isl_union_{map,set}_contains: rename "dim" argument to "space"

[isl.git] / isl_map_simplify.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2012-2013 Ecole Normale Superieure
 * Copyright 2014-2015 INRIA Rocquencourt
 * Copyright 2016      Sven Verdoolaege
 *
 * 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 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
 */

#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include "isl_equalities.h"
#include <isl/map.h>
#include <isl_seq.h>
#include "isl_tab.h"
#include <isl_space_private.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>

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

static void swap_equality(struct isl_basic_map *bmap, int a, int b)
{
        isl_int *t = bmap->eq[a];
        bmap->eq[a] = bmap->eq[b];
        bmap->eq[b] = t;
}

static void swap_inequality(struct isl_basic_map *bmap, int a, int b)
{
        if (a != b) {
                isl_int *t = bmap->ineq[a];
                bmap->ineq[a] = bmap->ineq[b];
                bmap->ineq[b] = t;
        }
}

static void constraint_drop_vars(isl_int *c, unsigned n, unsigned rem)
{
        isl_seq_cpy(c, c + n, rem);
        isl_seq_clr(c + rem, n);
}

/* Drop n dimensions starting at first.
 *
 * In principle, this frees up some extra variables as the number
 * of columns remains constant, but we would have to extend
 * the div array too as the number of rows in this array is assumed
 * to be equal to extra.
 */
struct isl_basic_set *isl_basic_set_drop_dims(
                struct isl_basic_set *bset, unsigned first, unsigned n)
{
        int i;

        if (!bset)
                goto error;

        isl_assert(bset->ctx, first + n <= bset->dim->n_out, goto error);

        if (n == 0 && !isl_space_get_tuple_name(bset->dim, isl_dim_set))
                return bset;

        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;

        for (i = 0; i < bset->n_eq; ++i)
                constraint_drop_vars(bset->eq[i]+1+bset->dim->nparam+first, n,
                                     (bset->dim->n_out-first-n)+bset->extra);

        for (i = 0; i < bset->n_ineq; ++i)
                constraint_drop_vars(bset->ineq[i]+1+bset->dim->nparam+first, n,
                                     (bset->dim->n_out-first-n)+bset->extra);

        for (i = 0; i < bset->n_div; ++i)
                constraint_drop_vars(bset->div[i]+1+1+bset->dim->nparam+first, n,
                                     (bset->dim->n_out-first-n)+bset->extra);

        bset->dim = isl_space_drop_outputs(bset->dim, first, n);
        if (!bset->dim)
                goto error;

        ISL_F_CLR(bset, ISL_BASIC_SET_NORMALIZED);
        bset = isl_basic_set_simplify(bset);
        return isl_basic_set_finalize(bset);
error:
        isl_basic_set_free(bset);
        return NULL;
}

struct isl_set *isl_set_drop_dims(
                struct isl_set *set, unsigned first, unsigned n)
{
        int i;

        if (!set)
                goto error;

        isl_assert(set->ctx, first + n <= set->dim->n_out, goto error);

        if (n == 0 && !isl_space_get_tuple_name(set->dim, isl_dim_set))
                return set;
        set = isl_set_cow(set);
        if (!set)
                goto error;
        set->dim = isl_space_drop_outputs(set->dim, first, n);
        if (!set->dim)
                goto error;

        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_drop_dims(set->p[i], first, n);
                if (!set->p[i])
                        goto error;
        }

        ISL_F_CLR(set, ISL_SET_NORMALIZED);
        return set;
error:
        isl_set_free(set);
        return NULL;
}

/* Move "n" divs starting at "first" to the end of the list of divs.
 */
static struct isl_basic_map *move_divs_last(struct isl_basic_map *bmap,
        unsigned first, unsigned n)
{
        isl_int **div;
        int i;

        if (first + n == bmap->n_div)
                return bmap;

        div = isl_alloc_array(bmap->ctx, isl_int *, n);
        if (!div)
                goto error;
        for (i = 0; i < n; ++i)
                div[i] = bmap->div[first + i];
        for (i = 0; i < bmap->n_div - first - n; ++i)
                bmap->div[first + i] = bmap->div[first + n + i];
        for (i = 0; i < n; ++i)
                bmap->div[bmap->n_div - n + i] = div[i];
        free(div);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Drop "n" dimensions of type "type" starting at "first".
 *
 * In principle, this frees up some extra variables as the number
 * of columns remains constant, but we would have to extend
 * the div array too as the number of rows in this array is assumed
 * to be equal to extra.
 */
struct isl_basic_map *isl_basic_map_drop(struct isl_basic_map *bmap,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;
        unsigned dim;
        unsigned offset;
        unsigned left;

        if (!bmap)
                goto error;

        dim = isl_basic_map_dim(bmap, type);
        isl_assert(bmap->ctx, first + n <= dim, goto error);

        if (n == 0 && !isl_space_is_named_or_nested(bmap->dim, type))
                return bmap;

        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;

        offset = isl_basic_map_offset(bmap, type) + first;
        left = isl_basic_map_total_dim(bmap) - (offset - 1) - n;
        for (i = 0; i < bmap->n_eq; ++i)
                constraint_drop_vars(bmap->eq[i]+offset, n, left);

        for (i = 0; i < bmap->n_ineq; ++i)
                constraint_drop_vars(bmap->ineq[i]+offset, n, left);

        for (i = 0; i < bmap->n_div; ++i)
                constraint_drop_vars(bmap->div[i]+1+offset, n, left);

        if (type == isl_dim_div) {
                bmap = move_divs_last(bmap, first, n);
                if (!bmap)
                        goto error;
                if (isl_basic_map_free_div(bmap, n) < 0)
                        return isl_basic_map_free(bmap);
        } else
                bmap->dim = isl_space_drop_dims(bmap->dim, type, first, n);
        if (!bmap->dim)
                goto error;

        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_drop(__isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return bset_from_bmap(isl_basic_map_drop(bset_to_bmap(bset),
                                                        type, first, n));
}

struct isl_basic_map *isl_basic_map_drop_inputs(
                struct isl_basic_map *bmap, unsigned first, unsigned n)
{
        return isl_basic_map_drop(bmap, isl_dim_in, first, n);
}

struct isl_map *isl_map_drop(struct isl_map *map,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

        if (!map)
                goto error;

        isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);

        if (n == 0 && !isl_space_get_tuple_name(map->dim, type))
                return map;
        map = isl_map_cow(map);
        if (!map)
                goto error;
        map->dim = isl_space_drop_dims(map->dim, type, first, n);
        if (!map->dim)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_drop(map->p[i], type, first, n);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);

        return map;
error:
        isl_map_free(map);
        return NULL;
}

struct isl_set *isl_set_drop(struct isl_set *set,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return set_from_map(isl_map_drop(set_to_map(set), type, first, n));
}

struct isl_map *isl_map_drop_inputs(
                struct isl_map *map, unsigned first, unsigned n)
{
        return isl_map_drop(map, isl_dim_in, first, n);
}

/*
 * We don't cow, as the div is assumed to be redundant.
 */
__isl_give isl_basic_map *isl_basic_map_drop_div(
        __isl_take isl_basic_map *bmap, unsigned div)
{
        int i;
        unsigned pos;

        if (!bmap)
                goto error;

        pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;

        isl_assert(bmap->ctx, div < bmap->n_div, goto error);

        for (i = 0; i < bmap->n_eq; ++i)
                constraint_drop_vars(bmap->eq[i]+pos, 1, bmap->extra-div-1);

        for (i = 0; i < bmap->n_ineq; ++i) {
                if (!isl_int_is_zero(bmap->ineq[i][pos])) {
                        isl_basic_map_drop_inequality(bmap, i);
                        --i;
                        continue;
                }
                constraint_drop_vars(bmap->ineq[i]+pos, 1, bmap->extra-div-1);
        }

        for (i = 0; i < bmap->n_div; ++i)
                constraint_drop_vars(bmap->div[i]+1+pos, 1, bmap->extra-div-1);

        if (div != bmap->n_div - 1) {
                int j;
                isl_int *t = bmap->div[div];

                for (j = div; j < bmap->n_div - 1; ++j)
                        bmap->div[j] = bmap->div[j+1];

                bmap->div[bmap->n_div - 1] = t;
        }
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        if (isl_basic_map_free_div(bmap, 1) < 0)
                return isl_basic_map_free(bmap);

        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_map *isl_basic_map_normalize_constraints(
        struct isl_basic_map *bmap)
{
        int i;
        isl_int gcd;
        unsigned total = isl_basic_map_total_dim(bmap);

        if (!bmap)
                return NULL;

        isl_int_init(gcd);
        for (i = bmap->n_eq - 1; i >= 0; --i) {
                isl_seq_gcd(bmap->eq[i]+1, total, &gcd);
                if (isl_int_is_zero(gcd)) {
                        if (!isl_int_is_zero(bmap->eq[i][0])) {
                                bmap = isl_basic_map_set_to_empty(bmap);
                                break;
                        }
                        isl_basic_map_drop_equality(bmap, i);
                        continue;
                }
                if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
                        isl_int_gcd(gcd, gcd, bmap->eq[i][0]);
                if (isl_int_is_one(gcd))
                        continue;
                if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) {
                        bmap = isl_basic_map_set_to_empty(bmap);
                        break;
                }
                isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total);
        }

        for (i = bmap->n_ineq - 1; i >= 0; --i) {
                isl_seq_gcd(bmap->ineq[i]+1, total, &gcd);
                if (isl_int_is_zero(gcd)) {
                        if (isl_int_is_neg(bmap->ineq[i][0])) {
                                bmap = isl_basic_map_set_to_empty(bmap);
                                break;
                        }
                        isl_basic_map_drop_inequality(bmap, i);
                        continue;
                }
                if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
                        isl_int_gcd(gcd, gcd, bmap->ineq[i][0]);
                if (isl_int_is_one(gcd))
                        continue;
                isl_int_fdiv_q(bmap->ineq[i][0], bmap->ineq[i][0], gcd);
                isl_seq_scale_down(bmap->ineq[i]+1, bmap->ineq[i]+1, gcd, total);
        }
        isl_int_clear(gcd);

        return bmap;
}

struct isl_basic_set *isl_basic_set_normalize_constraints(
        struct isl_basic_set *bset)
{
        isl_basic_map *bmap = bset_to_bmap(bset);
        return bset_from_bmap(isl_basic_map_normalize_constraints(bmap));
}

/* Reduce the coefficient of the variable at position "pos"
 * in integer division "div", such that it lies in the half-open
 * interval (1/2,1/2], extracting any excess value from this integer division.
 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
 * corresponds to the constant term.
 *
 * That is, the integer division is of the form
 *
 *      floor((... + (c * d + r) * x_pos + ...)/d)
 *
 * with -d < 2 * r <= d.
 * Replace it by
 *
 *      floor((... + r * x_pos + ...)/d) + c * x_pos
 *
 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
 * Otherwise, c = floor((c * d + r)/d) + 1.
 *
 * This is the same normalization that is performed by isl_aff_floor.
 */
static __isl_give isl_basic_map *reduce_coefficient_in_div(
        __isl_take isl_basic_map *bmap, int div, int pos)
{
        isl_int shift;
        int add_one;

        isl_int_init(shift);
        isl_int_fdiv_r(shift, bmap->div[div][1 + pos], bmap->div[div][0]);
        isl_int_mul_ui(shift, shift, 2);
        add_one = isl_int_gt(shift, bmap->div[div][0]);
        isl_int_fdiv_q(shift, bmap->div[div][1 + pos], bmap->div[div][0]);
        if (add_one)
                isl_int_add_ui(shift, shift, 1);
        isl_int_neg(shift, shift);
        bmap = isl_basic_map_shift_div(bmap, div, pos, shift);
        isl_int_clear(shift);

        return bmap;
}

/* Does the coefficient of the variable at position "pos"
 * in integer division "div" need to be reduced?
 * That is, does it lie outside the half-open interval (1/2,1/2]?
 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
 * 2 * c != d.
 */
static isl_bool needs_reduction(__isl_keep isl_basic_map *bmap, int div,
        int pos)
{
        isl_bool r;

        if (isl_int_is_zero(bmap->div[div][1 + pos]))
                return isl_bool_false;

        isl_int_mul_ui(bmap->div[div][1 + pos], bmap->div[div][1 + pos], 2);
        r = isl_int_abs_ge(bmap->div[div][1 + pos], bmap->div[div][0]) &&
            !isl_int_eq(bmap->div[div][1 + pos], bmap->div[div][0]);
        isl_int_divexact_ui(bmap->div[div][1 + pos],
                            bmap->div[div][1 + pos], 2);

        return r;
}

/* Reduce the coefficients (including the constant term) of
 * integer division "div", if needed.
 * In particular, make sure all coefficients lie in
 * the half-open interval (1/2,1/2].
 */
static __isl_give isl_basic_map *reduce_div_coefficients_of_div(
        __isl_take isl_basic_map *bmap, int div)
{
        int i;
        unsigned total = 1 + isl_basic_map_total_dim(bmap);

        for (i = 0; i < total; ++i) {
                isl_bool reduce;

                reduce = needs_reduction(bmap, div, i);
                if (reduce < 0)
                        return isl_basic_map_free(bmap);
                if (!reduce)
                        continue;
                bmap = reduce_coefficient_in_div(bmap, div, i);
                if (!bmap)
                        break;
        }

        return bmap;
}

/* Reduce the coefficients (including the constant term) of
 * the known integer divisions, if needed
 * In particular, make sure all coefficients lie in
 * the half-open interval (1/2,1/2].
 */
static __isl_give isl_basic_map *reduce_div_coefficients(
        __isl_take isl_basic_map *bmap)
{
        int i;

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

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                bmap = reduce_div_coefficients_of_div(bmap, i);
                if (!bmap)
                        break;
        }

        return bmap;
}

/* Remove any common factor in numerator and denominator of the div expression,
 * not taking into account the constant term.
 * That is, if the div is of the form
 *
 *      floor((a + m f(x))/(m d))
 *
 * then replace it by
 *
 *      floor((floor(a/m) + f(x))/d)
 *
 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
 * and can therefore not influence the result of the floor.
 */
static void normalize_div_expression(__isl_keep isl_basic_map *bmap, int div)
{
        unsigned total = isl_basic_map_total_dim(bmap);
        isl_ctx *ctx = bmap->ctx;

        if (isl_int_is_zero(bmap->div[div][0]))
                return;
        isl_seq_gcd(bmap->div[div] + 2, total, &ctx->normalize_gcd);
        isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, bmap->div[div][0]);
        if (isl_int_is_one(ctx->normalize_gcd))
                return;
        isl_int_fdiv_q(bmap->div[div][1], bmap->div[div][1],
                        ctx->normalize_gcd);
        isl_int_divexact(bmap->div[div][0], bmap->div[div][0],
                        ctx->normalize_gcd);
        isl_seq_scale_down(bmap->div[div] + 2, bmap->div[div] + 2,
                        ctx->normalize_gcd, total);
}

