isl_coalesce.c: extract out shared find function

[isl.git] / isl_coalesce.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2012-2013 Ecole Normale Superieure
 * Copyright 2014      INRIA Rocquencourt
 * Copyright 2016      INRIA Paris
 *
 * 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
 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
 * B.P. 105 - 78153 Le Chesnay, France
 * and Centre de Recherche Inria de Paris, 2 rue Simone Iff - Voie DQ12,
 * CS 42112, 75589 Paris Cedex 12, France
 */

#include <isl_ctx_private.h>
#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>
#include <isl_aff_private.h>
#include <isl_equalities.h>

#include <set_to_map.c>
#include <set_from_map.c>

#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;
}

/* Return the first position of "status" in the list "con" of length "len".
 * Return -1 if there is no such entry.
 */
static int find(int *con, unsigned len, int status)
{
        int i;

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

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;
}

/* Internal information associated to a basic map in a map
 * that is to be coalesced by isl_map_coalesce.
 *
 * "bmap" is the basic map itself (or NULL if "removed" is set)
 * "tab" is the corresponding tableau (or NULL if "removed" is set)
 * "hull_hash" identifies the affine space in which "bmap" lives.
 * "removed" is set if this basic map has been removed from the map
 * "simplify" is set if this basic map may have some unknown integer
 * divisions that were not present in the input basic maps.  The basic
 * map should then be simplified such that we may be able to find
 * a definition among the constraints.
 *
 * "eq" and "ineq" are only set if we are currently trying to coalesce
 * this basic map with another basic map, in which case they represent
 * the position of the inequalities of this basic map with respect to
 * the other basic map.  The number of elements in the "eq" array
 * is twice the number of equalities in the "bmap", corresponding
 * to the two inequalities that make up each equality.
 */
struct isl_coalesce_info {
        isl_basic_map *bmap;
        struct isl_tab *tab;
        uint32_t hull_hash;
        int removed;
        int simplify;
        int *eq;
        int *ineq;
};

/* Are all non-redundant constraints of the basic map represented by "info"
 * either valid or cut constraints with respect to the other basic map?
 */
static int all_valid_or_cut(struct isl_coalesce_info *info)
{
        int i;

        for (i = 0; i < 2 * info->bmap->n_eq; ++i) {
                if (info->eq[i] == STATUS_REDUNDANT)
                        continue;
                if (info->eq[i] == STATUS_VALID)
                        continue;
                if (info->eq[i] == STATUS_CUT)
                        continue;
                return 0;
        }

        for (i = 0; i < info->bmap->n_ineq; ++i) {
                if (info->ineq[i] == STATUS_REDUNDANT)
                        continue;
                if (info->ineq[i] == STATUS_VALID)
                        continue;
                if (info->ineq[i] == STATUS_CUT)
                        continue;
                return 0;
        }

        return 1;
}

/* Compute the hash of the (apparent) affine hull of info->bmap (with
 * the existentially quantified variables removed) and store it
 * in info->hash.
 */
static int coalesce_info_set_hull_hash(struct isl_coalesce_info *info)
{
        isl_basic_map *hull;
        unsigned n_div;

        hull = isl_basic_map_copy(info->bmap);
        hull = isl_basic_map_plain_affine_hull(hull);
        n_div = isl_basic_map_dim(hull, isl_dim_div);
        hull = isl_basic_map_drop_constraints_involving_dims(hull,
                                                        isl_dim_div, 0, n_div);
        info->hull_hash = isl_basic_map_get_hash(hull);
        isl_basic_map_free(hull);

        return hull ? 0 : -1;
}

/* Free all the allocated memory in an array
 * of "n" isl_coalesce_info elements.
 */
static void clear_coalesce_info(int n, struct isl_coalesce_info *info)
{
        int i;

        if (!info)
                return;

        for (i = 0; i < n; ++i) {
                isl_basic_map_free(info[i].bmap);
                isl_tab_free(info[i].tab);
        }

        free(info);
}

/* Drop the basic map represented by "info".
 * That is, clear the memory associated to the entry and
 * mark it as having been removed.
 */
static void drop(struct isl_coalesce_info *info)
{
        info->bmap = isl_basic_map_free(info->bmap);
        isl_tab_free(info->tab);
        info->tab = NULL;
        info->removed = 1;
}

/* Exchange the information in "info1" with that in "info2".
 */
static void exchange(struct isl_coalesce_info *info1,
        struct isl_coalesce_info *info2)
{
        struct isl_coalesce_info info;

        info = *info1;
        *info1 = *info2;
        *info2 = info;
}

/* This type represents the kind of change that has been performed
 * while trying to coalesce two basic maps.
 *
 * isl_change_none: nothing was changed
 * isl_change_drop_first: the first basic map was removed
 * isl_change_drop_second: the second basic map was removed
 * isl_change_fuse: the two basic maps were replaced by a new basic map.
 */
enum isl_change {
        isl_change_error = -1,
        isl_change_none = 0,
        isl_change_drop_first,
        isl_change_drop_second,
        isl_change_fuse,
};

/* Update "change" based on an interchange of the first and the second
 * basic map.  That is, interchange isl_change_drop_first and
 * isl_change_drop_second.
 */
static enum isl_change invert_change(enum isl_change change)
{
        switch (change) {
        case isl_change_error:
                return isl_change_error;
        case isl_change_none:
                return isl_change_none;
        case isl_change_drop_first:
                return isl_change_drop_second;
        case isl_change_drop_second:
                return isl_change_drop_first;
        case isl_change_fuse:
                return isl_change_fuse;
        }

        return isl_change_error;
}

/* Add the valid constraints of the basic map represented by "info"
 * to "bmap".  "len" is the size of the constraints.
 * If only one of the pair of inequalities that make up an equality
 * is valid, then add that inequality.
 */
static __isl_give isl_basic_map *add_valid_constraints(
        __isl_take isl_basic_map *bmap, struct isl_coalesce_info *info,
        unsigned len)
{
        int k, l;

        if (!bmap)
                return NULL;

        for (k = 0; k < info->bmap->n_eq; ++k) {
                if (info->eq[2 * k] == STATUS_VALID &&
                    info->eq[2 * k + 1] == STATUS_VALID) {
                        l = isl_basic_map_alloc_equality(bmap);
                        if (l < 0)
                                return isl_basic_map_free(bmap);
                        isl_seq_cpy(bmap->eq[l], info->bmap->eq[k], len);
                } else if (info->eq[2 * k] == STATUS_VALID) {
                        l = isl_basic_map_alloc_inequality(bmap);
                        if (l < 0)
                                return isl_basic_map_free(bmap);
                        isl_seq_neg(bmap->ineq[l], info->bmap->eq[k], len);
                } else if (info->eq[2 * k + 1] == STATUS_VALID) {
                        l = isl_basic_map_alloc_inequality(bmap);
                        if (l < 0)
                                return isl_basic_map_free(bmap);
                        isl_seq_cpy(bmap->ineq[l], info->bmap->eq[k], len);
                }
        }

        for (k = 0; k < info->bmap->n_ineq; ++k) {
                if (info->ineq[k] != STATUS_VALID)
                        continue;
                l = isl_basic_map_alloc_inequality(bmap);
                if (l < 0)
                        return isl_basic_map_free(bmap);
                isl_seq_cpy(bmap->ineq[l], info->bmap->ineq[k], len);
        }

        return bmap;
}

/* Is "bmap" defined by a number of (non-redundant) constraints that
 * is greater than the number of constraints of basic maps i and j combined?
 * Equalities are counted as two inequalities.
 */
static int number_of_constraints_increases(int i, int j,
        struct isl_coalesce_info *info,
        __isl_keep isl_basic_map *bmap, struct isl_tab *tab)
{
        int k, n_old, n_new;

        n_old = 2 * info[i].bmap->n_eq + info[i].bmap->n_ineq;
        n_old += 2 * info[j].bmap->n_eq + info[j].bmap->n_ineq;

        n_new = 2 * bmap->n_eq;
        for (k = 0; k < bmap->n_ineq; ++k)
                if (!isl_tab_is_redundant(tab, bmap->n_eq + k))
                        ++n_new;

        return n_new > n_old;
}

/* 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).
 * Place the fused basic map in the position that is the smallest of i and j.
 *
 * If "detect_equalities" is set, then look for equalities encoded
 * as pairs of inequalities.
 * If "check_number" is set, then the original basic maps are only
 * replaced if the total number of constraints does not increase.
 * While the number of integer divisions in the two basic maps
 * is assumed to be the same, the actual definitions may be different.
 * We only copy the definition from one of the basic map if it is
 * the same as that of the other basic map.  Otherwise, we mark
 * the integer division as unknown and simplify the basic map
 * in an attempt to recover the integer division definition.
 */
static enum isl_change fuse(int i, int j, struct isl_coalesce_info *info,
        __isl_keep isl_mat *extra, int detect_equalities, int check_number)
{
        int k, l;
        struct isl_basic_map *fused = NULL;
        struct isl_tab *fused_tab = NULL;
        unsigned total = isl_basic_map_total_dim(info[i].bmap);
        unsigned extra_rows = extra ? extra->n_row : 0;
        unsigned n_eq, n_ineq;
        int simplify = 0;

        if (j < i)
                return fuse(j, i, info, extra, detect_equalities, check_number);

        n_eq = info[i].bmap->n_eq + info[j].bmap->n_eq;
        n_ineq = info[i].bmap->n_ineq + info[j].bmap->n_ineq;
        fused = isl_basic_map_alloc_space(isl_space_copy(info[i].bmap->dim),
                    info[i].bmap->n_div, n_eq, n_eq + n_ineq + extra_rows);
        fused = add_valid_constraints(fused, &info[i], 1 + total);
        fused = add_valid_constraints(fused, &info[j], 1 + total);
        if (!fused)
                goto error;
        if (ISL_F_ISSET(info[i].bmap, ISL_BASIC_MAP_RATIONAL) &&
            ISL_F_ISSET(info[j].bmap, ISL_BASIC_MAP_RATIONAL))
                ISL_F_SET(fused, ISL_BASIC_MAP_RATIONAL);

        for (k = 0; k < info[i].bmap->n_div; ++k) {
                int l = isl_basic_map_alloc_div(fused);
                if (l < 0)
                        goto error;
                if (isl_seq_eq(info[i].bmap->div[k], info[j].bmap->div[k],
                                1 + 1 + total)) {
                        isl_seq_cpy(fused->div[l], info[i].bmap->div[k],
                                1 + 1 + total);
                } else {
                        isl_int_set_si(fused->div[l][0], 0);
                        simplify = 1;
                }
        }

        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);
        }

        if (detect_equalities)
                fused = isl_basic_map_detect_inequality_pairs(fused, NULL);
        fused = isl_basic_map_gauss(fused, NULL);
        if (simplify || info[j].simplify) {
                fused = isl_basic_map_simplify(fused);
                info[i].simplify = 0;
        }
        fused = isl_basic_map_finalize(fused);

