is already known
2906 * to be a multiple of g and therefore in particular of this power
2907 * of the prime factors.
2908 * Prime factors that appear with a smaller exponent in g cannot
2909 * be removed from m.
2910 * Let g' be the divisor of g containing all prime factors that
2911 * appear with the same exponent in m and g, then
2912 *
2913 * f(x) + m y_i = 0
2914 *
2915 * can be replaced by
2916 *
2917 * f(x) + m/g' y_i' = 0
2918 *
2919 * Note that (if g' != 1) this changes the explicit representation
2920 * of y_i to that of y_i', so the integer division at position i
2921 * is marked unknown and later recomputed by a call to
2922 * isl_basic_map_gauss.
2923 */
2926 {
2930 isl_int gcd;
2964 }
2971 error:
2975 }
2977 /* Simplify the stride constraints in "bmap" based on
2978 * the remaining equality constraints in "bmap" and all equality
2979 * constraints in "context".
2980 * Only do this if both "bmap" and "context" have stride constraints.
2981 *
2982 * First extract a copy of the stride constraints in "bmap" in a compressed
2983 * space exploiting all the other equality constraints and then
2984 * use this compressed copy to simplify the original stride constraints.
2985 */
2988 {
3010 }
3012 /* Return a basic map that has the same intersection with "context" as "bmap"
3013 * and that is as "simple" as possible.
3014 *
3015 * The core computation is performed on the pure constraints.
3016 * When we add back the meaning of the integer divisions, we need
3017 * to (re)introduce the div constraints. If we happen to have
3018 * discovered that some of these integer divisions are equal to
3019 * some affine combination of other variables, then these div
3020 * constraints may end up getting simplified in terms of the equalities,
3021 * resulting in extra inequalities on the other variables that
3022 * may have been removed already or that may not even have been
3023 * part of the input. We try and remove those constraints of
3024 * this form that are most obviously redundant with respect to
3025 * the context. We also remove those div constraints that are
3026 * redundant with respect to the other constraints in the result.
3027 *
3028 * The stride constraints among the equality constraints in "bmap" are
3029 * also simplified with respecting to the other equality constraints
3030 * in "bmap" and with respect to all equality constraints in "context".
3031 */
3034 {
3045 }
3051 }
3055 }
3077 }
3096 error:
3100 }
3102 /*
3103 * Assumes context has no implicit divs.
3104 */
3107 {
3118 }
3138 }
3139 }
3143 error:
3147 }
3149 /* Drop all inequalities from "bmap" that also appear in "context".
3150 * "context" is assumed to have only known local variables and
3151 * the initial local variables of "bmap" are assumed to be the same
3152 * as those of "context".
3153 * The constraints of both "bmap" and "context" are assumed
3154 * to have been sorted using isl_basic_map_sort_constraints.
3155 *
3156 * Run through the inequality constraints of "bmap" and "context"
3157 * in sorted order.
3158 * If a constraint of "bmap" involves variables not in "context",
3159 * then it cannot appear in "context".
3160 * If a matching constraint is found, it is removed from "bmap".
3161 */
3164 {
3183 }
3189 }
3193 }
3198 }
3201 }
3204 }
3206 /* Drop all equalities from "bmap" that also appear in "context".
3207 * "context" is assumed to have only known local variables and
3208 * the initial local variables of "bmap" are assumed to be the same
3209 * as those of "context".
3210 *
3211 * Run through the equality constraints of "bmap" and "context"
3212 * in sorted order.
3213 * If a constraint of "bmap" involves variables not in "context",
3214 * then it cannot appear in "context".
3215 * If a matching constraint is found, it is removed from "bmap".
3216 */
3219 {
3243 }
3247 }
3252 }
3255 }
3258 }
3260 /* Remove the constraints in "context" from "bmap".
3261 * "context" is assumed to have explicit representations
3262 * for all local variables.
3263 *
3264 * First align the divs of "bmap" to those of "context" and
3265 * sort the constraints. Then drop all constraints from "bmap"
3266 * that appear in "context".
3267 */
3270 {
3285 }
3304 error:
3308 }
3310 /* Replace "map" by the disjunct at position "pos" and free "context".
3311 */
3314 {
3321 }
3323 /* Remove the constraints in "context" from "map".
3324 * If any of the disjuncts in the result turns out to be the universe,
3325 * then return this universe.
3326 * "context" is assumed to have explicit representations
3327 * for all local variables.
