isl_coalesce.c: check_facets: update justification

[isl.git] / isl_coalesce.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2012-2013 Ecole Normale Superieure
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
 */

#include "isl_map_private.h"
#include <isl_seq.h>
#include <isl/options.h>
#include "isl_tab.h"
#include <isl_mat_private.h>
#include <isl_local_space_private.h>
#include <isl_vec_private.h>

#define STATUS_ERROR            -1
#define STATUS_REDUNDANT         1
#define STATUS_VALID             2
#define STATUS_SEPARATE          3
#define STATUS_CUT               4
#define STATUS_ADJ_EQ            5
#define STATUS_ADJ_INEQ          6

static int status_in(isl_int *ineq, struct isl_tab *tab)
{
        enum isl_ineq_type type = isl_tab_ineq_type(tab, ineq);
        switch (type) {
        default:
        case isl_ineq_error:            return STATUS_ERROR;
        case isl_ineq_redundant:        return STATUS_VALID;
        case isl_ineq_separate:         return STATUS_SEPARATE;
        case isl_ineq_cut:              return STATUS_CUT;
        case isl_ineq_adj_eq:           return STATUS_ADJ_EQ;
        case isl_ineq_adj_ineq:         return STATUS_ADJ_INEQ;
        }
}

/* Compute the position of the equalities of basic map "bmap_i"
 * with respect to the basic map represented by "tab_j".
 * The resulting array has twice as many entries as the number
 * of equalities corresponding to the two inequalties to which
 * each equality corresponds.
 */
static int *eq_status_in(__isl_keep isl_basic_map *bmap_i,
        struct isl_tab *tab_j)
{
        int k, l;
        int *eq = isl_calloc_array(bmap_i->ctx, int, 2 * bmap_i->n_eq);
        unsigned dim;

        if (!eq)
                return NULL;

        dim = isl_basic_map_total_dim(bmap_i);
        for (k = 0; k < bmap_i->n_eq; ++k) {
                for (l = 0; l < 2; ++l) {
                        isl_seq_neg(bmap_i->eq[k], bmap_i->eq[k], 1+dim);
                        eq[2 * k + l] = status_in(bmap_i->eq[k], tab_j);
                        if (eq[2 * k + l] == STATUS_ERROR)
                                goto error;
                }
                if (eq[2 * k] == STATUS_SEPARATE ||
                    eq[2 * k + 1] == STATUS_SEPARATE)
                        break;
        }

        return eq;
error:
        free(eq);
        return NULL;
}

/* Compute the position of the inequalities of basic map "bmap_i"
 * (also represented by "tab_i", if not NULL) with respect to the basic map
 * represented by "tab_j".
 */
static int *ineq_status_in(__isl_keep isl_basic_map *bmap_i,
        struct isl_tab *tab_i, struct isl_tab *tab_j)
{
        int k;
        unsigned n_eq = bmap_i->n_eq;
        int *ineq = isl_calloc_array(bmap_i->ctx, int, bmap_i->n_ineq);

        if (!ineq)
                return NULL;

        for (k = 0; k < bmap_i->n_ineq; ++k) {
                if (tab_i && isl_tab_is_redundant(tab_i, n_eq + k)) {
                        ineq[k] = STATUS_REDUNDANT;
                        continue;
                }
                ineq[k] = status_in(bmap_i->ineq[k], tab_j);
                if (ineq[k] == STATUS_ERROR)
                        goto error;
                if (ineq[k] == STATUS_SEPARATE)
                        break;
        }

        return ineq;
error:
        free(ineq);
        return NULL;
}

static int any(int *con, unsigned len, int status)
{
        int i;

        for (i = 0; i < len ; ++i)
                if (con[i] == status)
                        return 1;
        return 0;
}

static int count(int *con, unsigned len, int status)
{
        int i;
        int c = 0;

        for (i = 0; i < len ; ++i)
                if (con[i] == status)
                        c++;
        return c;
}

static int all(int *con, unsigned len, int status)
{
        int i;

        for (i = 0; i < len ; ++i) {
                if (con[i] == STATUS_REDUNDANT)
                        continue;
                if (con[i] != status)
                        return 0;
        }
        return 1;
}

static void drop(struct isl_map *map, int i, struct isl_tab **tabs)
{
        isl_basic_map_free(map->p[i]);
        isl_tab_free(tabs[i]);

        if (i != map->n - 1) {
                map->p[i] = map->p[map->n - 1];
                tabs[i] = tabs[map->n - 1];
        }
        tabs[map->n - 1] = NULL;
        map->n--;
}

/* Replace the pair of basic maps i and j by the basic map bounded
 * by the valid constraints in both basic maps and the constraints
 * in extra (if not NULL).
 */
static int fuse(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j,
        __isl_keep isl_mat *extra)
{
        int k, l;
        struct isl_basic_map *fused = NULL;
        struct isl_tab *fused_tab = NULL;
        unsigned total = isl_basic_map_total_dim(map->p[i]);
        unsigned extra_rows = extra ? extra->n_row : 0;

        fused = isl_basic_map_alloc_space(isl_space_copy(map->p[i]->dim),
                        map->p[i]->n_div,
                        map->p[i]->n_eq + map->p[j]->n_eq,
                        map->p[i]->n_ineq + map->p[j]->n_ineq + extra_rows);
        if (!fused)
                goto error;

        for (k = 0; k < map->p[i]->n_eq; ++k) {
                if (eq_i && (eq_i[2 * k] != STATUS_VALID ||
                             eq_i[2 * k + 1] != STATUS_VALID))
                        continue;
                l = isl_basic_map_alloc_equality(fused);
                if (l < 0)
                        goto error;
                isl_seq_cpy(fused->eq[l], map->p[i]->eq[k], 1 + total);
        }

        for (k = 0; k < map->p[j]->n_eq; ++k) {
                if (eq_j && (eq_j[2 * k] != STATUS_VALID ||
                             eq_j[2 * k + 1] != STATUS_VALID))
                        continue;
                l = isl_basic_map_alloc_equality(fused);
                if (l < 0)
                        goto error;
                isl_seq_cpy(fused->eq[l], map->p[j]->eq[k], 1 + total);
        }

        for (k = 0; k < map->p[i]->n_ineq; ++k) {
                if (ineq_i[k] != STATUS_VALID)
                        continue;
                l = isl_basic_map_alloc_inequality(fused);
                if (l < 0)
                        goto error;
                isl_seq_cpy(fused->ineq[l], map->p[i]->ineq[k], 1 + total);
        }

        for (k = 0; k < map->p[j]->n_ineq; ++k) {
                if (ineq_j[k] != STATUS_VALID)
                        continue;
                l = isl_basic_map_alloc_inequality(fused);
                if (l < 0)
                        goto error;
                isl_seq_cpy(fused->ineq[l], map->p[j]->ineq[k], 1 + total);
        }

        for (k = 0; k < map->p[i]->n_div; ++k) {
                int l = isl_basic_map_alloc_div(fused);
                if (l < 0)
                        goto error;
                isl_seq_cpy(fused->div[l], map->p[i]->div[k], 1 + 1 + total);
        }

        for (k = 0; k < extra_rows; ++k) {
                l = isl_basic_map_alloc_inequality(fused);
                if (l < 0)
                        goto error;
                isl_seq_cpy(fused->ineq[l], extra->row[k], 1 + total);
        }

        fused = isl_basic_map_gauss(fused, NULL);
        ISL_F_SET(fused, ISL_BASIC_MAP_FINAL);
        if (ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_RATIONAL) &&
            ISL_F_ISSET(map->p[j], ISL_BASIC_MAP_RATIONAL))
                ISL_F_SET(fused, ISL_BASIC_MAP_RATIONAL);

        fused_tab = isl_tab_from_basic_map(fused, 0);
        if (isl_tab_detect_redundant(fused_tab) < 0)
                goto error;

        isl_basic_map_free(map->p[i]);
        map->p[i] = fused;
        isl_tab_free(tabs[i]);
        tabs[i] = fused_tab;
        drop(map, j, tabs);

        return 1;
error:
        isl_tab_free(fused_tab);
        isl_basic_map_free(fused);
        return -1;
}