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

        if (check_number &&
            number_of_constraints_increases(i, j, info, fused, fused_tab)) {
                isl_tab_free(fused_tab);
                isl_basic_map_free(fused);
                return isl_change_none;
        }

        isl_basic_map_free(info[i].bmap);
        info[i].bmap = fused;
        isl_tab_free(info[i].tab);
        info[i].tab = fused_tab;
        drop(&info[j]);

        return isl_change_fuse;
error:
        isl_tab_free(fused_tab);
        isl_basic_map_free(fused);
        return isl_change_error;
}

/* 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 enum isl_change check_facets(int i, int j,
        struct isl_coalesce_info *info)
{
        int k, l;
        struct isl_tab_undo *snap, *snap2;
        unsigned n_eq = info[i].bmap->n_eq;

        snap = isl_tab_snap(info[i].tab);
        if (isl_tab_mark_rational(info[i].tab) < 0)
                return isl_change_error;
        snap2 = isl_tab_snap(info[i].tab);

        for (k = 0; k < info[i].bmap->n_ineq; ++k) {
                if (info[i].ineq[k] != STATUS_CUT)
                        continue;
                if (isl_tab_select_facet(info[i].tab, n_eq + k) < 0)
                        return isl_change_error;
                for (l = 0; l < info[j].bmap->n_ineq; ++l) {
                        int stat;
                        if (info[j].ineq[l] != STATUS_CUT)
                                continue;
                        stat = status_in(info[j].bmap->ineq[l], info[i].tab);
                        if (stat < 0)
                                return isl_change_error;
                        if (stat != STATUS_VALID)
                                break;
                }
                if (isl_tab_rollback(info[i].tab, snap2) < 0)
                        return isl_change_error;
                if (l < info[j].bmap->n_ineq)
                        break;
        }

        if (k < info[i].bmap->n_ineq) {
                if (isl_tab_rollback(info[i].tab, snap) < 0)
                        return isl_change_error;
                return isl_change_none;
        }
        return fuse(i, j, info, NULL, 0, 0);
}

/* Check if info->bmap contains the basic map represented
 * by the tableau "tab".
 * For each equality, we check both the constraint itself
 * (as an inequality) and its negation.  Make sure the
 * equality is returned to its original state before returning.
 */
static int contains(struct isl_coalesce_info *info, struct isl_tab *tab)
{
        int k;
        unsigned dim;
        isl_basic_map *bmap = info->bmap;

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

        for (k = 0; k < bmap->n_ineq; ++k) {
                int stat;
                if (info->ineq[k] == STATUS_REDUNDANT)
                        continue;
                stat = status_in(bmap->ineq[k], tab);
                if (stat < 0)
                        return -1;
                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 enum isl_change is_adj_ineq_extension(int i, int j,
        struct isl_coalesce_info *info)
{
        int k;
        struct isl_tab_undo *snap;
        unsigned n_eq = info[i].bmap->n_eq;
        unsigned total = isl_basic_map_total_dim(info[i].bmap);
        int r;
        int super;

        if (isl_tab_extend_cons(info[i].tab, 1 + info[j].bmap->n_ineq) < 0)
                return isl_change_error;

        k = find(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_INEQ);
        if (k < 0)
                isl_die(isl_basic_map_get_ctx(info[i].bmap), isl_error_internal,
                        "info[i].ineq should have exactly one STATUS_ADJ_INEQ",
                        return isl_change_error);

        snap = isl_tab_snap(info[i].tab);

        if (isl_tab_unrestrict(info[i].tab, n_eq + k) < 0)
                return isl_change_error;

        isl_seq_neg(info[i].bmap->ineq[k], info[i].bmap->ineq[k], 1 + total);
        isl_int_sub_ui(info[i].bmap->ineq[k][0], info[i].bmap->ineq[k][0], 1);
        r = isl_tab_add_ineq(info[i].tab, info[i].bmap->ineq[k]);
        isl_seq_neg(info[i].bmap->ineq[k], info[i].bmap->ineq[k], 1 + total);
        isl_int_sub_ui(info[i].bmap->ineq[k][0], info[i].bmap->ineq[k][0], 1);
        if (r < 0)
                return isl_change_error;

        for (k = 0; k < info[j].bmap->n_ineq; ++k) {
                if (info[j].ineq[k] != STATUS_VALID)
                        continue;
                if (isl_tab_add_ineq(info[i].tab, info[j].bmap->ineq[k]) < 0)
                        return isl_change_error;
        }

        super = contains(&info[j], info[i].tab);
        if (super < 0)
                return isl_change_error;
        if (super)
                return fuse(i, j, info, NULL, 0, 0);

        if (isl_tab_rollback(info[i].tab, snap) < 0)
                return isl_change_error;

        return isl_change_none;
}


/* 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 enum isl_change check_adj_ineq(int i, int j,
        struct isl_coalesce_info *info)
{
        int count_i, count_j;
        int cut_i, cut_j;

        count_i = count(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_INEQ);
        count_j = count(info[j].ineq, info[j].bmap->n_ineq, STATUS_ADJ_INEQ);

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

        cut_i = any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_CUT) ||
                any(info[i].ineq, info[i].bmap->n_ineq, STATUS_CUT);
        cut_j = any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_CUT) ||
                any(info[j].ineq, info[j].bmap->n_ineq, STATUS_CUT);

        if (!cut_i && !cut_j && count_i == 1 && count_j == 1)
                return fuse(i, j, info, NULL, 0, 0);

        if (count_i == 1 && !cut_i)
                return is_adj_ineq_extension(i, j, info);

        if (count_j == 1 && !cut_j)
                return is_adj_ineq_extension(j, i, info);

        return isl_change_none;
}

/* Given an affine transformation matrix "T", does row "row" represent
 * anything other than a unit vector (possibly shifted by a constant)
 * that is not involved in any of the other rows?
 *
 * That is, if a constraint involves the variable corresponding to
 * the row, then could its preimage by "T" have any coefficients
 * that are different from those in the original constraint?
 */
static int not_unique_unit_row(__isl_keep isl_mat *T, int row)
{
        int i, j;
        int len = T->n_col - 1;

        i = isl_seq_first_non_zero(T->row[row] + 1, len);
        if (i < 0)
                return 1;
        if (!isl_int_is_one(T->row[row][1 + i]) &&
            !isl_int_is_negone(T->row[row][1 + i]))
                return 1;

        j = isl_seq_first_non_zero(T->row[row] + 1 + i + 1, len - (i + 1));
        if (j >= 0)
                return 1;

        for (j = 1; j < T->n_row; ++j) {
                if (j == row)
                        continue;
                if (!isl_int_is_zero(T->row[j][1 + i]))
                        return 1;
        }

        return 0;
}

/* Does inequality constraint "ineq" of "bmap" involve any of
 * the variables marked in "affected"?
 * "total" is the total number of variables, i.e., the number
 * of entries in "affected".
 */
static int is_affected(__isl_keep isl_basic_map *bmap, int ineq, int *affected,
        int total)
{
        int i;

        for (i = 0; i < total; ++i) {
                if (!affected[i])
                        continue;
                if (!isl_int_is_zero(bmap->ineq[ineq][1 + i]))
                        return 1;
        }

        return 0;
}

/* Given the compressed version of inequality constraint "ineq"
 * of info->bmap in "v", check if the constraint can be tightened,
 * where the compression is based on an equality constraint valid
 * for info->tab.
 * If so, add the tightened version of the inequality constraint
 * to info->tab.  "v" may be modified by this function.
 *
 * That is, if the compressed constraint is of the form
 *
 *      m f() + c >= 0
 *
 * with 0 < c < m, then it is equivalent to
 *
 *      f() >= 0
 *
 * This means that c can also be subtracted from the original,
 * uncompressed constraint without affecting the integer points
 * in info->tab.  Add this tightened constraint as an extra row
 * to info->tab to make this information explicitly available.
 */
static __isl_give isl_vec *try_tightening(struct isl_coalesce_info *info,
        int ineq, __isl_take isl_vec *v)
{
        isl_ctx *ctx;
        int r;

        if (!v)
                return NULL;

        ctx = isl_vec_get_ctx(v);
        isl_seq_gcd(v->el + 1, v->size - 1, &ctx->normalize_gcd);
        if (isl_int_is_zero(ctx->normalize_gcd) ||
            isl_int_is_one(ctx->normalize_gcd)) {
                return v;
        }

        v = isl_vec_cow(v);
        if (!v)
                return NULL;

        isl_int_fdiv_r(v->el[0], v->el[0], ctx->normalize_gcd);
        if (isl_int_is_zero(v->el[0]))
                return v;

        if (isl_tab_extend_cons(info->tab, 1) < 0)
                return isl_vec_free(v);

        isl_int_sub(info->bmap->ineq[ineq][0],
                    info->bmap->ineq[ineq][0], v->el[0]);
        r = isl_tab_add_ineq(info->tab, info->bmap->ineq[ineq]);
        isl_int_add(info->bmap->ineq[ineq][0],
                    info->bmap->ineq[ineq][0], v->el[0]);

        if (r < 0)
                return isl_vec_free(v);

        return v;
}

/* Tighten the (non-redundant) constraints on the facet represented
 * by info->tab.
 * In particular, on input, info->tab represents the result
 * of replacing constraint k of info->bmap, i.e., f_k >= 0,
 * by the adjacent equality, i.e., f_k + 1 = 0.
 *
 * Compute a variable compression from the equality constraint f_k + 1 = 0
 * and use it to tighten the other constraints of info->bmap,
 * updating info->tab (and leaving info->bmap untouched).
 * The compression handles essentially two cases, one where a variable
 * is assigned a fixed value and can therefore be eliminated, and one
 * where one variable is a shifted multiple of some other variable and
 * can therefore be replaced by that multiple.
 * Gaussian elimination would also work for the first case, but for
 * the second case, the effectiveness would depend on the order
 * of the variables.
 * After compression, some of the constraints may have coefficients
 * with a common divisor.  If this divisor does not divide the constant
 * term, then the constraint can be tightened.
 * The tightening is performed on the tableau info->tab by introducing
 * extra (temporary) constraints.
 *
 * Only constraints that are possibly affected by the compression are
 * considered.  In particular, if the constraint only involves variables
 * that are directly mapped to a distinct set of other variables, then
 * no common divisor can be introduced and no tightening can occur.
 *
 * It is important to only consider the non-redundant constraints
 * since the facet constraint has been relaxed prior to the call
 * to this function, meaning that the constraints that were redundant
 * prior to the relaxation may no longer be redundant.
 * These constraints will be ignored in the fused result, so
 * the fusion detection should not exploit them.
 */
static isl_stat tighten_on_relaxed_facet(struct isl_coalesce_info *info,
        int k)
{
        unsigned total;
        isl_ctx *ctx;
        isl_vec *v = NULL;
        isl_mat *T;
        int i;
        int *affected;

        ctx = isl_basic_map_get_ctx(info->bmap);
        total = isl_basic_map_total_dim(info->bmap);
        isl_int_add_ui(info->bmap->ineq[k][0], info->bmap->ineq[k][0], 1);
        T = isl_mat_sub_alloc6(ctx, info->bmap->ineq, k, 1, 0, 1 + total);
        T = isl_mat_variable_compression(T, NULL);
        isl_int_sub_ui(info->bmap->ineq[k][0], info->bmap->ineq[k][0], 1);
        if (!T)
                return isl_stat_error;
        if (T->n_col == 0) {
                isl_mat_free(T);
                return isl_stat_ok;
        }

        affected = isl_alloc_array(ctx, int, total);
        if (!affected)
                goto error;

        for (i = 0; i < total; ++i)
                affected[i] = not_unique_unit_row(T, 1 + i);

        for (i = 0; i < info->bmap->n_ineq; ++i) {
                if (i == k)
                        continue;
                if (info->ineq[i] == STATUS_REDUNDANT)
                        continue;
                if (!is_affected(info->bmap, i, affected, total))
                        continue;
                v = isl_vec_alloc(ctx, 1 + total);
                if (!v)
                        goto error;
                isl_seq_cpy(v->el, info->bmap->ineq[i], 1 + total);
                v = isl_vec_mat_product(v, isl_mat_copy(T));
                v = try_tightening(info, i, v);
                isl_vec_free(v);
                if (!v)
                        goto error;
        }

        isl_mat_free(T);
        free(affected);
        return isl_stat_ok;
error:
        isl_mat_free(T);
        free(affected);
        return isl_stat_error;
}

