| blob | 955e08d8f942ff27b7f0d6f690277f2f02597437 |
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 }
48 {
51 }
53 /* Drop n dimensions starting at first.
54 *
55 * In principle, this frees up some extra variables as the number
56 * of columns remains constant, but we would have to extend
57 * the div array too as the number of rows in this array is assumed
58 * to be equal to extra.
59 */
62 {
96 error:
99 }
101 /* Move "n" divs starting at "first" to the end of the list of divs.
102 */
105 {
123 error:
126 }
128 /* Drop "n" dimensions of type "type" starting at "first".
129 *
130 * In principle, this frees up some extra variables as the number
131 * of columns remains constant, but we would have to extend
132 * the div array too as the number of rows in this array is assumed
133 * to be equal to extra.
134 */
137 {
181 error:
184 }
188 {
191 }
195 {
216 }
220 error:
223 }
227 {
229 }
231 /*
232 * We don't cow, as the div is assumed to be redundant.
233 */
236 {
255 }
257 }
270 }
276 error:
279 }
283 {
285 isl_int gcd;
298 }
301 }
309 }
311 }
319 }
322 }
329 }
333 }
337 {
340 }
342 /* Reduce the coefficient of the variable at position "pos"
343 * in integer division "div", such that it lies in the half-open
344 * interval (1/2,1/2], extracting any excess value from this integer division.
345 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
346 * corresponds to the constant term.
347 *
348 * That is, the integer division is of the form
349 *
350 * floor((... + (c * d + r) * x_pos + ...)/d)
351 *
352 * with -d < 2 * r <= d.
353 * Replace it by
354 *
355 * floor((... + r * x_pos + ...)/d) + c * x_pos
356 *
357 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
358 * Otherwise, c = floor((c * d + r)/d) + 1.
359 *
360 * This is the same normalization that is performed by isl_aff_floor.
361 */
364 {
365 isl_int shift;
380 }
382 /* Does the coefficient of the variable at position "pos"
383 * in integer division "div" need to be reduced?
384 * That is, does it lie outside the half-open interval (1/2,1/2]?
385 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
386 * 2 * c != d.
387 */
390 {
391 isl_bool r;
403 }
405 /* Reduce the coefficients (including the constant term) of
406 * integer division "div", if needed.
407 * In particular, make sure all coefficients lie in
408 * the half-open interval (1/2,1/2].
409 */
412 {
417 isl_bool reduce;
427 }
430 }
432 /* Reduce the coefficients (including the constant term) of
433 * the known integer divisions, if needed
434 * In particular, make sure all coefficients lie in
435 * the half-open interval (1/2,1/2].
436 */
439 {
453 }
456 }
458 /* Remove any common factor in numerator and denominator of the div expression,
459 * not taking into account the constant term.
460 * That is, if the div is of the form
461 *
462 * floor((a + m f(x))/(m d))
463 *
464 * then replace it by
465 *
466 * floor((floor(a/m) + f(x))/d)
467 *
468 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
469 * and can therefore not influence the result of the floor.
470 */
472 {
488 }
490 /* Remove any common factor in numerator and denominator of a div expression,
491 * not taking into account the constant term.
492 * That is, look for any div of the form
493 *
494 * floor((a + m f(x))/(m d))
495 *
496 * and replace it by
497 *
498 * floor((floor(a/m) + f(x))/d)
499 *
500 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
501 * and can therefore not influence the result of the floor.
502 */
505 {
517 }
519 /* Assumes divs have been ordered if keep_divs is set.
520 */
523 {
541 }
551 }
560 /* We need to be careful about circular definitions,
561 * so for now we just remove the definition of div k
562 * if the equality contains any divs.
563 * If keep_divs is set, then the divs have been ordered
564 * and we can keep the definition as long as the result
565 * is still ordered.
566 */
574 }
575 }
577 /* Assumes divs have been ordered if keep_divs is set.
578 */
581 {
589 }
591 /* Check if elimination of div "div" using equality "eq" would not
592 * result in a div depending on a later div.
593 */
596 {
611 }
614 }
616 /* Eliminate divs based on equalities
617 */
620 {
635 isl_bool ok;
651 }
652 }
656 }
658 /* Elimininate divs based on inequalities
659 */
662 {
692 }
694 }
696 /* Does the equality constraint at position "eq" in "bmap" involve
697 * any local variables in the range [first, first + n)
698 * that are not marked as having an explicit representation?
699 */
702 {
711 isl_bool unknown;
720 }
723 }
725 /* The last local variable involved in the equality constraint
726 * at position "eq" in "bmap" is the local variable at position "div".
727 * It can therefore be used to extract an explicit representation
728 * for that variable.
729 * Do so unless the local variable already has an explicit representation or
730 * the explicit representation would involve any other local variables
731 * that in turn do not have an explicit representation.
732 * An equality constraint involving local variables without an explicit
733 * representation can be used in isl_basic_map_drop_redundant_divs
734 * to separate out an independent local variable. Introducing
735 * an explicit representation here would block this transformation,
736 * while the partial explicit representation in itself is not very useful.
737 * Set *progress if anything is changed.
738 *
739 * The equality constraint is of the form
740 *
741 * f(x) + n e >= 0
742 *
743 * with n a positive number. The explicit representation derived from
744 * this constraint is
745 *
746 * floor((-f(x))/n)
747 */
750 {
752 isl_bool involves;
776 }
780 {
803 }
812 progress);
819 }
826 }
829 }
833 {
835 progress));
836 }
840 {
846 }
848 }
850 /* Hash table of inequalities in a basic map.
851 * "index" is an array of addresses of inequalities in the basic map, some
852 * of which are NULL. The inequalities are hashed on the coefficients
853 * except the constant term.
854 * "size" is the number of elements in the array and is always a power of two
855 * "bits" is the number of bits need to represent an index into the array.
856 * "total" is the total dimension of the basic map.
857 */
863 };
865 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
866 */
869 {
886 }
888 /* Free the memory allocated by create_constraint_index.
889 */
891 {
893 }
895 /* Return the position in ci->index that contains the address of
896 * an inequality that is equal to *ineq up to the constant term,
897 * provided this address is not identical to "ineq".
898 * If there is no such inequality, then return the position where
899 * such an inequality should be inserted.
900 */
902 {
910 }
912 /* Return the position in ci->index that contains the address of
913 * an inequality that is equal to the k'th inequality of "bmap"
914 * up to the constant term, provided it does not point to the very
915 * same inequality.
916 * If there is no such inequality, then return the position where
917 * such an inequality should be inserted.
918 */
921 {
923 }
927 {
929 }
931 /* Fill in the "ci" data structure with the inequalities of "bset".
932 */
935 {
944 }
947 }
949 /* Is the inequality ineq (obviously) redundant with respect
950 * to the constraints in "ci"?