/* Given a pair of basic maps i and j such that all constraints are either
 * "valid" or "cut", check if the facets corresponding to the "cut"
 * constraints of i lie entirely within basic map j.
 * If so, replace the pair by the basic map consisting of the valid
 * constraints in both basic maps.
 * Checking whether the facet lies entirely within basic map j
 * is performed by checking whether the constraints of basic map j
 * are valid for the facet.  These tests are performed on a rational
 * tableau to avoid the theoretical possibility that a constraint
 * that was considered to be a cut constraint for the entire basic map i
 * happens to be considered to be a valid constraint for the facet,
 * even though it cuts off the same rational points.
 *
 * To see that we are not introducing any extra points, call the
 * two basic maps A and B and the resulting map U and let x
 * be an element of U \setminus ( A \cup B ).
 * A line connecting x with an element of A \cup B meets a facet F
 * of either A or B.  Assume it is a facet of B and let c_1 be
 * the corresponding facet constraint.  We have c_1(x) < 0 and
 * so c_1 is a cut constraint.  This implies that there is some
 * (possibly rational) point x' satisfying the constraints of A
 * and the opposite of c_1 as otherwise c_1 would have been marked
 * valid for A.  The line connecting x and x' meets a facet of A
 * in a (possibly rational) point that also violates c_1, but this
 * is impossible since all cut constraints of B are valid for all
 * cut facets of A.
 * In case F is a facet of A rather than B, then we can apply the
 * above reasoning to find a facet of B separating x from A \cup B first.
 */
static int check_facets(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *ineq_i, int *ineq_j)
{
        int k, l;
        struct isl_tab_undo *snap, *snap2;
        unsigned n_eq = map->p[i]->n_eq;

        snap = isl_tab_snap(tabs[i]);
        if (isl_tab_mark_rational(tabs[i]) < 0)
                return -1;
        snap2 = isl_tab_snap(tabs[i]);

        for (k = 0; k < map->p[i]->n_ineq; ++k) {
                if (ineq_i[k] != STATUS_CUT)
                        continue;
                if (isl_tab_select_facet(tabs[i], n_eq + k) < 0)
                        return -1;
                for (l = 0; l < map->p[j]->n_ineq; ++l) {
                        int stat;
                        if (ineq_j[l] != STATUS_CUT)
                                continue;
                        stat = status_in(map->p[j]->ineq[l], tabs[i]);
                        if (stat != STATUS_VALID)
                                break;
                }
                if (isl_tab_rollback(tabs[i], snap2) < 0)
                        return -1;
                if (l < map->p[j]->n_ineq)
                        break;
        }

        if (k < map->p[i]->n_ineq) {
                if (isl_tab_rollback(tabs[i], snap) < 0)
                        return -1;
                return 0;
        }
        return fuse(map, i, j, tabs, NULL, ineq_i, NULL, ineq_j, NULL);
}

/* Check if basic map "i" contains the basic map represented
 * by the tableau "tab".
 */
static int contains(struct isl_map *map, int i, int *ineq_i,
        struct isl_tab *tab)
{
        int k, l;
        unsigned dim;

        dim = isl_basic_map_total_dim(map->p[i]);
        for (k = 0; k < map->p[i]->n_eq; ++k) {
                for (l = 0; l < 2; ++l) {
                        int stat;
                        isl_seq_neg(map->p[i]->eq[k], map->p[i]->eq[k], 1+dim);
                        stat = status_in(map->p[i]->eq[k], tab);
                        if (stat != STATUS_VALID)
                                return 0;
                }
        }

        for (k = 0; k < map->p[i]->n_ineq; ++k) {
                int stat;
                if (ineq_i[k] == STATUS_REDUNDANT)
                        continue;
                stat = status_in(map->p[i]->ineq[k], tab);
                if (stat != STATUS_VALID)
                        return 0;
        }
        return 1;
}

/* Basic map "i" has an inequality (say "k") that is adjacent
 * to some inequality of basic map "j".  All the other inequalities
 * are valid for "j".
 * Check if basic map "j" forms an extension of basic map "i".
 *
 * Note that this function is only called if some of the equalities or
 * inequalities of basic map "j" do cut basic map "i".  The function is
 * correct even if there are no such cut constraints, but in that case
 * the additional checks performed by this function are overkill.
 *
 * In particular, we replace constraint k, say f >= 0, by constraint
 * f <= -1, add the inequalities of "j" that are valid for "i"
 * and check if the result is a subset of basic map "j".
 * If so, then we know that this result is exactly equal to basic map "j"
 * since all its constraints are valid for basic map "j".
 * By combining the valid constraints of "i" (all equalities and all
 * inequalities except "k") and the valid constraints of "j" we therefore
 * obtain a basic map that is equal to their union.
 * In this case, there is no need to perform a rollback of the tableau
 * since it is going to be destroyed in fuse().
 *
 *
 *      |\__                    |\__
 *      |   \__                 |   \__
 *      |      \_       =>      |      \__
 *      |_______| _             |_________\
 *
 *
 *      |\                      |\
 *      | \                     | \
 *      |  \                    |  \
 *      |  |                    |   \
 *      |  ||\          =>      |    \
 *      |  || \                 |     \
 *      |  ||  |                |      |
 *      |__||_/                 |_____/
 */
static int is_adj_ineq_extension(__isl_keep isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int k;
        struct isl_tab_undo *snap;
        unsigned n_eq = map->p[i]->n_eq;
        unsigned total = isl_basic_map_total_dim(map->p[i]);
        int r;

        if (isl_tab_extend_cons(tabs[i], 1 + map->p[j]->n_ineq) < 0)
                return -1;

        for (k = 0; k < map->p[i]->n_ineq; ++k)
                if (ineq_i[k] == STATUS_ADJ_INEQ)
                        break;
        if (k >= map->p[i]->n_ineq)
                isl_die(isl_map_get_ctx(map), isl_error_internal,
                        "ineq_i should have exactly one STATUS_ADJ_INEQ",
                        return -1);

        snap = isl_tab_snap(tabs[i]);

        if (isl_tab_unrestrict(tabs[i], n_eq + k) < 0)
                return -1;

        isl_seq_neg(map->p[i]->ineq[k], map->p[i]->ineq[k], 1 + total);
        isl_int_sub_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
        r = isl_tab_add_ineq(tabs[i], map->p[i]->ineq[k]);
        isl_seq_neg(map->p[i]->ineq[k], map->p[i]->ineq[k], 1 + total);
        isl_int_sub_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
        if (r < 0)
                return -1;

        for (k = 0; k < map->p[j]->n_ineq; ++k) {
                if (ineq_j[k] != STATUS_VALID)
                        continue;
                if (isl_tab_add_ineq(tabs[i], map->p[j]->ineq[k]) < 0)
                        return -1;
        }

        if (contains(map, j, ineq_j, tabs[i]))
                return fuse(map, i, j, tabs, eq_i, ineq_i, eq_j, ineq_j, NULL);

        if (isl_tab_rollback(tabs[i], snap) < 0)
                return -1;