/* 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.
 * Before testing for inclusion, the constraints on the facet
 * are tightened to increase the chance of an inclusion being detected.
 * If the facet is included, 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.
 * Each integer division that does not have exactly the same
 * definition in "i" and "j" is marked unknown and the basic map
 * is scheduled to be simplified in an attempt to recover
 * the integer division definition.
 * Place this extension in the position that is the smallest of i and j.
 *        ____                    _____
 *       /    ||                 /     |
 *      /     ||                /      |
 *      \     ||        =>      \      |
 *       \    ||                 \     |
 *        \___||                  \____|
 */
static enum isl_change is_adj_eq_extension(int i, int j, int k,
        struct isl_coalesce_info *info)
{
        int change = isl_change_none;
        int super;
        struct isl_tab_undo *snap, *snap2;
        unsigned n_eq = info[i].bmap->n_eq;

        if (isl_tab_is_equality(info[i].tab, n_eq + k))
                return isl_change_none;

        snap = isl_tab_snap(info[i].tab);
        if (isl_tab_relax(info[i].tab, n_eq + k) < 0)
                return isl_change_error;
        snap2 = isl_tab_snap(info[i].tab);
        if (isl_tab_select_facet(info[i].tab, n_eq + k) < 0)
                return isl_change_error;
        if (tighten_on_relaxed_facet(&info[i], k) < 0)
                return isl_change_error;
        super = contains(&info[j], info[i].tab);
        if (super < 0)
                return isl_change_error;
        if (super) {
                int l;
                unsigned total;

                if (isl_tab_rollback(info[i].tab, snap2) < 0)
                        return isl_change_error;
                info[i].bmap = isl_basic_map_cow(info[i].bmap);
                if (!info[i].bmap)
                        return isl_change_error;
                total = isl_basic_map_total_dim(info[i].bmap);
                for (l = 0; l < info[i].bmap->n_div; ++l)
                        if (!isl_seq_eq(info[i].bmap->div[l],
                                        info[j].bmap->div[l], 1 + 1 + total)) {
                                isl_int_set_si(info[i].bmap->div[l][0], 0);
                                info[i].simplify = 1;
                        }
                isl_int_add_ui(info[i].bmap->ineq[k][0],
                                info[i].bmap->ineq[k][0], 1);
                ISL_F_SET(info[i].bmap, ISL_BASIC_MAP_FINAL);
                drop(&info[j]);
                if (j < i)
                        exchange(&info[i], &info[j]);
                change = isl_change_fuse;
        } else
                if (isl_tab_rollback(info[i].tab, snap) < 0)
                        return isl_change_error;

        return change;
}

/* 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 info->bmap that can be removed
 * if we end up applying wrapping.
 */
static void wraps_update_max(struct isl_wraps *wraps,
        struct isl_coalesce_info *info)
{
        int k;
        isl_int max_k;
        unsigned total = isl_basic_map_total_dim(info->bmap);

        isl_int_init(max_k);

        for (k = 0; k < info->bmap->n_eq; ++k) {
                if (info->eq[2 * k] == STATUS_VALID &&
                    info->eq[2 * k + 1] == STATUS_VALID)
                        continue;
                isl_seq_abs_max(info->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 < info->bmap->n_ineq; ++k) {
                if (info->ineq[k] == STATUS_VALID ||
                    info->ineq[k] == STATUS_REDUNDANT)
                        continue;
                isl_seq_abs_max(info->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,
        struct isl_coalesce_info *info, int i, int 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, &info[i]);
        wraps_update_max(wraps, &info[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;
}

/* Wrap "ineq" (or its opposite if "negate" is set) around "bound"
 * to include "set" and add the result in position "w" of "wraps".
 * "len" is the total number of coefficients in "bound" and "ineq".
 * Return 1 on success, 0 on failure and -1 on error.
 * Wrapping can fail if the result of wrapping is equal to "bound"
 * or if we want to bound the sizes of the coefficients and
 * the wrapped constraint does not satisfy this bound.
 */
static int add_wrap(struct isl_wraps *wraps, int w, isl_int *bound,
        isl_int *ineq, unsigned len, __isl_keep isl_set *set, int negate)
{
        isl_seq_cpy(wraps->mat->row[w], bound, len);
        if (negate) {
                isl_seq_neg(wraps->mat->row[w + 1], ineq, len);
                ineq = wraps->mat->row[w + 1];
        }
        if (!isl_set_wrap_facet(set, wraps->mat->row[w], ineq))
                return -1;
        if (isl_seq_eq(wraps->mat->row[w], bound, len))
                return 0;
        if (!allow_wrap(wraps, w))
                return 0;
        return 1;
}

/* For each constraint in info->bmap that is not redundant (as determined
 * by info->tab) and that is not a valid constraint for the other basic map,
 * wrap the constraint around "bound" such that it includes the whole
 * set "set" and append the resulting constraint to "wraps".
 * Note that the constraints that are valid for the other basic map
 * will be added to the combined basic map by default, so there is
 * no need to wrap them.
 * The caller wrap_in_facets even relies on this function not wrapping
 * any constraints that are already valid.
 * "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, struct isl_coalesce_info *info,
        isl_int *bound, __isl_keep isl_set *set)
{
        int l, m;
        int w;
        int added;
        isl_basic_map *bmap = info->bmap;
        unsigned len = 1 + isl_basic_map_total_dim(bmap);

        w = wraps->mat->n_row;

        for (l = 0; l < bmap->n_ineq; ++l) {
                if (info->ineq[l] == STATUS_VALID ||
                    info->ineq[l] == STATUS_REDUNDANT)
                        continue;
                if (isl_seq_is_neg(bound, bmap->ineq[l], len))
                        continue;
                if (isl_seq_eq(bound, bmap->ineq[l], len))
                        continue;
                if (isl_tab_is_redundant(info->tab, bmap->n_eq + l))
                        continue;

                added = add_wrap(wraps, w, bound, bmap->ineq[l], len, set, 0);
                if (added < 0)
                        return -1;
                if (!added)
                        goto unbounded;
                ++w;
        }
        for (l = 0; l < bmap->n_eq; ++l) {
                if (isl_seq_is_neg(bound, bmap->eq[l], len))
                        continue;
                if (isl_seq_eq(bound, bmap->eq[l], len))
                        continue;

                for (m = 0; m < 2; ++m) {
                        if (info->eq[2 * l + m] == STATUS_VALID)
                                continue;
                        added = add_wrap(wraps, w, bound, bmap->eq[l], len,
                                        set, !m);
                        if (added < 0)
                                return -1;
                        if (!added)
                                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.
 *
 * Update the underlying set to ensure that the dimension doesn't change.
 * Otherwise the integer divisions could get dropped if the tab
 * turns out to be empty.
 */
static __isl_give isl_set *set_from_updated_bmap(__isl_keep isl_basic_map *bmap,
        struct isl_tab *tab)
{
        isl_basic_set *bset;

        bmap = isl_basic_map_copy(bmap);
        bset = isl_basic_map_underlying_set(bmap);
        bset = isl_basic_set_cow(bset);
        bset = isl_basic_set_update_from_tab(bset, tab);
        return isl_set_from_basic_set(bset);
}

/* Wrap the constraints of info->bmap that bound the facet defined
 * by inequality "k" around (the opposite of) this inequality to
 * include "set".  "bound" may be used to store the negated inequality.
 * Since the wrapped constraints are not guaranteed to contain the whole
 * of info->bmap, we check them in check_wraps.
 * If any of the wrapped constraints turn out to be invalid, then
 * check_wraps will reset wrap->n_row to zero.
 */
static int add_wraps_around_facet(struct isl_wraps *wraps,
        struct isl_coalesce_info *info, int k, isl_int *bound,
        __isl_keep isl_set *set)
{
        struct isl_tab_undo *snap;
        int n;
        unsigned total = isl_basic_map_total_dim(info->bmap);

        snap = isl_tab_snap(info->tab);

        if (isl_tab_select_facet(info->tab, info->bmap->n_eq + k) < 0)
                return -1;
        if (isl_tab_detect_redundant(info->tab) < 0)
                return -1;

        isl_seq_neg(bound, info->bmap->ineq[k], 1 + total);

        n = wraps->mat->n_row;
        if (add_wraps(wraps, info, bound, set) < 0)
                return -1;

        if (isl_tab_rollback(info->tab, snap) < 0)
                return -1;
        if (check_wraps(wraps->mat, n, info->tab) < 0)
                return -1;

        return 0;
}

/* 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 (if "wrap_facet" is set) and basic map j
 * (always) 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 or
 * cut constraints for j.
 * Wrapping the cut constraints to include basic map j may result
 * in constraints that are no longer valid of basic map i
 * we have to check that the resulting wrapping constraints are valid for i.
 * If "wrap_facet" is not set, then all constraints of i (except k)
 * are assumed to be valid for j.
 *        ____                    _____
 *       /    |                  /     \
 *      /     ||                /      |
 *      \     ||        =>      \      |
 *       \    ||                 \     |
 *        \___||                  \____|
 *
 */
static enum isl_change can_wrap_in_facet(int i, int j, int k,
        struct isl_coalesce_info *info, int wrap_facet)
{
        enum isl_change change = isl_change_none;
        struct isl_wraps wraps;
        isl_ctx *ctx;
        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(info[i].bmap);

        set_i = set_from_updated_bmap(info[i].bmap, info[i].tab);
        set_j = set_from_updated_bmap(info[j].bmap, info[j].tab);
        ctx = isl_basic_map_get_ctx(info[i].bmap);
        mat = isl_mat_alloc(ctx, 2 * (info[i].bmap->n_eq + info[j].bmap->n_eq) +
                                    info[i].bmap->n_ineq + info[j].bmap->n_ineq,
                                    1 + total);
        wraps_init(&wraps, mat, info, i, j);
        bound = isl_vec_alloc(ctx, 1 + total);
        if (!set_i || !set_j || !wraps.mat || !bound)
                goto error;

        isl_seq_cpy(bound->el, info[i].bmap->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, &info[j], bound->el, set_i) < 0)
                goto error;
        if (!wraps.mat->n_row)
                goto unbounded;

        if (wrap_facet) {
                if (add_wraps_around_facet(&wraps, &info[i], k,
                                            bound->el, set_j) < 0)
                        goto error;
                if (!wraps.mat->n_row)
                        goto unbounded;
        }

        change = fuse(i, j, info, wraps.mat, 0, 0);

unbounded:
        wraps_free(&wraps);

        isl_set_free(set_i);
        isl_set_free(set_j);

        isl_vec_free(bound);

        return change;
error:
        wraps_free(&wraps);
        isl_vec_free(bound);
        isl_set_free(set_i);
        isl_set_free(set_j);
        return isl_change_error;
}

