| blob | 2412266e80354371cb9a9d6b59dcebcf49b1627f |
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 }
318 }
327 /* We need to be careful about circular definitions,
328 * so for now we just remove the definition of div k
329 * if the equality contains any divs.
330 * If keep_divs is set, then the divs have been ordered
331 * and we can keep the definition as long as the result
332 * is still ordered.
333 */
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;
541 }
545 {
568 }
577 progress);
584 }
591 }
594 }
598 {
600 progress));
601 }
605 {
611 }
613 }
615 /* Hash table of inequalities in a basic map.
616 * "index" is an array of addresses of inequalities in the basic map, some
617 * of which are NULL. The inequalities are hashed on the coefficients
618 * except the constant term.
619 * "size" is the number of elements in the array and is always a power of two
620 * "bits" is the number of bits need to represent an index into the array.
621 * "total" is the total dimension of the basic map.
622 */
628 };
630 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
631 */
634 {
651 }
653 /* Free the memory allocated by create_constraint_index.
654 */
656 {
658 }
660 /* Return the position in ci->index that contains the address of
661 * an inequality that is equal to *ineq up to the constant term,
662 * provided this address is not identical to "ineq".
663 * If there is no such inequality, then return the position where
664 * such an inequality should be inserted.
665 */
667 {
675 }
677 /* Return the position in ci->index that contains the address of
678 * an inequality that is equal to the k'th inequality of "bmap"
679 * up to the constant term, provided it does not point to the very
680 * same inequality.
681 * If there is no such inequality, then return the position where
682 * such an inequality should be inserted.
683 */
686 {
688 }
692 {
694 }
696 /* Fill in the "ci" data structure with the inequalities of "bset".
697 */
700 {
709 }
712 }
714 /* Is the inequality ineq (obviously) redundant with respect
715 * to the constraints in "ci"?
716 *
717 * Look for an inequality in "ci" with the same coefficients and then
718 * check if the contant term of "ineq" is greater than or equal
719 * to the constant term of that inequality. If so, "ineq" is clearly
720 * redundant.
721 *
722 * Note that hash_index_ineq ignores a stored constraint if it has
723 * the same address as the passed inequality. It is ok to pass
724 * the address of a local variable here since it will never be
725 * the same as the address of a constraint in "ci".
726 */
729 {
736 }
738 /* If we can eliminate more than one div, then we need to make
739 * sure we do it from last div to first div, in order not to
740 * change the position of the other divs that still need to
741 * be removed.
742 */
745 {
799 }
801 }
813 }
816 out:
820 }
823 {
835 }
837 }
839 /* Normalize divs that appear in equalities.
840 *
841 * In particular, we assume that bmap contains some equalities
842 * of the form
843 *
844 * a x = m * e_i
845 *
846 * and we want to replace the set of e_i by a minimal set and
847 * such that the new e_i have a canonical representation in terms
848 * of the vector x.
849 * If any of the equalities involves more than one divs, then
850 * we currently simply bail out.
851 *
852 * Let us first additionally assume that all equalities involve
853 * a div. The equalities then express modulo constraints on the
854 * remaining variables and we can use "parameter compression"
855 * to find a minimal set of constraints. The result is a transformation
856 *
857 * x = T(x') = x_0 + G x'
858 *
859 * with G a lower-triangular matrix with all elements below the diagonal
860 * non-negative and smaller than the diagonal element on the same row.
861 * We first normalize x_0 by making the same property hold in the affine
862 * T matrix.
863 * The rows i of G with a 1 on the diagonal do not impose any modulo
864 * constraint and simply express x_i = x'_i.
865 * For each of the remaining rows i, we introduce a div and a corresponding
866 * equality. In particular
867 *
868 * g_ii e_j = x_i - g_i(x')
869 *
870 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
871 * corresponding div (if g_kk != 1).
872 *
873 * If there are any equalities not involving any div, then we
874 * first apply a variable compression on the variables x:
875 *
876 * x = C x'' x'' = C_2 x
877 *
878 * and perform the above parameter compression on A C instead of on A.
879 * The resulting compression is then of the form
880 *
881 * x'' = T(x') = x_0 + G x'
882 *
883 * and in constructing the new divs and the corresponding equalities,
884 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
885 * by the corresponding row from C_2.
886 */
889 {
898 isl_int v;
930 }
931 }
940 }
946 }
956 }
963 }
968 /* We have to be careful because dropping equalities may reorder them */
978 }
979 }
985 else
986 needed++;
987 }
993 }
1003 else
1011 else
1014 }
1018 }
1025 done:
1029 error:
1035 }
1039 {
1049 }
1051 /* Check whether it is ok to define a div based on an inequality.
1052 * To avoid the introduction of circular definitions of divs, we
1053 * do not allow such a definition if the resulting expression would refer to
1054 * any other undefined divs or if any known div is defined in
1055 * terms of the unknown div.
1056 */
1059 {
1063 /* Not defined in terms of unknown divs */
1071 }
1073 /* No other div defined in terms of this one => avoid loops */
1081 }
1084 }
1086 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1087 * be a better expression than the current one?
1088 *
1089 * If we do not have any expression yet, then any expression would be better.
1090 * Otherwise we check if the last variable involved in the inequality
1091 * (disregarding the div that it would define) is in an earlier position
1092 * than the last variable involved in the current div expression.
1093 */
1096 {
1113 }
1115 /* Given two constraints "k" and "l" that are opposite to each other,
1116 * except for the constant term, check if we can use them
1117 * to obtain an expression for one of the hitherto unknown divs or
1118 * a "better" expression for a div for which we already have an expression.
1119 * "sum" is the sum of the constant terms of the constraints.
1120 * If this sum is strictly smaller than the coefficient of one
1121 * of the divs, then this pair can be used define the div.
1122 * To avoid the introduction of circular definitions of divs, we
1123 * do not use the pair if the resulting expression would refer to
1124 * any other undefined divs or if any known div is defined in
1125 * terms of the unknown div.
1126 */
1130 {
1135 isl_bool set_div;
1150 else
1155 }
1157 }
1161 {
1165 isl_int sum;
1180 }
1188 }
1203 }
1205 /* We need to break out of the loop after these
1206 * changes since the contents of the hash
1207 * will no longer be valid.
1208 * Plus, we probably we want to regauss first.