        return 0;
}


/* Both basic maps have at least one inequality with and adjacent
 * (but opposite) inequality in the other basic map.
 * Check that there are no cut constraints and that there is only
 * a single pair of adjacent inequalities.
 * If so, we can replace the pair by a single basic map described
 * by all but the pair of adjacent inequalities.
 * Any additional points introduced lie strictly between the two
 * adjacent hyperplanes and can therefore be integral.
 *
 *        ____                    _____
 *       /    ||\                /     \
 *      /     || \              /       \
 *      \     ||  \     =>      \        \
 *       \    ||  /              \       /
 *        \___||_/                \_____/
 *
 * The test for a single pair of adjancent inequalities is important
 * for avoiding the combination of two basic maps like the following
 *
 *       /|
 *      / |
 *     /__|
 *         _____
 *         |   |
 *         |   |
 *         |___|
 *
 * If there are some cut constraints on one side, then we may
 * still be able to fuse the two basic maps, but we need to perform
 * some additional checks in is_adj_ineq_extension.
 */
static int check_adj_ineq(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int count_i, count_j;
        int cut_i, cut_j;

        count_i = count(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_INEQ);
        count_j = count(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_INEQ);

        if (count_i != 1 && count_j != 1)
                return 0;

        cut_i = any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT) ||
                any(ineq_i, map->p[i]->n_ineq, STATUS_CUT);
        cut_j = any(eq_j, 2 * map->p[j]->n_eq, STATUS_CUT) ||
                any(ineq_j, map->p[j]->n_ineq, STATUS_CUT);

        if (!cut_i && !cut_j && count_i == 1 && count_j == 1)
                return fuse(map, i, j, tabs, NULL, ineq_i, NULL, ineq_j, NULL);

        if (count_i == 1 && !cut_i)
                return is_adj_ineq_extension(map, i, j, tabs,
                                                eq_i, ineq_i, eq_j, ineq_j);

        if (count_j == 1 && !cut_j)
                return is_adj_ineq_extension(map, j, i, tabs,
                                                eq_j, ineq_j, eq_i, ineq_i);

        return 0;
}

/* Basic map "i" has an inequality "k" that is adjacent to some equality
 * of basic map "j".  All the other inequalities are valid for "j".
 * Check if basic map "j" forms an extension of basic map "i".
 *
 * In particular, we relax constraint "k", compute the corresponding
 * facet and check whether it is included in the other basic map.
 * If so, we know that relaxing the constraint extends the basic
 * map with exactly the other basic map (we already know that this
 * other basic map is included in the extension, because there
 * were no "cut" inequalities in "i") and we can replace the
 * two basic maps by this extension.
 *        ____                    _____
 *       /    ||                 /     |
 *      /     ||                /      |
 *      \     ||        =>      \      |
 *       \    ||                 \     |
 *        \___||                  \____|
 */
static int is_adj_eq_extension(struct isl_map *map, int i, int j, int k,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int changed = 0;
        int super;
        struct isl_tab_undo *snap, *snap2;
        unsigned n_eq = map->p[i]->n_eq;

        if (isl_tab_is_equality(tabs[i], n_eq + k))
                return 0;

        snap = isl_tab_snap(tabs[i]);
        if (isl_tab_relax(tabs[i], n_eq + k) < 0)
                return -1;
        snap2 = isl_tab_snap(tabs[i]);
        if (isl_tab_select_facet(tabs[i], n_eq + k) < 0)
                return -1;
        super = contains(map, j, ineq_j, tabs[i]);
        if (super) {
                if (isl_tab_rollback(tabs[i], snap2) < 0)
                        return -1;
                map->p[i] = isl_basic_map_cow(map->p[i]);
                if (!map->p[i])
                        return -1;
                isl_int_add_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
                ISL_F_SET(map->p[i], ISL_BASIC_MAP_FINAL);
                drop(map, j, tabs);
                changed = 1;
        } else
                if (isl_tab_rollback(tabs[i], snap) < 0)
                        return -1;

        return changed;
}

/* Data structure that keeps track of the wrapping constraints
 * and of information to bound the coefficients of those constraints.
 *
 * bound is set if we want to apply a bound on the coefficients
 * mat contains the wrapping constraints
 * max is the bound on the coefficients (if bound is set)
 */
struct isl_wraps {
        int bound;
        isl_mat *mat;
        isl_int max;
};

/* Update wraps->max to be greater than or equal to the coefficients
 * in the equalities and inequalities of bmap that can be removed if we end up
 * applying wrapping.
 */
static void wraps_update_max(struct isl_wraps *wraps,
        __isl_keep isl_basic_map *bmap, int *eq, int *ineq)
{
        int k;
        isl_int max_k;
        unsigned total = isl_basic_map_total_dim(bmap);

        isl_int_init(max_k);

        for (k = 0; k < bmap->n_eq; ++k) {
                if (eq[2 * k] == STATUS_VALID &&
                    eq[2 * k + 1] == STATUS_VALID)
                        continue;
                isl_seq_abs_max(bmap->eq[k] + 1, total, &max_k);
                if (isl_int_abs_gt(max_k, wraps->max))
                        isl_int_set(wraps->max, max_k);
        }

        for (k = 0; k < bmap->n_ineq; ++k) {
                if (ineq[k] == STATUS_VALID || ineq[k] == STATUS_REDUNDANT)
                        continue;
                isl_seq_abs_max(bmap->ineq[k] + 1, total, &max_k);
                if (isl_int_abs_gt(max_k, wraps->max))
                        isl_int_set(wraps->max, max_k);
        }

        isl_int_clear(max_k);
}

/* Initialize the isl_wraps data structure.
 * If we want to bound the coefficients of the wrapping constraints,
 * we set wraps->max to the largest coefficient
 * in the equalities and inequalities that can be removed if we end up
 * applying wrapping.
 */
static void wraps_init(struct isl_wraps *wraps, __isl_take isl_mat *mat,
        __isl_keep isl_map *map, int i, int j,
        int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        isl_ctx *ctx;

        wraps->bound = 0;
        wraps->mat = mat;
        if (!mat)
                return;
        ctx = isl_mat_get_ctx(mat);
        wraps->bound = isl_options_get_coalesce_bounded_wrapping(ctx);
        if (!wraps->bound)
                return;
        isl_int_init(wraps->max);
        isl_int_set_si(wraps->max, 0);
        wraps_update_max(wraps, map->p[i], eq_i, ineq_i);
        wraps_update_max(wraps, map->p[j], eq_j, ineq_j);
}

/* Free the contents of the isl_wraps data structure.
 */
static void wraps_free(struct isl_wraps *wraps)
{
        isl_mat_free(wraps->mat);
        if (wraps->bound)
                isl_int_clear(wraps->max);
}