3328 */
3331 {
3341 }
3360 }
3367 error:
3371 }
3373 /* Replace "map" by a universe map in the same space and free "drop".
3374 */
3377 {
3384 }
3386 /* Return a map that has the same intersection with "context" as "map"
3387 * and that is as "simple" as possible.
3388 *
3389 * If "map" is already the universe, then we cannot make it any simpler.
3390 * Similarly, if "context" is the universe, then we cannot exploit it
3391 * to simplify "map"
3392 * If "map" and "context" are identical to each other, then we can
3393 * return the corresponding universe.
3394 *
3395 * If either "map" or "context" consists of multiple disjuncts,
3396 * then check if "context" happens to be a subset of "map",
3397 * in which case all constraints can be removed.
3398 * In case of multiple disjuncts, the standard procedure
3399 * may not be able to detect that all constraints can be removed.
3400 *
3401 * If none of these cases apply, we have to work a bit harder.
3402 * During this computation, we make use of a single disjunct context,
3403 * so if the original context consists of more than one disjunct
3404 * then we need to approximate the context by a single disjunct set.
3405 * Simply taking the simple hull may drop constraints that are
3406 * only implicitly available in each disjunct. We therefore also
3407 * look for constraints among those defining "map" that are valid
3408 * for the context. These can then be used to simplify away
3409 * the corresponding constraints in "map".
3410 */
3413 {
3417 isl_bool subset;
3428 }
3444 }
3460 list);
3461 }
3463 error:
3467 }
3471 {
3473 }
3477 {
3480 }
3484 {
3487 }
3491 {
3496 }
3500 {
3502 }
3504 /* Compute the gist of "bmap" with respect to the constraints "context"
3505 * on the domain.
3506 */
3509 {
3515 }
3519 {
3523 }
3527 {
3531 }
3535 {
3539 }
3543 {
3545 }
3547 /* Quick check to see if two basic maps are disjoint.
3548 * In particular, we reduce the equalities and inequalities of
3549 * one basic map in the context of the equalities of the other
3550 * basic map and check if we get a contradiction.
3551 */
3554 {
3586 }
3594 }
3603 }
3607 disjoint:
3611 error:
3615 }
3619 {
3622 }
3624 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3625 */
3629 {
3640 }
3641 }
3644 }
3646 /* Are "map1" and "map2" obviously disjoint, based on information
3647 * that can be derived without looking at the individual basic maps?
3648 *
3649 * In particular, if one of them is empty or if they live in different spaces
3650 * (ignoring parameters), then they are clearly disjoint.
3651 */
3654 {
3655 isl_bool disjoint;
3656 isl_bool match;
3680 }
3682 /* Are "map1" and "map2" obviously disjoint?
3683 *
3684 * If one of them is empty or if they live in different spaces (ignoring
3685 * parameters), then they are clearly disjoint.
3686 * This is checked by isl_map_plain_is_disjoint_global.
3687 *
3688 * If they have different parameters, then we skip any further tests.
3689 *
3690 * If they are obviously equal, but not obviously empty, then we will
3691 * not be able to detect if they are disjoint.
3692 *
3693 * Otherwise we check if each basic map in "map1" is obviously disjoint
3694 * from each basic map in "map2".
3695 */
3698 {
3699 isl_bool disjoint;
3700 isl_bool intersect;
3701 isl_bool match;
3716 }
3718 /* Are "map1" and "map2" disjoint?
3719 *
3720 * They are disjoint if they are "obviously disjoint" or if one of them
3721 * is empty. Otherwise, they are not disjoint if one of them is universal.
3722 * If the two inputs are (obviously) equal and not empty, then they are
3723 * not disjoint.
3724 * If none of these cases apply, then check if all pairs of basic maps
3725 * are disjoint.
3726 */
3728 {
3729 isl_bool disjoint;
3730 isl_bool intersect;
3757 }
3759 /* Are "bmap1" and "bmap2" disjoint?
3760 *
3761 * They are disjoint if they are "obviously disjoint" or if one of them
3762 * is empty. Otherwise, they are not disjoint if one of them is universal.
3763 * If none of these cases apply, we compute the intersection and see if
3764 * the result is empty.
3765 */
3768 {
3769 isl_bool disjoint;
3770 isl_bool intersect;
3799 }
3801 /* Are "bset1" and "bset2" disjoint?