1209 */
1217 }
1222 }
1224 /* Detect all pairs of inequalities that form an equality.
1225 *
1226 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1227 * Call it repeatedly while it is making progress.
1228 */
1231 {
1243 }
1245 /* Eliminate knowns divs from constraints where they appear with
1246 * a (positive or negative) unit coefficient.
1247 *
1248 * That is, replace
1249 *
1250 * floor(e/m) + f >= 0
1251 *
1252 * by
1253 *
1254 * e + m f >= 0
1255 *
1256 * and
1257 *
1258 * -floor(e/m) + f >= 0
1259 *
1260 * by
1261 *
1262 * -e + m f + m - 1 >= 0
1263 *
1264 * The first conversion is valid because floor(e/m) >= -f is equivalent
1265 * to e/m >= -f because -f is an integral expression.
1266 * The second conversion follows from the fact that
1267 *
1268 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1269 *
1270 *
1271 * Note that one of the div constraints may have been eliminated
1272 * due to being redundant with respect to the constraint that is
1273 * being modified by this function. The modified constraint may
1274 * no longer imply this div constraint, so we add it back to make
1275 * sure we do not lose any information.
1276 *
1277 * We skip integral divs, i.e., those with denominator 1, as we would
1278 * risk eliminating the div from the div constraints. We do not need
1279 * to handle those divs here anyway since the div constraints will turn
1280 * out to form an equality and this equality can then be used to eliminate
1281 * the div from all constraints.
1282 */
1285 {
1317 else
1327 }
1332 }
1333 }
1336 }
1339 {
1344 isl_bool empty;
1360 /* requires equalities in normal form */
1366 }
1368 }
1371 {
1373 }
1378 {
1410 }
1414 {
1416 }
1419 /* If the only constraints a div d=floor(f/m)
1420 * appears in are its two defining constraints
1421 *
1422 * f - m d >=0
1423 * -(f - (m - 1)) + m d >= 0
1424 *
1425 * then it can safely be removed.
1426 */
1428 {
1437 isl_bool red;
1444 }
1451 }
1454 }
1456 /*
1457 * Remove divs that don't occur in any of the constraints or other divs.
1458 * These can arise when dropping constraints from a basic map or
1459 * when the divs of a basic map have been temporarily aligned
1460 * with the divs of another basic map.
1461 */
1464 {
1473 isl_bool redundant;
1483 }
1485 }
1487 /* Mark "bmap" as final, without checking for obviously redundant
1488 * integer divisions. This function should be used when "bmap"
1489 * is known not to involve any such integer divisions.
1490 */
1493 {
1498 }
1500 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1501 */
1503 {
1507 }
1510 {
1512 }
1514 /* Remove definition of any div that is defined in terms of the given variable.
1515 * The div itself is not removed. Functions such as
1516 * eliminate_divs_ineq depend on the other divs remaining in place.
1517 */
1520 {
1534 }
1536 }
1538 /* Eliminate the specified variables from the constraints using
1539 * Fourier-Motzkin. The variables themselves are not removed.
1540 */
1543 {
1575 }
1582 n_lower++;
1584 n_upper++;
1585 }
1609 }
1612 }
1624 }
1625 }
1629 error:
1632 }
1636 {
1639 }
1641 /* Eliminate the specified n dimensions starting at first from the
1642 * constraints, without removing the dimensions from the space.
1643 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1644 * Otherwise, they are projected out and the original space is restored.
1645 */
1649 {
1665 }
1672 error:
1675 }
1680 {
1682 }
1684 /* Remove all constraints from "bmap" that reference any unknown local
1685 * variables (directly or indirectly).
1686 *
1687 * Dropping all constraints on a local variable will make it redundant,
1688 * so it will get removed implicitly by
1689 * isl_basic_map_drop_constraints_involving_dims. Some other local
1690 * variables may also end up becoming redundant if they only appear
1691 * in constraints together with the unknown local variable.
1692 * Therefore, start over after calling
1693 * isl_basic_map_drop_constraints_involving_dims.
1694 */
1697 {
1698 isl_bool known;
1723 }
1726 }
1728 /* Remove all constraints from "map" that reference any unknown local
1729 * variables (directly or indirectly).
1730 *
1731 * Since constraints may get dropped from the basic maps,
1732 * they may no longer be disjoint from each other.
1733 */
1736 {
1738 isl_bool known;
1756 }
1762 }
1764 /* Don't assume equalities are in order, because align_divs
1765 * may have changed the order of the divs.
1766 */
1768 {
1781 }
1782 }
1783 }
1787 {
1789 }
1793 {
1807 }
1809 }
1811 }
1815 {
1818 }
1822 {
1832 }
1853 error:
1857 }
1859 /* For each inequality in "ineq" that is a shifted (more relaxed)
1860 * copy of an inequality in "context", mark the corresponding entry
1861 * in "row" with -1.
1862 * If an inequality only has a non-negative constant term, then
1863 * mark it as well.
1864 */
1867 {
1884 isl_bool redundant;
1890 }
1897 }
1900 error:
1903 }
1907 {
1920 isl_bool redundant;
1932 }
1935 error:
1938 }
1940 /* Remove constraints from "bmap" that are identical to constraints
1941 * in "context" or that are more relaxed (greater constant term).
1942 *
1943 * We perform the test for shifted copies on the pure constraints
1944 * in remove_shifted_constraints.
1945 */
1948 {
1957 }
1970 error:
1974 }
1976 /* Does the (linear part of a) constraint "c" involve any of the "len"
1977 * "relevant" dimensions?
1978 */
1980 {
1988 }
1991 }
1993 /* Drop constraints from "bmap" that do not involve any of
1994 * the dimensions marked "relevant".
1995 */
1998 {
2013 }
2020 }
2023 }
2025 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2026 *
2027 * In particular, for any variable involved in the constraint,
2028 * find the actual group id from before and replace the group
2029 * of the corresponding variable by the minimal group of all
2030 * the variables involved in the constraint considered so far
2031 * (if this minimum is smaller) or replace the minimum by this group
2032 * (if the minimum is larger).
2033 *
2034 * At the end, all the variables in "c" will (indirectly) point
2035 * to the minimal of the groups that they referred to originally.
2036 */
2038 {
2055 }
2056 }
2058 /* Allocate an array of groups of variables, one for each variable
2059 * in "context", initialized to zero.