/* Given a cut constraint t(x) >= 0 of basic map i, stored in row "w"
 * of wrap.mat, replace it by its relaxed version t(x) + 1 >= 0, and
 * add wrapping constraints to wrap.mat for all constraints
 * of basic map j that bound the part of basic map j that sticks out
 * of the cut constraint.
 * "set_i" is the underlying set of basic map i.
 * If any wrapping fails, then wraps->mat.n_row is reset to zero.
 *
 * In particular, we first intersect basic map j with t(x) + 1 = 0.
 * If the result is empty, then t(x) >= 0 was actually a valid constraint
 * (with respect to the integer points), so we add t(x) >= 0 instead.
 * Otherwise, we wrap the constraints of basic map j that are not
 * redundant in this intersection and that are not already valid
 * for basic map i over basic map i.
 * Note that it is sufficient to wrap the constraints to include
 * basic map i, because we will only wrap the constraints that do
 * not include basic map i already.  The wrapped constraint will
 * therefore be more relaxed compared to the original constraint.
 * Since the original constraint is valid for basic map j, so is
 * the wrapped constraint.
 */
static isl_stat wrap_in_facet(struct isl_wraps *wraps, int w,
        struct isl_coalesce_info *info_j, __isl_keep isl_set *set_i,
        struct isl_tab_undo *snap)
{
        isl_int_add_ui(wraps->mat->row[w][0], wraps->mat->row[w][0], 1);
        if (isl_tab_add_eq(info_j->tab, wraps->mat->row[w]) < 0)
                return isl_stat_error;
        if (isl_tab_detect_redundant(info_j->tab) < 0)
                return isl_stat_error;

        if (info_j->tab->empty)
                isl_int_sub_ui(wraps->mat->row[w][0], wraps->mat->row[w][0], 1);
        else if (add_wraps(wraps, info_j, wraps->mat->row[w], set_i) < 0)
                return isl_stat_error;

        if (isl_tab_rollback(info_j->tab, snap) < 0)
                return isl_stat_error;

        return isl_stat_ok;
}

/* 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.
 * "set_i" is the underlying set of basic map i.
 * "wraps" has been initialized to be of the right size.
 *
 * For each cut constraint t(x) >= 0 of i, we add the relaxed version
 * t(x) + 1 >= 0, along with wrapping constraints for all constraints
 * of basic map j that bound the part of basic map j that sticks out
 * of the cut constraint.
 *
 * 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 enum isl_change try_wrap_in_facets(int i, int j,
        struct isl_coalesce_info *info, struct isl_wraps *wraps,
        __isl_keep isl_set *set_i)
{
        int k, l, w;
        unsigned total;
        struct isl_tab_undo *snap;

        total = isl_basic_map_total_dim(info[i].bmap);

        snap = isl_tab_snap(info[j].tab);

        wraps->mat->n_row = 0;

        for (k = 0; k < info[i].bmap->n_eq; ++k) {
                for (l = 0; l < 2; ++l) {
                        if (info[i].eq[2 * k + l] != STATUS_CUT)
                                continue;
                        w = wraps->mat->n_row++;
                        if (l == 0)
                                isl_seq_neg(wraps->mat->row[w],
                                            info[i].bmap->eq[k], 1 + total);
                        else
                                isl_seq_cpy(wraps->mat->row[w],
                                            info[i].bmap->eq[k], 1 + total);
                        if (wrap_in_facet(wraps, w, &info[j], set_i, snap) < 0)
                                return isl_change_error;

                        if (!wraps->mat->n_row)
                                return isl_change_none;
                }
        }

        for (k = 0; k < info[i].bmap->n_ineq; ++k) {
                if (info[i].ineq[k] != STATUS_CUT)
                        continue;
                w = wraps->mat->n_row++;
                isl_seq_cpy(wraps->mat->row[w],
                            info[i].bmap->ineq[k], 1 + total);
                if (wrap_in_facet(wraps, w, &info[j], set_i, snap) < 0)
                        return isl_change_error;

                if (!wraps->mat->n_row)
                        return isl_change_none;
        }

        return fuse(i, j, info, wraps->mat, 0, 1);
}

/* 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 core computation is performed by try_wrap_in_facets.
 * This function simply extracts an underlying set representation
 * of basic map i and initializes the data structure for keeping
 * track of wrapping constraints.
 */
static enum isl_change wrap_in_facets(int i, int j, int n,
        struct isl_coalesce_info *info)
{
        enum isl_change change = isl_change_none;
        struct isl_wraps wraps;
        isl_ctx *ctx;
        isl_mat *mat;
        isl_set *set_i = NULL;
        unsigned total = isl_basic_map_total_dim(info[i].bmap);
        int max_wrap;

        if (isl_tab_extend_cons(info[j].tab, 1) < 0)
                return isl_change_error;

        max_wrap = 1 + 2 * info[j].bmap->n_eq + info[j].bmap->n_ineq;
        max_wrap *= n;

        set_i = set_from_updated_bmap(info[i].bmap, info[i].tab);
        ctx = isl_basic_map_get_ctx(info[i].bmap);
        mat = isl_mat_alloc(ctx, max_wrap, 1 + total);
        wraps_init(&wraps, mat, info, i, j);
        if (!set_i || !wraps.mat)
                goto error;

        change = try_wrap_in_facets(i, j, info, &wraps, set_i);

        wraps_free(&wraps);
        isl_set_free(set_i);

        return change;
error:
        wraps_free(&wraps);
        isl_set_free(set_i);
        return isl_change_error;
}

/* Return the effect of inequality "ineq" on the tableau "tab",
 * after relaxing the constant term of "ineq" by one.
 */
static enum isl_ineq_type type_of_relaxed(struct isl_tab *tab, isl_int *ineq)
{
        enum isl_ineq_type type;

        isl_int_add_ui(ineq[0], ineq[0], 1);
        type = isl_tab_ineq_type(tab, ineq);
        isl_int_sub_ui(ineq[0], ineq[0], 1);

        return type;
}

/* Given two basic sets i and j,
 * check if relaxing all the cut constraints 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 enum isl_change can_wrap_in_set(int i, int j,
        struct isl_coalesce_info *info)
{
        int k, l;
        int n;
        unsigned total;

        if (ISL_F_ISSET(info[i].bmap, ISL_BASIC_MAP_RATIONAL) ||
            ISL_F_ISSET(info[j].bmap, ISL_BASIC_MAP_RATIONAL))
                return isl_change_none;

        n = count(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_CUT);
        n += count(info[i].ineq, info[i].bmap->n_ineq, STATUS_CUT);
        if (n == 0)
                return isl_change_none;

        total = isl_basic_map_total_dim(info[i].bmap);
        for (k = 0; k < info[i].bmap->n_eq; ++k) {
                for (l = 0; l < 2; ++l) {
                        enum isl_ineq_type type;

                        if (info[i].eq[2 * k + l] != STATUS_CUT)
                                continue;

                        if (l == 0)
                                isl_seq_neg(info[i].bmap->eq[k],
                                            info[i].bmap->eq[k], 1 + total);
                        type = type_of_relaxed(info[j].tab,
                                            info[i].bmap->eq[k]);
                        if (l == 0)
                                isl_seq_neg(info[i].bmap->eq[k],
                                            info[i].bmap->eq[k], 1 + total);
                        if (type == isl_ineq_error)
                                return isl_change_error;
                        if (type != isl_ineq_redundant)
                                return isl_change_none;
                }
        }

        for (k = 0; k < info[i].bmap->n_ineq; ++k) {
                enum isl_ineq_type type;

                if (info[i].ineq[k] != STATUS_CUT)
                        continue;

                type = type_of_relaxed(info[j].tab, info[i].bmap->ineq[k]);
                if (type == isl_ineq_error)
                        return isl_change_error;
                if (type != isl_ineq_redundant)
                        return isl_change_none;
        }

        return wrap_in_facets(i, j, n, info);
}

/* Check if either i or j has only cut constraints that can
 * be used to wrap in (a facet of) the other basic set.
 * if so, replace the pair by their union.
 */
static enum isl_change check_wrap(int i, int j, struct isl_coalesce_info *info)
{
        enum isl_change change = isl_change_none;

        change = can_wrap_in_set(i, j, info);
        if (change != isl_change_none)
                return change;

        change = can_wrap_in_set(j, i, info);
        return change;
}

/* 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.
 * If the other basic map does not have such an inequality, then
 * check if all its constraints are either valid or cut constraints
 * and, if so, try wrapping in the first map into the second.
 *
 * 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.  However, it may still be possible to wrap in the other
 * basic map.
 */
static enum isl_change check_adj_eq(int i, int j,
        struct isl_coalesce_info *info)
{
        enum isl_change change = isl_change_none;
        int k;
        int any_cut;

        if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ADJ_INEQ) &&
            any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_ADJ_INEQ))
                /* ADJ EQ TOO MANY */
                return isl_change_none;

        if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ADJ_INEQ))
                return check_adj_eq(j, i, info);