/* Remove any common factor in numerator and denominator of a div expression,
 * not taking into account the constant term.
 * That is, look for any div of the form
 *
 *      floor((a + m f(x))/(m d))
 *
 * and replace it by
 *
 *      floor((floor(a/m) + f(x))/d)
 *
 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
 * and can therefore not influence the result of the floor.
 */
static __isl_give isl_basic_map *normalize_div_expressions(
        __isl_take isl_basic_map *bmap)
{
        int i;

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

        for (i = 0; i < bmap->n_div; ++i)
                normalize_div_expression(bmap, i);

        return bmap;
}

/* Assumes divs have been ordered if keep_divs is set.
 */
static void eliminate_var_using_equality(struct isl_basic_map *bmap,
        unsigned pos, isl_int *eq, int keep_divs, int *progress)
{
        unsigned total;
        unsigned space_total;
        int k;
        int last_div;

        total = isl_basic_map_total_dim(bmap);
        space_total = isl_space_dim(bmap->dim, isl_dim_all);
        last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
        for (k = 0; k < bmap->n_eq; ++k) {
                if (bmap->eq[k] == eq)
                        continue;
                if (isl_int_is_zero(bmap->eq[k][1+pos]))
                        continue;
                if (progress)
                        *progress = 1;
                isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL);
                isl_seq_normalize(bmap->ctx, bmap->eq[k], 1 + total);
        }

        for (k = 0; k < bmap->n_ineq; ++k) {
                if (isl_int_is_zero(bmap->ineq[k][1+pos]))
                        continue;
                if (progress)
                        *progress = 1;
                isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL);
                isl_seq_normalize(bmap->ctx, bmap->ineq[k], 1 + total);
                ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        }

        for (k = 0; k < bmap->n_div; ++k) {
                if (isl_int_is_zero(bmap->div[k][0]))
                        continue;
                if (isl_int_is_zero(bmap->div[k][1+1+pos]))
                        continue;
                if (progress)
                        *progress = 1;
                /* We need to be careful about circular definitions,
                 * so for now we just remove the definition of div k
                 * if the equality contains any divs.
                 * If keep_divs is set, then the divs have been ordered
                 * and we can keep the definition as long as the result
                 * is still ordered.
                 */
                if (last_div == -1 || (keep_divs && last_div < k)) {
                        isl_seq_elim(bmap->div[k]+1, eq,
                                        1+pos, 1+total, &bmap->div[k][0]);
                        normalize_div_expression(bmap, k);
                } else
                        isl_seq_clr(bmap->div[k], 1 + total);
                ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        }
}

/* Assumes divs have been ordered if keep_divs is set.
 */
static __isl_give isl_basic_map *eliminate_div(__isl_take isl_basic_map *bmap,
        isl_int *eq, unsigned div, int keep_divs)
{
        unsigned pos = isl_space_dim(bmap->dim, isl_dim_all) + div;

        eliminate_var_using_equality(bmap, pos, eq, keep_divs, NULL);

        bmap = isl_basic_map_drop_div(bmap, div);

        return bmap;
}

/* Check if elimination of div "div" using equality "eq" would not
 * result in a div depending on a later div.
 */
static isl_bool ok_to_eliminate_div(struct isl_basic_map *bmap, isl_int *eq,
        unsigned div)
{
        int k;
        int last_div;
        unsigned space_total = isl_space_dim(bmap->dim, isl_dim_all);
        unsigned pos = space_total + div;

        last_div = isl_seq_last_non_zero(eq + 1 + space_total, bmap->n_div);
        if (last_div < 0 || last_div <= div)
                return isl_bool_true;

        for (k = 0; k <= last_div; ++k) {
                if (isl_int_is_zero(bmap->div[k][0]))
                        continue;
                if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos]))
                        return isl_bool_false;
        }

        return isl_bool_true;
}

/* Eliminate divs based on equalities
 */
static struct isl_basic_map *eliminate_divs_eq(
                struct isl_basic_map *bmap, int *progress)
{
        int d;
        int i;
        int modified = 0;
        unsigned off;

        bmap = isl_basic_map_order_divs(bmap);

        if (!bmap)
                return NULL;

        off = 1 + isl_space_dim(bmap->dim, isl_dim_all);

        for (d = bmap->n_div - 1; d >= 0 ; --d) {
                for (i = 0; i < bmap->n_eq; ++i) {
                        isl_bool ok;

                        if (!isl_int_is_one(bmap->eq[i][off + d]) &&
                            !isl_int_is_negone(bmap->eq[i][off + d]))
                                continue;
                        ok = ok_to_eliminate_div(bmap, bmap->eq[i], d);
                        if (ok < 0)
                                return isl_basic_map_free(bmap);
                        if (!ok)
                                continue;
                        modified = 1;
                        *progress = 1;
                        bmap = eliminate_div(bmap, bmap->eq[i], d, 1);
                        if (isl_basic_map_drop_equality(bmap, i) < 0)
                                return isl_basic_map_free(bmap);
                        break;
                }
        }
        if (modified)
                return eliminate_divs_eq(bmap, progress);
        return bmap;
}

/* Elimininate divs based on inequalities
 */
static struct isl_basic_map *eliminate_divs_ineq(
                struct isl_basic_map *bmap, int *progress)
{
        int d;
        int i;
        unsigned off;
        struct isl_ctx *ctx;

        if (!bmap)
                return NULL;

        ctx = bmap->ctx;
        off = 1 + isl_space_dim(bmap->dim, isl_dim_all);

        for (d = bmap->n_div - 1; d >= 0 ; --d) {
                for (i = 0; i < bmap->n_eq; ++i)
                        if (!isl_int_is_zero(bmap->eq[i][off + d]))
                                break;
                if (i < bmap->n_eq)
                        continue;
                for (i = 0; i < bmap->n_ineq; ++i)
                        if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one))
                                break;
                if (i < bmap->n_ineq)
                        continue;
                *progress = 1;
                bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1);
                if (!bmap || ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                        break;
                bmap = isl_basic_map_drop_div(bmap, d);
                if (!bmap)
                        break;
        }
        return bmap;
}

/* Does the equality constraint at position "eq" in "bmap" involve
 * any local variables in the range [first, first + n)
 * that are not marked as having an explicit representation?
 */
static isl_bool bmap_eq_involves_unknown_divs(__isl_keep isl_basic_map *bmap,
        int eq, unsigned first, unsigned n)
{
        unsigned o_div;
        int i;

        if (!bmap)
                return isl_bool_error;

        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        for (i = 0; i < n; ++i) {
                isl_bool unknown;

                if (isl_int_is_zero(bmap->eq[eq][o_div + first + i]))
                        continue;
                unknown = isl_basic_map_div_is_marked_unknown(bmap, first + i);
                if (unknown < 0)
                        return isl_bool_error;
                if (unknown)
                        return isl_bool_true;
        }

        return isl_bool_false;
}

/* The last local variable involved in the equality constraint
 * at position "eq" in "bmap" is the local variable at position "div".
 * It can therefore be used to extract an explicit representation
 * for that variable.
 * Do so unless the local variable already has an explicit representation or
 * the explicit representation would involve any other local variables
 * that in turn do not have an explicit representation.
 * An equality constraint involving local variables without an explicit
 * representation can be used in isl_basic_map_drop_redundant_divs
 * to separate out an independent local variable.  Introducing
 * an explicit representation here would block this transformation,
 * while the partial explicit representation in itself is not very useful.
 * Set *progress if anything is changed.
 *
 * The equality constraint is of the form
 *
 *      f(x) + n e >= 0
 *
 * with n a positive number.  The explicit representation derived from
 * this constraint is
 *
 *      floor((-f(x))/n)
 */
static __isl_give isl_basic_map *set_div_from_eq(__isl_take isl_basic_map *bmap,
        int div, int eq, int *progress)
{
        unsigned total, o_div;
        isl_bool involves;

        if (!bmap)
                return NULL;

        if (!isl_int_is_zero(bmap->div[div][0]))
                return bmap;

        involves = bmap_eq_involves_unknown_divs(bmap, eq, 0, div);
        if (involves < 0)
                return isl_basic_map_free(bmap);
        if (involves)
                return bmap;

        total = isl_basic_map_dim(bmap, isl_dim_all);
        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        isl_seq_neg(bmap->div[div] + 1, bmap->eq[eq], 1 + total);
        isl_int_set_si(bmap->div[div][1 + o_div + div], 0);
        isl_int_set(bmap->div[div][0], bmap->eq[eq][o_div + div]);
        if (progress)
                *progress = 1;
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);

        return bmap;
}

struct isl_basic_map *isl_basic_map_gauss(
        struct isl_basic_map *bmap, int *progress)
{
        int k;
        int done;
        int last_var;
        unsigned total_var;
        unsigned total;

        bmap = isl_basic_map_order_divs(bmap);

        if (!bmap)
                return NULL;

        total = isl_basic_map_total_dim(bmap);
        total_var = total - bmap->n_div;

        last_var = total - 1;
        for (done = 0; done < bmap->n_eq; ++done) {
                for (; last_var >= 0; --last_var) {
                        for (k = done; k < bmap->n_eq; ++k)
                                if (!isl_int_is_zero(bmap->eq[k][1+last_var]))
                                        break;
                        if (k < bmap->n_eq)
                                break;
                }
                if (last_var < 0)
                        break;
                if (k != done)
                        swap_equality(bmap, k, done);
                if (isl_int_is_neg(bmap->eq[done][1+last_var]))
                        isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total);

                eliminate_var_using_equality(bmap, last_var, bmap->eq[done], 1,
                                                progress);

                if (last_var >= total_var)
                        bmap = set_div_from_eq(bmap, last_var - total_var,
                                                done, progress);
                if (!bmap)
                        return NULL;
        }
        if (done == bmap->n_eq)
                return bmap;
        for (k = done; k < bmap->n_eq; ++k) {
                if (isl_int_is_zero(bmap->eq[k][0]))
                        continue;
                return isl_basic_map_set_to_empty(bmap);
        }
        isl_basic_map_free_equality(bmap, bmap->n_eq-done);
        return bmap;
}

struct isl_basic_set *isl_basic_set_gauss(
        struct isl_basic_set *bset, int *progress)
{
        return bset_from_bmap(isl_basic_map_gauss(bset_to_bmap(bset),
                                                        progress));
}


static unsigned int round_up(unsigned int v)
{
        int old_v = v;

        while (v) {
                old_v = v;
                v ^= v & -v;
        }
        return old_v << 1;
}

/* Hash table of inequalities in a basic map.
 * "index" is an array of addresses of inequalities in the basic map, some
 * of which are NULL.  The inequalities are hashed on the coefficients
 * except the constant term.
 * "size" is the number of elements in the array and is always a power of two
 * "bits" is the number of bits need to represent an index into the array.
 * "total" is the total dimension of the basic map.
 */
struct isl_constraint_index {
        unsigned int size;
        int bits;
        isl_int ***index;
        unsigned total;
};

/* Fill in the "ci" data structure for holding the inequalities of "bmap".
 */
static isl_stat create_constraint_index(struct isl_constraint_index *ci,
        __isl_keep isl_basic_map *bmap)
{
        isl_ctx *ctx;

        ci->index = NULL;
        if (!bmap)
                return isl_stat_error;
        ci->total = isl_basic_set_total_dim(bmap);
        if (bmap->n_ineq == 0)
                return isl_stat_ok;
        ci->size = round_up(4 * (bmap->n_ineq + 1) / 3 - 1);
        ci->bits = ffs(ci->size) - 1;
        ctx = isl_basic_map_get_ctx(bmap);
        ci->index = isl_calloc_array(ctx, isl_int **, ci->size);
        if (!ci->index)
                return isl_stat_error;

        return isl_stat_ok;
}

/* Free the memory allocated by create_constraint_index.
 */
static void constraint_index_free(struct isl_constraint_index *ci)
{
        free(ci->index);
}

/* Return the position in ci->index that contains the address of
 * an inequality that is equal to *ineq up to the constant term,
 * provided this address is not identical to "ineq".
 * If there is no such inequality, then return the position where
 * such an inequality should be inserted.
 */
static int hash_index_ineq(struct isl_constraint_index *ci, isl_int **ineq)
{
        int h;
        uint32_t hash = isl_seq_get_hash_bits((*ineq) + 1, ci->total, ci->bits);
        for (h = hash; ci->index[h]; h = (h+1) % ci->size)
                if (ineq != ci->index[h] &&
                    isl_seq_eq((*ineq) + 1, ci->index[h][0]+1, ci->total))
                        break;
        return h;
}

/* Return the position in ci->index that contains the address of
 * an inequality that is equal to the k'th inequality of "bmap"
 * up to the constant term, provided it does not point to the very
 * same inequality.
 * If there is no such inequality, then return the position where
 * such an inequality should be inserted.
 */
static int hash_index(struct isl_constraint_index *ci,
        __isl_keep isl_basic_map *bmap, int k)
{
        return hash_index_ineq(ci, &bmap->ineq[k]);
}

static int set_hash_index(struct isl_constraint_index *ci,
                          struct isl_basic_set *bset, int k)
{
        return hash_index(ci, bset, k);
}