2060 */
2062 {
2069 }
2071 /* Drop constraints from "bmap" that only involve variables that are
2072 * not related to any of the variables marked with a "-1" in "group".
2073 *
2074 * We construct groups of variables that collect variables that
2075 * (indirectly) appear in some common constraint of "bmap".
2076 * Each group is identified by the first variable in the group,
2077 * except for the special group of variables that was already identified
2078 * in the input as -1 (or are related to those variables).
2079 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2080 * otherwise the group of i is the group of group[i].
2081 *
2082 * We first initialize groups for the remaining variables.
2083 * Then we iterate over the constraints of "bmap" and update the
2084 * group of the variables in the constraint by the smallest group.
2085 * Finally, we resolve indirect references to groups by running over
2086 * the variables.
2087 *
2088 * After computing the groups, we drop constraints that do not involve
2089 * any variables in the -1 group.
2090 */
2093 {
2110 }
2128 }
2130 /* Drop constraints from "context" that are irrelevant for computing
2131 * the gist of "bset".
2132 *
2133 * In particular, drop constraints in variables that are not related
2134 * to any of the variables involved in the constraints of "bset"
2135 * in the sense that there is no sequence of constraints that connects them.
2136 *
2137 * We first mark all variables that appear in "bset" as belonging
2138 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2139 */
2142 {
2163 }
2169 }
2172 }
2174 /* Drop constraints from "context" that are irrelevant for computing
2175 * the gist of the inequalities "ineq".
2176 * Inequalities in "ineq" for which the corresponding element of row
2177 * is set to -1 have already been marked for removal and should be ignored.
2178 *
2179 * In particular, drop constraints in variables that are not related
2180 * to any of the variables involved in "ineq"
2181 * in the sense that there is no sequence of constraints that connects them.
2182 *
2183 * We first mark all variables that appear in "bset" as belonging
2184 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2185 */
2188 {
2209 }
2212 }
2215 }
2217 /* Do all "n" entries of "row" contain a negative value?
2218 */
2220 {
2228 }
2230 /* Update the inequalities in "bset" based on the information in "row"
2231 * and "tab".
2232 *
2233 * In particular, the array "row" contains either -1, meaning that
2234 * the corresponding inequality of "bset" is redundant, or the index
2235 * of an inequality in "tab".
2236 *
2237 * If the row entry is -1, then drop the inequality.
2238 * Otherwise, if the constraint is marked redundant in the tableau,
2239 * then drop the inequality. Similarly, if it is marked as an equality
2240 * in the tableau, then turn the inequality into an equality and
2241 * perform Gaussian elimination.
2242 */
2245 {
2262 }
2272 }
2273 }
2279 }
2281 /* Update the inequalities in "bset" based on the information in "row"
2282 * and "tab" and free all arguments (other than "bset").
2283 */
2288 {
2297 }
2299 /* Remove all information from bset that is redundant in the context
2300 * of context.
2301 * "ineq" contains the (possibly transformed) inequalities of "bset",
2302 * in the same order.
2303 * The (explicit) equalities of "bset" are assumed to have been taken
2304 * into account by the transformation such that only the inequalities
2305 * are relevant.
2306 * "context" is assumed not to be empty.
2307 *
2308 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2309 * A value of -1 means that the inequality is obviously redundant and may
2310 * not even appear in "tab".
2311 *
2312 * We first mark the inequalities of "bset"
2313 * that are obviously redundant with respect to some inequality in "context".
2314 * Then we remove those constraints from "context" that have become
2315 * irrelevant for computing the gist of "bset".
2316 * Note that this removal of constraints cannot be replaced by
2317 * a factorization because factors in "bset" may still be connected
2318 * to each other through constraints in "context".
2319 *
2320 * If there are any inequalities left, we construct a tableau for
2321 * the context and then add the inequalities of "bset".
2322 * Before adding these inequalities, we freeze all constraints such that
2323 * they won't be considered redundant in terms of the constraints of "bset".
2324 * Then we detect all redundant constraints (among the
2325 * constraints that weren't frozen), first by checking for redundancy in the
2326 * the tableau and then by checking if replacing a constraint by its negation
2327 * would lead to an empty set. This last step is fairly expensive
2328 * and could be optimized by more reuse of the tableau.
2329 * Finally, we update bset according to the results.
2330 */
2333 {
2348 }
2384 }
2408 }
2413 }
2417 error:
2425 }
2427 /* Extract the inequalities of "bset" as an isl_mat.
2428 */
2430 {
2444 }
2446 /* Remove all information from "bset" that is redundant in the context
2447 * of "context", for the case where both "bset" and "context" are
2448 * full-dimensional.
2449 */
2452 {
2457 }
2459 /* Remove all information from "bset" that is redundant in the context
2460 * of "context", for the case where the combined equalities of
2461 * "bset" and "context" allow for a compression that can be obtained
2462 * by preapplication of "T".
2463 *
2464 * "bset" itself is not transformed by "T". Instead, the inequalities
2465 * are extracted from "bset" and those are transformed by "T".
2466 * uset_gist_full then determines which of the transformed inequalities
2467 * are redundant with respect to the transformed "context" and removes
2468 * the corresponding inequalities from "bset".
2469 *
2470 * After preapplying "T" to the inequalities, any common factor is
2471 * removed from the coefficients. If this results in a tightening
2472 * of the constant term, then the same tightening is applied to
2473 * the corresponding untransformed inequality in "bset".
2474 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2475 *
2476 * g f'(x) + r >= 0
2477 *
2478 * with 0 <= r < g, then it is equivalent to
2479 *
2480 * f'(x) >= 0
2481 *
2482 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2483 * subspace compressed by T since the latter would be transformed to
2484 *
2485 * g f'(x) >= 0
2486 */
2490 {
2494 isl_int rem;
2506 }
2529 }
2533 error:
2538 }
2540 /* Project "bset" onto the variables that are involved in "template".
2541 */
2544 {
2553 isl_bool involved;
2562 }
2565 }
2567 /* Remove all information from bset that is redundant in the context
2568 * of context. In particular, equalities that are linear combinations
2569 * of those in context are removed. Then the inequalities that are
2570 * redundant in the context of the equalities and inequalities of
2571 * context are removed.
2572 *
2573 * First of all, we drop those constraints from "context"
2574 * that are irrelevant for computing the gist of "bset".