3802 */
3805 {
3807 }
3811 {
3813 }
3815 /* Are "set1" and "set2" disjoint?
3816 */
3818 {
3820 }
3822 /* Is "v" equal to 0, 1 or -1?
3823 */
3825 {
3827 }
3829 /* Check if we can combine a given div with lower bound l and upper
3830 * bound u with some other div and if so return that other div.
3831 * Otherwise return -1.
3832 *
3833 * We first check that
3834 * - the bounds are opposites of each other (except for the constant
3835 * term)
3836 * - the bounds do not reference any other div
3837 * - no div is defined in terms of this div
3838 *
3839 * Let m be the size of the range allowed on the div by the bounds.
3840 * That is, the bounds are of the form
3841 *
3842 * e <= a <= e + m - 1
3843 *
3844 * with e some expression in the other variables.
3845 * We look for another div b such that no third div is defined in terms
3846 * of this second div b and such that in any constraint that contains
3847 * a (except for the given lower and upper bound), also contains b
3848 * with a coefficient that is m times that of b.
3849 * That is, all constraints (execpt for the lower and upper bound)
3850 * are of the form
3851 *
3852 * e + f (a + m b) >= 0
3853 *
3854 * Furthermore, in the constraints that only contain b, the coefficient
3855 * of b should be equal to 1 or -1.
3856 * If so, we return b so that "a + m b" can be replaced by
3857 * a single div "c = a + m b".
3858 */
3861 {
3883 }
3893 }
3905 }
3916 }
3929 }
3934 }
3938 }
3940 /* Internal data structure used during the construction and/or evaluation of
3941 * an inequality that ensures that a pair of bounds always allows
3942 * for an integer value.
3943 *
3944 * "tab" is the tableau in which the inequality is evaluated. It may
3945 * be NULL until it is actually needed.
3946 * "v" contains the inequality coefficients.
3947 * "g", "fl" and "fu" are temporary scalars used during the construction and
3948 * evaluation.
3949 */
3953 isl_int g;
3954 isl_int fl;
3955 isl_int fu;
3956 };
3958 /* Free all the memory allocated by the fields of "data".
3959 */
3961 {
3967 }
3969 /* Is the inequality stored in data->v satisfied by "bmap"?
3970 * That is, does it only attain non-negative values?
3971 * data->tab is a tableau corresponding to "bmap".
3972 */
3975 {
3986 }
3988 /* Given a lower and an upper bound on div i, do they always allow
3989 * for an integer value of the given div?
3990 * Determine this property by constructing an inequality
3991 * such that the property is guaranteed when the inequality is nonnegative.
3992 * The lower bound is inequality l, while the upper bound is inequality u.
3993 * The constructed inequality is stored in data->v.
3994 *
3995 * Let the upper bound be
3996 *
3997 * -n_u a + e_u >= 0
3998 *
3999 * and the lower bound
4000 *
4001 * n_l a + e_l >= 0
4002 *
4003 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4004 * We have
4005 *
4006 * - f_u e_l <= f_u f_l g a <= f_l e_u
4007 *
4008 * Since all variables are integer valued, this is equivalent to
4009 *
4010 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4011 *
4012 * If this interval is at least f_u f_l g, then it contains at least
4013 * one integer value for a.
4014 * That is, the test constraint is
4015 *
4016 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4017 *
4018 * or
4019 *
4020 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4021 *
4022 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4023 * then the constraint can be scaled down by a factor g',
4024 * with the constant term replaced by
4025 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4026 * Note that the result of applying Fourier-Motzkin to this pair
4027 * of constraints is
4028 *
4029 * f_l e_u + f_u e_l >= 0
4030 *
4031 * If the constant term of the scaled down version of this constraint,
4032 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4033 * term of the scaled down test constraint, then the test constraint
4034 * is known to hold and no explicit evaluation is required.
4035 * This is essentially the Omega test.
4036 *
4037 * If the test constraint consists of only a constant term, then
4038 * it is sufficient to look at the sign of this constant term.
4039 */
4042 {
4067 }
4077 }
4079 /* Remove more kinds of divs that are not strictly needed.
4080 * In particular, if all pairs of lower and upper bounds on a div
4081 * are such that they allow at least one integer value of the div,
4082 * then we can eliminate the div using Fourier-Motzkin without
4083 * introducing any spurious solutions.