/* Fill in the "ci" data structure with the inequalities of "bset".
 */
static isl_stat setup_constraint_index(struct isl_constraint_index *ci,
        __isl_keep isl_basic_set *bset)
{
        int k, h;

        if (create_constraint_index(ci, bset) < 0)
                return isl_stat_error;

        for (k = 0; k < bset->n_ineq; ++k) {
                h = set_hash_index(ci, bset, k);
                ci->index[h] = &bset->ineq[k];
        }

        return isl_stat_ok;
}

/* Is the inequality ineq (obviously) redundant with respect
 * to the constraints in "ci"?
 *
 * Look for an inequality in "ci" with the same coefficients and then
 * check if the contant term of "ineq" is greater than or equal
 * to the constant term of that inequality.  If so, "ineq" is clearly
 * redundant.
 *
 * Note that hash_index_ineq ignores a stored constraint if it has
 * the same address as the passed inequality.  It is ok to pass
 * the address of a local variable here since it will never be
 * the same as the address of a constraint in "ci".
 */
static isl_bool constraint_index_is_redundant(struct isl_constraint_index *ci,
        isl_int *ineq)
{
        int h;

        h = hash_index_ineq(ci, &ineq);
        if (!ci->index[h])
                return isl_bool_false;
        return isl_int_ge(ineq[0], (*ci->index[h])[0]);
}

/* If we can eliminate more than one div, then we need to make
 * sure we do it from last div to first div, in order not to
 * change the position of the other divs that still need to
 * be removed.
 */
static struct isl_basic_map *remove_duplicate_divs(
        struct isl_basic_map *bmap, int *progress)
{
        unsigned int size;
        int *index;
        int *elim_for;
        int k, l, h;
        int bits;
        struct isl_blk eq;
        unsigned total_var;
        unsigned total;
        struct isl_ctx *ctx;

        bmap = isl_basic_map_order_divs(bmap);
        if (!bmap || bmap->n_div <= 1)
                return bmap;

        total_var = isl_space_dim(bmap->dim, isl_dim_all);
        total = total_var + bmap->n_div;

        ctx = bmap->ctx;
        for (k = bmap->n_div - 1; k >= 0; --k)
                if (!isl_int_is_zero(bmap->div[k][0]))
                        break;
        if (k <= 0)
                return bmap;

        size = round_up(4 * bmap->n_div / 3 - 1);
        if (size == 0)
                return bmap;
        elim_for = isl_calloc_array(ctx, int, bmap->n_div);
        bits = ffs(size) - 1;
        index = isl_calloc_array(ctx, int, size);
        if (!elim_for || !index)
                goto out;
        eq = isl_blk_alloc(ctx, 1+total);
        if (isl_blk_is_error(eq))
                goto out;

        isl_seq_clr(eq.data, 1+total);
        index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1;
        for (--k; k >= 0; --k) {
                uint32_t hash;

                if (isl_int_is_zero(bmap->div[k][0]))
                        continue;

                hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits);
                for (h = hash; index[h]; h = (h+1) % size)
                        if (isl_seq_eq(bmap->div[k],
                                       bmap->div[index[h]-1], 2+total))
                                break;
                if (index[h]) {
                        *progress = 1;
                        l = index[h] - 1;
                        elim_for[l] = k + 1;
                }
                index[h] = k+1;
        }
        for (l = bmap->n_div - 1; l >= 0; --l) {
                if (!elim_for[l])
                        continue;
                k = elim_for[l] - 1;
                isl_int_set_si(eq.data[1+total_var+k], -1);
                isl_int_set_si(eq.data[1+total_var+l], 1);
                bmap = eliminate_div(bmap, eq.data, l, 1);
                if (!bmap)
                        break;
                isl_int_set_si(eq.data[1+total_var+k], 0);
                isl_int_set_si(eq.data[1+total_var+l], 0);
        }

        isl_blk_free(ctx, eq);
out:
        free(index);
        free(elim_for);
        return bmap;
}

static int n_pure_div_eq(struct isl_basic_map *bmap)
{
        int i, j;
        unsigned total;

        total = isl_space_dim(bmap->dim, isl_dim_all);
        for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) {
                while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
                        --j;
                if (j < 0)
                        break;
                if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total, j) != -1)
                        return 0;
        }
        return i;
}

/* Normalize divs that appear in equalities.
 *
 * In particular, we assume that bmap contains some equalities
 * of the form
 *
 *      a x = m * e_i
 *
 * and we want to replace the set of e_i by a minimal set and
 * such that the new e_i have a canonical representation in terms
 * of the vector x.
 * If any of the equalities involves more than one divs, then
 * we currently simply bail out.
 *
 * Let us first additionally assume that all equalities involve
 * a div.  The equalities then express modulo constraints on the
 * remaining variables and we can use "parameter compression"
 * to find a minimal set of constraints.  The result is a transformation
 *
 *      x = T(x') = x_0 + G x'
 *
 * with G a lower-triangular matrix with all elements below the diagonal
 * non-negative and smaller than the diagonal element on the same row.
 * We first normalize x_0 by making the same property hold in the affine
 * T matrix.
 * The rows i of G with a 1 on the diagonal do not impose any modulo
 * constraint and simply express x_i = x'_i.
 * For each of the remaining rows i, we introduce a div and a corresponding
 * equality.  In particular
 *
 *      g_ii e_j = x_i - g_i(x')
 *
 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
 * corresponding div (if g_kk != 1).
 *
 * If there are any equalities not involving any div, then we
 * first apply a variable compression on the variables x:
 *
 *      x = C x''       x'' = C_2 x
 *
 * and perform the above parameter compression on A C instead of on A.
 * The resulting compression is then of the form
 *
 *      x'' = T(x') = x_0 + G x'
 *
 * and in constructing the new divs and the corresponding equalities,
 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
 * by the corresponding row from C_2.
 */
static struct isl_basic_map *normalize_divs(
        struct isl_basic_map *bmap, int *progress)
{
        int i, j, k;
        int total;
        int div_eq;
        struct isl_mat *B;
        struct isl_vec *d;
        struct isl_mat *T = NULL;
        struct isl_mat *C = NULL;
        struct isl_mat *C2 = NULL;
        isl_int v;
        int *pos = NULL;
        int dropped, needed;

        if (!bmap)
                return NULL;

        if (bmap->n_div == 0)
                return bmap;

        if (bmap->n_eq == 0)
                return bmap;

        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS))
                return bmap;

        total = isl_space_dim(bmap->dim, isl_dim_all);
        div_eq = n_pure_div_eq(bmap);
        if (div_eq == 0)
                return bmap;

        if (div_eq < bmap->n_eq) {
                B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, div_eq,
                                        bmap->n_eq - div_eq, 0, 1 + total);
                C = isl_mat_variable_compression(B, &C2);
                if (!C || !C2)
                        goto error;
                if (C->n_col == 0) {
                        bmap = isl_basic_map_set_to_empty(bmap);
                        isl_mat_free(C);
                        isl_mat_free(C2);
                        goto done;
                }
        }

        d = isl_vec_alloc(bmap->ctx, div_eq);
        if (!d)
                goto error;
        for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) {
                while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + total + j]))
                        --j;
                isl_int_set(d->block.data[i], bmap->eq[i][1 + total + j]);
        }
        B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + total);

        if (C) {
                B = isl_mat_product(B, C);
                C = NULL;
        }

        T = isl_mat_parameter_compression(B, d);
        if (!T)
                goto error;
        if (T->n_col == 0) {
                bmap = isl_basic_map_set_to_empty(bmap);
                isl_mat_free(C2);
                isl_mat_free(T);
                goto done;
        }
        isl_int_init(v);
        for (i = 0; i < T->n_row - 1; ++i) {
                isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]);
                if (isl_int_is_zero(v))
                        continue;
                isl_mat_col_submul(T, 0, v, 1 + i);
        }
        isl_int_clear(v);
        pos = isl_alloc_array(bmap->ctx, int, T->n_row);
        if (!pos)
                goto error;
        /* We have to be careful because dropping equalities may reorder them */
        dropped = 0;
        for (j = bmap->n_div - 1; j >= 0; --j) {
                for (i = 0; i < bmap->n_eq; ++i)
                        if (!isl_int_is_zero(bmap->eq[i][1 + total + j]))
                                break;
                if (i < bmap->n_eq) {
                        bmap = isl_basic_map_drop_div(bmap, j);
                        isl_basic_map_drop_equality(bmap, i);
                        ++dropped;
                }
        }
        pos[0] = 0;
        needed = 0;
        for (i = 1; i < T->n_row; ++i) {
                if (isl_int_is_one(T->row[i][i]))
                        pos[i] = i;
                else
                        needed++;
        }
        if (needed > dropped) {
                bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
                                needed, needed, 0);
                if (!bmap)
                        goto error;
        }
        for (i = 1; i < T->n_row; ++i) {
                if (isl_int_is_one(T->row[i][i]))
                        continue;
                k = isl_basic_map_alloc_div(bmap);
                pos[i] = 1 + total + k;
                isl_seq_clr(bmap->div[k] + 1, 1 + total + bmap->n_div);
                isl_int_set(bmap->div[k][0], T->row[i][i]);
                if (C2)
                        isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + total);
                else
                        isl_int_set_si(bmap->div[k][1 + i], 1);
                for (j = 0; j < i; ++j) {
                        if (isl_int_is_zero(T->row[i][j]))
                                continue;
                        if (pos[j] < T->n_row && C2)
                                isl_seq_submul(bmap->div[k] + 1, T->row[i][j],
                                                C2->row[pos[j]], 1 + total);
                        else
                                isl_int_neg(bmap->div[k][1 + pos[j]],
                                                                T->row[i][j]);
                }
                j = isl_basic_map_alloc_equality(bmap);
                isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+total+bmap->n_div);
                isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]);
        }
        free(pos);
        isl_mat_free(C2);
        isl_mat_free(T);

        if (progress)
                *progress = 1;
done:
        ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);

        return bmap;
error:
        free(pos);
        isl_mat_free(C);
        isl_mat_free(C2);
        isl_mat_free(T);
        return bmap;
}

static struct isl_basic_map *set_div_from_lower_bound(
        struct isl_basic_map *bmap, int div, int ineq)
{
        unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);

        isl_seq_neg(bmap->div[div] + 1, bmap->ineq[ineq], total + bmap->n_div);
        isl_int_set(bmap->div[div][0], bmap->ineq[ineq][total + div]);
        isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]);
        isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
        isl_int_set_si(bmap->div[div][1 + total + div], 0);

        return bmap;
}

/* Check whether it is ok to define a div based on an inequality.
 * To avoid the introduction of circular definitions of divs, we
 * do not allow such a definition if the resulting expression would refer to
 * any other undefined divs or if any known div is defined in
 * terms of the unknown div.
 */
static isl_bool ok_to_set_div_from_bound(struct isl_basic_map *bmap,
        int div, int ineq)
{
        int j;
        unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);

        /* Not defined in terms of unknown divs */
        for (j = 0; j < bmap->n_div; ++j) {
                if (div == j)
                        continue;
                if (isl_int_is_zero(bmap->ineq[ineq][total + j]))
                        continue;
                if (isl_int_is_zero(bmap->div[j][0]))
                        return isl_bool_false;
        }

        /* No other div defined in terms of this one => avoid loops */
        for (j = 0; j < bmap->n_div; ++j) {
                if (div == j)
                        continue;
                if (isl_int_is_zero(bmap->div[j][0]))
                        continue;
                if (!isl_int_is_zero(bmap->div[j][1 + total + div]))
                        return isl_bool_false;
        }

        return isl_bool_true;
}

/* Would an expression for div "div" based on inequality "ineq" of "bmap"
 * be a better expression than the current one?
 *
 * If we do not have any expression yet, then any expression would be better.
 * Otherwise we check if the last variable involved in the inequality
 * (disregarding the div that it would define) is in an earlier position
 * than the last variable involved in the current div expression.
 */
static isl_bool better_div_constraint(__isl_keep isl_basic_map *bmap,
        int div, int ineq)
{
        unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);
        int last_div;
        int last_ineq;

        if (isl_int_is_zero(bmap->div[div][0]))
                return isl_bool_true;

        if (isl_seq_last_non_zero(bmap->ineq[ineq] + total + div + 1,
                                  bmap->n_div - (div + 1)) >= 0)
                return isl_bool_false;

        last_ineq = isl_seq_last_non_zero(bmap->ineq[ineq], total + div);
        last_div = isl_seq_last_non_zero(bmap->div[div] + 1,
                                         total + bmap->n_div);

        return last_ineq < last_div;
}

/* Given two constraints "k" and "l" that are opposite to each other,
 * except for the constant term, check if we can use them
 * to obtain an expression for one of the hitherto unknown divs or
 * a "better" expression for a div for which we already have an expression.
 * "sum" is the sum of the constant terms of the constraints.
 * If this sum is strictly smaller than the coefficient of one
 * of the divs, then this pair can be used define the div.
 * To avoid the introduction of circular definitions of divs, we
 * do not use the pair if the resulting expression would refer to
 * any other undefined divs or if any known div is defined in
 * terms of the unknown div.
 */
static struct isl_basic_map *check_for_div_constraints(
        struct isl_basic_map *bmap, int k, int l, isl_int sum, int *progress)
{
        int i;
        unsigned total = 1 + isl_space_dim(bmap->dim, isl_dim_all);

        for (i = 0; i < bmap->n_div; ++i) {
                isl_bool set_div;

                if (isl_int_is_zero(bmap->ineq[k][total + i]))
                        continue;
                if (isl_int_abs_ge(sum, bmap->ineq[k][total + i]))
                        continue;
                set_div = better_div_constraint(bmap, i, k);
                if (set_div >= 0 && set_div)
                        set_div = ok_to_set_div_from_bound(bmap, i, k);
                if (set_div < 0)
                        return isl_basic_map_free(bmap);
                if (!set_div)
                        break;
                if (isl_int_is_pos(bmap->ineq[k][total + i]))
                        bmap = set_div_from_lower_bound(bmap, i, k);
                else
                        bmap = set_div_from_lower_bound(bmap, i, l);
                if (progress)
                        *progress = 1;
                break;
        }
        return bmap;
}