2575 * Alternatively, we could factorize the intersection of "context" and "bset".
2576 *
2577 * We first compute the intersection of the integer affine hulls
2578 * of "bset" and "context",
2579 * compute the gist inside this intersection and then reduce
2580 * the constraints with respect to the equalities of the context
2581 * that only involve variables already involved in the input.
2582 *
2583 * If two constraints are mutually redundant, then uset_gist_full
2584 * will remove the second of those constraints. We therefore first
2585 * sort the constraints so that constraints not involving existentially
2586 * quantified variables are given precedence over those that do.
2587 * We have to perform this sorting before the variable compression,
2588 * because that may effect the order of the variables.
2589 */
2592 {
2617 }
2622 }
2632 }
2643 }
2646 error:
2650 }
2652 /* Return the number of equality constraints in "bmap" that involve
2653 * local variables. This function assumes that Gaussian elimination
2654 * has been applied to the equality constraints.
2655 */
2657 {
2677 }
2679 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2680 * The constraints are assumed not to involve any local variables.
2681 */
2684 {
2702 }
2707 error:
2712 }
2714 /* Construct and return a variable compression based on the equality
2715 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2716 * "n1" is the number of (initial) equality constraints in "bmap1"
2717 * that do involve local variables.
2718 * "n2" is the number of (initial) equality constraints in "bmap2"
2719 * that do involve local variables.
2720 * "total" is the total number of other variables.
2721 * This function assumes that Gaussian elimination
2722 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2723 * such that the equality constraints not involving local variables
2724 * are those that start at "n1" or "n2".
2725 *
2726 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2727 * then simply compute the compression based on the equality constraints
2728 * in the other basic map.
2729 * Otherwise, combine the equality constraints from both into a new
2730 * basic map such that Gaussian elimination can be applied to this combination
2731 * and then construct a variable compression from the resulting
2732 * equality constraints.
2733 */
2737 {
2747 }
2752 }
2767 }
2769 /* Extract the stride constraints from "bmap", compressed
2770 * with respect to both the stride constraints in "context" and
2771 * the remaining equality constraints in both "bmap" and "context".
2772 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2773 * "context_n_eq" is the number of (initial) stride constraints in "context".
2774 *
2775 * Let x be all variables in "bmap" (and "context") other than the local
2776 * variables. First compute a variable compression
2777 *
2778 * x = V x'
2779 *
2780 * based on the non-stride equality constraints in "bmap" and "context".
2781 * Consider the stride constraints of "context",
2782 *
2783 * A(x) + B(y) = 0
2784 *
2785 * with y the local variables and plug in the variable compression,
2786 * resulting in
2787 *
2788 * A(V x') + B(y) = 0
2789 *
2790 * Use these constraints to compute a parameter compression on x'
2791 *
2792 * x' = T x''
2793 *
2794 * Now consider the stride constraints of "bmap"
2795 *
2796 * C(x) + D(y) = 0
2797 *
2798 * and plug in x = V*T x''.
2799 * That is, return A = [C*V*T D].
2800 */
2804 {
2833 }
2835 /* Remove the prime factors from *g that have an exponent that
2836 * is strictly smaller than the exponent in "c".
2837 * All exponents in *g are known to be smaller than or equal
2838 * to those in "c".
2839 *
2840 * That is, if *g is equal to
2841 *
2842 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2843 *
2844 * and "c" is equal to
2845 *
2846 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2847 *
2848 * then update *g to
2849 *
2850 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2851 * p_n^{e_n * (e_n = f_n)}
2852 *
2853 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2854 * neither does the gcd of *g and c / *g.
2855 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2856 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2857 * Dividing *g by this gcd therefore strictly reduces the exponent
2858 * of the prime factors that need to be removed, while leaving the
2859 * other prime factors untouched.
2860 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2861 * removes all undesired factors, without removing any others.
2862 */
2864 {
2865 isl_int t;
2874 }
2876 }
2878 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2879 * of the same stride constraints in a compressed space that exploits
2880 * all equalities in the context and the other equalities in "bmap".
2881 *
2882 * If the stride constraints of "bmap" are of the form
2883 *
2884 * C(x) + D(y) = 0
2885 *
2886 * then A is of the form
2887 *
2888 * B(x') + D(y) = 0
2889 *
2890 * If any of these constraints involves only a single local variable y,
2891 * then the constraint appears as
2892 *
2893 * f(x) + m y_i = 0
2894 *
2895 * in "bmap" and as
2896 *
2897 * h(x') + m y_i = 0
2898 *
2899 * in "A".
2900 *
2901 * Let g be the gcd of m and the coefficients of h.
2902 * Then, in particular, g is a divisor of the coefficients of h and
2903 *
2904 * f(x) = h(x')
2905 *
2906 * is known to be a multiple of g.
2907 * If some prime factor in m appears with the same exponent in g,
2908 * then it can be removed from m because f(x) is already known
2909 * to be a multiple of g and therefore in particular of this power
2910 * of the prime factors.
2911 * Prime factors that appear with a smaller exponent in g cannot
2912 * be removed from m.
2913 * Let g' be the divisor of g containing all prime factors that
2914 * appear with the same exponent in m and g, then
2915 *
2916 * f(x) + m y_i = 0
2917 *
2918 * can be replaced by
2919 *
2920 * f(x) + m/g' y_i' = 0
2921 *
2922 * Note that (if g' != 1) this changes the explicit representation
2923 * of y_i to that of y_i', so the integer division at position i
2924 * is marked unknown and later recomputed by a call to
2925 * isl_basic_map_gauss.
2926 */
2929 {
2933 isl_int gcd;
2967 }
2974 error:
2978 }
2980 /* Simplify the stride constraints in "bmap" based on
2981 * the remaining equality constraints in "bmap" and all equality
2982 * constraints in "context".
2983 * Only do this if both "bmap" and "context" have stride constraints.
2984 *
2985 * First extract a copy of the stride constraints in "bmap" in a compressed
2986 * space exploiting all the other equality constraints and then
2987 * use this compressed copy to simplify the original stride constraints.
2988 */
2991 {
3013 }
3015 /* Return a basic map that has the same intersection with "context" as "bmap"
3016 * and that is as "simple" as possible.
3017 *
3018 * The core computation is performed on the pure constraints.