/* Is the wrapping constraint in row "row" allowed?
 *
 * If wraps->bound is set, we check that none of the coefficients
 * is greater than wraps->max.
 */
static int allow_wrap(struct isl_wraps *wraps, int row)
{
        int i;

        if (!wraps->bound)
                return 1;

        for (i = 1; i < wraps->mat->n_col; ++i)
                if (isl_int_abs_gt(wraps->mat->row[row][i], wraps->max))
                        return 0;

        return 1;
}

/* For each non-redundant constraint in "bmap" (as determined by "tab"),
 * wrap the constraint around "bound" such that it includes the whole
 * set "set" and append the resulting constraint to "wraps".
 * "wraps" is assumed to have been pre-allocated to the appropriate size.
 * wraps->n_row is the number of actual wrapped constraints that have
 * been added.
 * If any of the wrapping problems results in a constraint that is
 * identical to "bound", then this means that "set" is unbounded in such
 * way that no wrapping is possible.  If this happens then wraps->n_row
 * is reset to zero.
 * Similarly, if we want to bound the coefficients of the wrapping
 * constraints and a newly added wrapping constraint does not
 * satisfy the bound, then wraps->n_row is also reset to zero.
 */
static int add_wraps(struct isl_wraps *wraps, __isl_keep isl_basic_map *bmap,
        struct isl_tab *tab, isl_int *bound, __isl_keep isl_set *set)
{
        int l;
        int w;
        unsigned total = isl_basic_map_total_dim(bmap);

        w = wraps->mat->n_row;

        for (l = 0; l < bmap->n_ineq; ++l) {
                if (isl_seq_is_neg(bound, bmap->ineq[l], 1 + total))
                        continue;
                if (isl_seq_eq(bound, bmap->ineq[l], 1 + total))
                        continue;
                if (isl_tab_is_redundant(tab, bmap->n_eq + l))
                        continue;

                isl_seq_cpy(wraps->mat->row[w], bound, 1 + total);
                if (!isl_set_wrap_facet(set, wraps->mat->row[w], bmap->ineq[l]))
                        return -1;
                if (isl_seq_eq(wraps->mat->row[w], bound, 1 + total))
                        goto unbounded;
                if (!allow_wrap(wraps, w))
                        goto unbounded;
                ++w;
        }
        for (l = 0; l < bmap->n_eq; ++l) {
                if (isl_seq_is_neg(bound, bmap->eq[l], 1 + total))
                        continue;
                if (isl_seq_eq(bound, bmap->eq[l], 1 + total))
                        continue;

                isl_seq_cpy(wraps->mat->row[w], bound, 1 + total);
                isl_seq_neg(wraps->mat->row[w + 1], bmap->eq[l], 1 + total);
                if (!isl_set_wrap_facet(set, wraps->mat->row[w],
                                        wraps->mat->row[w + 1]))
                        return -1;
                if (isl_seq_eq(wraps->mat->row[w], bound, 1 + total))
                        goto unbounded;
                if (!allow_wrap(wraps, w))
                        goto unbounded;
                ++w;

                isl_seq_cpy(wraps->mat->row[w], bound, 1 + total);
                if (!isl_set_wrap_facet(set, wraps->mat->row[w], bmap->eq[l]))
                        return -1;
                if (isl_seq_eq(wraps->mat->row[w], bound, 1 + total))
                        goto unbounded;
                if (!allow_wrap(wraps, w))
                        goto unbounded;
                ++w;
        }

        wraps->mat->n_row = w;
        return 0;
unbounded:
        wraps->mat->n_row = 0;
        return 0;
}

/* Check if the constraints in "wraps" from "first" until the last
 * are all valid for the basic set represented by "tab".
 * If not, wraps->n_row is set to zero.
 */
static int check_wraps(__isl_keep isl_mat *wraps, int first,
        struct isl_tab *tab)
{
        int i;

        for (i = first; i < wraps->n_row; ++i) {
                enum isl_ineq_type type;
                type = isl_tab_ineq_type(tab, wraps->row[i]);
                if (type == isl_ineq_error)
                        return -1;
                if (type == isl_ineq_redundant)
                        continue;
                wraps->n_row = 0;
                return 0;
        }

        return 0;
}

/* Return a set that corresponds to the non-redundant constraints
 * (as recorded in tab) of bmap.
 *
 * It's important to remove the redundant constraints as some
 * of the other constraints may have been modified after the
 * constraints were marked redundant.
 * In particular, a constraint may have been relaxed.
 * Redundant constraints are ignored when a constraint is relaxed
 * and should therefore continue to be ignored ever after.
 * Otherwise, the relaxation might be thwarted by some of
 * these constraints.
 */
static __isl_give isl_set *set_from_updated_bmap(__isl_keep isl_basic_map *bmap,
        struct isl_tab *tab)
{
        bmap = isl_basic_map_copy(bmap);
        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_update_from_tab(bmap, tab);
        return isl_set_from_basic_set(isl_basic_map_underlying_set(bmap));
}

/* Given a basic set i with a constraint k that is adjacent to
 * basic set j, check if we can wrap
 * both the facet corresponding to k and basic map j
 * around their ridges to include the other set.
 * If so, replace the pair of basic sets by their union.
 *
 * All constraints of i (except k) are assumed to be valid for j.
 * This means that there is no real need to wrap the ridges of
 * the faces of basic map i around basic map j but since we do,
 * we have to check that the resulting wrapping constraints are valid for i.
 *        ____                    _____
 *       /    |                  /     \
 *      /     ||                /      |
 *      \     ||        =>      \      |
 *       \    ||                 \     |
 *        \___||                  \____|
 *
 */
static int can_wrap_in_facet(struct isl_map *map, int i, int j, int k,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int changed = 0;
        struct isl_wraps wraps;
        isl_mat *mat;
        struct isl_set *set_i = NULL;
        struct isl_set *set_j = NULL;
        struct isl_vec *bound = NULL;
        unsigned total = isl_basic_map_total_dim(map->p[i]);
        struct isl_tab_undo *snap;
        int n;

        set_i = set_from_updated_bmap(map->p[i], tabs[i]);
        set_j = set_from_updated_bmap(map->p[j], tabs[j]);
        mat = isl_mat_alloc(map->ctx, 2 * (map->p[i]->n_eq + map->p[j]->n_eq) +
                                        map->p[i]->n_ineq + map->p[j]->n_ineq,
                                        1 + total);
        wraps_init(&wraps, mat, map, i, j, eq_i, ineq_i, eq_j, ineq_j);
        bound = isl_vec_alloc(map->ctx, 1 + total);
        if (!set_i || !set_j || !wraps.mat || !bound)
                goto error;

        isl_seq_cpy(bound->el, map->p[i]->ineq[k], 1 + total);
        isl_int_add_ui(bound->el[0], bound->el[0], 1);

        isl_seq_cpy(wraps.mat->row[0], bound->el, 1 + total);
        wraps.mat->n_row = 1;

        if (add_wraps(&wraps, map->p[j], tabs[j], bound->el, set_i) < 0)
                goto error;
        if (!wraps.mat->n_row)
                goto unbounded;

        snap = isl_tab_snap(tabs[i]);

        if (isl_tab_select_facet(tabs[i], map->p[i]->n_eq + k) < 0)
                goto error;
        if (isl_tab_detect_redundant(tabs[i]) < 0)
                goto error;

        isl_seq_neg(bound->el, map->p[i]->ineq[k], 1 + total);

        n = wraps.mat->n_row;
        if (add_wraps(&wraps, map->p[i], tabs[i], bound->el, set_j) < 0)
                goto error;

        if (isl_tab_rollback(tabs[i], snap) < 0)
                goto error;
        if (check_wraps(wraps.mat, n, tabs[i]) < 0)
                goto error;
        if (!wraps.mat->n_row)
                goto unbounded;

        changed = fuse(map, i, j, tabs, eq_i, ineq_i, eq_j, ineq_j, wraps.mat);

unbounded:
        wraps_free(&wraps);

        isl_set_free(set_i);
        isl_set_free(set_j);

        isl_vec_free(bound);

        return changed;
error:
        wraps_free(&wraps);
        isl_vec_free(bound);
        isl_set_free(set_i);
        isl_set_free(set_j);
        return -1;
}

/* Set the is_redundant property of the "n" constraints in "cuts",
 * except "k" to "v".
 * This is a fairly tricky operation as it bypasses isl_tab.c.
 * The reason we want to temporarily mark some constraints redundant
 * is that we want to ignore them in add_wraps.
 *
 * Initially all cut constraints are non-redundant, but the
 * selection of a facet right before the call to this function
 * may have made some of them redundant.
 * Likewise, the same constraints are marked non-redundant
 * in the second call to this function, before they are officially
 * made non-redundant again in the subsequent rollback.
 */
static void set_is_redundant(struct isl_tab *tab, unsigned n_eq,
        int *cuts, int n, int k, int v)
{
        int l;

        for (l = 0; l < n; ++l) {
                if (l == k)
                        continue;
                tab->con[n_eq + cuts[l]].is_redundant = v;
        }
}