__isl_give isl_basic_map *isl_basic_map_remove_duplicate_constraints(
        __isl_take isl_basic_map *bmap, int *progress, int detect_divs)
{
        struct isl_constraint_index ci;
        int k, l, h;
        unsigned total = isl_basic_map_total_dim(bmap);
        isl_int sum;

        if (!bmap || bmap->n_ineq <= 1)
                return bmap;

        if (create_constraint_index(&ci, bmap) < 0)
                return bmap;

        h = isl_seq_get_hash_bits(bmap->ineq[0] + 1, total, ci.bits);
        ci.index[h] = &bmap->ineq[0];
        for (k = 1; k < bmap->n_ineq; ++k) {
                h = hash_index(&ci, bmap, k);
                if (!ci.index[h]) {
                        ci.index[h] = &bmap->ineq[k];
                        continue;
                }
                if (progress)
                        *progress = 1;
                l = ci.index[h] - &bmap->ineq[0];
                if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0]))
                        swap_inequality(bmap, k, l);
                isl_basic_map_drop_inequality(bmap, k);
                --k;
        }
        isl_int_init(sum);
        for (k = 0; k < bmap->n_ineq-1; ++k) {
                isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
                h = hash_index(&ci, bmap, k);
                isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
                if (!ci.index[h])
                        continue;
                l = ci.index[h] - &bmap->ineq[0];
                isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]);
                if (isl_int_is_pos(sum)) {
                        if (detect_divs)
                                bmap = check_for_div_constraints(bmap, k, l,
                                                                 sum, progress);
                        continue;
                }
                if (isl_int_is_zero(sum)) {
                        /* We need to break out of the loop after these
                         * changes since the contents of the hash
                         * will no longer be valid.
                         * Plus, we probably we want to regauss first.
                         */
                        if (progress)
                                *progress = 1;
                        isl_basic_map_drop_inequality(bmap, l);
                        isl_basic_map_inequality_to_equality(bmap, k);
                } else
                        bmap = isl_basic_map_set_to_empty(bmap);
                break;
        }
        isl_int_clear(sum);

        constraint_index_free(&ci);
        return bmap;
}

/* Detect all pairs of inequalities that form an equality.
 *
 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
 * Call it repeatedly while it is making progress.
 */
__isl_give isl_basic_map *isl_basic_map_detect_inequality_pairs(
        __isl_take isl_basic_map *bmap, int *progress)
{
        int duplicate;

        do {
                duplicate = 0;
                bmap = isl_basic_map_remove_duplicate_constraints(bmap,
                                                                &duplicate, 0);
                if (progress && duplicate)
                        *progress = 1;
        } while (duplicate);

        return bmap;
}

/* Eliminate knowns divs from constraints where they appear with
 * a (positive or negative) unit coefficient.
 *
 * That is, replace
 *
 *      floor(e/m) + f >= 0
 *
 * by
 *
 *      e + m f >= 0
 *
 * and
 *
 *      -floor(e/m) + f >= 0
 *
 * by
 *
 *      -e + m f + m - 1 >= 0
 *
 * The first conversion is valid because floor(e/m) >= -f is equivalent
 * to e/m >= -f because -f is an integral expression.
 * The second conversion follows from the fact that
 *
 *      -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
 *
 *
 * Note that one of the div constraints may have been eliminated
 * due to being redundant with respect to the constraint that is
 * being modified by this function.  The modified constraint may
 * no longer imply this div constraint, so we add it back to make
 * sure we do not lose any information.
 *
 * We skip integral divs, i.e., those with denominator 1, as we would
 * risk eliminating the div from the div constraints.  We do not need
 * to handle those divs here anyway since the div constraints will turn
 * out to form an equality and this equality can then be used to eliminate
 * the div from all constraints.
 */
static __isl_give isl_basic_map *eliminate_unit_divs(
        __isl_take isl_basic_map *bmap, int *progress)
{
        int i, j;
        isl_ctx *ctx;
        unsigned total;

        if (!bmap)
                return NULL;

        ctx = isl_basic_map_get_ctx(bmap);
        total = 1 + isl_space_dim(bmap->dim, isl_dim_all);

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (isl_int_is_one(bmap->div[i][0]))
                        continue;
                for (j = 0; j < bmap->n_ineq; ++j) {
                        int s;

                        if (!isl_int_is_one(bmap->ineq[j][total + i]) &&
                            !isl_int_is_negone(bmap->ineq[j][total + i]))
                                continue;

                        *progress = 1;

                        s = isl_int_sgn(bmap->ineq[j][total + i]);
                        isl_int_set_si(bmap->ineq[j][total + i], 0);
                        if (s < 0)
                                isl_seq_combine(bmap->ineq[j],
                                        ctx->negone, bmap->div[i] + 1,
                                        bmap->div[i][0], bmap->ineq[j],
                                        total + bmap->n_div);
                        else
                                isl_seq_combine(bmap->ineq[j],
                                        ctx->one, bmap->div[i] + 1,
                                        bmap->div[i][0], bmap->ineq[j],
                                        total + bmap->n_div);
                        if (s < 0) {
                                isl_int_add(bmap->ineq[j][0],
                                        bmap->ineq[j][0], bmap->div[i][0]);
                                isl_int_sub_ui(bmap->ineq[j][0],
                                        bmap->ineq[j][0], 1);
                        }

                        bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
                        if (isl_basic_map_add_div_constraint(bmap, i, s) < 0)
                                return isl_basic_map_free(bmap);
                }
        }

        return bmap;
}

struct isl_basic_map *isl_basic_map_simplify(struct isl_basic_map *bmap)
{
        int progress = 1;
        if (!bmap)
                return NULL;
        while (progress) {
                isl_bool empty;

                progress = 0;
                empty = isl_basic_map_plain_is_empty(bmap);
                if (empty < 0)
                        return isl_basic_map_free(bmap);
                if (empty)
                        break;
                bmap = isl_basic_map_normalize_constraints(bmap);
                bmap = reduce_div_coefficients(bmap);
                bmap = normalize_div_expressions(bmap);
                bmap = remove_duplicate_divs(bmap, &progress);
                bmap = eliminate_unit_divs(bmap, &progress);
                bmap = eliminate_divs_eq(bmap, &progress);
                bmap = eliminate_divs_ineq(bmap, &progress);
                bmap = isl_basic_map_gauss(bmap, &progress);
                /* requires equalities in normal form */
                bmap = normalize_divs(bmap, &progress);
                bmap = isl_basic_map_remove_duplicate_constraints(bmap,
                                                                &progress, 1);
                if (bmap && progress)
                        ISL_F_CLR(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS);
        }
        return bmap;
}

struct isl_basic_set *isl_basic_set_simplify(struct isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_simplify(bset_to_bmap(bset)));
}


isl_bool isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
        isl_int *constraint, unsigned div)
{
        unsigned pos;

        if (!bmap)
                return isl_bool_error;

        pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;

        if (isl_int_eq(constraint[pos], bmap->div[div][0])) {
                int neg;
                isl_int_sub(bmap->div[div][1],
                                bmap->div[div][1], bmap->div[div][0]);
                isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
                neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos);
                isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
                isl_int_add(bmap->div[div][1],
                                bmap->div[div][1], bmap->div[div][0]);
                if (!neg)
                        return isl_bool_false;
                if (isl_seq_first_non_zero(constraint+pos+1,
                                            bmap->n_div-div-1) != -1)
                        return isl_bool_false;
        } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
                if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
                        return isl_bool_false;
                if (isl_seq_first_non_zero(constraint+pos+1,
                                            bmap->n_div-div-1) != -1)
                        return isl_bool_false;
        } else
                return isl_bool_false;

        return isl_bool_true;
}

isl_bool isl_basic_set_is_div_constraint(__isl_keep isl_basic_set *bset,
        isl_int *constraint, unsigned div)
{
        return isl_basic_map_is_div_constraint(bset, constraint, div);
}


/* If the only constraints a div d=floor(f/m)
 * appears in are its two defining constraints
 *
 *      f - m d >=0
 *      -(f - (m - 1)) + m d >= 0
 *
 * then it can safely be removed.
 */
static isl_bool div_is_redundant(struct isl_basic_map *bmap, int div)
{
        int i;
        unsigned pos = 1 + isl_space_dim(bmap->dim, isl_dim_all) + div;

        for (i = 0; i < bmap->n_eq; ++i)
                if (!isl_int_is_zero(bmap->eq[i][pos]))
                        return isl_bool_false;

        for (i = 0; i < bmap->n_ineq; ++i) {
                isl_bool red;

                if (isl_int_is_zero(bmap->ineq[i][pos]))
                        continue;
                red = isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div);
                if (red < 0 || !red)
                        return red;
        }

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (!isl_int_is_zero(bmap->div[i][1+pos]))
                        return isl_bool_false;
        }

        return isl_bool_true;
}

/*
 * Remove divs that don't occur in any of the constraints or other divs.
 * These can arise when dropping constraints from a basic map or
 * when the divs of a basic map have been temporarily aligned
 * with the divs of another basic map.
 */
static struct isl_basic_map *remove_redundant_divs(struct isl_basic_map *bmap)
{
        int i;

        if (!bmap)
                return NULL;

        for (i = bmap->n_div-1; i >= 0; --i) {
                isl_bool redundant;

                redundant = div_is_redundant(bmap, i);
                if (redundant < 0)
                        return isl_basic_map_free(bmap);
                if (!redundant)
                        continue;
                bmap = isl_basic_map_drop_div(bmap, i);
        }
        return bmap;
}

/* Mark "bmap" as final, without checking for obviously redundant
 * integer divisions.  This function should be used when "bmap"
 * is known not to involve any such integer divisions.
 */
__isl_give isl_basic_map *isl_basic_map_mark_final(
        __isl_take isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;
        ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
        return bmap;
}

/* Mark "bmap" as final, after removing obviously redundant integer divisions.
 */
struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
{
        bmap = remove_redundant_divs(bmap);
        bmap = isl_basic_map_mark_final(bmap);
        return bmap;
}

struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_finalize(bset_to_bmap(bset)));
}

struct isl_set *isl_set_finalize(struct isl_set *set)
{
        int i;

        if (!set)
                return NULL;
        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_finalize(set->p[i]);
                if (!set->p[i])
                        goto error;
        }
        return set;
error:
        isl_set_free(set);
        return NULL;
}

struct isl_map *isl_map_finalize(struct isl_map *map)
{
        int i;

        if (!map)
                return NULL;
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_finalize(map->p[i]);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}


/* Remove definition of any div that is defined in terms of the given variable.
 * The div itself is not removed.  Functions such as
 * eliminate_divs_ineq depend on the other divs remaining in place.
 */
static struct isl_basic_map *remove_dependent_vars(struct isl_basic_map *bmap,
                                                                        int pos)
{
        int i;

        if (!bmap)
                return NULL;

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (isl_int_is_zero(bmap->div[i][1+1+pos]))
                        continue;
                bmap = isl_basic_map_mark_div_unknown(bmap, i);
                if (!bmap)
                        return NULL;
        }
        return bmap;
}

/* Eliminate the specified variables from the constraints using
 * Fourier-Motzkin.  The variables themselves are not removed.
 */
struct isl_basic_map *isl_basic_map_eliminate_vars(
        struct isl_basic_map *bmap, unsigned pos, unsigned n)
{
        int d;
        int i, j, k;
        unsigned total;
        int need_gauss = 0;

        if (n == 0)
                return bmap;
        if (!bmap)
                return NULL;
        total = isl_basic_map_total_dim(bmap);

        bmap = isl_basic_map_cow(bmap);
        for (d = pos + n - 1; d >= 0 && d >= pos; --d)
                bmap = remove_dependent_vars(bmap, d);
        if (!bmap)
                return NULL;

        for (d = pos + n - 1;
             d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
                isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
        for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
                int n_lower, n_upper;
                if (!bmap)
                        return NULL;
                for (i = 0; i < bmap->n_eq; ++i) {
                        if (isl_int_is_zero(bmap->eq[i][1+d]))
                                continue;
                        eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL);
                        isl_basic_map_drop_equality(bmap, i);
                        need_gauss = 1;
                        break;
                }
                if (i < bmap->n_eq)
                        continue;
                n_lower = 0;
                n_upper = 0;
                for (i = 0; i < bmap->n_ineq; ++i) {
                        if (isl_int_is_pos(bmap->ineq[i][1+d]))
                                n_lower++;
                        else if (isl_int_is_neg(bmap->ineq[i][1+d]))
                                n_upper++;
                }
                bmap = isl_basic_map_extend_constraints(bmap,
                                0, n_lower * n_upper);
                if (!bmap)
                        goto error;
                for (i = bmap->n_ineq - 1; i >= 0; --i) {
                        int last;
                        if (isl_int_is_zero(bmap->ineq[i][1+d]))
                                continue;
                        last = -1;
                        for (j = 0; j < i; ++j) {
                                if (isl_int_is_zero(bmap->ineq[j][1+d]))
                                        continue;
                                last = j;
                                if (isl_int_sgn(bmap->ineq[i][1+d]) ==
                                    isl_int_sgn(bmap->ineq[j][1+d]))
                                        continue;
                                k = isl_basic_map_alloc_inequality(bmap);
                                if (k < 0)
                                        goto error;
                                isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
                                                1+total);
                                isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
                                                1+d, 1+total, NULL);
                        }
                        isl_basic_map_drop_inequality(bmap, i);
                        i = last + 1;
                }
                if (n_lower > 0 && n_upper > 0) {
                        bmap = isl_basic_map_normalize_constraints(bmap);
                        bmap = isl_basic_map_remove_duplicate_constraints(bmap,
                                                                    NULL, 0);
                        bmap = isl_basic_map_gauss(bmap, NULL);
                        bmap = isl_basic_map_remove_redundancies(bmap);
                        need_gauss = 0;
                        if (!bmap)
                                goto error;
                        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                                break;
                }
        }
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        if (need_gauss)
                bmap = isl_basic_map_gauss(bmap, NULL);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_set *isl_basic_set_eliminate_vars(
        struct isl_basic_set *bset, unsigned pos, unsigned n)
{
        return bset_from_bmap(isl_basic_map_eliminate_vars(bset_to_bmap(bset),
                                                                pos, n));
}

/* Eliminate the specified n dimensions starting at first from the
 * constraints, without removing the dimensions from the space.
 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
 * Otherwise, they are projected out and the original space is restored.
 */
__isl_give isl_basic_map *isl_basic_map_eliminate(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        isl_space *space;

        if (!bmap)
                return NULL;
        if (n == 0)
                return bmap;

        if (first + n > isl_basic_map_dim(bmap, type) || first + n < first)
                isl_die(bmap->ctx, isl_error_invalid,
                        "index out of bounds", goto error);

        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
                first += isl_basic_map_offset(bmap, type) - 1;
                bmap = isl_basic_map_eliminate_vars(bmap, first, n);
                return isl_basic_map_finalize(bmap);
        }

        space = isl_basic_map_get_space(bmap);
        bmap = isl_basic_map_project_out(bmap, type, first, n);
        bmap = isl_basic_map_insert_dims(bmap, type, first, n);
        bmap = isl_basic_map_reset_space(bmap, space);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_eliminate(
        __isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return isl_basic_map_eliminate(bset, type, first, n);
}