3019 * When we add back the meaning of the integer divisions, we need
3020 * to (re)introduce the div constraints. If we happen to have
3021 * discovered that some of these integer divisions are equal to
3022 * some affine combination of other variables, then these div
3023 * constraints may end up getting simplified in terms of the equalities,
3024 * resulting in extra inequalities on the other variables that
3025 * may have been removed already or that may not even have been
3026 * part of the input. We try and remove those constraints of
3027 * this form that are most obviously redundant with respect to
3028 * the context. We also remove those div constraints that are
3029 * redundant with respect to the other constraints in the result.
3030 *
3031 * The stride constraints among the equality constraints in "bmap" are
3032 * also simplified with respecting to the other equality constraints
3033 * in "bmap" and with respect to all equality constraints in "context".
3034 */
3037 {
3048 }
3054 }
3058 }
3080 }
3099 error:
3103 }
3105 /*
3106 * Assumes context has no implicit divs.
3107 */
3110 {
3121 }
3141 }
3142 }
3146 error:
3150 }
3152 /* Drop all inequalities from "bmap" that also appear in "context".
3153 * "context" is assumed to have only known local variables and
3154 * the initial local variables of "bmap" are assumed to be the same
3155 * as those of "context".
3156 * The constraints of both "bmap" and "context" are assumed
3157 * to have been sorted using isl_basic_map_sort_constraints.
3158 *
3159 * Run through the inequality constraints of "bmap" and "context"
3160 * in sorted order.
3161 * If a constraint of "bmap" involves variables not in "context",
3162 * then it cannot appear in "context".
3163 * If a matching constraint is found, it is removed from "bmap".
3164 */
3167 {
3186 }
3192 }
3196 }
3201 }
3204 }
3207 }
3209 /* Drop all equalities from "bmap" that also appear in "context".
3210 * "context" is assumed to have only known local variables and
3211 * the initial local variables of "bmap" are assumed to be the same
3212 * as those of "context".
3213 *
3214 * Run through the equality constraints of "bmap" and "context"
3215 * in sorted order.
3216 * If a constraint of "bmap" involves variables not in "context",
3217 * then it cannot appear in "context".
3218 * If a matching constraint is found, it is removed from "bmap".
3219 */
3222 {
3246 }
3250 }
3255 }
3258 }
3261 }
3263 /* Remove the constraints in "context" from "bmap".
3264 * "context" is assumed to have explicit representations
3265 * for all local variables.
3266 *
3267 * First align the divs of "bmap" to those of "context" and
3268 * sort the constraints. Then drop all constraints from "bmap"
3269 * that appear in "context".
3270 */
3273 {
3288 }
3307 error:
3311 }
3313 /* Replace "map" by the disjunct at position "pos" and free "context".
3314 */
3317 {
3324 }
3326 /* Remove the constraints in "context" from "map".
3327 * If any of the disjuncts in the result turns out to be the universe,
3328 * then return this universe.
3329 * "context" is assumed to have explicit representations
3330 * for all local variables.
3331 */
3334 {
3344 }
3363 }
3370 error:
3374 }
3376 /* Remove the constraints in "context" from "set".
3377 * If any of the disjuncts in the result turns out to be the universe,
3378 * then return this universe.
3379 * "context" is assumed to have explicit representations
3380 * for all local variables.
3381 */
3384 {
3387 }
3389 /* Remove the constraints in "context" from "map".
3390 * If any of the disjuncts in the result turns out to be the universe,
3391 * then return this universe.
3392 * "context" is assumed to consist of a single disjunct and
3393 * to have explicit representations for all local variables.
3394 */
3397 {
3402 }
3404 /* Replace "map" by a universe map in the same space and free "drop".
3405 */
3408 {
3415 }
3417 /* Return a map that has the same intersection with "context" as "map"
3418 * and that is as "simple" as possible.
3419 *
3420 * If "map" is already the universe, then we cannot make it any simpler.
3421 * Similarly, if "context" is the universe, then we cannot exploit it
3422 * to simplify "map"
3423 * If "map" and "context" are identical to each other, then we can
3424 * return the corresponding universe.
3425 *
3426 * If either "map" or "context" consists of multiple disjuncts,
3427 * then check if "context" happens to be a subset of "map",
3428 * in which case all constraints can be removed.
3429 * In case of multiple disjuncts, the standard procedure
3430 * may not be able to detect that all constraints can be removed.
3431 *
3432 * If none of these cases apply, we have to work a bit harder.
3433 * During this computation, we make use of a single disjunct context,
3434 * so if the original context consists of more than one disjunct
3435 * then we need to approximate the context by a single disjunct set.
3436 * Simply taking the simple hull may drop constraints that are
3437 * only implicitly available in each disjunct. We therefore also
3438 * look for constraints among those defining "map" that are valid
3439 * for the context. These can then be used to simplify away
3440 * the corresponding constraints in "map".
3441 */
3444 {
3448 isl_bool subset;
3459 }
3475 }
3491 list);
3492 }
3494 error:
3498 }
3502 {
3504 }
3508 {
3511 }
3515 {
3518 }
3522 {
3527 }
3531 {
3533 }
3535 /* Compute the gist of "bmap" with respect to the constraints "context"
3536 * on the domain.
3537 */
3540 {
3546 }
3550 {
3554 }
3558 {
3562 }
3566 {
3570 }
3574 {
3576 }
3578 /* Quick check to see if two basic maps are disjoint.
3579 * In particular, we reduce the equalities and inequalities of
3580 * one basic map in the context of the equalities of the other
3581 * basic map and check if we get a contradiction.
3582 */
3585 {
3617 }
3625 }
3634 }
3638 disjoint:
3642 error:
3646 }
3650 {
3653 }
3655 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3656 */
3660 {
3671 }
3672 }
3675 }
3677 /* Are "map1" and "map2" obviously disjoint, based on information
3678 * that can be derived without looking at the individual basic maps?
3679 *
3680 * In particular, if one of them is empty or if they live in different spaces
3681 * (ignoring parameters), then they are clearly disjoint.
3682 */
3685 {
3686 isl_bool disjoint;
3687 isl_bool match;
3711 }
3713 /* Are "map1" and "map2" obviously disjoint?
3714 *
3715 * If one of them is empty or if they live in different spaces (ignoring
3716 * parameters), then they are clearly disjoint.
3717 * This is checked by isl_map_plain_is_disjoint_global.
3718 *
3719 * If they have different parameters, then we skip any further tests.