951 *
952 * Look for an inequality in "ci" with the same coefficients and then
953 * check if the contant term of "ineq" is greater than or equal
954 * to the constant term of that inequality. If so, "ineq" is clearly
955 * redundant.
956 *
957 * Note that hash_index_ineq ignores a stored constraint if it has
958 * the same address as the passed inequality. It is ok to pass
959 * the address of a local variable here since it will never be
960 * the same as the address of a constraint in "ci".
961 */
964 {
971 }
973 /* If we can eliminate more than one div, then we need to make
974 * sure we do it from last div to first div, in order not to
975 * change the position of the other divs that still need to
976 * be removed.
977 */
980 {
1034 }
1036 }
1048 }
1051 out:
1055 }
1058 {
1070 }
1072 }
1074 /* Normalize divs that appear in equalities.
1075 *
1076 * In particular, we assume that bmap contains some equalities
1077 * of the form
1078 *
1079 * a x = m * e_i
1080 *
1081 * and we want to replace the set of e_i by a minimal set and
1082 * such that the new e_i have a canonical representation in terms
1083 * of the vector x.
1084 * If any of the equalities involves more than one divs, then
1085 * we currently simply bail out.
1086 *
1087 * Let us first additionally assume that all equalities involve
1088 * a div. The equalities then express modulo constraints on the
1089 * remaining variables and we can use "parameter compression"
1090 * to find a minimal set of constraints. The result is a transformation
1091 *
1092 * x = T(x') = x_0 + G x'
1093 *
1094 * with G a lower-triangular matrix with all elements below the diagonal
1095 * non-negative and smaller than the diagonal element on the same row.
1096 * We first normalize x_0 by making the same property hold in the affine
1097 * T matrix.
1098 * The rows i of G with a 1 on the diagonal do not impose any modulo
1099 * constraint and simply express x_i = x'_i.
1100 * For each of the remaining rows i, we introduce a div and a corresponding
1101 * equality. In particular
1102 *
1103 * g_ii e_j = x_i - g_i(x')
1104 *
1105 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
1106 * corresponding div (if g_kk != 1).
1107 *
1108 * If there are any equalities not involving any div, then we
1109 * first apply a variable compression on the variables x:
1110 *
1111 * x = C x'' x'' = C_2 x
1112 *
1113 * and perform the above parameter compression on A C instead of on A.
1114 * The resulting compression is then of the form
1115 *
1116 * x'' = T(x') = x_0 + G x'
1117 *
1118 * and in constructing the new divs and the corresponding equalities,
1119 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
1120 * by the corresponding row from C_2.
1121 */
1124 {
1133 isl_int v;
1165 }
1166 }
1175 }
1181 }
1191 }
1198 }
1203 /* We have to be careful because dropping equalities may reorder them */
1213 }
1214 }
1220 else
1221 needed++;
1222 }
1228 }
1238 else
1246 else
1249 }
1253 }
1260 done:
1264 error:
1270 }
1274 {
1284 }
1286 /* Check whether it is ok to define a div based on an inequality.
1287 * To avoid the introduction of circular definitions of divs, we
1288 * do not allow such a definition if the resulting expression would refer to
1289 * any other undefined divs or if any known div is defined in
1290 * terms of the unknown div.
1291 */
1294 {
1298 /* Not defined in terms of unknown divs */
1306 }
1308 /* No other div defined in terms of this one => avoid loops */
1316 }
1319 }
1321 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1322 * be a better expression than the current one?
1323 *
1324 * If we do not have any expression yet, then any expression would be better.
1325 * Otherwise we check if the last variable involved in the inequality
1326 * (disregarding the div that it would define) is in an earlier position
1327 * than the last variable involved in the current div expression.
1328 */
1331 {
1348 }
1350 /* Given two constraints "k" and "l" that are opposite to each other,
1351 * except for the constant term, check if we can use them
1352 * to obtain an expression for one of the hitherto unknown divs or
1353 * a "better" expression for a div for which we already have an expression.
1354 * "sum" is the sum of the constant terms of the constraints.
1355 * If this sum is strictly smaller than the coefficient of one
1356 * of the divs, then this pair can be used define the div.
1357 * To avoid the introduction of circular definitions of divs, we
1358 * do not use the pair if the resulting expression would refer to
1359 * any other undefined divs or if any known div is defined in
1360 * terms of the unknown div.
1361 */
1364 {
1369 isl_bool set_div;
1384 else
1389 }
1391 }
1395 {
1399 isl_int sum;
1414 }
1422 }
1437 }
1439 /* We need to break out of the loop after these
1440 * changes since the contents of the hash
1441 * will no longer be valid.
1442 * Plus, we probably we want to regauss first.
1443 */
1451 }
1456 }
1458 /* Detect all pairs of inequalities that form an equality.
1459 *
1460 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1461 * Call it repeatedly while it is making progress.
1462 */
1465 {
1477 }
1479 /* Eliminate knowns divs from constraints where they appear with
1480 * a (positive or negative) unit coefficient.
1481 *
1482 * That is, replace
1483 *
1484 * floor(e/m) + f >= 0
1485 *
1486 * by
1487 *
1488 * e + m f >= 0
1489 *
1490 * and
1491 *
1492 * -floor(e/m) + f >= 0
1493 *
1494 * by
1495 *
1496 * -e + m f + m - 1 >= 0
1497 *
1498 * The first conversion is valid because floor(e/m) >= -f is equivalent
1499 * to e/m >= -f because -f is an integral expression.
1500 * The second conversion follows from the fact that
1501 *
1502 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1503 *
1504 *
1505 * Note that one of the div constraints may have been eliminated
1506 * due to being redundant with respect to the constraint that is
1507 * being modified by this function. The modified constraint may
1508 * no longer imply this div constraint, so we add it back to make
1509 * sure we do not lose any information.
1510 *
1511 * We skip integral divs, i.e., those with denominator 1, as we would
1512 * risk eliminating the div from the div constraints. We do not need
1513 * to handle those divs here anyway since the div constraints will turn
1514 * out to form an equality and this equality can then be used to eliminate
1515 * the div from all constraints.
1516 */
1519 {
1551 else
1561 }
1566 }
1567 }
1570 }
1573 {
1578 isl_bool empty;
1594 /* requires equalities in normal form */
1600 }
1602 }
1605 {
1607 }
1612 {
1644 }
1648 {
1650 }
1653 /* If the only constraints a div d=floor(f/m)
1654 * appears in are its two defining constraints
1655 *
1656 * f - m d >=0
1657 * -(f - (m - 1)) + m d >= 0
1658 *
1659 * then it can safely be removed.
1660 */
1662 {
1671 isl_bool red;
1678 }
1685 }
1688 }
1690 /*
1691 * Remove divs that don't occur in any of the constraints or other divs.
1692 * These can arise when dropping constraints from a basic map or
1693 * when the divs of a basic map have been temporarily aligned
1694 * with the divs of another basic map.