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

        if (count(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_EQ) != 1) {
                if (all_valid_or_cut(&info[i]))
                        return can_wrap_in_set(i, j, info);
                return isl_change_none;
        }
        if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_CUT))
                return isl_change_none;
        any_cut = any(info[i].ineq, info[i].bmap->n_ineq, STATUS_CUT);
        if (any(info[j].ineq, info[j].bmap->n_ineq, STATUS_ADJ_EQ) ||
            any(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_INEQ) ||
            any(info[j].ineq, info[j].bmap->n_ineq, STATUS_ADJ_INEQ))
                /* ADJ EQ TOO MANY */
                return isl_change_none;

        k = find(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_EQ);

        if (!any_cut) {
                change = is_adj_eq_extension(i, j, k, info);
                if (change != isl_change_none)
                        return change;
        }

        change = can_wrap_in_facet(i, j, k, info, any_cut);

        return change;
}

/* 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.
 *                              _
 *      \\                      \\
 *       \\             =>       \\
 *        \                       \|
 *
 * If there is more than one equality of "i" adjacent to an equality of "j",
 * then the result will satisfy one or more equalities that are a linear
 * combination of these equalities.  These will be encoded as pairs
 * of inequalities in the wrapping constraints and need to be made
 * explicit.
 */
static enum isl_change check_eq_adj_eq(int i, int j,
        struct isl_coalesce_info *info)
{
        int k;
        enum isl_change change = isl_change_none;
        int detect_equalities = 0;
        struct isl_wraps wraps;
        isl_ctx *ctx;
        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(info[i].bmap);

        if (count(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ADJ_EQ) != 1)
                detect_equalities = 1;

        k = find(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ADJ_EQ);

        set_i = set_from_updated_bmap(info[i].bmap, info[i].tab);
        set_j = set_from_updated_bmap(info[j].bmap, info[j].tab);
        ctx = isl_basic_map_get_ctx(info[i].bmap);
        mat = isl_mat_alloc(ctx, 2 * (info[i].bmap->n_eq + info[j].bmap->n_eq) +
                                    info[i].bmap->n_ineq + info[j].bmap->n_ineq,
                                    1 + total);
        wraps_init(&wraps, mat, info, i, j);
        bound = isl_vec_alloc(ctx, 1 + total);
        if (!set_i || !set_j || !wraps.mat || !bound)
                goto error;

        if (k % 2 == 0)
                isl_seq_neg(bound->el, info[i].bmap->eq[k / 2], 1 + total);
        else
                isl_seq_cpy(bound->el, info[i].bmap->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, &info[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, &info[i], bound->el, set_j) < 0)
                goto error;
        if (!wraps.mat->n_row)
                goto unbounded;

        change = fuse(i, j, info, wraps.mat, detect_equalities, 0);

        if (0) {
error:          change = isl_change_error;
        }
unbounded:

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

        return change;
}

/* Initialize the "eq" and "ineq" fields of "info".
 */
static void init_status(struct isl_coalesce_info *info)
{
        info->eq = info->ineq = NULL;
}

/* Set info->eq to the positions of the equalities of info->bmap
 * with respect to the basic map represented by "tab".
 * If info->eq has already been computed, then do not compute it again.
 */
static void set_eq_status_in(struct isl_coalesce_info *info,
        struct isl_tab *tab)
{
        if (info->eq)
                return;
        info->eq = eq_status_in(info->bmap, tab);
}

/* Set info->ineq to the positions of the inequalities of info->bmap
 * with respect to the basic map represented by "tab".
 * If info->ineq has already been computed, then do not compute it again.
 */
static void set_ineq_status_in(struct isl_coalesce_info *info,
        struct isl_tab *tab)
{
        if (info->ineq)
                return;
        info->ineq = ineq_status_in(info->bmap, info->tab, tab);
}

/* Free the memory allocated by the "eq" and "ineq" fields of "info".
 * This function assumes that init_status has been called on "info" first,
 * after which the "eq" and "ineq" fields may or may not have been
 * assigned a newly allocated array.
 */
static void clear_status(struct isl_coalesce_info *info)
{
        free(info->eq);
        free(info->ineq);
}

/* 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
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 * The two basic maps are assumed to live in the same local space.
 * The "eq" and "ineq" fields of info[i] and info[j] are assumed
 * to have been initialized by the caller, either to NULL or
 * to valid information.
 *
 * 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 enum isl_change coalesce_local_pair_reuse(int i, int j,
        struct isl_coalesce_info *info)
{
        enum isl_change change = isl_change_none;

        set_eq_status_in(&info[i], info[j].tab);
        if (info[i].bmap->n_eq && !info[i].eq)
                goto error;
        if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ERROR))
                goto error;
        if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_SEPARATE))
                goto done;

        set_eq_status_in(&info[j], info[i].tab);
        if (info[j].bmap->n_eq && !info[j].eq)
                goto error;
        if (any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_ERROR))
                goto error;
        if (any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_SEPARATE))
                goto done;

        set_ineq_status_in(&info[i], info[j].tab);
        if (info[i].bmap->n_ineq && !info[i].ineq)
                goto error;
        if (any(info[i].ineq, info[i].bmap->n_ineq, STATUS_ERROR))
                goto error;
        if (any(info[i].ineq, info[i].bmap->n_ineq, STATUS_SEPARATE))
                goto done;

        set_ineq_status_in(&info[j], info[i].tab);
        if (info[j].bmap->n_ineq && !info[j].ineq)
                goto error;
        if (any(info[j].ineq, info[j].bmap->n_ineq, STATUS_ERROR))
                goto error;
        if (any(info[j].ineq, info[j].bmap->n_ineq, STATUS_SEPARATE))
                goto done;

        if (all(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_VALID) &&
            all(info[i].ineq, info[i].bmap->n_ineq, STATUS_VALID)) {
                drop(&info[j]);
                change = isl_change_drop_second;
        } else if (all(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_VALID) &&
                   all(info[j].ineq, info[j].bmap->n_ineq, STATUS_VALID)) {
                drop(&info[i]);
                change = isl_change_drop_first;
        } else if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ADJ_EQ)) {
                change = check_eq_adj_eq(i, j, info);
        } else if (any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_ADJ_EQ)) {
                change = check_eq_adj_eq(j, i, info);
        } else if (any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_ADJ_INEQ) ||
                   any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_ADJ_INEQ)) {
                change = check_adj_eq(i, j, info);
        } else if (any(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_EQ) ||
                   any(info[j].ineq, info[j].bmap->n_ineq, STATUS_ADJ_EQ)) {
                /* Can't happen */
                /* BAD ADJ INEQ */
        } else if (any(info[i].ineq, info[i].bmap->n_ineq, STATUS_ADJ_INEQ) ||
                   any(info[j].ineq, info[j].bmap->n_ineq, STATUS_ADJ_INEQ)) {
                change = check_adj_ineq(i, j, info);
        } else {
                if (!any(info[i].eq, 2 * info[i].bmap->n_eq, STATUS_CUT) &&
                    !any(info[j].eq, 2 * info[j].bmap->n_eq, STATUS_CUT))
                        change = check_facets(i, j, info);
                if (change == isl_change_none)
                        change = check_wrap(i, j, info);
        }

done:
        clear_status(&info[i]);
        clear_status(&info[j]);
        return change;
error:
        clear_status(&info[i]);
        clear_status(&info[j]);
        return isl_change_error;
}

/* 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
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 * The two basic maps are assumed to live in the same local space.
 */
static enum isl_change coalesce_local_pair(int i, int j,
        struct isl_coalesce_info *info)
{
        init_status(&info[i]);
        init_status(&info[j]);
        return coalesce_local_pair_reuse(i, j, info);
}

/* Shift the integer division at position "div" of the basic map
 * represented by "info" by "shift".
 *
 * That is, if the integer division has the form
 *
 *      floor(f(x)/d)
 *
 * then replace it by
 *
 *      floor((f(x) + shift * d)/d) - shift
 */
static int shift_div(struct isl_coalesce_info *info, int div, isl_int shift)
{
        unsigned total;

        info->bmap = isl_basic_map_shift_div(info->bmap, div, 0, shift);
        if (!info->bmap)
                return -1;

        total = isl_basic_map_dim(info->bmap, isl_dim_all);
        total -= isl_basic_map_dim(info->bmap, isl_dim_div);
        if (isl_tab_shift_var(info->tab, total + div, shift) < 0)
                return -1;

        return 0;
}

/* Check if some of the divs in the basic map represented by "info1"
 * are shifts of the corresponding divs in the basic map represented
 * by "info2".  If so, align them with those of "info2".
 * Only do this if "info1" and "info2" have the same number
 * of integer divisions.
 *
 * An integer division is considered to be a shift of another integer
 * division if one is equal to the other plus a constant.
 *
 * In particular, for each pair of integer divisions, if both are known,
 * have identical coefficients (apart from the constant term) and
 * if the difference between the constant terms (taking into account
 * the denominator) is an integer, then move the difference outside.
 * That is, if one integer division is of the form
 *
 *      floor((f(x) + c_1)/d)
 *
 * while the other is of the form
 *
 *      floor((f(x) + c_2)/d)
 *
 * and n = (c_2 - c_1)/d is an integer, then replace the first
 * integer division by
 *
 *      floor((f(x) + c_1 + n * d)/d) - n = floor((f(x) + c_2)/d) - n
 */
static int harmonize_divs(struct isl_coalesce_info *info1,
        struct isl_coalesce_info *info2)
{
        int i;
        int total;

        if (!info1->bmap || !info2->bmap)
                return -1;

        if (info1->bmap->n_div != info2->bmap->n_div)
                return 0;
        if (info1->bmap->n_div == 0)
                return 0;

        total = isl_basic_map_total_dim(info1->bmap);
        for (i = 0; i < info1->bmap->n_div; ++i) {
                isl_int d;
                int r = 0;

                if (isl_int_is_zero(info1->bmap->div[i][0]) ||
                    isl_int_is_zero(info2->bmap->div[i][0]))
                        continue;
                if (isl_int_ne(info1->bmap->div[i][0], info2->bmap->div[i][0]))
                        continue;
                if (isl_int_eq(info1->bmap->div[i][1], info2->bmap->div[i][1]))
                        continue;
                if (!isl_seq_eq(info1->bmap->div[i] + 2,
                                info2->bmap->div[i] + 2, total))
                        continue;
                isl_int_init(d);
                isl_int_sub(d, info2->bmap->div[i][1], info1->bmap->div[i][1]);
                if (isl_int_is_divisible_by(d, info1->bmap->div[i][0])) {
                        isl_int_divexact(d, d, info1->bmap->div[i][0]);
                        r = shift_div(info1, i, d);
                }
                isl_int_clear(d);
                if (r < 0)
                        return -1;
        }

        return 0;
}

/* 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;
}

/* Expand info->tab in the same way info->bmap was expanded in
 * isl_basic_map_expand_divs using the expansion "exp" and
 * update info->ineq with respect to the redundant constraints
 * in the resulting tableau. "bmap" is the original version
 * of info->bmap, i.e., the one that corresponds to the current
 * state of info->tab.  The number of constraints in "bmap"
 * is assumed to be the same as the number of constraints
 * in info->tab.  This is required to be able to detect
 * the extra constraints in info->bmap.
 *
 * In particular, introduce extra variables corresponding
 * to the extra integer divisions and add the div constraints
 * that were added to info->bmap after info->tab was created
 * from the original info->bmap.
 * info->ineq was computed without a tableau and therefore
 * does not take into account the redundant constraints
 * in the tableau.  Mark them here.
 */
static isl_stat expand_tab(struct isl_coalesce_info *info, int *exp,
        __isl_keep isl_basic_map *bmap)
{
        unsigned total, pos, n_div;
        int extra_var;
        int i, n, j, n_ineq;
        unsigned n_eq;