/* Given a pair of basic maps i and j such that j sticks out
 * of i at n cut constraints, each time by at most one,
 * try to compute wrapping constraints and replace the two
 * basic maps by a single basic map.
 * The other constraints of i are assumed to be valid for j.
 *
 * The facets of i corresponding to the cut constraints are
 * wrapped around their ridges, except those ridges determined
 * by any of the other cut constraints.
 * The intersections of cut constraints need to be ignored
 * as the result of wrapping one cut constraint around another
 * would result in a constraint cutting the union.
 * In each case, the facets are wrapped to include the union
 * of the two basic maps.
 *
 * The pieces of j that lie at an offset of exactly one from
 * one of the cut constraints of i are wrapped around their edges.
 * Here, there is no need to ignore intersections because we
 * are wrapping around the union of the two basic maps.
 *
 * If any wrapping fails, i.e., if we cannot wrap to touch
 * the union, then we give up.
 * Otherwise, the pair of basic maps is replaced by their union.
 */
static int wrap_in_facets(struct isl_map *map, int i, int j,
        int *cuts, int n, struct isl_tab **tabs,
        int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int changed = 0;
        struct isl_wraps wraps;
        isl_mat *mat;
        isl_set *set = NULL;
        isl_vec *bound = NULL;
        unsigned total = isl_basic_map_total_dim(map->p[i]);
        int max_wrap;
        int k;
        struct isl_tab_undo *snap_i, *snap_j;

        if (isl_tab_extend_cons(tabs[j], 1) < 0)
                goto error;

        max_wrap = 2 * (map->p[i]->n_eq + map->p[j]->n_eq) +
                    map->p[i]->n_ineq + map->p[j]->n_ineq;
        max_wrap *= n;

        set = isl_set_union(set_from_updated_bmap(map->p[i], tabs[i]),
                            set_from_updated_bmap(map->p[j], tabs[j]));
        mat = isl_mat_alloc(map->ctx, max_wrap, 1 + total);
        wraps_init(&wraps, mat, map, i, j, eq_i, ineq_i, eq_j, ineq_j);
        bound = isl_vec_alloc(map->ctx, 1 + total);
        if (!set || !wraps.mat || !bound)
                goto error;

        snap_i = isl_tab_snap(tabs[i]);
        snap_j = isl_tab_snap(tabs[j]);

        wraps.mat->n_row = 0;

        for (k = 0; k < n; ++k) {
                if (isl_tab_select_facet(tabs[i], map->p[i]->n_eq + cuts[k]) < 0)
                        goto error;
                if (isl_tab_detect_redundant(tabs[i]) < 0)
                        goto error;
                set_is_redundant(tabs[i], map->p[i]->n_eq, cuts, n, k, 1);

                isl_seq_neg(bound->el, map->p[i]->ineq[cuts[k]], 1 + total);
                if (!tabs[i]->empty &&
                    add_wraps(&wraps, map->p[i], tabs[i], bound->el, set) < 0)
                        goto error;

                set_is_redundant(tabs[i], map->p[i]->n_eq, cuts, n, k, 0);
                if (isl_tab_rollback(tabs[i], snap_i) < 0)
                        goto error;

                if (tabs[i]->empty)
                        break;
                if (!wraps.mat->n_row)
                        break;

                isl_seq_cpy(bound->el, map->p[i]->ineq[cuts[k]], 1 + total);
                isl_int_add_ui(bound->el[0], bound->el[0], 1);
                if (isl_tab_add_eq(tabs[j], bound->el) < 0)
                        goto error;
                if (isl_tab_detect_redundant(tabs[j]) < 0)
                        goto error;

                if (!tabs[j]->empty &&
                    add_wraps(&wraps, map->p[j], tabs[j], bound->el, set) < 0)
                        goto error;

                if (isl_tab_rollback(tabs[j], snap_j) < 0)
                        goto error;

                if (!wraps.mat->n_row)
                        break;
        }

        if (k == n)
                changed = fuse(map, i, j, tabs,
                                eq_i, ineq_i, eq_j, ineq_j, wraps.mat);

        isl_vec_free(bound);
        wraps_free(&wraps);
        isl_set_free(set);

        return changed;
error:
        isl_vec_free(bound);
        wraps_free(&wraps);
        isl_set_free(set);
        return -1;
}

/* Given two basic sets i and j such that i has no cut equalities,
 * check if relaxing all the cut inequalities of i by one turns
 * them into valid constraint for j and check if we can wrap in
 * the bits that are sticking out.
 * If so, replace the pair by their union.
 *
 * We first check if all relaxed cut inequalities of i are valid for j
 * and then try to wrap in the intersections of the relaxed cut inequalities
 * with j.
 *
 * During this wrapping, we consider the points of j that lie at a distance
 * of exactly 1 from i.  In particular, we ignore the points that lie in
 * between this lower-dimensional space and the basic map i.
 * We can therefore only apply this to integer maps.
 *        ____                    _____
 *       / ___|_                 /     \
 *      / |    |                /      |
 *      \ |    |        =>      \      |
 *       \|____|                 \     |
 *        \___|                   \____/
 *
 *       _____                   ______
 *      | ____|_                |      \
 *      | |     |               |       |
 *      | |     |       =>      |       |
 *      |_|     |               |       |
 *        |_____|                \______|
 *
 *       _______
 *      |       |
 *      |  |\   |
 *      |  | \  |
 *      |  |  \ |
 *      |  |   \|
 *      |  |    \
 *      |  |_____\
 *      |       |
 *      |_______|
 *
 * Wrapping can fail if the result of wrapping one of the facets
 * around its edges does not produce any new facet constraint.
 * In particular, this happens when we try to wrap in unbounded sets.
 *
 *       _______________________________________________________________________
 *      |
 *      |  ___
 *      | |   |
 *      |_|   |_________________________________________________________________
 *        |___|
 *
 * The following is not an acceptable result of coalescing the above two
 * sets as it includes extra integer points.
 *       _______________________________________________________________________
 *      |
 *      |     
 *      |      
 *      |
 *       \______________________________________________________________________
 */
static int can_wrap_in_set(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int changed = 0;
        int k, m;
        int n;
        int *cuts = NULL;

        if (ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_RATIONAL) ||
            ISL_F_ISSET(map->p[j], ISL_BASIC_MAP_RATIONAL))
                return 0;

        n = count(ineq_i, map->p[i]->n_ineq, STATUS_CUT);
        if (n == 0)
                return 0;

        cuts = isl_alloc_array(map->ctx, int, n);
        if (!cuts)
                return -1;

        for (k = 0, m = 0; m < n; ++k) {
                enum isl_ineq_type type;

                if (ineq_i[k] != STATUS_CUT)
                        continue;

                isl_int_add_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
                type = isl_tab_ineq_type(tabs[j], map->p[i]->ineq[k]);
                isl_int_sub_ui(map->p[i]->ineq[k][0], map->p[i]->ineq[k][0], 1);
                if (type == isl_ineq_error)
                        goto error;
                if (type != isl_ineq_redundant)
                        break;
                cuts[m] = k;
                ++m;
        }

        if (m == n)
                changed = wrap_in_facets(map, i, j, cuts, n, tabs,
                                         eq_i, ineq_i, eq_j, ineq_j);

        free(cuts);

        return changed;
error:
        free(cuts);
        return -1;
}

/* Check if either i or j has only cut inequalities that can
 * be used to wrap in (a facet of) the other basic set.
 * if so, replace the pair by their union.
 */
static int check_wrap(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int changed = 0;

        if (!any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT))
                changed = can_wrap_in_set(map, i, j, tabs,
                                            eq_i, ineq_i, eq_j, ineq_j);
        if (changed)
                return changed;

        if (!any(eq_j, 2 * map->p[j]->n_eq, STATUS_CUT))
                changed = can_wrap_in_set(map, j, i, tabs,
                                            eq_j, ineq_j, eq_i, ineq_i);
        return changed;
}