1695 */
1697 {
1704 isl_bool redundant;
1712 }
1714 }
1716 /* Mark "bmap" as final, without checking for obviously redundant
1717 * integer divisions. This function should be used when "bmap"
1718 * is known not to involve any such integer divisions.
1719 */
1722 {
1727 }
1729 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1730 */
1732 {
1736 }
1739 {
1741 }
1743 /* Remove definition of any div that is defined in terms of the given variable.
1744 * The div itself is not removed. Functions such as
1745 * eliminate_divs_ineq depend on the other divs remaining in place.
1746 */
1749 {
1763 }
1765 }
1767 /* Eliminate the specified variables from the constraints using
1768 * Fourier-Motzkin. The variables themselves are not removed.
1769 */
1772 {
1804 }
1811 n_lower++;
1813 n_upper++;
1814 }
1838 }
1841 }
1853 }
1854 }
1859 error:
1862 }
1866 {
1869 }
1871 /* Eliminate the specified n dimensions starting at first from the
1872 * constraints, without removing the dimensions from the space.
1873 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1874 * Otherwise, they are projected out and the original space is restored.
1875 */
1879 {
1895 }
1902 error:
1905 }
1910 {
1912 }
1914 /* Remove all constraints from "bmap" that reference any unknown local
1915 * variables (directly or indirectly).
1916 *
1917 * Dropping all constraints on a local variable will make it redundant,
1918 * so it will get removed implicitly by
1919 * isl_basic_map_drop_constraints_involving_dims. Some other local
1920 * variables may also end up becoming redundant if they only appear
1921 * in constraints together with the unknown local variable.
1922 * Therefore, start over after calling
1923 * isl_basic_map_drop_constraints_involving_dims.
1924 */
1927 {
1928 isl_bool known;
1953 }
1956 }
1958 /* Remove all constraints from "map" that reference any unknown local
1959 * variables (directly or indirectly).
1960 *
1961 * Since constraints may get dropped from the basic maps,
1962 * they may no longer be disjoint from each other.
1963 */
1966 {
1968 isl_bool known;
1986 }
1992 }
1994 /* Don't assume equalities are in order, because align_divs
1995 * may have changed the order of the divs.
1996 */
1998 {
2011 }
2012 }
2013 }
2016 {
2018 }
2022 {
2036 }
2038 }
2040 }
2044 {
2047 }
2051 {
2061 }
2082 error:
2086 }
2088 /* For each inequality in "ineq" that is a shifted (more relaxed)
2089 * copy of an inequality in "context", mark the corresponding entry
2090 * in "row" with -1.
2091 * If an inequality only has a non-negative constant term, then
2092 * mark it as well.
2093 */
2096 {
2113 isl_bool redundant;
2119 }
2126 }
2129 error:
2132 }
2136 {
2149 isl_bool redundant;
2161 }
2164 error:
2167 }
2169 /* Remove constraints from "bmap" that are identical to constraints
2170 * in "context" or that are more relaxed (greater constant term).
2171 *
2172 * We perform the test for shifted copies on the pure constraints
2173 * in remove_shifted_constraints.
2174 */
2177 {
2186 }
2199 error:
2203 }
2205 /* Does the (linear part of a) constraint "c" involve any of the "len"
2206 * "relevant" dimensions?
2207 */
2209 {
2217 }
2220 }
2222 /* Drop constraints from "bmap" that do not involve any of
2223 * the dimensions marked "relevant".
2224 */
2227 {
2242 }
2249 }
2252 }
2254 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2255 *
2256 * In particular, for any variable involved in the constraint,
2257 * find the actual group id from before and replace the group
2258 * of the corresponding variable by the minimal group of all
2259 * the variables involved in the constraint considered so far
2260 * (if this minimum is smaller) or replace the minimum by this group
2261 * (if the minimum is larger).
2262 *
2263 * At the end, all the variables in "c" will (indirectly) point
2264 * to the minimal of the groups that they referred to originally.
2265 */
2267 {
2284 }
2285 }
2287 /* Allocate an array of groups of variables, one for each variable
2288 * in "context", initialized to zero.
2289 */
2291 {
2298 }
2300 /* Drop constraints from "bmap" that only involve variables that are
2301 * not related to any of the variables marked with a "-1" in "group".
2302 *
2303 * We construct groups of variables that collect variables that
2304 * (indirectly) appear in some common constraint of "bmap".
2305 * Each group is identified by the first variable in the group,
2306 * except for the special group of variables that was already identified
2307 * in the input as -1 (or are related to those variables).
2308 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2309 * otherwise the group of i is the group of group[i].
2310 *
2311 * We first initialize groups for the remaining variables.
2312 * Then we iterate over the constraints of "bmap" and update the
2313 * group of the variables in the constraint by the smallest group.
2314 * Finally, we resolve indirect references to groups by running over
2315 * the variables.
2316 *
2317 * After computing the groups, we drop constraints that do not involve
2318 * any variables in the -1 group.
2319 */
2322 {
2339 }
2357 }
2359 /* Drop constraints from "context" that are irrelevant for computing
2360 * the gist of "bset".
2361 *
2362 * In particular, drop constraints in variables that are not related
2363 * to any of the variables involved in the constraints of "bset"
2364 * in the sense that there is no sequence of constraints that connects them.
2365 *
2366 * We first mark all variables that appear in "bset" as belonging
2367 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2368 */
2371 {
2392 }
2398 }
2401 }
2403 /* Drop constraints from "context" that are irrelevant for computing
2404 * the gist of the inequalities "ineq".
2405 * Inequalities in "ineq" for which the corresponding element of row
2406 * is set to -1 have already been marked for removal and should be ignored.
2407 *
2408 * In particular, drop constraints in variables that are not related
2409 * to any of the variables involved in "ineq"
2410 * in the sense that there is no sequence of constraints that connects them.
2411 *
2412 * We first mark all variables that appear in "bset" as belonging
2413 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2414 */
2417 {
2438 }
2441 }
2444 }
2446 /* Do all "n" entries of "row" contain a negative value?
2447 */
2449 {
2457 }
2459 /* Update the inequalities in "bset" based on the information in "row"
2460 * and "tab".
2461 *
2462 * In particular, the array "row" contains either -1, meaning that
2463 * the corresponding inequality of "bset" is redundant, or the index
2464 * of an inequality in "tab".
2465 *
2466 * If the row entry is -1, then drop the inequality.
2467 * Otherwise, if the constraint is marked redundant in the tableau,
2468 * then drop the inequality. Similarly, if it is marked as an equality
2469 * in the tableau, then turn the inequality into an equality and
2470 * perform Gaussian elimination.
2471 */
2474 {
2491 }
2501 }
2502 }
2508 }
2510 /* Update the inequalities in "bset" based on the information in "row"
2511 * and "tab" and free all arguments (other than "bset").