        if (!bmap)
                return isl_stat_error;
        if (bmap->n_eq + bmap->n_ineq != info->tab->n_con)
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_internal,
                        "original tableau does not correspond "
                        "to original basic map", return isl_stat_error);

        total = isl_basic_map_dim(info->bmap, isl_dim_all);
        n_div = isl_basic_map_dim(info->bmap, isl_dim_div);
        pos = total - n_div;
        extra_var = total - info->tab->n_var;
        n = n_div - extra_var;

        if (isl_tab_extend_vars(info->tab, extra_var) < 0)
                return isl_stat_error;
        if (isl_tab_extend_cons(info->tab, 2 * extra_var) < 0)
                return isl_stat_error;

        i = 0;
        for (j = 0; j < n_div; ++j) {
                if (i < n && exp[i] == j) {
                        ++i;
                        continue;
                }
                if (isl_tab_insert_var(info->tab, pos + j) < 0)
                        return isl_stat_error;
        }

        n_ineq = info->tab->n_con - info->tab->n_eq;
        for (i = n_ineq; i < info->bmap->n_ineq; ++i)
                if (isl_tab_add_ineq(info->tab, info->bmap->ineq[i]) < 0)
                        return isl_stat_error;

        n_eq = info->bmap->n_eq;
        for (i = 0; i < info->bmap->n_ineq; ++i) {
                if (isl_tab_is_redundant(info->tab, n_eq + i))
                        info->ineq[i] = STATUS_REDUNDANT;
        }

        return isl_stat_ok;
}

/* Check if the union of the basic maps represented by info[i] and info[j]
 * can be represented by a single basic map,
 * after expanding the divs of info[i] to match those of info[j].
 * If so, replace the pair by the single basic map and return
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 *
 * The caller has already checked for info[j] being a subset of info[i].
 * If some of the divs of info[j] are unknown, then the expanded info[i]
 * will not have the corresponding div constraints.  The other patterns
 * therefore cannot apply.  Skip the computation in this case.
 *
 * The expansion is performed using the divs "div" and expansion "exp"
 * computed by the caller.
 * info[i].bmap has already been expanded and the result is passed in
 * as "bmap".
 * The "eq" and "ineq" fields of info[i] reflect the status of
 * the constraints of the expanded "bmap" with respect to info[j].tab.
 * However, inequality constraints that are redundant in info[i].tab
 * have not yet been marked as such because no tableau was available.
 *
 * Replace info[i].bmap by "bmap" and expand info[i].tab as well,
 * updating info[i].ineq with respect to the redundant constraints.
 * Then try and coalesce the expanded info[i] with info[j],
 * reusing the information in info[i].eq and info[i].ineq.
 * If this does not result in any coalescing or if it results in info[j]
 * getting dropped (which should not happen in practice, since the case
 * of info[j] being a subset of info[i] has already been checked by
 * the caller), then revert info[i] to its original state.
 */
static enum isl_change coalesce_expand_tab_divs(__isl_take isl_basic_map *bmap,
        int i, int j, struct isl_coalesce_info *info, __isl_keep isl_mat *div,
        int *exp)
{
        isl_bool known;
        isl_basic_map *bmap_i;
        struct isl_tab_undo *snap;
        enum isl_change change = isl_change_none;

        known = isl_basic_map_divs_known(info[j].bmap);
        if (known < 0 || !known) {
                clear_status(&info[i]);
                isl_basic_map_free(bmap);
                return known < 0 ? isl_change_error : isl_change_none;
        }

        bmap_i = info[i].bmap;
        info[i].bmap = isl_basic_map_copy(bmap);
        snap = isl_tab_snap(info[i].tab);
        if (!info[i].bmap || expand_tab(&info[i], exp, bmap_i) < 0)
                change = isl_change_error;

        init_status(&info[j]);
        if (change == isl_change_none)
                change = coalesce_local_pair_reuse(i, j, info);
        else
                clear_status(&info[i]);
        if (change != isl_change_none && change != isl_change_drop_second) {
                isl_basic_map_free(bmap_i);
        } else {
                isl_basic_map_free(info[i].bmap);
                info[i].bmap = bmap_i;

                if (isl_tab_rollback(info[i].tab, snap) < 0)
                        change = isl_change_error;
        }

        isl_basic_map_free(bmap);
        return change;
}

/* Check if the union of "bmap" and the basic map represented by info[j]
 * can be represented by a single basic map,
 * after expanding the divs of "bmap" to match those of info[j].
 * If so, replace the pair by the single basic map and return
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 *
 * In particular, check if the expanded "bmap" contains the basic map
 * represented by the tableau info[j].tab.
 * The expansion is performed using the divs "div" and expansion "exp"
 * computed by the caller.
 * Then we check if all constraints of the expanded "bmap" are valid for
 * info[j].tab.
 *
 * If "i" is not equal to -1, then "bmap" is equal to info[i].bmap.
 * In this case, the positions of the constraints of info[i].bmap
 * with respect to the basic map represented by info[j] are stored
 * in info[i].
 *
 * If the expanded "bmap" does not contain the basic map
 * represented by the tableau info[j].tab and if "i" is not -1,
 * i.e., if the original "bmap" is info[i].bmap, then expand info[i].tab
 * as well and check if that results in coalescing.
 */
static enum isl_change coalesce_with_expanded_divs(
        __isl_keep isl_basic_map *bmap, int i, int j,
        struct isl_coalesce_info *info, __isl_keep isl_mat *div, int *exp)
{
        enum isl_change change = isl_change_none;
        struct isl_coalesce_info info_local, *info_i;

        info_i = i >= 0 ? &info[i] : &info_local;
        init_status(info_i);
        bmap = isl_basic_map_copy(bmap);
        bmap = isl_basic_map_expand_divs(bmap, isl_mat_copy(div), exp);
        bmap = isl_basic_map_mark_final(bmap);

        if (!bmap)
                goto error;

        info_i->eq = eq_status_in(bmap, info[j].tab);
        if (bmap->n_eq && !info_i->eq)
                goto error;
        if (any(info_i->eq, 2 * bmap->n_eq, STATUS_ERROR))
                goto error;
        if (any(info_i->eq, 2 * bmap->n_eq, STATUS_SEPARATE))
                goto done;

        info_i->ineq = ineq_status_in(bmap, NULL, info[j].tab);
        if (bmap->n_ineq && !info_i->ineq)
                goto error;
        if (any(info_i->ineq, bmap->n_ineq, STATUS_ERROR))
                goto error;
        if (any(info_i->ineq, bmap->n_ineq, STATUS_SEPARATE))
                goto done;

        if (all(info_i->eq, 2 * bmap->n_eq, STATUS_VALID) &&
            all(info_i->ineq, bmap->n_ineq, STATUS_VALID)) {
                drop(&info[j]);
                change = isl_change_drop_second;
        }

        if (change == isl_change_none && i != -1)
                return coalesce_expand_tab_divs(bmap, i, j, info, div, exp);

done:
        isl_basic_map_free(bmap);
        clear_status(info_i);
        return change;
error:
        isl_basic_map_free(bmap);
        clear_status(info_i);
        return isl_change_error;
}

/* Check if the union of "bmap_i" and the basic map represented by info[j]
 * can be represented by a single basic map,
 * after aligning the divs of "bmap_i" to match those of info[j].
 * If so, replace the pair by the single basic map and return
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 *
 * In particular, check if "bmap_i" contains the basic map represented by
 * info[j] after aligning the divs of "bmap_i" to those of info[j].
 * Note that this can only succeed if the number of divs of "bmap_i"
 * is smaller than (or equal to) the number of divs of info[j].
 *
 * We first check if the divs of "bmap_i" are all known and form a subset
 * of those of info[j].bmap.  If so, we pass control over to
 * coalesce_with_expanded_divs.
 *
 * If "i" is not equal to -1, then "bmap" is equal to info[i].bmap.
 */
static enum isl_change coalesce_after_aligning_divs(
        __isl_keep isl_basic_map *bmap_i, int i, int j,
        struct isl_coalesce_info *info)
{
        int known;
        isl_mat *div_i, *div_j, *div;
        int *exp1 = NULL;
        int *exp2 = NULL;
        isl_ctx *ctx;
        enum isl_change change;

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

        ctx = isl_basic_map_get_ctx(bmap_i);

        div_i = isl_basic_map_get_divs(bmap_i);
        div_j = isl_basic_map_get_divs(info[j].bmap);

        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)
                change = coalesce_with_expanded_divs(bmap_i,
                                                        i, j, info, div, exp1);
        else
                change = isl_change_none;

        isl_mat_free(div);

        isl_mat_free(div_i);
        isl_mat_free(div_j);

        free(exp2);
        free(exp1);

        return change;
error:
        isl_mat_free(div_i);
        isl_mat_free(div_j);
        free(exp1);
        free(exp2);
        return isl_change_error;
}

/* Check if basic map "j" is a subset of basic map "i" after
 * exploiting the extra equalities of "j" to simplify the divs of "i".
 * If so, remove basic map "j" and return isl_change_drop_second.
 *
 * If "j" does not have any equalities or if they are the same
 * as those of "i", then we cannot exploit them to simplify the divs.
 * Similarly, if there are no divs in "i", then they cannot be simplified.
 * If, on the other hand, the affine hulls of "i" and "j" do not intersect,
 * then "j" cannot be a subset of "i".
 *
 * Otherwise, we intersect "i" with the affine hull of "j" and then
 * check if "j" is a subset of the result after aligning the divs.
 * If so, then "j" is definitely a subset of "i" and can be removed.
 * Note that if after intersection with the affine hull of "j".
 * "i" still has more divs than "j", then there is no way we can
 * align the divs of "i" to those of "j".
 */
static enum isl_change coalesce_subset_with_equalities(int i, int j,
        struct isl_coalesce_info *info)
{
        isl_basic_map *hull_i, *hull_j, *bmap_i;
        int equal, empty;
        enum isl_change change;

        if (info[j].bmap->n_eq == 0)
                return isl_change_none;
        if (info[i].bmap->n_div == 0)
                return isl_change_none;

        hull_i = isl_basic_map_copy(info[i].bmap);
        hull_i = isl_basic_map_plain_affine_hull(hull_i);
        hull_j = isl_basic_map_copy(info[j].bmap);
        hull_j = isl_basic_map_plain_affine_hull(hull_j);

        hull_j = isl_basic_map_intersect(hull_j, isl_basic_map_copy(hull_i));
        equal = isl_basic_map_plain_is_equal(hull_i, hull_j);
        empty = isl_basic_map_plain_is_empty(hull_j);
        isl_basic_map_free(hull_i);

        if (equal < 0 || equal || empty < 0 || empty) {
                isl_basic_map_free(hull_j);
                if (equal < 0 || empty < 0)
                        return isl_change_error;
                return isl_change_none;
        }

        bmap_i = isl_basic_map_copy(info[i].bmap);
        bmap_i = isl_basic_map_intersect(bmap_i, hull_j);
        if (!bmap_i)
                return isl_change_error;