/* Remove all constraints from "bmap" that reference any unknown local
 * variables (directly or indirectly).
 *
 * Dropping all constraints on a local variable will make it redundant,
 * so it will get removed implicitly by
 * isl_basic_map_drop_constraints_involving_dims.  Some other local
 * variables may also end up becoming redundant if they only appear
 * in constraints together with the unknown local variable.
 * Therefore, start over after calling
 * isl_basic_map_drop_constraints_involving_dims.
 */
__isl_give isl_basic_map *isl_basic_map_drop_constraint_involving_unknown_divs(
        __isl_take isl_basic_map *bmap)
{
        isl_bool known;
        int i, n_div, o_div;

        known = isl_basic_map_divs_known(bmap);
        if (known < 0)
                return isl_basic_map_free(bmap);
        if (known)
                return bmap;

        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        o_div = isl_basic_map_offset(bmap, isl_dim_div) - 1;

        for (i = 0; i < n_div; ++i) {
                known = isl_basic_map_div_is_known(bmap, i);
                if (known < 0)
                        return isl_basic_map_free(bmap);
                if (known)
                        continue;
                bmap = remove_dependent_vars(bmap, o_div + i);
                bmap = isl_basic_map_drop_constraints_involving_dims(bmap,
                                                            isl_dim_div, i, 1);
                if (!bmap)
                        return NULL;
                n_div = isl_basic_map_dim(bmap, isl_dim_div);
                i = -1;
        }

        return bmap;
}

/* Remove all constraints from "map" that reference any unknown local
 * variables (directly or indirectly).
 *
 * Since constraints may get dropped from the basic maps,
 * they may no longer be disjoint from each other.
 */
__isl_give isl_map *isl_map_drop_constraint_involving_unknown_divs(
        __isl_take isl_map *map)
{
        int i;
        isl_bool known;

        known = isl_map_divs_known(map);
        if (known < 0)
                return isl_map_free(map);
        if (known)
                return map;

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

        for (i = 0; i < map->n; ++i) {
                map->p[i] =
                    isl_basic_map_drop_constraint_involving_unknown_divs(
                                                                    map->p[i]);
                if (!map->p[i])
                        return isl_map_free(map);
        }

        if (map->n > 1)
                ISL_F_CLR(map, ISL_MAP_DISJOINT);

        return map;
}

/* Don't assume equalities are in order, because align_divs
 * may have changed the order of the divs.
 */
static void compute_elimination_index(struct isl_basic_map *bmap, int *elim)
{
        int d, i;
        unsigned total;

        total = isl_space_dim(bmap->dim, isl_dim_all);
        for (d = 0; d < total; ++d)
                elim[d] = -1;
        for (i = 0; i < bmap->n_eq; ++i) {
                for (d = total - 1; d >= 0; --d) {
                        if (isl_int_is_zero(bmap->eq[i][1+d]))
                                continue;
                        elim[d] = i;
                        break;
                }
        }
}

static void set_compute_elimination_index(struct isl_basic_set *bset, int *elim)
{
        compute_elimination_index(bset_to_bmap(bset), elim);
}

static int reduced_using_equalities(isl_int *dst, isl_int *src,
        struct isl_basic_map *bmap, int *elim)
{
        int d;
        int copied = 0;
        unsigned total;

        total = isl_space_dim(bmap->dim, isl_dim_all);
        for (d = total - 1; d >= 0; --d) {
                if (isl_int_is_zero(src[1+d]))
                        continue;
                if (elim[d] == -1)
                        continue;
                if (!copied) {
                        isl_seq_cpy(dst, src, 1 + total);
                        copied = 1;
                }
                isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
        }
        return copied;
}

static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
        struct isl_basic_set *bset, int *elim)
{
        return reduced_using_equalities(dst, src,
                                        bset_to_bmap(bset), elim);
}

static struct isl_basic_set *isl_basic_set_reduce_using_equalities(
        struct isl_basic_set *bset, struct isl_basic_set *context)
{
        int i;
        int *elim;

        if (!bset || !context)
                goto error;

        if (context->n_eq == 0) {
                isl_basic_set_free(context);
                return bset;
        }

        bset = isl_basic_set_cow(bset);
        if (!bset)
                goto error;

        elim = isl_alloc_array(bset->ctx, int, isl_basic_set_n_dim(bset));
        if (!elim)
                goto error;
        set_compute_elimination_index(context, elim);
        for (i = 0; i < bset->n_eq; ++i)
                set_reduced_using_equalities(bset->eq[i], bset->eq[i],
                                                        context, elim);
        for (i = 0; i < bset->n_ineq; ++i)
                set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
                                                        context, elim);
        isl_basic_set_free(context);
        free(elim);
        bset = isl_basic_set_simplify(bset);
        bset = isl_basic_set_finalize(bset);
        return bset;
error:
        isl_basic_set_free(bset);
        isl_basic_set_free(context);
        return NULL;
}

/* For each inequality in "ineq" that is a shifted (more relaxed)
 * copy of an inequality in "context", mark the corresponding entry
 * in "row" with -1.
 * If an inequality only has a non-negative constant term, then
 * mark it as well.
 */
static isl_stat mark_shifted_constraints(__isl_keep isl_mat *ineq,
        __isl_keep isl_basic_set *context, int *row)
{
        struct isl_constraint_index ci;
        int n_ineq;
        unsigned total;
        int k;

        if (!ineq || !context)
                return isl_stat_error;
        if (context->n_ineq == 0)
                return isl_stat_ok;
        if (setup_constraint_index(&ci, context) < 0)
                return isl_stat_error;

        n_ineq = isl_mat_rows(ineq);
        total = isl_mat_cols(ineq) - 1;
        for (k = 0; k < n_ineq; ++k) {
                int l;
                isl_bool redundant;

                l = isl_seq_first_non_zero(ineq->row[k] + 1, total);
                if (l < 0 && isl_int_is_nonneg(ineq->row[k][0])) {
                        row[k] = -1;
                        continue;
                }
                redundant = constraint_index_is_redundant(&ci, ineq->row[k]);
                if (redundant < 0)
                        goto error;
                if (!redundant)
                        continue;
                row[k] = -1;
        }
        constraint_index_free(&ci);
        return isl_stat_ok;
error:
        constraint_index_free(&ci);
        return isl_stat_error;
}

static struct isl_basic_set *remove_shifted_constraints(
        struct isl_basic_set *bset, struct isl_basic_set *context)
{
        struct isl_constraint_index ci;
        int k;

        if (!bset || !context)
                return bset;

        if (context->n_ineq == 0)
                return bset;
        if (setup_constraint_index(&ci, context) < 0)
                return bset;

        for (k = 0; k < bset->n_ineq; ++k) {
                isl_bool redundant;

                redundant = constraint_index_is_redundant(&ci, bset->ineq[k]);
                if (redundant < 0)
                        goto error;
                if (!redundant)
                        continue;
                bset = isl_basic_set_cow(bset);
                if (!bset)
                        goto error;
                isl_basic_set_drop_inequality(bset, k);
                --k;
        }
        constraint_index_free(&ci);
        return bset;
error:
        constraint_index_free(&ci);
        return bset;
}

/* Remove constraints from "bmap" that are identical to constraints
 * in "context" or that are more relaxed (greater constant term).
 *
 * We perform the test for shifted copies on the pure constraints
 * in remove_shifted_constraints.
 */
static __isl_give isl_basic_map *isl_basic_map_remove_shifted_constraints(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context)
{
        isl_basic_set *bset, *bset_context;

        if (!bmap || !context)
                goto error;

        if (bmap->n_ineq == 0 || context->n_ineq == 0) {
                isl_basic_map_free(context);
                return bmap;
        }

        context = isl_basic_map_align_divs(context, bmap);
        bmap = isl_basic_map_align_divs(bmap, context);

        bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
        bset_context = isl_basic_map_underlying_set(context);
        bset = remove_shifted_constraints(bset, bset_context);
        isl_basic_set_free(bset_context);

        bmap = isl_basic_map_overlying_set(bset, bmap);

        return bmap;
error:
        isl_basic_map_free(bmap);
        isl_basic_map_free(context);
        return NULL;
}

/* Does the (linear part of a) constraint "c" involve any of the "len"
 * "relevant" dimensions?
 */
static int is_related(isl_int *c, int len, int *relevant)
{
        int i;

        for (i = 0; i < len; ++i) {
                if (!relevant[i])
                        continue;
                if (!isl_int_is_zero(c[i]))
                        return 1;
        }

        return 0;
}

/* Drop constraints from "bmap" that do not involve any of
 * the dimensions marked "relevant".
 */
static __isl_give isl_basic_map *drop_unrelated_constraints(
        __isl_take isl_basic_map *bmap, int *relevant)
{
        int i, dim;

        dim = isl_basic_map_dim(bmap, isl_dim_all);
        for (i = 0; i < dim; ++i)
                if (!relevant[i])
                        break;
        if (i >= dim)
                return bmap;

        for (i = bmap->n_eq - 1; i >= 0; --i)
                if (!is_related(bmap->eq[i] + 1, dim, relevant)) {
                        bmap = isl_basic_map_cow(bmap);
                        if (isl_basic_map_drop_equality(bmap, i) < 0)
                                return isl_basic_map_free(bmap);
                }

        for (i = bmap->n_ineq - 1; i >= 0; --i)
                if (!is_related(bmap->ineq[i] + 1, dim, relevant)) {
                        bmap = isl_basic_map_cow(bmap);
                        if (isl_basic_map_drop_inequality(bmap, i) < 0)
                                return isl_basic_map_free(bmap);
                }

        return bmap;
}

/* Update the groups in "group" based on the (linear part of a) constraint "c".
 *
 * In particular, for any variable involved in the constraint,
 * find the actual group id from before and replace the group
 * of the corresponding variable by the minimal group of all
 * the variables involved in the constraint considered so far
 * (if this minimum is smaller) or replace the minimum by this group
 * (if the minimum is larger).
 *
 * At the end, all the variables in "c" will (indirectly) point
 * to the minimal of the groups that they referred to originally.
 */
static void update_groups(int dim, int *group, isl_int *c)
{
        int j;
        int min = dim;

        for (j = 0; j < dim; ++j) {
                if (isl_int_is_zero(c[j]))
                        continue;
                while (group[j] >= 0 && group[group[j]] != group[j])
                        group[j] = group[group[j]];
                if (group[j] == min)
                        continue;
                if (group[j] < min) {
                        if (min >= 0 && min < dim)
                                group[min] = group[j];
                        min = group[j];
                } else
                        group[group[j]] = min;
        }
}

/* Allocate an array of groups of variables, one for each variable
 * in "context", initialized to zero.
 */
static int *alloc_groups(__isl_keep isl_basic_set *context)
{
        isl_ctx *ctx;
        int dim;

        dim = isl_basic_set_dim(context, isl_dim_set);
        ctx = isl_basic_set_get_ctx(context);
        return isl_calloc_array(ctx, int, dim);
}

/* Drop constraints from "bmap" that only involve variables that are
 * not related to any of the variables marked with a "-1" in "group".
 *
 * We construct groups of variables that collect variables that
 * (indirectly) appear in some common constraint of "bmap".
 * Each group is identified by the first variable in the group,
 * except for the special group of variables that was already identified
 * in the input as -1 (or are related to those variables).
 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
 * otherwise the group of i is the group of group[i].
 *
 * We first initialize groups for the remaining variables.
 * Then we iterate over the constraints of "bmap" and update the
 * group of the variables in the constraint by the smallest group.
 * Finally, we resolve indirect references to groups by running over
 * the variables.
 *
 * After computing the groups, we drop constraints that do not involve
 * any variables in the -1 group.
 */
__isl_give isl_basic_map *isl_basic_map_drop_unrelated_constraints(
        __isl_take isl_basic_map *bmap, __isl_take int *group)
{
        int dim;
        int i;
        int last;

        if (!bmap)
                return NULL;

        dim = isl_basic_map_dim(bmap, isl_dim_all);

        last = -1;
        for (i = 0; i < dim; ++i)
                if (group[i] >= 0)
                        last = group[i] = i;
        if (last < 0) {
                free(group);
                return bmap;
        }

        for (i = 0; i < bmap->n_eq; ++i)
                update_groups(dim, group, bmap->eq[i] + 1);
        for (i = 0; i < bmap->n_ineq; ++i)
                update_groups(dim, group, bmap->ineq[i] + 1);

        for (i = 0; i < dim; ++i)
                if (group[i] >= 0)
                        group[i] = group[group[i]];

        for (i = 0; i < dim; ++i)
                group[i] = group[i] == -1;

        bmap = drop_unrelated_constraints(bmap, group);

        free(group);
        return bmap;
}

/* Drop constraints from "context" that are irrelevant for computing
 * the gist of "bset".
 *
 * In particular, drop constraints in variables that are not related
 * to any of the variables involved in the constraints of "bset"
 * in the sense that there is no sequence of constraints that connects them.
 *
 * We first mark all variables that appear in "bset" as belonging
 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
 */
static __isl_give isl_basic_set *drop_irrelevant_constraints(
        __isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset)
{
        int *group;
        int dim;
        int i, j;

        if (!context || !bset)
                return isl_basic_set_free(context);

        group = alloc_groups(context);

        if (!group)
                return isl_basic_set_free(context);

        dim = isl_basic_set_dim(bset, isl_dim_set);
        for (i = 0; i < dim; ++i) {
                for (j = 0; j < bset->n_eq; ++j)
                        if (!isl_int_is_zero(bset->eq[j][1 + i]))
                                break;
                if (j < bset->n_eq) {
                        group[i] = -1;
                        continue;
                }
                for (j = 0; j < bset->n_ineq; ++j)
                        if (!isl_int_is_zero(bset->ineq[j][1 + i]))
                                break;
                if (j < bset->n_ineq)
                        group[i] = -1;
        }