2512 */
2517 {
2526 }
2528 /* Remove all information from bset that is redundant in the context
2529 * of context.
2530 * "ineq" contains the (possibly transformed) inequalities of "bset",
2531 * in the same order.
2532 * The (explicit) equalities of "bset" are assumed to have been taken
2533 * into account by the transformation such that only the inequalities
2534 * are relevant.
2535 * "context" is assumed not to be empty.
2536 *
2537 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2538 * A value of -1 means that the inequality is obviously redundant and may
2539 * not even appear in "tab".
2540 *
2541 * We first mark the inequalities of "bset"
2542 * that are obviously redundant with respect to some inequality in "context".
2543 * Then we remove those constraints from "context" that have become
2544 * irrelevant for computing the gist of "bset".
2545 * Note that this removal of constraints cannot be replaced by
2546 * a factorization because factors in "bset" may still be connected
2547 * to each other through constraints in "context".
2548 *
2549 * If there are any inequalities left, we construct a tableau for
2550 * the context and then add the inequalities of "bset".
2551 * Before adding these inequalities, we freeze all constraints such that
2552 * they won't be considered redundant in terms of the constraints of "bset".
2553 * Then we detect all redundant constraints (among the
2554 * constraints that weren't frozen), first by checking for redundancy in the
2555 * the tableau and then by checking if replacing a constraint by its negation
2556 * would lead to an empty set. This last step is fairly expensive
2557 * and could be optimized by more reuse of the tableau.
2558 * Finally, we update bset according to the results.
2559 */
2562 {
2577 }
2613 }
2637 }
2642 }
2646 error:
2654 }
2656 /* Extract the inequalities of "bset" as an isl_mat.
2657 */
2659 {
2673 }
2675 /* Remove all information from "bset" that is redundant in the context
2676 * of "context", for the case where both "bset" and "context" are
2677 * full-dimensional.
2678 */
2681 {
2686 }
2688 /* Remove all information from "bset" that is redundant in the context
2689 * of "context", for the case where the combined equalities of
2690 * "bset" and "context" allow for a compression that can be obtained
2691 * by preapplication of "T".
2692 *
2693 * "bset" itself is not transformed by "T". Instead, the inequalities
2694 * are extracted from "bset" and those are transformed by "T".
2695 * uset_gist_full then determines which of the transformed inequalities
2696 * are redundant with respect to the transformed "context" and removes
2697 * the corresponding inequalities from "bset".
2698 *
2699 * After preapplying "T" to the inequalities, any common factor is
2700 * removed from the coefficients. If this results in a tightening
2701 * of the constant term, then the same tightening is applied to
2702 * the corresponding untransformed inequality in "bset".
2703 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2704 *
2705 * g f'(x) + r >= 0
2706 *
2707 * with 0 <= r < g, then it is equivalent to
2708 *
2709 * f'(x) >= 0
2710 *
2711 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2712 * subspace compressed by T since the latter would be transformed to
2713 *
2714 * g f'(x) >= 0
2715 */
2719 {
2723 isl_int rem;
2735 }
2758 }
2762 error:
2767 }
2769 /* Project "bset" onto the variables that are involved in "template".
2770 */
2773 {
2782 isl_bool involved;
2791 }
2794 }
2796 /* Remove all information from bset that is redundant in the context
2797 * of context. In particular, equalities that are linear combinations
2798 * of those in context are removed. Then the inequalities that are
2799 * redundant in the context of the equalities and inequalities of
2800 * context are removed.
2801 *
2802 * First of all, we drop those constraints from "context"
2803 * that are irrelevant for computing the gist of "bset".
2804 * Alternatively, we could factorize the intersection of "context" and "bset".
2805 *
2806 * We first compute the intersection of the integer affine hulls
2807 * of "bset" and "context",
2808 * compute the gist inside this intersection and then reduce
2809 * the constraints with respect to the equalities of the context
2810 * that only involve variables already involved in the input.
2811 *
2812 * If two constraints are mutually redundant, then uset_gist_full
2813 * will remove the second of those constraints. We therefore first
2814 * sort the constraints so that constraints not involving existentially
2815 * quantified variables are given precedence over those that do.
2816 * We have to perform this sorting before the variable compression,
2817 * because that may effect the order of the variables.
2818 */
2821 {
2846 }
2851 }
2861 }
2872 }
2875 error:
2879 }
2881 /* Return the number of equality constraints in "bmap" that involve
2882 * local variables. This function assumes that Gaussian elimination
2883 * has been applied to the equality constraints.
2884 */
2886 {
2906 }
2908 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2909 * The constraints are assumed not to involve any local variables.
2910 */
2913 {
2931 }
2936 error:
2941 }
2943 /* Construct and return a variable compression based on the equality
2944 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2945 * "n1" is the number of (initial) equality constraints in "bmap1"
2946 * that do involve local variables.
2947 * "n2" is the number of (initial) equality constraints in "bmap2"
2948 * that do involve local variables.
2949 * "total" is the total number of other variables.
2950 * This function assumes that Gaussian elimination
2951 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2952 * such that the equality constraints not involving local variables
2953 * are those that start at "n1" or "n2".
2954 *
2955 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2956 * then simply compute the compression based on the equality constraints
2957 * in the other basic map.
2958 * Otherwise, combine the equality constraints from both into a new
2959 * basic map such that Gaussian elimination can be applied to this combination
2960 * and then construct a variable compression from the resulting
2961 * equality constraints.
2962 */
2966 {
2976 }
2981 }
2996 }
2998 /* Extract the stride constraints from "bmap", compressed
2999 * with respect to both the stride constraints in "context" and
3000 * the remaining equality constraints in both "bmap" and "context".
3001 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
3002 * "context_n_eq" is the number of (initial) stride constraints in "context".
3003 *
3004 * Let x be all variables in "bmap" (and "context") other than the local
3005 * variables. First compute a variable compression
3006 *
3007 * x = V x'
3008 *
3009 * based on the non-stride equality constraints in "bmap" and "context".
3010 * Consider the stride constraints of "context",
3011 *
3012 * A(x) + B(y) = 0
3013 *
3014 * with y the local variables and plug in the variable compression,
3015 * resulting in
3016 *
3017 * A(V x') + B(y) = 0
3018 *
3019 * Use these constraints to compute a parameter compression on x'
3020 *
3021 * x' = T x''
3022 *
3023 * Now consider the stride constraints of "bmap"
3024 *
3025 * C(x) + D(y) = 0
3026 *
3027 * and plug in x = V*T x''.
3028 * That is, return A = [C*V*T D].
3029 */
3033 {
3062 }
3064 /* Remove the prime factors from *g that have an exponent that
3065 * is strictly smaller than the exponent in "c".
3066 * All exponents in *g are known to be smaller than or equal
3067 * to those in "c".