        if (bmap_i->n_div > info[j].bmap->n_div) {
                isl_basic_map_free(bmap_i);
                return isl_change_none;
        }

        change = coalesce_after_aligning_divs(bmap_i, -1, j, info);

        isl_basic_map_free(bmap_i);

        return change;
}

/* Check if the union of and the basic maps represented by info[i] and info[j]
 * can be represented by a single basic map, by aligning or equating
 * their integer divisions.
 * If so, replace the pair by the single basic map and return
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 *
 * Note that we only perform any test if the number of divs is different
 * in the two basic maps.  In case the number of divs is the same,
 * we have already established that the divs are different
 * in the two basic maps.
 * In particular, if the number of divs of basic map i is smaller than
 * the number of divs of basic map j, then we check if j is a subset of i
 * and vice versa.
 */
static enum isl_change coalesce_divs(int i, int j,
        struct isl_coalesce_info *info)
{
        enum isl_change change = isl_change_none;

        if (info[i].bmap->n_div < info[j].bmap->n_div)
                change = coalesce_after_aligning_divs(info[i].bmap, i, j, info);
        if (change != isl_change_none)
                return change;

        if (info[j].bmap->n_div < info[i].bmap->n_div)
                change = coalesce_after_aligning_divs(info[j].bmap, j, i, info);
        if (change != isl_change_none)
                return invert_change(change);

        change = coalesce_subset_with_equalities(i, j, info);
        if (change != isl_change_none)
                return change;

        change = coalesce_subset_with_equalities(j, i, info);
        if (change != isl_change_none)
                return invert_change(change);

        return isl_change_none;
}

/* Does "bmap" involve any divs that themselves refer to divs?
 */
static int has_nested_div(__isl_keep isl_basic_map *bmap)
{
        int i;
        unsigned total;
        unsigned n_div;

        total = isl_basic_map_dim(bmap, isl_dim_all);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        total -= n_div;

        for (i = 0; i < n_div; ++i)
                if (isl_seq_first_non_zero(bmap->div[i] + 2 + total,
                                            n_div) != -1)
                        return 1;

        return 0;
}

/* Return a list of affine expressions, one for each integer division
 * in "bmap_i".  For each integer division that also appears in "bmap_j",
 * the affine expression is set to NaN.  The number of NaNs in the list
 * is equal to the number of integer divisions in "bmap_j".
 * For the other integer divisions of "bmap_i", the corresponding
 * element in the list is a purely affine expression equal to the integer
 * division in "hull".
 * If no such list can be constructed, then the number of elements
 * in the returned list is smaller than the number of integer divisions
 * in "bmap_i".
 */
static __isl_give isl_aff_list *set_up_substitutions(
        __isl_keep isl_basic_map *bmap_i, __isl_keep isl_basic_map *bmap_j,
        __isl_take isl_basic_map *hull)
{
        unsigned n_div_i, n_div_j, total;
        isl_ctx *ctx;
        isl_local_space *ls;
        isl_basic_set *wrap_hull;
        isl_aff *aff_nan;
        isl_aff_list *list;
        int i, j;

        if (!hull)
                return NULL;

        ctx = isl_basic_map_get_ctx(hull);

        n_div_i = isl_basic_map_dim(bmap_i, isl_dim_div);
        n_div_j = isl_basic_map_dim(bmap_j, isl_dim_div);
        total = isl_basic_map_total_dim(bmap_i) - n_div_i;

        ls = isl_basic_map_get_local_space(bmap_i);
        ls = isl_local_space_wrap(ls);
        wrap_hull = isl_basic_map_wrap(hull);

        aff_nan = isl_aff_nan_on_domain(isl_local_space_copy(ls));
        list = isl_aff_list_alloc(ctx, n_div_i);

        j = 0;
        for (i = 0; i < n_div_i; ++i) {
                isl_aff *aff;

                if (j < n_div_j &&
                    isl_seq_eq(bmap_i->div[i], bmap_j->div[j], 2 + total)) {
                        ++j;
                        list = isl_aff_list_add(list, isl_aff_copy(aff_nan));
                        continue;
                }
                if (n_div_i - i <= n_div_j - j)
                        break;

                aff = isl_local_space_get_div(ls, i);
                aff = isl_aff_substitute_equalities(aff,
                                                isl_basic_set_copy(wrap_hull));
                aff = isl_aff_floor(aff);
                if (!aff)
                        goto error;
                if (isl_aff_dim(aff, isl_dim_div) != 0) {
                        isl_aff_free(aff);
                        break;
                }

                list = isl_aff_list_add(list, aff);
        }

        isl_aff_free(aff_nan);
        isl_local_space_free(ls);
        isl_basic_set_free(wrap_hull);

        return list;
error:
        isl_aff_free(aff_nan);
        isl_local_space_free(ls);
        isl_basic_set_free(wrap_hull);
        isl_aff_list_free(list);
        return NULL;
}

/* Add variables to info->bmap and info->tab corresponding to the elements
 * in "list" that are not set to NaN.
 * "extra_var" is the number of these elements.
 * "dim" is the offset in the variables of "tab" where we should
 * start considering the elements in "list".
 * When this function returns, the total number of variables in "tab"
 * is equal to "dim" plus the number of elements in "list".
 *
 * The newly added existentially quantified variables are not given
 * an explicit representation because the corresponding div constraints
 * do not appear in info->bmap.  These constraints are not added
 * to info->bmap because for internal consistency, they would need to
 * be added to info->tab as well, where they could combine with the equality
 * that is added later to result in constraints that do not hold
 * in the original input.
 */
static int add_sub_vars(struct isl_coalesce_info *info,
        __isl_keep isl_aff_list *list, int dim, int extra_var)
{
        int i, j, n, d;
        isl_space *space;

        space = isl_basic_map_get_space(info->bmap);
        info->bmap = isl_basic_map_cow(info->bmap);
        info->bmap = isl_basic_map_extend_space(info->bmap, space,
                                                extra_var, 0, 0);
        if (!info->bmap)
                return -1;
        n = isl_aff_list_n_aff(list);
        for (i = 0; i < n; ++i) {
                int is_nan;
                isl_aff *aff;

                aff = isl_aff_list_get_aff(list, i);
                is_nan = isl_aff_is_nan(aff);
                isl_aff_free(aff);
                if (is_nan < 0)
                        return -1;
                if (is_nan)
                        continue;

                if (isl_tab_insert_var(info->tab, dim + i) < 0)
                        return -1;
                d = isl_basic_map_alloc_div(info->bmap);
                if (d < 0)
                        return -1;
                info->bmap = isl_basic_map_mark_div_unknown(info->bmap, d);
                if (!info->bmap)
                        return -1;
                for (j = d; j > i; --j)
                        isl_basic_map_swap_div(info->bmap, j - 1, j);
        }

        return 0;
}

/* For each element in "list" that is not set to NaN, fix the corresponding
 * variable in "tab" to the purely affine expression defined by the element.
 * "dim" is the offset in the variables of "tab" where we should
 * start considering the elements in "list".
 *
 * This function assumes that a sufficient number of rows and
 * elements in the constraint array are available in the tableau.
 */
static int add_sub_equalities(struct isl_tab *tab,
        __isl_keep isl_aff_list *list, int dim)
{
        int i, n;
        isl_ctx *ctx;
        isl_vec *sub;
        isl_aff *aff;

        n = isl_aff_list_n_aff(list);

        ctx = isl_tab_get_ctx(tab);
        sub = isl_vec_alloc(ctx, 1 + dim + n);
        if (!sub)
                return -1;
        isl_seq_clr(sub->el + 1 + dim, n);

        for (i = 0; i < n; ++i) {
                aff = isl_aff_list_get_aff(list, i);
                if (!aff)
                        goto error;
                if (isl_aff_is_nan(aff)) {
                        isl_aff_free(aff);
                        continue;
                }
                isl_seq_cpy(sub->el, aff->v->el + 1, 1 + dim);
                isl_int_neg(sub->el[1 + dim + i], aff->v->el[0]);
                if (isl_tab_add_eq(tab, sub->el) < 0)
                        goto error;
                isl_int_set_si(sub->el[1 + dim + i], 0);
                isl_aff_free(aff);
        }

        isl_vec_free(sub);
        return 0;
error:
        isl_aff_free(aff);
        isl_vec_free(sub);
        return -1;
}

/* Add variables to info->tab and info->bmap corresponding to the elements
 * in "list" that are not set to NaN.  The value of the added variable
 * in info->tab is fixed to the purely affine expression defined by the element.
 * "dim" is the offset in the variables of info->tab where we should
 * start considering the elements in "list".
 * When this function returns, the total number of variables in info->tab
 * is equal to "dim" plus the number of elements in "list".
 */
static int add_subs(struct isl_coalesce_info *info,
        __isl_keep isl_aff_list *list, int dim)
{
        int extra_var;
        int n;

        if (!list)
                return -1;

        n = isl_aff_list_n_aff(list);
        extra_var = n - (info->tab->n_var - dim);

        if (isl_tab_extend_vars(info->tab, extra_var) < 0)
                return -1;
        if (isl_tab_extend_cons(info->tab, 2 * extra_var) < 0)
                return -1;
        if (add_sub_vars(info, list, dim, extra_var) < 0)
                return -1;

        return add_sub_equalities(info->tab, list, dim);
}

/* Coalesce basic map "j" into basic map "i" after adding the extra integer
 * divisions in "i" but not in "j" to basic map "j", with values
 * specified by "list".  The total number of elements in "list"
 * is equal to the number of integer divisions in "i", while the number
 * of NaN elements in the list is equal to the number of integer divisions
 * in "j".
 *
 * If no coalescing can be performed, then we need to revert basic map "j"
 * to its original state.  We do the same if basic map "i" gets dropped
 * during the coalescing, even though this should not happen in practice
 * since we have already checked for "j" being a subset of "i"
 * before we reach this stage.
 */
static enum isl_change coalesce_with_subs(int i, int j,
        struct isl_coalesce_info *info, __isl_keep isl_aff_list *list)
{
        isl_basic_map *bmap_j;
        struct isl_tab_undo *snap;
        unsigned dim;
        enum isl_change change;

        bmap_j = isl_basic_map_copy(info[j].bmap);
        snap = isl_tab_snap(info[j].tab);

        dim = isl_basic_map_dim(bmap_j, isl_dim_all);
        dim -= isl_basic_map_dim(bmap_j, isl_dim_div);
        if (add_subs(&info[j], list, dim) < 0)
                goto error;

        change = coalesce_local_pair(i, j, info);
        if (change != isl_change_none && change != isl_change_drop_first) {
                isl_basic_map_free(bmap_j);
        } else {
                isl_basic_map_free(info[j].bmap);
                info[j].bmap = bmap_j;

                if (isl_tab_rollback(info[j].tab, snap) < 0)
                        return isl_change_error;
        }

        return change;
error:
        isl_basic_map_free(bmap_j);
        return isl_change_error;
}