/* At least one of the basic maps has an equality that is adjacent
 * to inequality.  Make sure that only one of the basic maps has
 * such an equality and that the other basic map has exactly one
 * inequality adjacent to an equality.
 * We call the basic map that has the inequality "i" and the basic
 * map that has the equality "j".
 * If "i" has any "cut" (in)equality, then relaxing the inequality
 * by one would not result in a basic map that contains the other
 * basic map.
 */
static int check_adj_eq(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int changed = 0;
        int k;

        if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_INEQ) &&
            any(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_INEQ))
                /* ADJ EQ TOO MANY */
                return 0;

        if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_INEQ))
                return check_adj_eq(map, j, i, tabs,
                                        eq_j, ineq_j, eq_i, ineq_i);

        /* j has an equality adjacent to an inequality in i */

        if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT))
                return 0;
        if (any(ineq_i, map->p[i]->n_ineq, STATUS_CUT))
                /* ADJ EQ CUT */
                return 0;
        if (count(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_EQ) != 1 ||
            any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_EQ) ||
            any(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_INEQ) ||
            any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_INEQ))
                /* ADJ EQ TOO MANY */
                return 0;

        for (k = 0; k < map->p[i]->n_ineq; ++k)
                if (ineq_i[k] == STATUS_ADJ_EQ)
                        break;

        changed = is_adj_eq_extension(map, i, j, k, tabs,
                                        eq_i, ineq_i, eq_j, ineq_j);
        if (changed)
                return changed;

        if (count(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_INEQ) != 1)
                return 0;

        changed = can_wrap_in_facet(map, i, j, k, tabs, eq_i, ineq_i, eq_j, ineq_j);

        return changed;
}

/* The two basic maps lie on adjacent hyperplanes.  In particular,
 * basic map "i" has an equality that lies parallel to basic map "j".
 * Check if we can wrap the facets around the parallel hyperplanes
 * to include the other set.
 *
 * We perform basically the same operations as can_wrap_in_facet,
 * except that we don't need to select a facet of one of the sets.
 *                              _
 *      \\                      \\
 *       \\             =>       \\
 *        \                       \|
 *
 * We only allow one equality of "i" to be adjacent to an equality of "j"
 * to avoid coalescing
 *
 *      [m, n] -> { [x, y] -> [x, 1 + y] : x >= 1 and y >= 1 and
 *                                          x <= 10 and y <= 10;
 *                  [x, y] -> [1 + x, y] : x >= 1 and x <= 20 and
 *                                          y >= 5 and y <= 15 }
 *
 * to
 *
 *      [m, n] -> { [x, y] -> [x2, y2] : x >= 1 and 10y2 <= 20 - x + 10y and
 *                                      4y2 >= 5 + 3y and 5y2 <= 15 + 4y and
 *                                      y2 <= 1 + x + y - x2 and y2 >= y and
 *                                      y2 >= 1 + x + y - x2 }
 */
static int check_eq_adj_eq(struct isl_map *map, int i, int j,
        struct isl_tab **tabs, int *eq_i, int *ineq_i, int *eq_j, int *ineq_j)
{
        int k;
        int changed = 0;
        struct isl_wraps wraps;
        isl_mat *mat;
        struct isl_set *set_i = NULL;
        struct isl_set *set_j = NULL;
        struct isl_vec *bound = NULL;
        unsigned total = isl_basic_map_total_dim(map->p[i]);

        if (count(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_EQ) != 1)
                return 0;

        for (k = 0; k < 2 * map->p[i]->n_eq ; ++k)
                if (eq_i[k] == STATUS_ADJ_EQ)
                        break;

        set_i = set_from_updated_bmap(map->p[i], tabs[i]);
        set_j = set_from_updated_bmap(map->p[j], tabs[j]);
        mat = isl_mat_alloc(map->ctx, 2 * (map->p[i]->n_eq + map->p[j]->n_eq) +
                                        map->p[i]->n_ineq + map->p[j]->n_ineq,
                                        1 + total);
        wraps_init(&wraps, mat, map, i, j, eq_i, ineq_i, eq_j, ineq_j);
        bound = isl_vec_alloc(map->ctx, 1 + total);
        if (!set_i || !set_j || !wraps.mat || !bound)
                goto error;

        if (k % 2 == 0)
                isl_seq_neg(bound->el, map->p[i]->eq[k / 2], 1 + total);
        else
                isl_seq_cpy(bound->el, map->p[i]->eq[k / 2], 1 + total);
        isl_int_add_ui(bound->el[0], bound->el[0], 1);

        isl_seq_cpy(wraps.mat->row[0], bound->el, 1 + total);
        wraps.mat->n_row = 1;

        if (add_wraps(&wraps, map->p[j], tabs[j], bound->el, set_i) < 0)
                goto error;
        if (!wraps.mat->n_row)
                goto unbounded;

        isl_int_sub_ui(bound->el[0], bound->el[0], 1);
        isl_seq_neg(bound->el, bound->el, 1 + total);

        isl_seq_cpy(wraps.mat->row[wraps.mat->n_row], bound->el, 1 + total);
        wraps.mat->n_row++;

        if (add_wraps(&wraps, map->p[i], tabs[i], bound->el, set_j) < 0)
                goto error;
        if (!wraps.mat->n_row)
                goto unbounded;

        changed = fuse(map, i, j, tabs, eq_i, ineq_i, eq_j, ineq_j, wraps.mat);

        if (0) {
error:          changed = -1;
        }
unbounded:

        wraps_free(&wraps);
        isl_set_free(set_i);
        isl_set_free(set_j);
        isl_vec_free(bound);