3720 *
3721 * If they are obviously equal, but not obviously empty, then we will
3722 * not be able to detect if they are disjoint.
3723 *
3724 * Otherwise we check if each basic map in "map1" is obviously disjoint
3725 * from each basic map in "map2".
3726 */
3729 {
3730 isl_bool disjoint;
3731 isl_bool intersect;
3732 isl_bool match;
3747 }
3749 /* Are "map1" and "map2" disjoint?
3750 * The parameters are assumed to have been aligned.
3751 *
3752 * In particular, check whether all pairs of basic maps are disjoint.
3753 */
3756 {
3758 }
3760 /* Are "map1" and "map2" disjoint?
3761 *
3762 * They are disjoint if they are "obviously disjoint" or if one of them
3763 * is empty. Otherwise, they are not disjoint if one of them is universal.
3764 * If the two inputs are (obviously) equal and not empty, then they are
3765 * not disjoint.
3766 * If none of these cases apply, then check if all pairs of basic maps
3767 * are disjoint after aligning the parameters.
3768 */
3770 {
3771 isl_bool disjoint;
3772 isl_bool intersect;
3800 }
3802 /* Are "bmap1" and "bmap2" disjoint?
3803 *
3804 * They are disjoint if they are "obviously disjoint" or if one of them
3805 * is empty. Otherwise, they are not disjoint if one of them is universal.
3806 * If none of these cases apply, we compute the intersection and see if
3807 * the result is empty.
3808 */
3811 {
3812 isl_bool disjoint;
3813 isl_bool intersect;
3842 }
3844 /* Are "bset1" and "bset2" disjoint?
3845 */
3848 {
3850 }
3854 {
3856 }
3858 /* Are "set1" and "set2" disjoint?
3859 */
3861 {
3863 }
3865 /* Is "v" equal to 0, 1 or -1?
3866 */
3868 {
3870 }
3872 /* Check if we can combine a given div with lower bound l and upper
3873 * bound u with some other div and if so return that other div.
3874 * Otherwise return -1.
3875 *
3876 * We first check that
3877 * - the bounds are opposites of each other (except for the constant
3878 * term)
3879 * - the bounds do not reference any other div
3880 * - no div is defined in terms of this div
3881 *
3882 * Let m be the size of the range allowed on the div by the bounds.
3883 * That is, the bounds are of the form
3884 *
3885 * e <= a <= e + m - 1
3886 *
3887 * with e some expression in the other variables.
3888 * We look for another div b such that no third div is defined in terms
3889 * of this second div b and such that in any constraint that contains
3890 * a (except for the given lower and upper bound), also contains b
3891 * with a coefficient that is m times that of b.
3892 * That is, all constraints (except for the lower and upper bound)
3893 * are of the form
3894 *
3895 * e + f (a + m b) >= 0
3896 *
3897 * Furthermore, in the constraints that only contain b, the coefficient
3898 * of b should be equal to 1 or -1.
3899 * If so, we return b so that "a + m b" can be replaced by
3900 * a single div "c = a + m b".
3901 */
3904 {
3926 }
3936 }
3948 }
3959 }
3972 }
3977 }
3981 }
3983 /* Internal data structure used during the construction and/or evaluation of
3984 * an inequality that ensures that a pair of bounds always allows
3985 * for an integer value.
3986 *
3987 * "tab" is the tableau in which the inequality is evaluated. It may
3988 * be NULL until it is actually needed.
3989 * "v" contains the inequality coefficients.
3990 * "g", "fl" and "fu" are temporary scalars used during the construction and
3991 * evaluation.
3992 */
3996 isl_int g;
3997 isl_int fl;
3998 isl_int fu;
3999 };
4001 /* Free all the memory allocated by the fields of "data".
4002 */
4004 {
4010 }
4012 /* Is the inequality stored in data->v satisfied by "bmap"?
4013 * That is, does it only attain non-negative values?
4014 * data->tab is a tableau corresponding to "bmap".
4015 */
4018 {
4029 }
4031 /* Given a lower and an upper bound on div i, do they always allow
4032 * for an integer value of the given div?
4033 * Determine this property by constructing an inequality
4034 * such that the property is guaranteed when the inequality is nonnegative.
4035 * The lower bound is inequality l, while the upper bound is inequality u.
4036 * The constructed inequality is stored in data->v.
4037 *
4038 * Let the upper bound be
4039 *
4040 * -n_u a + e_u >= 0
4041 *
4042 * and the lower bound
4043 *
4044 * n_l a + e_l >= 0
4045 *
4046 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4047 * We have
4048 *
4049 * - f_u e_l <= f_u f_l g a <= f_l e_u
4050 *
4051 * Since all variables are integer valued, this is equivalent to
4052 *
4053 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4054 *
4055 * If this interval is at least f_u f_l g, then it contains at least
4056 * one integer value for a.
4057 * That is, the test constraint is
4058 *
4059 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4060 *
4061 * or
4062 *
4063 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4064 *
4065 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4066 * then the constraint can be scaled down by a factor g',
4067 * with the constant term replaced by
4068 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4069 * Note that the result of applying Fourier-Motzkin to this pair
4070 * of constraints is
4071 *
4072 * f_l e_u + f_u e_l >= 0
4073 *
4074 * If the constant term of the scaled down version of this constraint,
4075 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4076 * term of the scaled down test constraint, then the test constraint
4077 * is known to hold and no explicit evaluation is required.
4078 * This is essentially the Omega test.
4079 *
4080 * If the test constraint consists of only a constant term, then
4081 * it is sufficient to look at the sign of this constant term.
4082 */
4085 {
4110 }
4120 }
4122 /* Remove more kinds of divs that are not strictly needed.
4123 * In particular, if all pairs of lower and upper bounds on a div
4124 * are such that they allow at least one integer value of the div,
4125 * then we can eliminate the div using Fourier-Motzkin without
4126 * introducing any spurious solutions.
4127 *
4128 * If at least one of the two constraints has a unit coefficient for the div,
4129 * then the presence of such a value is guaranteed so there is no need to check.
4130 * In particular, the value attained by the bound with unit coefficient
4131 * can serve as this intermediate value.