4084 *
4085 * If at least one of the two constraints has a unit coefficient for the div,
4086 * then the presence of such a value is guaranteed so there is no need to check.
4087 * In particular, the value attained by the bound with unit coefficient
4088 * can serve as this intermediate value.
4089 */
4092 {
4115 isl_bool has_int;
4123 }
4144 }
4147 }
4151 }
4155 }
4158 }
4169 error:
4174 }
4176 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4177 * and the upper bound u, div1 always occurs together with div2 in the form
4178 * (div1 + m div2), where m is the constant range on the variable div1
4179 * allowed by l and u, replace the pair div1 and div2 by a single
4180 * div that is equal to div1 + m div2.
4181 *
4182 * The new div will appear in the location that contains div2.
4183 * We need to modify all constraints that contain
4184 * div2 = (div - div1) / m
4185 * The coefficient of div2 is known to be equal to 1 or -1.
4186 * (If a constraint does not contain div2, it will also not contain div1.)
4187 * If the constraint also contains div1, then we know they appear
4188 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4189 * i.e., the coefficient of div is f.
4190 *
4191 * Otherwise, we first need to introduce div1 into the constraint.
4192 * Let the l be
4193 *
4194 * div1 + f >=0
4195 *
4196 * and u
4197 *
4198 * -div1 + f' >= 0
4199 *
4200 * A lower bound on div2
4201 *
4202 * div2 + t >= 0
4203 *
4204 * can be replaced by
4205 *
4206 * m div2 + div1 + m t + f >= 0
4207 *
4208 * An upper bound
4209 *
4210 * -div2 + t >= 0
4211 *
4212 * can be replaced by
4213 *
4214 * -(m div2 + div1) + m t + f' >= 0
4215 *
4216 * These constraint are those that we would obtain from eliminating
4217 * div1 using Fourier-Motzkin.
4218 *
4219 * After all constraints have been modified, we drop the lower and upper
4220 * bound and then drop div1.
4221 * Since the new div is only placed in the same location that used
4222 * to store div2, but otherwise has a different meaning, any possible
4223 * explicit representation of the original div2 is removed.
4224 */
4227 {
4229 isl_int m;
4251 else
4254 }
4258 }
4267 }
4271 }
4273 /* First check if we can coalesce any pair of divs and
4274 * then continue with dropping more redundant divs.
4275 *
4276 * We loop over all pairs of lower and upper bounds on a div
4277 * with coefficient 1 and -1, respectively, check if there
4278 * is any other div "c" with which we can coalesce the div
4279 * and if so, perform the coalescing.
4280 */
4283 {
4306 }
4307 }
4308 }
4313 }
4316 }
4318 /* Are the "n" coefficients starting at "first" of inequality constraints
4319 * "i" and "j" of "bmap" equal to each other?
4320 */
4323 {
4325 }
4327 /* Are the "n" coefficients starting at "first" of inequality constraints
4328 * "i" and "j" of "bmap" opposite to each other?
4329 */
4332 {
4334 }
4336 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4337 * apart from the constant term?
4338 */
4340 {
4345 }
4347 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4348 * apart from the constant term and the coefficient at position "pos"?
4349 */
4352 {
4358 }
4360 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4361 * apart from the constant term and the coefficient at position "pos"?
4362 */
4365 {
4371 }
4373 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4374 * been modified, simplying it if "simplify" is set.
4375 * Free the temporary data structure "pairs" that was associated
4376 * to the old version of "bmap".
4377 */
4380 {
4385 }
4387 /* Is "div" the single unknown existentially quantified variable
4388 * in inequality constraint "ineq" of "bmap"?
4389 * "div" is known to have a non-zero coefficient in "ineq".
4390 */
4393 {
4396 isl_bool known;
4406 isl_bool known;
4415 }
4418 }
4420 /* Does integer division "div" have coefficient 1 in inequality constraint
4421 * "ineq" of "map"?
4422 */
4424 {
4432 }
4434 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4435 * then try and drop redundant divs again,
4436 * freeing the temporary data structure "pairs" that was associated
4437 * to the old version of "bmap".
4438 */
4441 {
4445 }
4447 /* Drop the integer division at position "div", along with the two
4448 * inequality constraints "ineq1" and "ineq2" in which it appears
4449 * from "bmap" and then try and drop redundant divs again,
4450 * freeing the temporary data structure "pairs" that was associated
4451 * to the old version of "bmap".