        return isl_basic_map_drop_unrelated_constraints(context, group);
}

/* Drop constraints from "context" that are irrelevant for computing
 * the gist of the inequalities "ineq".
 * Inequalities in "ineq" for which the corresponding element of row
 * is set to -1 have already been marked for removal and should be ignored.
 *
 * In particular, drop constraints in variables that are not related
 * to any of the variables involved in "ineq"
 * in the sense that there is no sequence of constraints that connects them.
 *
 * We first mark all variables that appear in "bset" as belonging
 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
 */
static __isl_give isl_basic_set *drop_irrelevant_constraints_marked(
        __isl_take isl_basic_set *context, __isl_keep isl_mat *ineq, int *row)
{
        int *group;
        int dim;
        int i, j, n;

        if (!context || !ineq)
                return isl_basic_set_free(context);

        group = alloc_groups(context);

        if (!group)
                return isl_basic_set_free(context);

        dim = isl_basic_set_dim(context, isl_dim_set);
        n = isl_mat_rows(ineq);
        for (i = 0; i < dim; ++i) {
                for (j = 0; j < n; ++j) {
                        if (row[j] < 0)
                                continue;
                        if (!isl_int_is_zero(ineq->row[j][1 + i]))
                                break;
                }
                if (j < n)
                        group[i] = -1;
        }

        return isl_basic_map_drop_unrelated_constraints(context, group);
}

/* Do all "n" entries of "row" contain a negative value?
 */
static int all_neg(int *row, int n)
{
        int i;

        for (i = 0; i < n; ++i)
                if (row[i] >= 0)
                        return 0;

        return 1;
}

/* Update the inequalities in "bset" based on the information in "row"
 * and "tab".
 *
 * In particular, the array "row" contains either -1, meaning that
 * the corresponding inequality of "bset" is redundant, or the index
 * of an inequality in "tab".
 *
 * If the row entry is -1, then drop the inequality.
 * Otherwise, if the constraint is marked redundant in the tableau,
 * then drop the inequality.  Similarly, if it is marked as an equality
 * in the tableau, then turn the inequality into an equality and
 * perform Gaussian elimination.
 */
static __isl_give isl_basic_set *update_ineq(__isl_take isl_basic_set *bset,
        __isl_keep int *row, struct isl_tab *tab)
{
        int i;
        unsigned n_ineq;
        unsigned n_eq;
        int found_equality = 0;

        if (!bset)
                return NULL;
        if (tab && tab->empty)
                return isl_basic_set_set_to_empty(bset);

        n_ineq = bset->n_ineq;
        for (i = n_ineq - 1; i >= 0; --i) {
                if (row[i] < 0) {
                        if (isl_basic_set_drop_inequality(bset, i) < 0)
                                return isl_basic_set_free(bset);
                        continue;
                }
                if (!tab)
                        continue;
                n_eq = tab->n_eq;
                if (isl_tab_is_equality(tab, n_eq + row[i])) {
                        isl_basic_map_inequality_to_equality(bset, i);
                        found_equality = 1;
                } else if (isl_tab_is_redundant(tab, n_eq + row[i])) {
                        if (isl_basic_set_drop_inequality(bset, i) < 0)
                                return isl_basic_set_free(bset);
                }
        }

        if (found_equality)
                bset = isl_basic_set_gauss(bset, NULL);
        bset = isl_basic_set_finalize(bset);
        return bset;
}

/* Update the inequalities in "bset" based on the information in "row"
 * and "tab" and free all arguments (other than "bset").
 */
static __isl_give isl_basic_set *update_ineq_free(
        __isl_take isl_basic_set *bset, __isl_take isl_mat *ineq,
        __isl_take isl_basic_set *context, __isl_take int *row,
        struct isl_tab *tab)
{
        isl_mat_free(ineq);
        isl_basic_set_free(context);

        bset = update_ineq(bset, row, tab);

        free(row);
        isl_tab_free(tab);
        return bset;
}

/* Remove all information from bset that is redundant in the context
 * of context.
 * "ineq" contains the (possibly transformed) inequalities of "bset",
 * in the same order.
 * The (explicit) equalities of "bset" are assumed to have been taken
 * into account by the transformation such that only the inequalities
 * are relevant.
 * "context" is assumed not to be empty.
 *
 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
 * A value of -1 means that the inequality is obviously redundant and may
 * not even appear in  "tab".
 *
 * We first mark the inequalities of "bset"
 * that are obviously redundant with respect to some inequality in "context".
 * Then we remove those constraints from "context" that have become
 * irrelevant for computing the gist of "bset".
 * Note that this removal of constraints cannot be replaced by
 * a factorization because factors in "bset" may still be connected
 * to each other through constraints in "context".
 *
 * If there are any inequalities left, we construct a tableau for
 * the context and then add the inequalities of "bset".
 * Before adding these inequalities, we freeze all constraints such that
 * they won't be considered redundant in terms of the constraints of "bset".
 * Then we detect all redundant constraints (among the
 * constraints that weren't frozen), first by checking for redundancy in the
 * the tableau and then by checking if replacing a constraint by its negation
 * would lead to an empty set.  This last step is fairly expensive
 * and could be optimized by more reuse of the tableau.
 * Finally, we update bset according to the results.
 */
static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset,
        __isl_take isl_mat *ineq, __isl_take isl_basic_set *context)
{
        int i, r;
        int *row = NULL;
        isl_ctx *ctx;
        isl_basic_set *combined = NULL;
        struct isl_tab *tab = NULL;
        unsigned n_eq, context_ineq;
        unsigned total;

        if (!bset || !ineq || !context)
                goto error;

        if (bset->n_ineq == 0 || isl_basic_set_plain_is_universe(context)) {
                isl_basic_set_free(context);
                isl_mat_free(ineq);
                return bset;
        }

        ctx = isl_basic_set_get_ctx(context);
        row = isl_calloc_array(ctx, int, bset->n_ineq);
        if (!row)
                goto error;

        if (mark_shifted_constraints(ineq, context, row) < 0)
                goto error;
        if (all_neg(row, bset->n_ineq))
                return update_ineq_free(bset, ineq, context, row, NULL);

        context = drop_irrelevant_constraints_marked(context, ineq, row);
        if (!context)
                goto error;
        if (isl_basic_set_plain_is_universe(context))
                return update_ineq_free(bset, ineq, context, row, NULL);

        n_eq = context->n_eq;
        context_ineq = context->n_ineq;
        combined = isl_basic_set_cow(isl_basic_set_copy(context));
        combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq);
        tab = isl_tab_from_basic_set(combined, 0);
        for (i = 0; i < context_ineq; ++i)
                if (isl_tab_freeze_constraint(tab, n_eq + i) < 0)
                        goto error;
        if (isl_tab_extend_cons(tab, bset->n_ineq) < 0)
                goto error;
        r = context_ineq;
        for (i = 0; i < bset->n_ineq; ++i) {
                if (row[i] < 0)
                        continue;
                combined = isl_basic_set_add_ineq(combined, ineq->row[i]);
                if (isl_tab_add_ineq(tab, ineq->row[i]) < 0)
                        goto error;
                row[i] = r++;
        }
        if (isl_tab_detect_implicit_equalities(tab) < 0)
                goto error;
        if (isl_tab_detect_redundant(tab) < 0)
                goto error;
        total = isl_basic_set_total_dim(bset);
        for (i = bset->n_ineq - 1; i >= 0; --i) {
                isl_basic_set *test;
                int is_empty;

                if (row[i] < 0)
                        continue;
                r = row[i];
                if (tab->con[n_eq + r].is_redundant)
                        continue;
                test = isl_basic_set_dup(combined);
                if (isl_inequality_negate(test, r) < 0)
                        test = isl_basic_set_free(test);
                test = isl_basic_set_update_from_tab(test, tab);
                is_empty = isl_basic_set_is_empty(test);
                isl_basic_set_free(test);
                if (is_empty < 0)
                        goto error;
                if (is_empty)
                        tab->con[n_eq + r].is_redundant = 1;
        }
        bset = update_ineq_free(bset, ineq, context, row, tab);
        if (bset) {
                ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
                ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
        }

        isl_basic_set_free(combined);
        return bset;
error:
        free(row);
        isl_mat_free(ineq);
        isl_tab_free(tab);
        isl_basic_set_free(combined);
        isl_basic_set_free(context);
        isl_basic_set_free(bset);
        return NULL;
}

/* Extract the inequalities of "bset" as an isl_mat.
 */
static __isl_give isl_mat *extract_ineq(__isl_keep isl_basic_set *bset)
{
        unsigned total;
        isl_ctx *ctx;
        isl_mat *ineq;

        if (!bset)
                return NULL;

        ctx = isl_basic_set_get_ctx(bset);
        total = isl_basic_set_total_dim(bset);
        ineq = isl_mat_sub_alloc6(ctx, bset->ineq, 0, bset->n_ineq,
                                    0, 1 + total);

        return ineq;
}

/* Remove all information from "bset" that is redundant in the context
 * of "context", for the case where both "bset" and "context" are
 * full-dimensional.
 */
static __isl_give isl_basic_set *uset_gist_uncompressed(
        __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context)
{
        isl_mat *ineq;

        ineq = extract_ineq(bset);
        return uset_gist_full(bset, ineq, context);
}

/* Remove all information from "bset" that is redundant in the context
 * of "context", for the case where the combined equalities of
 * "bset" and "context" allow for a compression that can be obtained
 * by preapplication of "T".
 *
 * "bset" itself is not transformed by "T".  Instead, the inequalities
 * are extracted from "bset" and those are transformed by "T".
 * uset_gist_full then determines which of the transformed inequalities
 * are redundant with respect to the transformed "context" and removes
 * the corresponding inequalities from "bset".
 *
 * After preapplying "T" to the inequalities, any common factor is
 * removed from the coefficients.  If this results in a tightening
 * of the constant term, then the same tightening is applied to
 * the corresponding untransformed inequality in "bset".
 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
 *
 *      g f'(x) + r >= 0
 *
 * with 0 <= r < g, then it is equivalent to
 *
 *      f'(x) >= 0
 *
 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
 * subspace compressed by T since the latter would be transformed to
 *
 *      g f'(x) >= 0
 */
static __isl_give isl_basic_set *uset_gist_compressed(
        __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context,
        __isl_take isl_mat *T)
{
        isl_ctx *ctx;
        isl_mat *ineq;
        int i, n_row, n_col;
        isl_int rem;

        ineq = extract_ineq(bset);
        ineq = isl_mat_product(ineq, isl_mat_copy(T));
        context = isl_basic_set_preimage(context, T);

        if (!ineq || !context)
                goto error;
        if (isl_basic_set_plain_is_empty(context)) {
                isl_mat_free(ineq);
                isl_basic_set_free(context);
                return isl_basic_set_set_to_empty(bset);
        }

        ctx = isl_mat_get_ctx(ineq);
        n_row = isl_mat_rows(ineq);
        n_col = isl_mat_cols(ineq);
        isl_int_init(rem);
        for (i = 0; i < n_row; ++i) {
                isl_seq_gcd(ineq->row[i] + 1, n_col - 1, &ctx->normalize_gcd);
                if (isl_int_is_zero(ctx->normalize_gcd))
                        continue;
                if (isl_int_is_one(ctx->normalize_gcd))
                        continue;
                isl_seq_scale_down(ineq->row[i] + 1, ineq->row[i] + 1,
                                    ctx->normalize_gcd, n_col - 1);
                isl_int_fdiv_r(rem, ineq->row[i][0], ctx->normalize_gcd);
                isl_int_fdiv_q(ineq->row[i][0],
                                ineq->row[i][0], ctx->normalize_gcd);
                if (isl_int_is_zero(rem))
                        continue;
                bset = isl_basic_set_cow(bset);
                if (!bset)
                        break;
                isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], rem);
        }
        isl_int_clear(rem);

        return uset_gist_full(bset, ineq, context);
error:
        isl_mat_free(ineq);
        isl_basic_set_free(context);
        isl_basic_set_free(bset);
        return NULL;
}

/* Project "bset" onto the variables that are involved in "template".
 */
static __isl_give isl_basic_set *project_onto_involved(
        __isl_take isl_basic_set *bset, __isl_keep isl_basic_set *template)
{
        int i, n;

        if (!bset || !template)
                return isl_basic_set_free(bset);

        n = isl_basic_set_dim(template, isl_dim_set);

        for (i = 0; i < n; ++i) {
                isl_bool involved;

                involved = isl_basic_set_involves_dims(template,
                                                        isl_dim_set, i, 1);
                if (involved < 0)
                        return isl_basic_set_free(bset);
                if (involved)
                        continue;
                bset = isl_basic_set_eliminate_vars(bset, i, 1);
        }

        return bset;
}

/* Remove all information from bset that is redundant in the context
 * of context.  In particular, equalities that are linear combinations
 * of those in context are removed.  Then the inequalities that are
 * redundant in the context of the equalities and inequalities of
 * context are removed.
 *
 * First of all, we drop those constraints from "context"
 * that are irrelevant for computing the gist of "bset".
 * Alternatively, we could factorize the intersection of "context" and "bset".
 *
 * We first compute the intersection of the integer affine hulls
 * of "bset" and "context",
 * compute the gist inside this intersection and then reduce
 * the constraints with respect to the equalities of the context
 * that only involve variables already involved in the input.
 *
 * If two constraints are mutually redundant, then uset_gist_full
 * will remove the second of those constraints.  We therefore first
 * sort the constraints so that constraints not involving existentially
 * quantified variables are given precedence over those that do.
 * We have to perform this sorting before the variable compression,
 * because that may effect the order of the variables.
 */
static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset,
        __isl_take isl_basic_set *context)
{
        isl_mat *eq;
        isl_mat *T;
        isl_basic_set *aff;
        isl_basic_set *aff_context;
        unsigned total;

        if (!bset || !context)
                goto error;

        context = drop_irrelevant_constraints(context, bset);

        bset = isl_basic_set_detect_equalities(bset);
        aff = isl_basic_set_copy(bset);
        aff = isl_basic_set_plain_affine_hull(aff);
        context = isl_basic_set_detect_equalities(context);
        aff_context = isl_basic_set_copy(context);
        aff_context = isl_basic_set_plain_affine_hull(aff_context);
        aff = isl_basic_set_intersect(aff, aff_context);
        if (!aff)
                goto error;
        if (isl_basic_set_plain_is_empty(aff)) {
                isl_basic_set_free(bset);
                isl_basic_set_free(context);
                return aff;
        }
        bset = isl_basic_set_sort_constraints(bset);
        if (aff->n_eq == 0) {
                isl_basic_set_free(aff);
                return uset_gist_uncompressed(bset, context);
        }
        total = isl_basic_set_total_dim(bset);
        eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total);
        eq = isl_mat_cow(eq);
        T = isl_mat_variable_compression(eq, NULL);
        isl_basic_set_free(aff);
        if (T && T->n_col == 0) {
                isl_mat_free(T);
                isl_basic_set_free(context);
                return isl_basic_set_set_to_empty(bset);
        }

        aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));
        aff_context = project_onto_involved(aff_context, bset);

        bset = uset_gist_compressed(bset, context, T);
        bset = isl_basic_set_reduce_using_equalities(bset, aff_context);

        if (bset) {
                ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
                ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
        }

        return bset;
error:
        isl_basic_set_free(bset);
        isl_basic_set_free(context);
        return NULL;
}