4132 */
4135 {
4158 isl_bool has_int;
4166 }
4187 }
4190 }
4194 }
4198 }
4201 }
4212 error:
4217 }
4219 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4220 * and the upper bound u, div1 always occurs together with div2 in the form
4221 * (div1 + m div2), where m is the constant range on the variable div1
4222 * allowed by l and u, replace the pair div1 and div2 by a single
4223 * div that is equal to div1 + m div2.
4224 *
4225 * The new div will appear in the location that contains div2.
4226 * We need to modify all constraints that contain
4227 * div2 = (div - div1) / m
4228 * The coefficient of div2 is known to be equal to 1 or -1.
4229 * (If a constraint does not contain div2, it will also not contain div1.)
4230 * If the constraint also contains div1, then we know they appear
4231 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4232 * i.e., the coefficient of div is f.
4233 *
4234 * Otherwise, we first need to introduce div1 into the constraint.
4235 * Let l be
4236 *
4237 * div1 + f >=0
4238 *
4239 * and u
4240 *
4241 * -div1 + f' >= 0
4242 *
4243 * A lower bound on div2
4244 *
4245 * div2 + t >= 0
4246 *
4247 * can be replaced by
4248 *
4249 * m div2 + div1 + m t + f >= 0
4250 *
4251 * An upper bound
4252 *
4253 * -div2 + t >= 0
4254 *
4255 * can be replaced by
4256 *
4257 * -(m div2 + div1) + m t + f' >= 0
4258 *
4259 * These constraint are those that we would obtain from eliminating
4260 * div1 using Fourier-Motzkin.
4261 *
4262 * After all constraints have been modified, we drop the lower and upper
4263 * bound and then drop div1.
4264 * Since the new div is only placed in the same location that used
4265 * to store div2, but otherwise has a different meaning, any possible
4266 * explicit representation of the original div2 is removed.
4267 */
4270 {
4272 isl_int m;
4294 else
4297 }
4301 }
4310 }
4314 }
4316 /* First check if we can coalesce any pair of divs and
4317 * then continue with dropping more redundant divs.
4318 *
4319 * We loop over all pairs of lower and upper bounds on a div
4320 * with coefficient 1 and -1, respectively, check if there
4321 * is any other div "c" with which we can coalesce the div
4322 * and if so, perform the coalescing.
4323 */
4326 {
4349 }
4350 }
4351 }
4356 }
4359 }
4361 /* Are the "n" coefficients starting at "first" of inequality constraints
4362 * "i" and "j" of "bmap" equal to each other?
4363 */
4366 {
4368 }
4370 /* Are the "n" coefficients starting at "first" of inequality constraints
4371 * "i" and "j" of "bmap" opposite to each other?
4372 */
4375 {
4377 }
4379 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4380 * apart from the constant term?
4381 */
4383 {
4388 }
4390 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4391 * apart from the constant term and the coefficient at position "pos"?
4392 */
4395 {
4401 }
4403 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4404 * apart from the constant term and the coefficient at position "pos"?
4405 */
4408 {
4414 }
4416 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4417 * been modified, simplying it if "simplify" is set.
4418 * Free the temporary data structure "pairs" that was associated
4419 * to the old version of "bmap".
4420 */
4423 {
4428 }
4430 /* Is "div" the single unknown existentially quantified variable
4431 * in inequality constraint "ineq" of "bmap"?
4432 * "div" is known to have a non-zero coefficient in "ineq".
4433 */
4436 {
4439 isl_bool known;
4449 isl_bool known;
4458 }
4461 }
4463 /* Does integer division "div" have coefficient 1 in inequality constraint
4464 * "ineq" of "map"?
4465 */
4467 {
4475 }
4477 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4478 * then try and drop redundant divs again,
4479 * freeing the temporary data structure "pairs" that was associated
4480 * to the old version of "bmap".
4481 */
4484 {
4488 }
4490 /* Drop the integer division at position "div", along with the two
4491 * inequality constraints "ineq1" and "ineq2" in which it appears
4492 * from "bmap" and then try and drop redundant divs again,
4493 * freeing the temporary data structure "pairs" that was associated
4494 * to the old version of "bmap".
4495 */
4499 {
4506 }
4509 }
4511 /* Given two inequality constraints
4512 *
4513 * f(x) + n d + c >= 0, (ineq)
4514 *
4515 * with d the variable at position "pos", and
4516 *
4517 * f(x) + c0 >= 0, (lower)
4518 *
4519 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4520 * determined by the first constraint.
4521 * That is, store
4522 *
4523 * ceil((c0 - c)/n)
4524 *
4525 * in *l.
4526 */
4529 {
4533 }
4535 /* Given two inequality constraints
4536 *
4537 * f(x) + n d + c >= 0, (ineq)
4538 *
4539 * with d the variable at position "pos", and
4540 *
4541 * -f(x) - c0 >= 0, (upper)
4542 *
4543 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4544 * determined by the first constraint.
4545 * That is, store
4546 *
4547 * ceil((-c1 - c)/n)
4548 *
4549 * in *u.
4550 */
4553 {
4557 }
4559 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4560 * does the corresponding lower bound have a fixed value in "bmap"?
4561 *
4562 * In particular, "ineq" is of the form
4563 *
4564 * f(x) + n d + c >= 0
4565 *
4566 * with n > 0, c the constant term and
4567 * d the existentially quantified variable "div".
4568 * That is, the lower bound is
4569 *
4570 * ceil((-f(x) - c)/n)
4571 *
4572 * Look for a pair of constraints
4573 *
4574 * f(x) + c0 >= 0
4575 * -f(x) + c1 >= 0
4576 *
4577 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4578 * That is, check that
4579 *
4580 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4581 *
4582 * If so, return the index of inequality f(x) + c0 >= 0.
4583 * Otherwise, return -1.
4584 */
4586 {
4603 }
4607 }
4608 }
4625 }
4627 /* Given a lower bound constraint "ineq" on the existentially quantified
4628 * variable "div", such that the corresponding lower bound has
4629 * a fixed value in "bmap", assign this fixed value to the variable and
4630 * then try and drop redundant divs again,
4631 * freeing the temporary data structure "pairs" that was associated
4632 * to the old version of "bmap".
4633 * "lower" determines the constant value for the lower bound.
4634 *
4635 * In particular, "ineq" is of the form
4636 *
4637 * f(x) + n d + c >= 0,
4638 *
4639 * while "lower" is of the form
4640 *
4641 * f(x) + c0 >= 0
4642 *
4643 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4644 * is ceil((c0 - c)/n).