        return changed;
}

/* Check if the union of the given pair of basic maps
 * can be represented by a single basic map.
 * If so, replace the pair by the single basic map and return 1.
 * Otherwise, return 0;
 * The two basic maps are assumed to live in the same local space.
 *
 * We first check the effect of each constraint of one basic map
 * on the other basic map.
 * The constraint may be
 *      redundant       the constraint is redundant in its own
 *                      basic map and should be ignore and removed
 *                      in the end
 *      valid           all (integer) points of the other basic map
 *                      satisfy the constraint
 *      separate        no (integer) point of the other basic map
 *                      satisfies the constraint
 *      cut             some but not all points of the other basic map
 *                      satisfy the constraint
 *      adj_eq          the given constraint is adjacent (on the outside)
 *                      to an equality of the other basic map
 *      adj_ineq        the given constraint is adjacent (on the outside)
 *                      to an inequality of the other basic map
 *
 * We consider seven cases in which we can replace the pair by a single
 * basic map.  We ignore all "redundant" constraints.
 *
 *      1. all constraints of one basic map are valid
 *              => the other basic map is a subset and can be removed
 *
 *      2. all constraints of both basic maps are either "valid" or "cut"
 *         and the facets corresponding to the "cut" constraints
 *         of one of the basic maps lies entirely inside the other basic map
 *              => the pair can be replaced by a basic map consisting
 *                 of the valid constraints in both basic maps
 *
 *      3. there is a single pair of adjacent inequalities
 *         (all other constraints are "valid")
 *              => the pair can be replaced by a basic map consisting
 *                 of the valid constraints in both basic maps
 *
 *      4. one basic map has a single adjacent inequality, while the other
 *         constraints are "valid".  The other basic map has some
 *         "cut" constraints, but replacing the adjacent inequality by
 *         its opposite and adding the valid constraints of the other
 *         basic map results in a subset of the other basic map
 *              => the pair can be replaced by a basic map consisting
 *                 of the valid constraints in both basic maps
 *
 *      5. there is a single adjacent pair of an inequality and an equality,
 *         the other constraints of the basic map containing the inequality are
 *         "valid".  Moreover, if the inequality the basic map is relaxed
 *         and then turned into an equality, then resulting facet lies
 *         entirely inside the other basic map
 *              => the pair can be replaced by the basic map containing
 *                 the inequality, with the inequality relaxed.
 *
 *      6. there is a single adjacent pair of an inequality and an equality,
 *         the other constraints of the basic map containing the inequality are
 *         "valid".  Moreover, the facets corresponding to both
 *         the inequality and the equality can be wrapped around their
 *         ridges to include the other basic map
 *              => the pair can be replaced by a basic map consisting
 *                 of the valid constraints in both basic maps together
 *                 with all wrapping constraints
 *
 *      7. one of the basic maps extends beyond the other by at most one.
 *         Moreover, the facets corresponding to the cut constraints and
 *         the pieces of the other basic map at offset one from these cut
 *         constraints can be wrapped around their ridges to include
 *         the union of the two basic maps
 *              => the pair can be replaced by a basic map consisting
 *                 of the valid constraints in both basic maps together
 *                 with all wrapping constraints
 *
 *      8. the two basic maps live in adjacent hyperplanes.  In principle
 *         such sets can always be combined through wrapping, but we impose
 *         that there is only one such pair, to avoid overeager coalescing.
 *
 * Throughout the computation, we maintain a collection of tableaus
 * corresponding to the basic maps.  When the basic maps are dropped
 * or combined, the tableaus are modified accordingly.
 */
static int coalesce_local_pair(__isl_keep isl_map *map, int i, int j,
        struct isl_tab **tabs)
{
        int changed = 0;
        int *eq_i = NULL;
        int *eq_j = NULL;
        int *ineq_i = NULL;
        int *ineq_j = NULL;

        eq_i = eq_status_in(map->p[i], tabs[j]);
        if (map->p[i]->n_eq && !eq_i)
                goto error;
        if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ERROR))
                goto error;
        if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_SEPARATE))
                goto done;

        eq_j = eq_status_in(map->p[j], tabs[i]);
        if (map->p[j]->n_eq && !eq_j)
                goto error;
        if (any(eq_j, 2 * map->p[j]->n_eq, STATUS_ERROR))
                goto error;
        if (any(eq_j, 2 * map->p[j]->n_eq, STATUS_SEPARATE))
                goto done;

        ineq_i = ineq_status_in(map->p[i], tabs[i], tabs[j]);
        if (map->p[i]->n_ineq && !ineq_i)
                goto error;
        if (any(ineq_i, map->p[i]->n_ineq, STATUS_ERROR))
                goto error;
        if (any(ineq_i, map->p[i]->n_ineq, STATUS_SEPARATE))
                goto done;

        ineq_j = ineq_status_in(map->p[j], tabs[j], tabs[i]);
        if (map->p[j]->n_ineq && !ineq_j)
                goto error;
        if (any(ineq_j, map->p[j]->n_ineq, STATUS_ERROR))
                goto error;
        if (any(ineq_j, map->p[j]->n_ineq, STATUS_SEPARATE))
                goto done;

        if (all(eq_i, 2 * map->p[i]->n_eq, STATUS_VALID) &&
            all(ineq_i, map->p[i]->n_ineq, STATUS_VALID)) {
                drop(map, j, tabs);
                changed = 1;
        } else if (all(eq_j, 2 * map->p[j]->n_eq, STATUS_VALID) &&
                   all(ineq_j, map->p[j]->n_ineq, STATUS_VALID)) {
                drop(map, i, tabs);
                changed = 1;
        } else if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_EQ)) {
                changed = check_eq_adj_eq(map, i, j, tabs,
                                        eq_i, ineq_i, eq_j, ineq_j);
        } else if (any(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_EQ)) {
                changed = check_eq_adj_eq(map, j, i, tabs,
                                        eq_j, ineq_j, eq_i, ineq_i);
        } else if (any(eq_i, 2 * map->p[i]->n_eq, STATUS_ADJ_INEQ) ||
                   any(eq_j, 2 * map->p[j]->n_eq, STATUS_ADJ_INEQ)) {
                changed = check_adj_eq(map, i, j, tabs,
                                        eq_i, ineq_i, eq_j, ineq_j);
        } else if (any(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_EQ) ||
                   any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_EQ)) {
                /* Can't happen */
                /* BAD ADJ INEQ */
        } else if (any(ineq_i, map->p[i]->n_ineq, STATUS_ADJ_INEQ) ||
                   any(ineq_j, map->p[j]->n_ineq, STATUS_ADJ_INEQ)) {
                changed = check_adj_ineq(map, i, j, tabs,
                                        eq_i, ineq_i, eq_j, ineq_j);
        } else {
                if (!any(eq_i, 2 * map->p[i]->n_eq, STATUS_CUT) &&
                    !any(eq_j, 2 * map->p[j]->n_eq, STATUS_CUT))
                        changed = check_facets(map, i, j, tabs, ineq_i, ineq_j);
                if (!changed)
                        changed = check_wrap(map, i, j, tabs,
                                                eq_i, ineq_i, eq_j, ineq_j);
        }

done:
        free(eq_i);
        free(eq_j);
        free(ineq_i);
        free(ineq_j);
        return changed;
error:
        free(eq_i);
        free(eq_j);
        free(ineq_i);
        free(ineq_j);
        return -1;
}

/* Do the two basic maps live in the same local space, i.e.,
 * do they have the same (known) divs?
 * If either basic map has any unknown divs, then we can only assume
 * that they do not live in the same local space.
 */
static int same_divs(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        int i;
        int known;
        int total;

        if (!bmap1 || !bmap2)
                return -1;
        if (bmap1->n_div != bmap2->n_div)
                return 0;

        if (bmap1->n_div == 0)
                return 1;

        known = isl_basic_map_divs_known(bmap1);
        if (known < 0 || !known)
                return known;
        known = isl_basic_map_divs_known(bmap2);
        if (known < 0 || !known)
                return known;

        total = isl_basic_map_total_dim(bmap1);
        for (i = 0; i < bmap1->n_div; ++i)
                if (!isl_seq_eq(bmap1->div[i], bmap2->div[i], 2 + total))
                        return 0;

        return 1;
}

/* Given two basic maps "i" and "j", where the divs of "i" form a subset
 * of those of "j", check if basic map "j" is a subset of basic map "i"
 * and, if so, drop basic map "j".
 *
 * We first expand the divs of basic map "i" to match those of basic map "j",
 * using the divs and expansion computed by the caller.
 * Then we check if all constraints of the expanded "i" are valid for "j".
 */
static int coalesce_subset(__isl_keep isl_map *map, int i, int j,
        struct isl_tab **tabs, __isl_keep isl_mat *div, int *exp)
{
        isl_basic_map *bmap;
        int changed = 0;
        int *eq_i = NULL;
        int *ineq_i = NULL;

        bmap = isl_basic_map_copy(map->p[i]);
        bmap = isl_basic_set_expand_divs(bmap, isl_mat_copy(div), exp);