/* Check if we can coalesce basic map "j" into basic map "i" after copying
 * those extra integer divisions in "i" that can be simplified away
 * using the extra equalities in "j".
 * All divs are assumed to be known and not contain any nested divs.
 *
 * We first check if there are any extra equalities in "j" that we
 * can exploit.  Then we check if every integer division in "i"
 * either already appears in "j" or can be simplified using the
 * extra equalities to a purely affine expression.
 * If these tests succeed, then we try to coalesce the two basic maps
 * by introducing extra dimensions in "j" corresponding to
 * the extra integer divsisions "i" fixed to the corresponding
 * purely affine expression.
 */
static enum isl_change check_coalesce_into_eq(int i, int j,
        struct isl_coalesce_info *info)
{
        unsigned n_div_i, n_div_j;
        isl_basic_map *hull_i, *hull_j;
        int equal, empty;
        isl_aff_list *list;
        enum isl_change change;

        n_div_i = isl_basic_map_dim(info[i].bmap, isl_dim_div);
        n_div_j = isl_basic_map_dim(info[j].bmap, isl_dim_div);
        if (n_div_i <= n_div_j)
                return isl_change_none;
        if (info[j].bmap->n_eq == 0)
                return isl_change_none;

        hull_i = isl_basic_map_copy(info[i].bmap);
        hull_i = isl_basic_map_plain_affine_hull(hull_i);
        hull_j = isl_basic_map_copy(info[j].bmap);
        hull_j = isl_basic_map_plain_affine_hull(hull_j);

        hull_j = isl_basic_map_intersect(hull_j, isl_basic_map_copy(hull_i));
        equal = isl_basic_map_plain_is_equal(hull_i, hull_j);
        empty = isl_basic_map_plain_is_empty(hull_j);
        isl_basic_map_free(hull_i);

        if (equal < 0 || empty < 0)
                goto error;
        if (equal || empty) {
                isl_basic_map_free(hull_j);
                return isl_change_none;
        }

        list = set_up_substitutions(info[i].bmap, info[j].bmap, hull_j);
        if (!list)
                return isl_change_error;
        if (isl_aff_list_n_aff(list) < n_div_i)
                change = isl_change_none;
        else
                change = coalesce_with_subs(i, j, info, list);

        isl_aff_list_free(list);

        return change;
error:
        isl_basic_map_free(hull_j);
        return isl_change_error;
}

/* Check if we can coalesce basic maps "i" and "j" after copying
 * those extra integer divisions in one of the basic maps that can
 * be simplified away using the extra equalities in the other basic map.
 * We require all divs to be known in both basic maps.
 * Furthermore, to simplify the comparison of div expressions,
 * we do not allow any nested integer divisions.
 */
static enum isl_change check_coalesce_eq(int i, int j,
        struct isl_coalesce_info *info)
{
        int known, nested;
        enum isl_change change;

        known = isl_basic_map_divs_known(info[i].bmap);
        if (known < 0 || !known)
                return known < 0 ? isl_change_error : isl_change_none;
        known = isl_basic_map_divs_known(info[j].bmap);
        if (known < 0 || !known)
                return known < 0 ? isl_change_error : isl_change_none;
        nested = has_nested_div(info[i].bmap);
        if (nested < 0 || nested)
                return nested < 0 ? isl_change_error : isl_change_none;
        nested = has_nested_div(info[j].bmap);
        if (nested < 0 || nested)
                return nested < 0 ? isl_change_error : isl_change_none;

        change = check_coalesce_into_eq(i, j, info);
        if (change != isl_change_none)
                return change;
        change = check_coalesce_into_eq(j, i, info);
        if (change != isl_change_none)
                return invert_change(change);

        return isl_change_none;
}

/* 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
 * isl_change_drop_first, isl_change_drop_second or isl_change_fuse.
 * Otherwise, return isl_change_none.
 *
 * We first check if the two basic maps live in the same local space,
 * after aligning the divs that differ by only an integer constant.
 * If so, we do the complete check.  Otherwise, we check if they have
 * the same number of integer divisions and can be coalesced, if one is
 * an obvious subset of the other or if the extra integer divisions
 * of one basic map can be simplified away using the extra equalities
 * of the other basic map.
 */
static enum isl_change coalesce_pair(int i, int j,
        struct isl_coalesce_info *info)
{
        int same;
        enum isl_change change;

        if (harmonize_divs(&info[i], &info[j]) < 0)
                return isl_change_error;
        same = same_divs(info[i].bmap, info[j].bmap);
        if (same < 0)
                return isl_change_error;
        if (same)
                return coalesce_local_pair(i, j, info);

        if (info[i].bmap->n_div == info[j].bmap->n_div) {
                change = coalesce_local_pair(i, j, info);
                if (change != isl_change_none)
                        return change;
        }

        change = coalesce_divs(i, j, info);
        if (change != isl_change_none)
                return change;

        return check_coalesce_eq(i, j, info);
}

/* Return the maximum of "a" and "b".
 */
static int isl_max(int a, int b)
{
        return a > b ? a : b;
}

/* Pairwise coalesce the basic maps in the range [start1, end1[ of "info"
 * with those in the range [start2, end2[, skipping basic maps
 * that have been removed (either before or within this function).
 *
 * For each basic map i in the first range, we check if it can be coalesced
 * with respect to any previously considered basic map j in the second range.
 * If i gets dropped (because it was a subset of some j), then
 * we can move on to the next basic map.
 * If j gets dropped, we need to continue checking against the other
 * previously considered basic maps.
 * If the two basic maps got fused, then we recheck the fused basic map
 * against the previously considered basic maps, starting at i + 1
 * (even if start2 is greater than i + 1).
 */
static int coalesce_range(isl_ctx *ctx, struct isl_coalesce_info *info,
        int start1, int end1, int start2, int end2)
{
        int i, j;

        for (i = end1 - 1; i >= start1; --i) {
                if (info[i].removed)
                        continue;
                for (j = isl_max(i + 1, start2); j < end2; ++j) {
                        enum isl_change changed;

                        if (info[j].removed)
                                continue;
                        if (info[i].removed)
                                isl_die(ctx, isl_error_internal,
                                        "basic map unexpectedly removed",
                                        return -1);
                        changed = coalesce_pair(i, j, info);
                        switch (changed) {
                        case isl_change_error:
                                return -1;
                        case isl_change_none:
                        case isl_change_drop_second:
                                continue;
                        case isl_change_drop_first:
                                j = end2;
                                break;
                        case isl_change_fuse:
                                j = i;
                                break;
                        }
                }
        }

        return 0;
}

/* Pairwise coalesce the basic maps described by the "n" elements of "info".
 *
 * We consider groups of basic maps that live in the same apparent
 * affine hull and we first coalesce within such a group before we
 * coalesce the elements in the group with elements of previously
 * considered groups.  If a fuse happens during the second phase,
 * then we also reconsider the elements within the group.
 */
static int coalesce(isl_ctx *ctx, int n, struct isl_coalesce_info *info)
{
        int start, end;

        for (end = n; end > 0; end = start) {
                start = end - 1;
                while (start >= 1 &&
                    info[start - 1].hull_hash == info[start].hull_hash)
                        start--;
                if (coalesce_range(ctx, info, start, end, start, end) < 0)
                        return -1;
                if (coalesce_range(ctx, info, start, end, end, n) < 0)
                        return -1;
        }

        return 0;
}

/* Update the basic maps in "map" based on the information in "info".
 * In particular, remove the basic maps that have been marked removed and
 * update the others based on the information in the corresponding tableau.
 * Since we detected implicit equalities without calling
 * isl_basic_map_gauss, we need to do it now.
 * Also call isl_basic_map_simplify if we may have lost the definition
 * of one or more integer divisions.
 */
static __isl_give isl_map *update_basic_maps(__isl_take isl_map *map,
        int n, struct isl_coalesce_info *info)
{
        int i;

        if (!map)
                return NULL;

        for (i = n - 1; i >= 0; --i) {
                if (info[i].removed) {
                        isl_basic_map_free(map->p[i]);
                        if (i != map->n - 1)
                                map->p[i] = map->p[map->n - 1];
                        map->n--;
                        continue;
                }

                info[i].bmap = isl_basic_map_update_from_tab(info[i].bmap,
                                                        info[i].tab);
                info[i].bmap = isl_basic_map_gauss(info[i].bmap, NULL);
                if (info[i].simplify)
                        info[i].bmap = isl_basic_map_simplify(info[i].bmap);
                info[i].bmap = isl_basic_map_finalize(info[i].bmap);
                if (!info[i].bmap)
                        return isl_map_free(map);
                ISL_F_SET(info[i].bmap, ISL_BASIC_MAP_NO_IMPLICIT);
                ISL_F_SET(info[i].bmap, ISL_BASIC_MAP_NO_REDUNDANT);
                isl_basic_map_free(map->p[i]);
                map->p[i] = info[i].bmap;
                info[i].bmap = NULL;
        }

        return map;
}

/* 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.
 *
 * We factor out any (hidden) common factor from the constraint
 * coefficients to improve the detection of adjacent constraints.
 *
 * 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 (in update_basic_maps) to ensure that
 * the basic maps are not left in an unexpected state.
 * For each basic map, we also compute the hash of the apparent affine hull
 * for use in coalesce.
 */
struct isl_map *isl_map_coalesce(struct isl_map *map)
{
        int i;
        unsigned n;
        isl_ctx *ctx;
        struct isl_coalesce_info *info = NULL;

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

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

        ctx = isl_map_get_ctx(map);
        map = isl_map_sort_divs(map);
        map = isl_map_cow(map);

        if (!map)
                return NULL;

        n = map->n;

        info = isl_calloc_array(map->ctx, struct isl_coalesce_info, n);
        if (!info)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_reduce_coefficients(map->p[i]);
                if (!map->p[i])
                        goto error;
                info[i].bmap = isl_basic_map_copy(map->p[i]);
                info[i].tab = isl_tab_from_basic_map(info[i].bmap, 0);
                if (!info[i].tab)
                        goto error;
                if (!ISL_F_ISSET(info[i].bmap, ISL_BASIC_MAP_NO_IMPLICIT))
                        if (isl_tab_detect_implicit_equalities(info[i].tab) < 0)
                                goto error;
                info[i].bmap = isl_tab_make_equalities_explicit(info[i].tab,
                                                                info[i].bmap);
                if (!info[i].bmap)
                        goto error;
                if (!ISL_F_ISSET(info[i].bmap, ISL_BASIC_MAP_NO_REDUNDANT))
                        if (isl_tab_detect_redundant(info[i].tab) < 0)
                                goto error;
                if (coalesce_info_set_hull_hash(&info[i]) < 0)
                        goto error;
        }
        for (i = map->n - 1; i >= 0; --i)
                if (info[i].tab->empty)
                        drop(&info[i]);

        if (coalesce(ctx, n, info) < 0)
                goto error;

        map = update_basic_maps(map, n, info);

        clear_coalesce_info(n, info);

        return map;
error:
        clear_coalesce_info(n, info);
        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 set_from_map(isl_map_coalesce(set_to_map(set)));
}