4645 */
4648 {
4649 isl_int c;
4662 }
4664 /* Remove divs that are not strictly needed based on the inequality
4665 * constraints.
4666 * In particular, if a div only occurs positively (or negatively)
4667 * in constraints, then it can simply be dropped.
4668 * Also, if a div occurs in only two constraints and if moreover
4669 * those two constraints are opposite to each other, except for the constant
4670 * term and if the sum of the constant terms is such that for any value
4671 * of the other values, there is always at least one integer value of the
4672 * div, i.e., if one plus this sum is greater than or equal to
4673 * the (absolute value) of the coefficient of the div in the constraints,
4674 * then we can also simply drop the div.
4675 *
4676 * If an existentially quantified variable does not have an explicit
4677 * representation, appears in only a single lower bound that does not
4678 * involve any other such existentially quantified variables and appears
4679 * in this lower bound with coefficient 1,
4680 * then fix the variable to the value of the lower bound. That is,
4681 * turn the inequality into an equality.
4682 * If for any value of the other variables, there is any value
4683 * for the existentially quantified variable satisfying the constraints,
4684 * then this lower bound also satisfies the constraints.
4685 * It is therefore safe to pick this lower bound.
4686 *
4687 * The same reasoning holds even if the coefficient is not one.
4688 * However, fixing the variable to the value of the lower bound may
4689 * in general introduce an extra integer division, in which case
4690 * it may be better to pick another value.
4691 * If this integer division has a known constant value, then plugging
4692 * in this constant value removes the existentially quantified variable
4693 * completely. In particular, if the lower bound is of the form
4694 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4695 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4696 * then the existentially quantified variable can be assigned this
4697 * shared value.
4698 *
4699 * We skip divs that appear in equalities or in the definition of other divs.
4700 * Divs that appear in the definition of other divs usually occur in at least
4701 * 4 constraints, but the constraints may have been simplified.
4702 *
4703 * If any divs are left after these simple checks then we move on
4704 * to more complicated cases in drop_more_redundant_divs.
4705 */
4708 {
4748 }
4752 }
4753 }
4761 }
4764 else
4784 pairs);
4788 pairs);
4790 }
4807 else
4814 }
4817 }
4824 error:
4828 }
4830 /* Consider the coefficients at "c" as a row vector and replace
4831 * them with their product with "T". "T" is assumed to be a square matrix.
4832 */
4834 {
4856 }
4858 /* Plug in T for the variables in "bmap" starting at "pos".
4859 * T is a linear unimodular matrix, i.e., without constant term.
4860 */
4863 {
4892 }
4896 error:
4900 }
4902 /* Remove divs that are not strictly needed.
4903 *
4904 * First look for an equality constraint involving two or more
4905 * existentially quantified variables without an explicit
4906 * representation. Replace the combination that appears
4907 * in the equality constraint by a single existentially quantified
4908 * variable such that the equality can be used to derive
4909 * an explicit representation for the variable.
4910 * If there are no more such equality constraints, then continue
4911 * with isl_basic_map_drop_redundant_divs_ineq.
4912 *
4913 * In particular, if the equality constraint is of the form
4914 *
4915 * f(x) + \sum_i c_i a_i = 0
4916 *
4917 * with a_i existentially quantified variable without explicit
4918 * representation, then apply a transformation on the existentially
4919 * quantified variables to turn the constraint into
4920 *
4921 * f(x) + g a_1' = 0
4922 *
4923 * with g the gcd of the c_i.
4924 * In order to easily identify which existentially quantified variables
4925 * have a complete explicit representation, i.e., without being defined
4926 * in terms of other existentially quantified variables without
4927 * an explicit representation, the existentially quantified variables
4928 * are first sorted.
4929 *
4930 * The variable transformation is computed by extending the row
4931 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4932 *
4933 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4934 * [a_2'] [ a_2 ]
4935 * ... = U ....
4936 * [a_n'] [ a_n ]
4937 *
4938 * with [c_1/g ... c_n/g] representing the first row of U.
4939 * The inverse of U is then plugged into the original constraints.
4940 * The call to isl_basic_map_simplify makes sure the explicit
4941 * representation for a_1' is extracted from the equality constraint.
4942 */
4945 {
4980 }
4999 }
5001 /* Does "bmap" satisfy any equality that involves more than 2 variables
5002 * and/or has coefficients different from -1 and 1?
5003 */
5005 {
5034 }
5037 }
5039 /* Remove any common factor g from the constraint coefficients in "v".
5040 * The constant term is stored in the first position and is replaced
5041 * by floor(c/g). If any common factor is removed and if this results
5042 * in a tightening of the constraint, then set *tightened.
5043 */
5046 {
5066 }
5068 /* If "bmap" is an integer set that satisfies any equality involving
5069 * more than 2 variables and/or has coefficients different from -1 and 1,
5070 * then use variable compression to reduce the coefficients by removing
5071 * any (hidden) common factor.
5072 * In particular, apply the variable compression to each constraint,
5073 * factor out any common factor in the non-constant coefficients and
5074 * then apply the inverse of the compression.
5075 * At the end, we mark the basic map as having reduced constants.
5076 * If this flag is still set on the next invocation of this function,
5077 * then we skip the computation.
5078 *
5079 * Removing a common factor may result in a tightening of some of
5080 * the constraints. If this happens, then we may end up with two
5081 * opposite inequalities that can be replaced by an equality.
5082 * We therefore call isl_basic_map_detect_inequality_pairs,
5083 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5084 * and isl_basic_map_gauss if such a pair was found.
5085 *
5086 * Note that this function may leave the result in an inconsistent state.
5087 * In particular, the constraints may not be gaussed.
5088 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5089 * for some of the test cases to pass successfully.
5090 * Any potential modification of the representation is therefore only
5091 * performed on a single copy of the basic map.
5092 */
5095 {
5129 }
5144 }
5159 }
5160 }
5163 error:
5168 }
5170 /* Shift the integer division at position "div" of "bmap"
5171 * by "shift" times the variable at position "pos".
5172 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5173 * corresponds to the constant term.
5174 *
5175 * That is, if the integer division has the form
5176 *
5177 * floor(f(x)/d)
5178 *
5179 * then replace it by
5180 *
5181 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5182 */
5185 {
5204 }
5210 }
5218 }
5221 }