3068 *
3069 * That is, if *g is equal to
3070 *
3071 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
3072 *
3073 * and "c" is equal to
3074 *
3075 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
3076 *
3077 * then update *g to
3078 *
3079 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
3080 * p_n^{e_n * (e_n = f_n)}
3081 *
3082 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
3083 * neither does the gcd of *g and c / *g.
3084 * If e_i < f_i, then the gcd of *g and c / *g has a positive
3085 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
3086 * Dividing *g by this gcd therefore strictly reduces the exponent
3087 * of the prime factors that need to be removed, while leaving the
3088 * other prime factors untouched.
3089 * Repeating this process until gcd(*g, c / *g) = 1 therefore
3090 * removes all undesired factors, without removing any others.
3091 */
3093 {
3094 isl_int t;
3103 }
3105 }
3107 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
3108 * of the same stride constraints in a compressed space that exploits
3109 * all equalities in the context and the other equalities in "bmap".
3110 *
3111 * If the stride constraints of "bmap" are of the form
3112 *
3113 * C(x) + D(y) = 0
3114 *
3115 * then A is of the form
3116 *
3117 * B(x') + D(y) = 0
3118 *
3119 * If any of these constraints involves only a single local variable y,
3120 * then the constraint appears as
3121 *
3122 * f(x) + m y_i = 0
3123 *
3124 * in "bmap" and as
3125 *
3126 * h(x') + m y_i = 0
3127 *
3128 * in "A".
3129 *
3130 * Let g be the gcd of m and the coefficients of h.
3131 * Then, in particular, g is a divisor of the coefficients of h and
3132 *
3133 * f(x) = h(x')
3134 *
3135 * is known to be a multiple of g.
3136 * If some prime factor in m appears with the same exponent in g,
3137 * then it can be removed from m because f(x) is already known
3138 * to be a multiple of g and therefore in particular of this power
3139 * of the prime factors.
3140 * Prime factors that appear with a smaller exponent in g cannot
3141 * be removed from m.
3142 * Let g' be the divisor of g containing all prime factors that
3143 * appear with the same exponent in m and g, then
3144 *
3145 * f(x) + m y_i = 0
3146 *
3147 * can be replaced by
3148 *
3149 * f(x) + m/g' y_i' = 0
3150 *
3151 * Note that (if g' != 1) this changes the explicit representation
3152 * of y_i to that of y_i', so the integer division at position i
3153 * is marked unknown and later recomputed by a call to
3154 * isl_basic_map_gauss.
3155 */
3158 {
3162 isl_int gcd;
3196 }
3203 error:
3207 }
3209 /* Simplify the stride constraints in "bmap" based on
3210 * the remaining equality constraints in "bmap" and all equality
3211 * constraints in "context".
3212 * Only do this if both "bmap" and "context" have stride constraints.
3213 *
3214 * First extract a copy of the stride constraints in "bmap" in a compressed
3215 * space exploiting all the other equality constraints and then
3216 * use this compressed copy to simplify the original stride constraints.
3217 */
3220 {
3242 }
3244 /* Return a basic map that has the same intersection with "context" as "bmap"
3245 * and that is as "simple" as possible.
3246 *
3247 * The core computation is performed on the pure constraints.
3248 * When we add back the meaning of the integer divisions, we need
3249 * to (re)introduce the div constraints. If we happen to have
3250 * discovered that some of these integer divisions are equal to
3251 * some affine combination of other variables, then these div
3252 * constraints may end up getting simplified in terms of the equalities,
3253 * resulting in extra inequalities on the other variables that
3254 * may have been removed already or that may not even have been
3255 * part of the input. We try and remove those constraints of
3256 * this form that are most obviously redundant with respect to
3257 * the context. We also remove those div constraints that are
3258 * redundant with respect to the other constraints in the result.
3259 *
3260 * The stride constraints among the equality constraints in "bmap" are
3261 * also simplified with respecting to the other equality constraints
3262 * in "bmap" and with respect to all equality constraints in "context".
3263 */
3266 {
3277 }
3283 }
3287 }
3309 }
3328 error:
3332 }
3334 /*
3335 * Assumes context has no implicit divs.
3336 */
3339 {
3350 }
3370 }
3371 }
3375 error:
3379 }
3381 /* Drop all inequalities from "bmap" that also appear in "context".
3382 * "context" is assumed to have only known local variables and
3383 * the initial local variables of "bmap" are assumed to be the same
3384 * as those of "context".
3385 * The constraints of both "bmap" and "context" are assumed
3386 * to have been sorted using isl_basic_map_sort_constraints.
3387 *
3388 * Run through the inequality constraints of "bmap" and "context"
3389 * in sorted order.
3390 * If a constraint of "bmap" involves variables not in "context",
3391 * then it cannot appear in "context".
3392 * If a matching constraint is found, it is removed from "bmap".
3393 */
3396 {
3415 }
3421 }
3425 }
3430 }
3433 }
3436 }
3438 /* Drop all equalities from "bmap" that also appear in "context".
3439 * "context" is assumed to have only known local variables and
3440 * the initial local variables of "bmap" are assumed to be the same
3441 * as those of "context".
3442 *
3443 * Run through the equality constraints of "bmap" and "context"
3444 * in sorted order.
3445 * If a constraint of "bmap" involves variables not in "context",
3446 * then it cannot appear in "context".
3447 * If a matching constraint is found, it is removed from "bmap".
3448 */
3451 {
3475 }
3479 }
3484 }
3487 }
3490 }
3492 /* Remove the constraints in "context" from "bmap".
3493 * "context" is assumed to have explicit representations
3494 * for all local variables.
3495 *
3496 * First align the divs of "bmap" to those of "context" and
3497 * sort the constraints. Then drop all constraints from "bmap"
3498 * that appear in "context".
3499 */
3502 {
3517 }
3536 error:
3540 }
3542 /* Replace "map" by the disjunct at position "pos" and free "context".
3543 */
3546 {
3553 }
3555 /* Remove the constraints in "context" from "map".
3556 * If any of the disjuncts in the result turns out to be the universe,
3557 * then return this universe.
3558 * "context" is assumed to have explicit representations
3559 * for all local variables.
3560 */
3563 {
3573 }
3592 }
3599 error:
3603 }
3605 /* Replace "map" by a universe map in the same space and free "drop".
3606 */
3609 {
3616 }
3618 /* Return a map that has the same intersection with "context" as "map"
3619 * and that is as "simple" as possible.
3620 *
3621 * If "map" is already the universe, then we cannot make it any simpler.
3622 * Similarly, if "context" is the universe, then we cannot exploit it
3623 * to simplify "map"
3624 * If "map" and "context" are identical to each other, then we can
3625 * return the corresponding universe.
3626 *
3627 * If either "map" or "context" consists of multiple disjuncts,
3628 * then check if "context" happens to be a subset of "map",
3629 * in which case all constraints can be removed.