4452 */
4456 {
4463 }
4466 }
4468 /* Given two inequality constraints
4469 *
4470 * f(x) + n d + c >= 0, (ineq)
4471 *
4472 * with d the variable at position "pos", and
4473 *
4474 * f(x) + c0 >= 0, (lower)
4475 *
4476 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4477 * determined by the first constraint.
4478 * That is, store
4479 *
4480 * ceil((c0 - c)/n)
4481 *
4482 * in *l.
4483 */
4486 {
4490 }
4492 /* Given two inequality constraints
4493 *
4494 * f(x) + n d + c >= 0, (ineq)
4495 *
4496 * with d the variable at position "pos", and
4497 *
4498 * -f(x) - c0 >= 0, (upper)
4499 *
4500 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4501 * determined by the first constraint.
4502 * That is, store
4503 *
4504 * ceil((-c1 - c)/n)
4505 *
4506 * in *u.
4507 */
4510 {
4514 }
4516 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4517 * does the corresponding lower bound have a fixed value in "bmap"?
4518 *
4519 * In particular, "ineq" is of the form
4520 *
4521 * f(x) + n d + c >= 0
4522 *
4523 * with n > 0, c the constant term and
4524 * d the existentially quantified variable "div".
4525 * That is, the lower bound is
4526 *
4527 * ceil((-f(x) - c)/n)
4528 *
4529 * Look for a pair of constraints
4530 *
4531 * f(x) + c0 >= 0
4532 * -f(x) + c1 >= 0
4533 *
4534 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4535 * That is, check that
4536 *
4537 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4538 *
4539 * If so, return the index of inequality f(x) + c0 >= 0.
4540 * Otherwise, return -1.
4541 */
4543 {
4560 }
4564 }
4565 }
4582 }
4584 /* Given a lower bound constraint "ineq" on the existentially quantified
4585 * variable "div", such that the corresponding lower bound has
4586 * a fixed value in "bmap", assign this fixed value to the variable and
4587 * then try and drop redundant divs again,
4588 * freeing the temporary data structure "pairs" that was associated
4589 * to the old version of "bmap".
4590 * "lower" determines the constant value for the lower bound.
4591 *
4592 * In particular, "ineq" is of the form
4593 *
4594 * f(x) + n d + c >= 0,
4595 *
4596 * while "lower" is of the form
4597 *
4598 * f(x) + c0 >= 0
4599 *
4600 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4601 * is ceil((c0 - c)/n).
4602 */
4605 {
4606 isl_int c;
4619 }
4621 /* Remove divs that are not strictly needed based on the inequality
4622 * constraints.
4623 * In particular, if a div only occurs positively (or negatively)
4624 * in constraints, then it can simply be dropped.
4625 * Also, if a div occurs in only two constraints and if moreover
4626 * those two constraints are opposite to each other, except for the constant
4627 * term and if the sum of the constant terms is such that for any value
4628 * of the other values, there is always at least one integer value of the
4629 * div, i.e., if one plus this sum is greater than or equal to
4630 * the (absolute value) of the coefficient of the div in the constraints,
4631 * then we can also simply drop the div.
4632 *
4633 * If an existentially quantified variable does not have an explicit
4634 * representation, appears in only a single lower bound that does not
4635 * involve any other such existentially quantified variables and appears
4636 * in this lower bound with coefficient 1,
4637 * then fix the variable to the value of the lower bound. That is,
4638 * turn the inequality into an equality.
4639 * If for any value of the other variables, there is any value
4640 * for the existentially quantified variable satisfying the constraints,
4641 * then this lower bound also satisfies the constraints.
4642 * It is therefore safe to pick this lower bound.
4643 *
4644 * The same reasoning holds even if the coefficient is not one.
4645 * However, fixing the variable to the value of the lower bound may
4646 * in general introduce an extra integer division, in which case
4647 * it may be better to pick another value.
4648 * If this integer division has a known constant value, then plugging
4649 * in this constant value removes the existentially quantified variable
4650 * completely. In particular, if the lower bound is of the form
4651 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4652 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4653 * then the existentially quantified variable can be assigned this
4654 * shared value.
4655 *
4656 * We skip divs that appear in equalities or in the definition of other divs.
4657 * Divs that appear in the definition of other divs usually occur in at least
4658 * 4 constraints, but the constraints may have been simplified.
4659 *
4660 * If any divs are left after these simple checks then we move on
4661 * to more complicated cases in drop_more_redundant_divs.