/* Return the number of equality constraints in "bmap" that involve
 * local variables.  This function assumes that Gaussian elimination
 * has been applied to the equality constraints.
 */
static int n_div_eq(__isl_keep isl_basic_map *bmap)
{
        int i;
        int total, n_div;

        if (!bmap)
                return -1;

        if (bmap->n_eq == 0)
                return 0;

        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 < bmap->n_eq; ++i)
                if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total,
                                            n_div) == -1)
                        return i;

        return bmap->n_eq;
}

/* Construct a basic map in "space" defined by the equality constraints in "eq".
 * The constraints are assumed not to involve any local variables.
 */
static __isl_give isl_basic_map *basic_map_from_equalities(
        __isl_take isl_space *space, __isl_take isl_mat *eq)
{
        int i, k;
        isl_basic_map *bmap = NULL;

        if (!space || !eq)
                goto error;

        if (1 + isl_space_dim(space, isl_dim_all) != eq->n_col)
                isl_die(isl_space_get_ctx(space), isl_error_internal,
                        "unexpected number of columns", goto error);

        bmap = isl_basic_map_alloc_space(isl_space_copy(space),
                                            0, eq->n_row, 0);
        for (i = 0; i < eq->n_row; ++i) {
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_cpy(bmap->eq[k], eq->row[i], eq->n_col);
        }

        isl_space_free(space);
        isl_mat_free(eq);
        return bmap;
error:
        isl_space_free(space);
        isl_mat_free(eq);
        isl_basic_map_free(bmap);
        return NULL;
}

/* Construct and return a variable compression based on the equality
 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
 * "n1" is the number of (initial) equality constraints in "bmap1"
 * that do involve local variables.
 * "n2" is the number of (initial) equality constraints in "bmap2"
 * that do involve local variables.
 * "total" is the total number of other variables.
 * This function assumes that Gaussian elimination
 * has been applied to the equality constraints in both "bmap1" and "bmap2"
 * such that the equality constraints not involving local variables
 * are those that start at "n1" or "n2".
 *
 * If either of "bmap1" and "bmap2" does not have such equality constraints,
 * then simply compute the compression based on the equality constraints
 * in the other basic map.
 * Otherwise, combine the equality constraints from both into a new
 * basic map such that Gaussian elimination can be applied to this combination
 * and then construct a variable compression from the resulting
 * equality constraints.
 */
static __isl_give isl_mat *combined_variable_compression(
        __isl_keep isl_basic_map *bmap1, int n1,
        __isl_keep isl_basic_map *bmap2, int n2, int total)
{
        isl_ctx *ctx;
        isl_mat *E1, *E2, *V;
        isl_basic_map *bmap;

        ctx = isl_basic_map_get_ctx(bmap1);
        if (bmap1->n_eq == n1) {
                E2 = isl_mat_sub_alloc6(ctx, bmap2->eq,
                                        n2, bmap2->n_eq - n2, 0, 1 + total);
                return isl_mat_variable_compression(E2, NULL);
        }
        if (bmap2->n_eq == n2) {
                E1 = isl_mat_sub_alloc6(ctx, bmap1->eq,
                                        n1, bmap1->n_eq - n1, 0, 1 + total);
                return isl_mat_variable_compression(E1, NULL);
        }
        E1 = isl_mat_sub_alloc6(ctx, bmap1->eq,
                                n1, bmap1->n_eq - n1, 0, 1 + total);
        E2 = isl_mat_sub_alloc6(ctx, bmap2->eq,
                                n2, bmap2->n_eq - n2, 0, 1 + total);
        E1 = isl_mat_concat(E1, E2);
        bmap = basic_map_from_equalities(isl_basic_map_get_space(bmap1), E1);
        bmap = isl_basic_map_gauss(bmap, NULL);
        if (!bmap)
                return NULL;
        E1 = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap->n_eq, 0, 1 + total);
        V = isl_mat_variable_compression(E1, NULL);
        isl_basic_map_free(bmap);

        return V;
}

/* Extract the stride constraints from "bmap", compressed
 * with respect to both the stride constraints in "context" and
 * the remaining equality constraints in both "bmap" and "context".
 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
 * "context_n_eq" is the number of (initial) stride constraints in "context".
 *
 * Let x be all variables in "bmap" (and "context") other than the local
 * variables.  First compute a variable compression
 *
 *      x = V x'
 *
 * based on the non-stride equality constraints in "bmap" and "context".
 * Consider the stride constraints of "context",
 *
 *      A(x) + B(y) = 0
 *
 * with y the local variables and plug in the variable compression,
 * resulting in
 *
 *      A(V x') + B(y) = 0
 *
 * Use these constraints to compute a parameter compression on x'
 *
 *      x' = T x''
 *
 * Now consider the stride constraints of "bmap"
 *
 *      C(x) + D(y) = 0
 *
 * and plug in x = V*T x''.
 * That is, return A = [C*V*T D].
 */
static __isl_give isl_mat *extract_compressed_stride_constraints(
        __isl_keep isl_basic_map *bmap, int bmap_n_eq,
        __isl_keep isl_basic_map *context, int context_n_eq)
{
        int total, n_div;
        isl_ctx *ctx;
        isl_mat *A, *B, *T, *V;

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

        ctx = isl_basic_map_get_ctx(bmap);

        V = combined_variable_compression(bmap, bmap_n_eq,
                                                context, context_n_eq, total);

        A = isl_mat_sub_alloc6(ctx, context->eq, 0, context_n_eq, 0, 1 + total);
        B = isl_mat_sub_alloc6(ctx, context->eq,
                                0, context_n_eq, 1 + total, n_div);
        A = isl_mat_product(A, isl_mat_copy(V));
        T = isl_mat_parameter_compression_ext(A, B);
        T = isl_mat_product(V, T);

        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        T = isl_mat_diagonal(T, isl_mat_identity(ctx, n_div));

        A = isl_mat_sub_alloc6(ctx, bmap->eq,
                                0, bmap_n_eq, 0, 1 + total + n_div);
        A = isl_mat_product(A, T);

        return A;
}

/* Remove the prime factors from *g that have an exponent that
 * is strictly smaller than the exponent in "c".
 * All exponents in *g are known to be smaller than or equal
 * to those in "c".
 *
 * That is, if *g is equal to
 *
 *      p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
 *
 * and "c" is equal to
 *
 *      p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
 *
 * then update *g to
 *
 *      p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
 *              p_n^{e_n * (e_n = f_n)}
 *
 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
 * neither does the gcd of *g and c / *g.
 * If e_i < f_i, then the gcd of *g and c / *g has a positive
 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
 * Dividing *g by this gcd therefore strictly reduces the exponent
 * of the prime factors that need to be removed, while leaving the
 * other prime factors untouched.
 * Repeating this process until gcd(*g, c / *g) = 1 therefore
 * removes all undesired factors, without removing any others.
 */
static void remove_incomplete_powers(isl_int *g, isl_int c)
{
        isl_int t;

        isl_int_init(t);
        for (;;) {
                isl_int_divexact(t, c, *g);
                isl_int_gcd(t, t, *g);
                if (isl_int_is_one(t))
                        break;
                isl_int_divexact(*g, *g, t);
        }
        isl_int_clear(t);
}

/* Reduce the "n" stride constraints in "bmap" based on a copy "A"
 * of the same stride constraints in a compressed space that exploits
 * all equalities in the context and the other equalities in "bmap".
 *
 * If the stride constraints of "bmap" are of the form
 *
 *      C(x) + D(y) = 0
 *
 * then A is of the form
 *
 *      B(x') + D(y) = 0
 *
 * If any of these constraints involves only a single local variable y,
 * then the constraint appears as
 *
 *      f(x) + m y_i = 0
 *
 * in "bmap" and as
 *
 *      h(x') + m y_i = 0
 *
 * in "A".
 *
 * Let g be the gcd of m and the coefficients of h.
 * Then, in particular, g is a divisor of the coefficients of h and
 *
 *      f(x) = h(x')
 *
 * is known to be a multiple of g.
 * If some prime factor in m appears with the same exponent in g,
 * then it can be removed from m because f(x) is already known
 * to be a multiple of g and therefore in particular of this power
 * of the prime factors.
 * Prime factors that appear with a smaller exponent in g cannot
 * be removed from m.
 * Let g' be the divisor of g containing all prime factors that
 * appear with the same exponent in m and g, then
 *
 *      f(x) + m y_i = 0
 *
 * can be replaced by
 *
 *      f(x) + m/g' y_i' = 0
 *
 * Note that (if g' != 1) this changes the explicit representation
 * of y_i to that of y_i', so the integer division at position i
 * is marked unknown and later recomputed by a call to
 * isl_basic_map_gauss.
 */
static __isl_give isl_basic_map *reduce_stride_constraints(
        __isl_take isl_basic_map *bmap, int n, __isl_keep isl_mat *A)
{
        int i;
        int total, n_div;
        int any = 0;
        isl_int gcd;

        if (!bmap || !A)
                return isl_basic_map_free(bmap);

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

        isl_int_init(gcd);
        for (i = 0; i < n; ++i) {
                int div;

                div = isl_seq_first_non_zero(bmap->eq[i] + 1 + total, n_div);
                if (div < 0)
                        isl_die(isl_basic_map_get_ctx(bmap), isl_error_internal,
                                "equality constraints modified unexpectedly",
                                goto error);
                if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total + div + 1,
                                                n_div - div - 1) != -1)
                        continue;
                if (isl_mat_row_gcd(A, i, &gcd) < 0)
                        goto error;
                if (isl_int_is_one(gcd))
                        continue;
                remove_incomplete_powers(&gcd, bmap->eq[i][1 + total + div]);
                if (isl_int_is_one(gcd))
                        continue;
                isl_int_divexact(bmap->eq[i][1 + total + div],
                                bmap->eq[i][1 + total + div], gcd);
                bmap = isl_basic_map_mark_div_unknown(bmap, div);
                if (!bmap)
                        goto error;
                any = 1;
        }
        isl_int_clear(gcd);

        if (any)
                bmap = isl_basic_map_gauss(bmap, NULL);

        return bmap;
error:
        isl_int_clear(gcd);
        isl_basic_map_free(bmap);
        return NULL;
}

/* Simplify the stride constraints in "bmap" based on
 * the remaining equality constraints in "bmap" and all equality
 * constraints in "context".
 * Only do this if both "bmap" and "context" have stride constraints.
 *
 * First extract a copy of the stride constraints in "bmap" in a compressed
 * space exploiting all the other equality constraints and then
 * use this compressed copy to simplify the original stride constraints.
 */
static __isl_give isl_basic_map *gist_strides(__isl_take isl_basic_map *bmap,
        __isl_keep isl_basic_map *context)
{
        int bmap_n_eq, context_n_eq;
        isl_mat *A;

        if (!bmap || !context)
                return isl_basic_map_free(bmap);

        bmap_n_eq = n_div_eq(bmap);
        context_n_eq = n_div_eq(context);

        if (bmap_n_eq < 0 || context_n_eq < 0)
                return isl_basic_map_free(bmap);
        if (bmap_n_eq == 0 || context_n_eq == 0)
                return bmap;

        A = extract_compressed_stride_constraints(bmap, bmap_n_eq,
                                                    context, context_n_eq);
        bmap = reduce_stride_constraints(bmap, bmap_n_eq, A);

        isl_mat_free(A);

        return bmap;
}

/* Return a basic map that has the same intersection with "context" as "bmap"
 * and that is as "simple" as possible.
 *
 * The core computation is performed on the pure constraints.
 * When we add back the meaning of the integer divisions, we need
 * to (re)introduce the div constraints.  If we happen to have
 * discovered that some of these integer divisions are equal to
 * some affine combination of other variables, then these div
 * constraints may end up getting simplified in terms of the equalities,
 * resulting in extra inequalities on the other variables that
 * may have been removed already or that may not even have been
 * part of the input.  We try and remove those constraints of
 * this form that are most obviously redundant with respect to
 * the context.  We also remove those div constraints that are
 * redundant with respect to the other constraints in the result.
 *
 * The stride constraints among the equality constraints in "bmap" are
 * also simplified with respecting to the other equality constraints
 * in "bmap" and with respect to all equality constraints in "context".
 */
struct isl_basic_map *isl_basic_map_gist(struct isl_basic_map *bmap,
        struct isl_basic_map *context)
{
        isl_basic_set *bset, *eq;
        isl_basic_map *eq_bmap;
        unsigned total, n_div, extra, n_eq, n_ineq;

        if (!bmap || !context)
                goto error;

        if (isl_basic_map_plain_is_universe(bmap)) {
                isl_basic_map_free(context);
                return bmap;
        }
        if (isl_basic_map_plain_is_empty(context)) {
                isl_space *space = isl_basic_map_get_space(bmap);
                isl_basic_map_free(bmap);
                isl_basic_map_free(context);
                return isl_basic_map_universe(space);
        }
        if (isl_basic_map_plain_is_empty(bmap)) {
                isl_basic_map_free(context);
                return bmap;
        }

        bmap = isl_basic_map_remove_redundancies(bmap);
        context = isl_basic_map_remove_redundancies(context);
        if (!context)
                goto error;