3630 * In case of multiple disjuncts, the standard procedure
3631 * may not be able to detect that all constraints can be removed.
3632 *
3633 * If none of these cases apply, we have to work a bit harder.
3634 * During this computation, we make use of a single disjunct context,
3635 * so if the original context consists of more than one disjunct
3636 * then we need to approximate the context by a single disjunct set.
3637 * Simply taking the simple hull may drop constraints that are
3638 * only implicitly available in each disjunct. We therefore also
3639 * look for constraints among those defining "map" that are valid
3640 * for the context. These can then be used to simplify away
3641 * the corresponding constraints in "map".
3642 */
3645 {
3649 isl_bool subset;
3660 }
3676 }
3692 list);
3693 }
3695 error:
3699 }
3703 {
3705 }
3709 {
3712 }
3716 {
3719 }
3723 {
3728 }
3732 {
3734 }
3736 /* Compute the gist of "bmap" with respect to the constraints "context"
3737 * on the domain.
3738 */
3741 {
3747 }
3751 {
3755 }
3759 {
3763 }
3767 {
3771 }
3775 {
3777 }
3779 /* Quick check to see if two basic maps are disjoint.
3780 * In particular, we reduce the equalities and inequalities of
3781 * one basic map in the context of the equalities of the other
3782 * basic map and check if we get a contradiction.
3783 */
3786 {
3818 }
3826 }
3835 }
3839 disjoint:
3843 error:
3847 }
3851 {
3854 }
3856 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3857 */
3861 {
3872 }
3873 }
3876 }
3878 /* Are "map1" and "map2" obviously disjoint, based on information
3879 * that can be derived without looking at the individual basic maps?
3880 *
3881 * In particular, if one of them is empty or if they live in different spaces
3882 * (ignoring parameters), then they are clearly disjoint.
3883 */
3886 {
3887 isl_bool disjoint;
3888 isl_bool match;
3912 }
3914 /* Are "map1" and "map2" obviously disjoint?
3915 *
3916 * If one of them is empty or if they live in different spaces (ignoring
3917 * parameters), then they are clearly disjoint.
3918 * This is checked by isl_map_plain_is_disjoint_global.
3919 *
3920 * If they have different parameters, then we skip any further tests.
3921 *
3922 * If they are obviously equal, but not obviously empty, then we will
3923 * not be able to detect if they are disjoint.
3924 *
3925 * Otherwise we check if each basic map in "map1" is obviously disjoint
3926 * from each basic map in "map2".
3927 */
3930 {
3931 isl_bool disjoint;
3932 isl_bool intersect;
3933 isl_bool match;
3949 }
3951 /* Are "map1" and "map2" disjoint?
3952 *
3953 * They are disjoint if they are "obviously disjoint" or if one of them
3954 * is empty. Otherwise, they are not disjoint if one of them is universal.
3955 * If the two inputs are (obviously) equal and not empty, then they are
3956 * not disjoint.
3957 * If none of these cases apply, then check if all pairs of basic maps
3958 * are disjoint.
3959 */
3961 {
3962 isl_bool disjoint;
3963 isl_bool intersect;
3990 }
3992 /* Are "bmap1" and "bmap2" disjoint?
3993 *
3994 * They are disjoint if they are "obviously disjoint" or if one of them
3995 * is empty. Otherwise, they are not disjoint if one of them is universal.
3996 * If none of these cases apply, we compute the intersection and see if
3997 * the result is empty.
3998 */
4001 {
4002 isl_bool disjoint;
4003 isl_bool intersect;
4032 }
4034 /* Are "bset1" and "bset2" disjoint?
4035 */
4038 {
4040 }
4044 {
4046 }
4048 /* Are "set1" and "set2" disjoint?
4049 */
4051 {
4053 }
4055 /* Is "v" equal to 0, 1 or -1?
4056 */
4058 {
4060 }
4062 /* Check if we can combine a given div with lower bound l and upper
4063 * bound u with some other div and if so return that other div.
4064 * Otherwise return -1.
4065 *
4066 * We first check that
4067 * - the bounds are opposites of each other (except for the constant
4068 * term)
4069 * - the bounds do not reference any other div
4070 * - no div is defined in terms of this div
4071 *
4072 * Let m be the size of the range allowed on the div by the bounds.
4073 * That is, the bounds are of the form
4074 *
4075 * e <= a <= e + m - 1
4076 *
4077 * with e some expression in the other variables.
4078 * We look for another div b such that no third div is defined in terms
4079 * of this second div b and such that in any constraint that contains
4080 * a (except for the given lower and upper bound), also contains b
4081 * with a coefficient that is m times that of b.
4082 * That is, all constraints (execpt for the lower and upper bound)
4083 * are of the form
4084 *
4085 * e + f (a + m b) >= 0
4086 *
4087 * Furthermore, in the constraints that only contain b, the coefficient
4088 * of b should be equal to 1 or -1.
4089 * If so, we return b so that "a + m b" can be replaced by
4090 * a single div "c = a + m b".
4091 */
4094 {
4116 }
4126 }
4138 }
4149 }
4162 }
4167 }
4171 }
4173 /* Internal data structure used during the construction and/or evaluation of
4174 * an inequality that ensures that a pair of bounds always allows
4175 * for an integer value.
4176 *
4177 * "tab" is the tableau in which the inequality is evaluated. It may
4178 * be NULL until it is actually needed.
4179 * "v" contains the inequality coefficients.
4180 * "g", "fl" and "fu" are temporary scalars used during the construction and
4181 * evaluation.
4182 */
4186 isl_int g;
4187 isl_int fl;
4188 isl_int fu;
4189 };
4191 /* Free all the memory allocated by the fields of "data".
4192 */
4194 {
4200 }
4202 /* Is the inequality stored in data->v satisfied by "bmap"?
4203 * That is, does it only attain non-negative values?
4204 * data->tab is a tableau corresponding to "bmap".
4205 */
4208 {
4219 }
4221 /* Given a lower and an upper bound on div i, do they always allow
4222 * for an integer value of the given div?
4223 * Determine this property by constructing an inequality
4224 * such that the property is guaranteed when the inequality is nonnegative.
4225 * The lower bound is inequality l, while the upper bound is inequality u.
4226 * The constructed inequality is stored in data->v.
4227 *
4228 * Let the upper bound be
4229 *
4230 * -n_u a + e_u >= 0
4231 *
4232 * and the lower bound
4233 *
4234 * n_l a + e_l >= 0
4235 *
4236 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4237 * We have
4238 *
4239 * - f_u e_l <= f_u f_l g a <= f_l e_u
4240 *
4241 * Since all variables are integer valued, this is equivalent to
4242 *
4243 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4244 *
4245 * If this interval is at least f_u f_l g, then it contains at least
4246 * one integer value for a.