4662 */
4665 {
4705 }
4709 }
4710 }
4718 }
4721 else
4741 pairs);
4745 pairs);
4747 }
4764 else
4771 }
4774 }
4781 error:
4785 }
4787 /* Consider the coefficients at "c" as a row vector and replace
4788 * them with their product with "T". "T" is assumed to be a square matrix.
4789 */
4791 {
4813 }
4815 /* Plug in T for the variables in "bmap" starting at "pos".
4816 * T is a linear unimodular matrix, i.e., without constant term.
4817 */
4820 {
4849 }
4853 error:
4857 }
4859 /* Remove divs that are not strictly needed.
4860 *
4861 * First look for an equality constraint involving two or more
4862 * existentially quantified variables without an explicit
4863 * representation. Replace the combination that appears
4864 * in the equality constraint by a single existentially quantified
4865 * variable such that the equality can be used to derive
4866 * an explicit representation for the variable.
4867 * If there are no more such equality constraints, then continue
4868 * with isl_basic_map_drop_redundant_divs_ineq.
4869 *
4870 * In particular, if the equality constraint is of the form
4871 *
4872 * f(x) + \sum_i c_i a_i = 0
4873 *
4874 * with a_i existentially quantified variable without explicit
4875 * representation, then apply a transformation on the existentially
4876 * quantified variables to turn the constraint into
4877 *
4878 * f(x) + g a_1' = 0
4879 *
4880 * with g the gcd of the c_i.
4881 * In order to easily identify which existentially quantified variables
4882 * have a complete explicit representation, i.e., without being defined
4883 * in terms of other existentially quantified variables without
4884 * an explicit representation, the existentially quantified variables
4885 * are first sorted.
4886 *
4887 * The variable transformation is computed by extending the row
4888 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4889 *
4890 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4891 * [a_2'] [ a_2 ]
4892 * ... = U ....
4893 * [a_n'] [ a_n ]
4894 *
4895 * with [c_1/g ... c_n/g] representing the first row of U.
4896 * The inverse of U is then plugged into the original constraints.
4897 * The call to isl_basic_map_simplify makes sure the explicit
4898 * representation for a_1' is extracted from the equality constraint.
4899 */
4902 {
4937 }
4956 }
4958 /* Does "bmap" satisfy any equality that involves more than 2 variables
4959 * and/or has coefficients different from -1 and 1?
4960 */
4962 {
4991 }
4994 }
4996 /* Remove any common factor g from the constraint coefficients in "v".
4997 * The constant term is stored in the first position and is replaced
4998 * by floor(c/g). If any common factor is removed and if this results
4999 * in a tightening of the constraint, then set *tightened.
5000 */
5003 {
5023 }
5025 /* If "bmap" is an integer set that satisfies any equality involving
5026 * more than 2 variables and/or has coefficients different from -1 and 1,
5027 * then use variable compression to reduce the coefficients by removing
5028 * any (hidden) common factor.
5029 * In particular, apply the variable compression to each constraint,
5030 * factor out any common factor in the non-constant coefficients and
5031 * then apply the inverse of the compression.
5032 * At the end, we mark the basic map as having reduced constants.
5033 * If this flag is still set on the next invocation of this function,
5034 * then we skip the computation.
5035 *
5036 * Removing a common factor may result in a tightening of some of
5037 * the constraints. If this happens, then we may end up with two
5038 * opposite inequalities that can be replaced by an equality.
5039 * We therefore call isl_basic_map_detect_inequality_pairs,
5040 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5041 * and isl_basic_map_gauss if such a pair was found.
5042 *
5043 * Note that this function may leave the result in an inconsistent state.
5044 * In particular, the constraints may not be gaussed.
5045 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5046 * for some of the test cases to pass successfully.
5047 * Any potential modification of the representation is therefore only
5048 * performed on a single copy of the basic map.
5049 */
5052 {
5086 }
5101 }
5116 }
5117 }
5120 error:
5125 }
5127 /* Shift the integer division at position "div" of "bmap"
5128 * by "shift" times the variable at position "pos".
5129 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5130 * corresponds to the constant term.
5131 *
5132 * That is, if the integer division has the form
5133 *
5134 * floor(f(x)/d)
5135 *
5136 * then replace it by
5137 *
5138 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5139 */
5142 {
5161 }
5167 }
5175 }
5178 }