        if (!bmap)
                goto error;

        eq_i = eq_status_in(bmap, tabs[j]);
        if (bmap->n_eq && !eq_i)
                goto error;
        if (any(eq_i, 2 * bmap->n_eq, STATUS_ERROR))
                goto error;
        if (any(eq_i, 2 * bmap->n_eq, STATUS_SEPARATE))
                goto done;

        ineq_i = ineq_status_in(bmap, NULL, tabs[j]);
        if (bmap->n_ineq && !ineq_i)
                goto error;
        if (any(ineq_i, bmap->n_ineq, STATUS_ERROR))
                goto error;
        if (any(ineq_i, bmap->n_ineq, STATUS_SEPARATE))
                goto done;

        if (all(eq_i, 2 * map->p[i]->n_eq, STATUS_VALID) &&
            all(ineq_i, map->p[i]->n_ineq, STATUS_VALID)) {
                drop(map, j, tabs);
                changed = 1;
        }

done:
        isl_basic_map_free(bmap);
        free(eq_i);
        free(ineq_i);
        return 0;
error:
        isl_basic_map_free(bmap);
        free(eq_i);
        free(ineq_i);
        return -1;
}

/* Check if the basic map "j" is a subset of basic map "i",
 * assuming that "i" has fewer divs that "j".
 * If not, then we change the order.
 *
 * If the two basic maps have the same number of divs, then
 * they must necessarily be different.  Otherwise, we would have
 * called coalesce_local_pair.  We therefore don't try anything
 * in this case.
 *
 * We first check if the divs of "i" are all known and form a subset
 * of those of "j".  If so, we pass control over to coalesce_subset.
 */
static int check_coalesce_subset(__isl_keep isl_map *map, int i, int j,
        struct isl_tab **tabs)
{
        int known;
        isl_mat *div_i, *div_j, *div;
        int *exp1 = NULL;
        int *exp2 = NULL;
        isl_ctx *ctx;
        int subset;

        if (map->p[i]->n_div == map->p[j]->n_div)
                return 0;
        if (map->p[j]->n_div < map->p[i]->n_div)
                return check_coalesce_subset(map, j, i, tabs);

        known = isl_basic_map_divs_known(map->p[i]);
        if (known < 0 || !known)
                return known;

        ctx = isl_map_get_ctx(map);

        div_i = isl_basic_map_get_divs(map->p[i]);
        div_j = isl_basic_map_get_divs(map->p[j]);

        if (!div_i || !div_j)
                goto error;

        exp1 = isl_alloc_array(ctx, int, div_i->n_row);
        exp2 = isl_alloc_array(ctx, int, div_j->n_row);
        if ((div_i->n_row && !exp1) || (div_j->n_row && !exp2))
                goto error;

        div = isl_merge_divs(div_i, div_j, exp1, exp2);
        if (!div)
                goto error;

        if (div->n_row == div_j->n_row)
                subset = coalesce_subset(map, i, j, tabs, div, exp1);
        else
                subset = 0;

        isl_mat_free(div);

        isl_mat_free(div_i);
        isl_mat_free(div_j);

        free(exp2);
        free(exp1);

        return subset;
error:
        isl_mat_free(div_i);
        isl_mat_free(div_j);
        free(exp1);
        free(exp2);
        return -1;
}

/* Check if the union of the given pair of basic maps
 * can be represented by a single basic map.
 * If so, replace the pair by the single basic map and return 1.
 * Otherwise, return 0;
 *
 * We first check if the two basic maps live in the same local space.
 * If so, we do the complete check.  Otherwise, we check if one is
 * an obvious subset of the other.
 */
static int coalesce_pair(__isl_keep isl_map *map, int i, int j,
        struct isl_tab **tabs)
{
        int same;

        same = same_divs(map->p[i], map->p[j]);
        if (same < 0)
                return -1;
        if (same)
                return coalesce_local_pair(map, i, j, tabs);

        return check_coalesce_subset(map, i, j, tabs);
}

static struct isl_map *coalesce(struct isl_map *map, struct isl_tab **tabs)
{
        int i, j;

        for (i = map->n - 2; i >= 0; --i)
restart:
                for (j = i + 1; j < map->n; ++j) {
                        int changed;
                        changed = coalesce_pair(map, i, j, tabs);
                        if (changed < 0)
                                goto error;
                        if (changed)
                                goto restart;
                }
        return map;
error:
        isl_map_free(map);
        return NULL;
}

/* For each pair of basic maps in the map, check if the union of the two
 * can be represented by a single basic map.
 * If so, replace the pair by the single basic map and start over.
 *
 * Since we are constructing the tableaus of the basic maps anyway,
 * we exploit them to detect implicit equalities and redundant constraints.
 * This also helps the coalescing as it can ignore the redundant constraints.
 * In order to avoid confusion, we make all implicit equalities explicit
 * in the basic maps.  We don't call isl_basic_map_gauss, though,
 * as that may affect the number of constraints.
 * This means that we have to call isl_basic_map_gauss at the end
 * of the computation to ensure that the basic maps are not left
 * in an unexpected state.
 */
struct isl_map *isl_map_coalesce(struct isl_map *map)
{
        int i;
        unsigned n;
        struct isl_tab **tabs = NULL;

        map = isl_map_remove_empty_parts(map);
        if (!map)
                return NULL;

        if (map->n <= 1)
                return map;

        map = isl_map_sort_divs(map);
        map = isl_map_cow(map);

        if (!map)
                return NULL;

        tabs = isl_calloc_array(map->ctx, struct isl_tab *, map->n);
        if (!tabs)
                goto error;

        n = map->n;
        for (i = 0; i < map->n; ++i) {
                tabs[i] = isl_tab_from_basic_map(map->p[i], 0);
                if (!tabs[i])
                        goto error;
                if (!ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_NO_IMPLICIT))
                        if (isl_tab_detect_implicit_equalities(tabs[i]) < 0)
                                goto error;
                map->p[i] = isl_tab_make_equalities_explicit(tabs[i],
                                                                map->p[i]);
                if (!map->p[i])
                        goto error;
                if (!ISL_F_ISSET(map->p[i], ISL_BASIC_MAP_NO_REDUNDANT))
                        if (isl_tab_detect_redundant(tabs[i]) < 0)
                                goto error;
        }
        for (i = map->n - 1; i >= 0; --i)
                if (tabs[i]->empty)
                        drop(map, i, tabs);

        map = coalesce(map, tabs);

        if (map)
                for (i = 0; i < map->n; ++i) {
                        map->p[i] = isl_basic_map_update_from_tab(map->p[i],
                                                                    tabs[i]);
                        map->p[i] = isl_basic_map_gauss(map->p[i], NULL);
                        map->p[i] = isl_basic_map_finalize(map->p[i]);
                        if (!map->p[i])
                                goto error;
                        ISL_F_SET(map->p[i], ISL_BASIC_MAP_NO_IMPLICIT);
                        ISL_F_SET(map->p[i], ISL_BASIC_MAP_NO_REDUNDANT);
                }

        for (i = 0; i < n; ++i)
                isl_tab_free(tabs[i]);

        free(tabs);

        return map;
error:
        if (tabs)
                for (i = 0; i < n; ++i)
                        isl_tab_free(tabs[i]);
        free(tabs);
        isl_map_free(map);
        return NULL;
}

/* For each pair of basic sets in the set, check if the union of the two
 * can be represented by a single basic set.
 * If so, replace the pair by the single basic set and start over.
 */
struct isl_set *isl_set_coalesce(struct isl_set *set)
{
        return (struct isl_set *)isl_map_coalesce((struct isl_map *)set);
}