4247 * That is, the test constraint is
4248 *
4249 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4250 *
4251 * or
4252 *
4253 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4254 *
4255 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4256 * then the constraint can be scaled down by a factor g',
4257 * with the constant term replaced by
4258 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4259 * Note that the result of applying Fourier-Motzkin to this pair
4260 * of constraints is
4261 *
4262 * f_l e_u + f_u e_l >= 0
4263 *
4264 * If the constant term of the scaled down version of this constraint,
4265 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4266 * term of the scaled down test constraint, then the test constraint
4267 * is known to hold and no explicit evaluation is required.
4268 * This is essentially the Omega test.
4269 *
4270 * If the test constraint consists of only a constant term, then
4271 * it is sufficient to look at the sign of this constant term.
4272 */
4275 {
4300 }
4310 }
4312 /* Remove more kinds of divs that are not strictly needed.
4313 * In particular, if all pairs of lower and upper bounds on a div
4314 * are such that they allow at least one integer value of the div,
4315 * then we can eliminate the div using Fourier-Motzkin without
4316 * introducing any spurious solutions.
4317 *
4318 * If at least one of the two constraints has a unit coefficient for the div,
4319 * then the presence of such a value is guaranteed so there is no need to check.
4320 * In particular, the value attained by the bound with unit coefficient
4321 * can serve as this intermediate value.
4322 */
4325 {
4348 isl_bool has_int;
4356 }
4377 }
4380 }
4384 }
4388 }
4391 }
4402 error:
4407 }
4409 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4410 * and the upper bound u, div1 always occurs together with div2 in the form
4411 * (div1 + m div2), where m is the constant range on the variable div1
4412 * allowed by l and u, replace the pair div1 and div2 by a single
4413 * div that is equal to div1 + m div2.
4414 *
4415 * The new div will appear in the location that contains div2.
4416 * We need to modify all constraints that contain
4417 * div2 = (div - div1) / m
4418 * The coefficient of div2 is known to be equal to 1 or -1.
4419 * (If a constraint does not contain div2, it will also not contain div1.)
4420 * If the constraint also contains div1, then we know they appear
4421 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4422 * i.e., the coefficient of div is f.
4423 *
4424 * Otherwise, we first need to introduce div1 into the constraint.
4425 * Let the l be
4426 *
4427 * div1 + f >=0
4428 *
4429 * and u
4430 *
4431 * -div1 + f' >= 0
4432 *
4433 * A lower bound on div2
4434 *
4435 * div2 + t >= 0
4436 *
4437 * can be replaced by
4438 *
4439 * m div2 + div1 + m t + f >= 0
4440 *
4441 * An upper bound
4442 *
4443 * -div2 + t >= 0
4444 *
4445 * can be replaced by
4446 *
4447 * -(m div2 + div1) + m t + f' >= 0
4448 *
4449 * These constraint are those that we would obtain from eliminating
4450 * div1 using Fourier-Motzkin.
4451 *
4452 * After all constraints have been modified, we drop the lower and upper
4453 * bound and then drop div1.
4454 */
4457 {
4459 isl_int m;
4481 else
4484 }
4488 }
4497 }
4500 }
4502 /* First check if we can coalesce any pair of divs and
4503 * then continue with dropping more redundant divs.
4504 *
4505 * We loop over all pairs of lower and upper bounds on a div
4506 * with coefficient 1 and -1, respectively, check if there
4507 * is any other div "c" with which we can coalesce the div
4508 * and if so, perform the coalescing.
4509 */
4512 {
4535 }
4536 }
4537 }
4542 }
4545 }
4547 /* Are the "n" coefficients starting at "first" of inequality constraints
4548 * "i" and "j" of "bmap" equal to each other?
4549 */
4552 {
4554 }
4556 /* Are the "n" coefficients starting at "first" of inequality constraints
4557 * "i" and "j" of "bmap" opposite to each other?
4558 */
4561 {
4563 }
4565 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4566 * apart from the constant term?
4567 */
4569 {
4574 }
4576 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4577 * apart from the constant term and the coefficient at position "pos"?
4578 */
4581 {
4587 }
4589 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4590 * apart from the constant term and the coefficient at position "pos"?
4591 */
4594 {
4600 }
4602 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4603 * been modified, simplying it if "simplify" is set.
4604 * Free the temporary data structure "pairs" that was associated
4605 * to the old version of "bmap".
4606 */
4609 {
4614 }
4616 /* Is "div" the single unknown existentially quantified variable
4617 * in inequality constraint "ineq" of "bmap"?
4618 * "div" is known to have a non-zero coefficient in "ineq".
4619 */
4622 {
4625 isl_bool known;
4635 isl_bool known;
4644 }
4647 }
4649 /* Does integer division "div" have coefficient 1 in inequality constraint
4650 * "ineq" of "map"?
4651 */
4653 {
4661 }
4663 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4664 * then try and drop redundant divs again,
4665 * freeing the temporary data structure "pairs" that was associated
4666 * to the old version of "bmap".
4667 */
4670 {
4674 }
4676 /* Drop the integer division at position "div", along with the two
4677 * inequality constraints "ineq1" and "ineq2" in which it appears
4678 * from "bmap" and then try and drop redundant divs again,
4679 * freeing the temporary data structure "pairs" that was associated
4680 * to the old version of "bmap".
4681 */
4685 {
4692 }
4695 }
4697 /* Given two inequality constraints
4698 *
4699 * f(x) + n d + c >= 0, (ineq)
4700 *
4701 * with d the variable at position "pos", and
4702 *
4703 * f(x) + c0 >= 0, (lower)
4704 *
4705 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4706 * determined by the first constraint.
4707 * That is, store
4708 *
4709 * ceil((c0 - c)/n)
4710 *
4711 * in *l.
4712 */
4715 {
4719 }
4721 /* Given two inequality constraints
4722 *
4723 * f(x) + n d + c >= 0, (ineq)
4724 *
4725 * with d the variable at position "pos", and
4726 *
4727 * -f(x) - c0 >= 0, (upper)
4728 *
4729 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4730 * determined by the first constraint.
4731 * That is, store
4732 *
4733 * ceil((-c1 - c)/n)
4734 *
4735 * in *u.
4736 */
4739 {
4743 }
4745 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4746 * does the corresponding lower bound have a fixed value in "bmap"?
4747 *
4748 * In particular, "ineq" is of the form
4749 *
4750 * f(x) + n d + c >= 0
4751 *
4752 * with n > 0, c the constant term and
4753 * d the existentially quantified variable "div".
4754 * That is, the lower bound is
4755 *
4756 * ceil((-f(x) - c)/n)
4757 *
4758 * Look for a pair of constraints
4759 *
4760 * f(x) + c0 >= 0
4761 * -f(x) + c1 >= 0
4762 *
4763 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4764 * That is, check that
4765 *
4766 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4767 *
4768 * If so, return the index of inequality f(x) + c0 >= 0.
4769 * Otherwise, return -1.
4770 */
4772 {
4789 }
4793 }
4794 }
4811 }
4813 /* Given a lower bound constraint "ineq" on the existentially quantified
4814 * variable "div", such that the corresponding lower bound has
4815 * a fixed value in "bmap", assign this fixed value to the variable and
4816 * then try and drop redundant divs again,
4817 * freeing the temporary data structure "pairs" that was associated
4818 * to the old version of "bmap".
4819 * "lower" determines the constant value for the lower bound.
4820 *
4821 * In particular, "ineq" is of the form
4822 *
4823 * f(x) + n d + c >= 0,
4824 *
4825 * while "lower" is of the form
4826 *
4827 * f(x) + c0 >= 0
4828 *
4829 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4830 * is ceil((c0 - c)/n).
4831 */
4834 {
4835 isl_int c;
4848 }
4850 /* Remove divs that are not strictly needed based on the inequality
4851 * constraints.
4852 * In particular, if a div only occurs positively (or negatively)
4853 * in constraints, then it can simply be dropped.
4854 * Also, if a div occurs in only two constraints and if moreover
4855 * those two constraints are opposite to each other, except for the constant
4856 * term and if the sum of the constant terms is such that for any value
4857 * of the other values, there is always at least one integer value of the
4858 * div, i.e., if one plus this sum is greater than or equal to
4859 * the (absolute value) of the coefficient of the div in the constraints,
4860 * then we can also simply drop the div.
4861 *
4862 * If an existentially quantified variable does not have an explicit
4863 * representation, appears in only a single lower bound that does not
4864 * involve any other such existentially quantified variables and appears
4865 * in this lower bound with coefficient 1,
4866 * then fix the variable to the value of the lower bound. That is,
4867 * turn the inequality into an equality.
4868 * If for any value of the other variables, there is any value
4869 * for the existentially quantified variable satisfying the constraints,
4870 * then this lower bound also satisfies the constraints.
4871 * It is therefore safe to pick this lower bound.
4872 *
4873 * The same reasoning holds even if the coefficient is not one.
4874 * However, fixing the variable to the value of the lower bound may
4875 * in general introduce an extra integer division, in which case
4876 * it may be better to pick another value.
4877 * If this integer division has a known constant value, then plugging
4878 * in this constant value removes the existentially quantified variable
4879 * completely. In particular, if the lower bound is of the form
4880 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4881 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4882 * then the existentially quantified variable can be assigned this
4883 * shared value.
4884 *
4885 * We skip divs that appear in equalities or in the definition of other divs.
4886 * Divs that appear in the definition of other divs usually occur in at least
4887 * 4 constraints, but the constraints may have been simplified.
4888 *
4889 * If any divs are left after these simple checks then we move on
4890 * to more complicated cases in drop_more_redundant_divs.
4891 */
4894 {
4934 }
4938 }
4939 }
4947 }
4950 else
4970 pairs);
4974 pairs);
4976 }
4993 else
5000 }
5003 }
5010 error:
5014 }
5016 /* Consider the coefficients at "c" as a row vector and replace
5017 * them with their product with "T". "T" is assumed to be a square matrix.
5018 */
5020 {
5042 }
5044 /* Plug in T for the variables in "bmap" starting at "pos".
5045 * T is a linear unimodular matrix, i.e., without constant term.
5046 */
5049 {
5078 }
5082 error:
5086 }
5088 /* Remove divs that are not strictly needed.
5089 *
5090 * First look for an equality constraint involving two or more
5091 * existentially quantified variables without an explicit
5092 * representation. Replace the combination that appears
5093 * in the equality constraint by a single existentially quantified
5094 * variable such that the equality can be used to derive
5095 * an explicit representation for the variable.
5096 * If there are no more such equality constraints, then continue
5097 * with isl_basic_map_drop_redundant_divs_ineq.
5098 *
5099 * In particular, if the equality constraint is of the form
5100 *
5101 * f(x) + \sum_i c_i a_i = 0
5102 *
5103 * with a_i existentially quantified variable without explicit
5104 * representation, then apply a transformation on the existentially
5105 * quantified variables to turn the constraint into
5106 *
5107 * f(x) + g a_1' = 0
5108 *
5109 * with g the gcd of the c_i.
5110 * In order to easily identify which existentially quantified variables
5111 * have a complete explicit representation, i.e., without being defined
5112 * in terms of other existentially quantified variables without
5113 * an explicit representation, the existentially quantified variables
5114 * are first sorted.
5115 *
5116 * The variable transformation is computed by extending the row
5117 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
5118 *
5119 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
5120 * [a_2'] [ a_2 ]
5121 * ... = U ....
5122 * [a_n'] [ a_n ]
5123 *
5124 * with [c_1/g ... c_n/g] representing the first row of U.
5125 * The inverse of U is then plugged into the original constraints.
5126 * The call to isl_basic_map_simplify makes sure the explicit
5127 * representation for a_1' is extracted from the equality constraint.
5128 */
5131 {
5166 }
5185 }
5189 {
5192 }
5194 /* Does "bmap" satisfy any equality that involves more than 2 variables
5195 * and/or has coefficients different from -1 and 1?
5196 */
5198 {
5227 }
5230 }
5232 /* Remove any common factor g from the constraint coefficients in "v".
5233 * The constant term is stored in the first position and is replaced
5234 * by floor(c/g). If any common factor is removed and if this results
5235 * in a tightening of the constraint, then set *tightened.
5236 */
5239 {
5259 }
5261 /* If "bmap" is an integer set that satisfies any equality involving
5262 * more than 2 variables and/or has coefficients different from -1 and 1,
5263 * then use variable compression to reduce the coefficients by removing
5264 * any (hidden) common factor.
5265 * In particular, apply the variable compression to each constraint,
5266 * factor out any common factor in the non-constant coefficients and
5267 * then apply the inverse of the compression.
5268 * At the end, we mark the basic map as having reduced constants.
5269 * If this flag is still set on the next invocation of this function,
5270 * then we skip the computation.
5271 *
5272 * Removing a common factor may result in a tightening of some of
5273 * the constraints. If this happens, then we may end up with two
5274 * opposite inequalities that can be replaced by an equality.
5275 * We therefore call isl_basic_map_detect_inequality_pairs,
5276 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5277 * and isl_basic_map_gauss if such a pair was found.
5278 */
5281 {
5315 }
5326 }
5341 }
5342 }
5345 error:
5350 }
5352 /* Shift the integer division at position "div" of "bmap"
5353 * by "shift" times the variable at position "pos".
5354 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5355 * corresponds to the constant term.
5356 *
5357 * That is, if the integer division has the form
5358 *
5359 * floor(f(x)/d)
5360 *
5361 * then replace it by
5362 *
5363 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5364 */
5367 {
5386 }
5392 }
5400 }
5403 }