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

        bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
        bset = isl_basic_set_add_dims(bset, isl_dim_set, extra);
        bset = uset_gist(bset,
                    isl_basic_map_underlying_set(isl_basic_map_copy(context)));
        bset = isl_basic_set_project_out(bset, isl_dim_set, total, extra);

        if (!bset || bset->n_eq == 0 || n_div == 0 ||
            isl_basic_set_plain_is_empty(bset)) {
                isl_basic_map_free(context);
                return isl_basic_map_overlying_set(bset, bmap);
        }

        n_eq = bset->n_eq;
        n_ineq = bset->n_ineq;
        eq = isl_basic_set_copy(bset);
        eq = isl_basic_set_cow(eq);
        if (isl_basic_set_free_inequality(eq, n_ineq) < 0)
                eq = isl_basic_set_free(eq);
        if (isl_basic_set_free_equality(bset, n_eq) < 0)
                bset = isl_basic_set_free(bset);

        eq_bmap = isl_basic_map_overlying_set(eq, isl_basic_map_copy(bmap));
        eq_bmap = gist_strides(eq_bmap, context);
        eq_bmap = isl_basic_map_remove_shifted_constraints(eq_bmap, context);
        bmap = isl_basic_map_overlying_set(bset, bmap);
        bmap = isl_basic_map_intersect(bmap, eq_bmap);
        bmap = isl_basic_map_remove_redundancies(bmap);

        return bmap;
error:
        isl_basic_map_free(bmap);
        isl_basic_map_free(context);
        return NULL;
}

/*
 * Assumes context has no implicit divs.
 */
__isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
        __isl_take isl_basic_map *context)
{
        int i;

        if (!map || !context)
                goto error;

        if (isl_basic_map_plain_is_empty(context)) {
                isl_space *space = isl_map_get_space(map);
                isl_map_free(map);
                isl_basic_map_free(context);
                return isl_map_universe(space);
        }

        context = isl_basic_map_remove_redundancies(context);
        map = isl_map_cow(map);
        if (!map || !context)
                goto error;
        isl_assert(map->ctx, isl_space_is_equal(map->dim, context->dim), goto error);
        map = isl_map_compute_divs(map);
        if (!map)
                goto error;
        for (i = map->n - 1; i >= 0; --i) {
                map->p[i] = isl_basic_map_gist(map->p[i],
                                                isl_basic_map_copy(context));
                if (!map->p[i])
                        goto error;
                if (isl_basic_map_plain_is_empty(map->p[i])) {
                        isl_basic_map_free(map->p[i]);
                        if (i != map->n - 1)
                                map->p[i] = map->p[map->n - 1];
                        map->n--;
                }
        }
        isl_basic_map_free(context);
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        isl_basic_map_free(context);
        return NULL;
}

/* Drop all inequalities from "bmap" that also appear in "context".
 * "context" is assumed to have only known local variables and
 * the initial local variables of "bmap" are assumed to be the same
 * as those of "context".
 * The constraints of both "bmap" and "context" are assumed
 * to have been sorted using isl_basic_map_sort_constraints.
 *
 * Run through the inequality constraints of "bmap" and "context"
 * in sorted order.
 * If a constraint of "bmap" involves variables not in "context",
 * then it cannot appear in "context".
 * If a matching constraint is found, it is removed from "bmap".
 */
static __isl_give isl_basic_map *drop_inequalities(
        __isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context)
{
        int i1, i2;
        unsigned total, extra;

        if (!bmap || !context)
                return isl_basic_map_free(bmap);

        total = isl_basic_map_total_dim(context);
        extra = isl_basic_map_total_dim(bmap) - total;

        i1 = bmap->n_ineq - 1;
        i2 = context->n_ineq - 1;
        while (bmap && i1 >= 0 && i2 >= 0) {
                int cmp;

                if (isl_seq_first_non_zero(bmap->ineq[i1] + 1 + total,
                                            extra) != -1) {
                        --i1;
                        continue;
                }
                cmp = isl_basic_map_constraint_cmp(context, bmap->ineq[i1],
                                                        context->ineq[i2]);
                if (cmp < 0) {
                        --i2;
                        continue;
                }
                if (cmp > 0) {
                        --i1;
                        continue;
                }
                if (isl_int_eq(bmap->ineq[i1][0], context->ineq[i2][0])) {
                        bmap = isl_basic_map_cow(bmap);
                        if (isl_basic_map_drop_inequality(bmap, i1) < 0)
                                bmap = isl_basic_map_free(bmap);
                }
                --i1;
                --i2;
        }

        return bmap;
}

/* Drop all equalities from "bmap" that also appear in "context".
 * "context" is assumed to have only known local variables and
 * the initial local variables of "bmap" are assumed to be the same
 * as those of "context".
 *
 * Run through the equality constraints of "bmap" and "context"
 * in sorted order.
 * If a constraint of "bmap" involves variables not in "context",
 * then it cannot appear in "context".
 * If a matching constraint is found, it is removed from "bmap".
 */
static __isl_give isl_basic_map *drop_equalities(
        __isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context)
{
        int i1, i2;
        unsigned total, extra;

        if (!bmap || !context)
                return isl_basic_map_free(bmap);

        total = isl_basic_map_total_dim(context);
        extra = isl_basic_map_total_dim(bmap) - total;

        i1 = bmap->n_eq - 1;
        i2 = context->n_eq - 1;

        while (bmap && i1 >= 0 && i2 >= 0) {
                int last1, last2;

                if (isl_seq_first_non_zero(bmap->eq[i1] + 1 + total,
                                            extra) != -1)
                        break;
                last1 = isl_seq_last_non_zero(bmap->eq[i1] + 1, total);
                last2 = isl_seq_last_non_zero(context->eq[i2] + 1, total);
                if (last1 > last2) {
                        --i2;
                        continue;
                }
                if (last1 < last2) {
                        --i1;
                        continue;
                }
                if (isl_seq_eq(bmap->eq[i1], context->eq[i2], 1 + total)) {
                        bmap = isl_basic_map_cow(bmap);
                        if (isl_basic_map_drop_equality(bmap, i1) < 0)
                                bmap = isl_basic_map_free(bmap);
                }
                --i1;
                --i2;
        }

        return bmap;
}

/* Remove the constraints in "context" from "bmap".
 * "context" is assumed to have explicit representations
 * for all local variables.
 *
 * First align the divs of "bmap" to those of "context" and
 * sort the constraints.  Then drop all constraints from "bmap"
 * that appear in "context".
 */
__isl_give isl_basic_map *isl_basic_map_plain_gist(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context)
{
        isl_bool done, known;

        done = isl_basic_map_plain_is_universe(context);
        if (done == isl_bool_false)
                done = isl_basic_map_plain_is_universe(bmap);
        if (done == isl_bool_false)
                done = isl_basic_map_plain_is_empty(context);
        if (done == isl_bool_false)
                done = isl_basic_map_plain_is_empty(bmap);
        if (done < 0)
                goto error;
        if (done) {
                isl_basic_map_free(context);
                return bmap;
        }
        known = isl_basic_map_divs_known(context);
        if (known < 0)
                goto error;
        if (!known)
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "context has unknown divs", goto error);

        bmap = isl_basic_map_align_divs(bmap, context);
        bmap = isl_basic_map_gauss(bmap, NULL);
        bmap = isl_basic_map_sort_constraints(bmap);
        context = isl_basic_map_sort_constraints(context);

        bmap = drop_inequalities(bmap, context);
        bmap = drop_equalities(bmap, context);

        isl_basic_map_free(context);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_basic_map_free(bmap);
        isl_basic_map_free(context);
        return NULL;
}

/* Replace "map" by the disjunct at position "pos" and free "context".
 */
static __isl_give isl_map *replace_by_disjunct(__isl_take isl_map *map,
        int pos, __isl_take isl_basic_map *context)
{
        isl_basic_map *bmap;

        bmap = isl_basic_map_copy(map->p[pos]);
        isl_map_free(map);
        isl_basic_map_free(context);
        return isl_map_from_basic_map(bmap);
}

/* Remove the constraints in "context" from "map".
 * If any of the disjuncts in the result turns out to be the universe,
 * then return this universe.
 * "context" is assumed to have explicit representations
 * for all local variables.
 */
__isl_give isl_map *isl_map_plain_gist_basic_map(__isl_take isl_map *map,
        __isl_take isl_basic_map *context)
{
        int i;
        isl_bool univ, known;

        univ = isl_basic_map_plain_is_universe(context);
        if (univ < 0)
                goto error;
        if (univ) {
                isl_basic_map_free(context);
                return map;
        }
        known = isl_basic_map_divs_known(context);
        if (known < 0)
                goto error;
        if (!known)
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "context has unknown divs", goto error);

        map = isl_map_cow(map);
        if (!map)
                goto error;
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_plain_gist(map->p[i],
                                                isl_basic_map_copy(context));
                univ = isl_basic_map_plain_is_universe(map->p[i]);
                if (univ < 0)
                        goto error;
                if (univ && map->n > 1)
                        return replace_by_disjunct(map, i, context);
        }

        isl_basic_map_free(context);
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        if (map->n > 1)
                ISL_F_CLR(map, ISL_MAP_DISJOINT);
        return map;
error:
        isl_map_free(map);
        isl_basic_map_free(context);
        return NULL;
}

/* Replace "map" by a universe map in the same space and free "drop".
 */
static __isl_give isl_map *replace_by_universe(__isl_take isl_map *map,
        __isl_take isl_map *drop)
{
        isl_map *res;

        res = isl_map_universe(isl_map_get_space(map));
        isl_map_free(map);
        isl_map_free(drop);
        return res;
}

/* Return a map that has the same intersection with "context" as "map"
 * and that is as "simple" as possible.
 *
 * If "map" is already the universe, then we cannot make it any simpler.
 * Similarly, if "context" is the universe, then we cannot exploit it
 * to simplify "map"
 * If "map" and "context" are identical to each other, then we can
 * return the corresponding universe.
 *
 * If either "map" or "context" consists of multiple disjuncts,
 * then check if "context" happens to be a subset of "map",
 * in which case all constraints can be removed.
 * In case of multiple disjuncts, the standard procedure
 * may not be able to detect that all constraints can be removed.
 *
 * If none of these cases apply, we have to work a bit harder.
 * During this computation, we make use of a single disjunct context,
 * so if the original context consists of more than one disjunct
 * then we need to approximate the context by a single disjunct set.
 * Simply taking the simple hull may drop constraints that are
 * only implicitly available in each disjunct.  We therefore also
 * look for constraints among those defining "map" that are valid
 * for the context.  These can then be used to simplify away
 * the corresponding constraints in "map".
 */
static __isl_give isl_map *map_gist(__isl_take isl_map *map,
        __isl_take isl_map *context)
{
        int equal;
        int is_universe;
        int single_disjunct_map, single_disjunct_context;
        isl_bool subset;
        isl_basic_map *hull;

        is_universe = isl_map_plain_is_universe(map);
        if (is_universe >= 0 && !is_universe)
                is_universe = isl_map_plain_is_universe(context);
        if (is_universe < 0)
                goto error;
        if (is_universe) {
                isl_map_free(context);
                return map;
        }

        equal = isl_map_plain_is_equal(map, context);
        if (equal < 0)
                goto error;
        if (equal)
                return replace_by_universe(map, context);

        single_disjunct_map = isl_map_n_basic_map(map) == 1;
        single_disjunct_context = isl_map_n_basic_map(context) == 1;
        if (!single_disjunct_map || !single_disjunct_context) {
                subset = isl_map_is_subset(context, map);
                if (subset < 0)
                        goto error;
                if (subset)
                        return replace_by_universe(map, context);
        }

        context = isl_map_compute_divs(context);
        if (!context)
                goto error;
        if (single_disjunct_context) {
                hull = isl_map_simple_hull(context);
        } else {
                isl_ctx *ctx;
                isl_map_list *list;

                ctx = isl_map_get_ctx(map);
                list = isl_map_list_alloc(ctx, 2);
                list = isl_map_list_add(list, isl_map_copy(context));
                list = isl_map_list_add(list, isl_map_copy(map));
                hull = isl_map_unshifted_simple_hull_from_map_list(context,
                                                                    list);
        }
        return isl_map_gist_basic_map(map, hull);
error:
        isl_map_free(map);
        isl_map_free(context);
        return NULL;
}

__isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
        __isl_take isl_map *context)
{
        return isl_map_align_params_map_map_and(map, context, &map_gist);
}

struct isl_basic_set *isl_basic_set_gist(struct isl_basic_set *bset,
                                                struct isl_basic_set *context)
{
        return bset_from_bmap(isl_basic_map_gist(bset_to_bmap(bset),
                                                bset_to_bmap(context)));
}

__isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
        __isl_take isl_basic_set *context)
{
        return set_from_map(isl_map_gist_basic_map(set_to_map(set),
                                        bset_to_bmap(context)));
}

__isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set,
        __isl_take isl_basic_set *context)
{
        isl_space *space = isl_set_get_space(set);
        isl_basic_set *dom_context = isl_basic_set_universe(space);
        dom_context = isl_basic_set_intersect_params(dom_context, context);
        return isl_set_gist_basic_set(set, dom_context);
}

__isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
        __isl_take isl_set *context)
{
        return set_from_map(isl_map_gist(set_to_map(set), set_to_map(context)));
}

/* Compute the gist of "bmap" with respect to the constraints "context"
 * on the domain.
 */
__isl_give isl_basic_map *isl_basic_map_gist_domain(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *context)
{
        isl_space *space = isl_basic_map_get_space(bmap);
        isl_basic_map *bmap_context = isl_basic_map_universe(space);

        bmap_context = isl_basic_map_intersect_domain(bmap_context, context);
        return isl_basic_map_gist(bmap, bmap_context);
}

__isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map,
        __isl_take isl_set *context)
{
        isl_map *map_context = isl_map_universe(isl_map_get_space(map));
        map_context = isl_map_intersect_domain(map_context, context);
        return isl_map_gist(map, map_context);
}

__isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map,
        __isl_take isl_set *context)
{
        isl_map *map_context = isl_map_universe(isl_map_get_space(map));
        map_context = isl_map_intersect_range(map_context, context);
        return isl_map_gist(map, map_context);
}

__isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map,
        __isl_take isl_set *context)
{
        isl_map *map_context = isl_map_universe(isl_map_get_space(map));
        map_context = isl_map_intersect_params(map_context, context);
        return isl_map_gist(map, map_context);
}

__isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set,
        __isl_take isl_set *context)
{
        return isl_map_gist_params(set, context);
}

/* Quick check to see if two basic maps are disjoint.
 * In particular, we reduce the equalities and inequalities of
 * one basic map in the context of the equalities of the other
 * basic map and check if we get a contradiction.
 */
isl_bool isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        struct isl_vec *v = NULL;
        int *elim = NULL;
        unsigned total;
        int i;

        if (!bmap1 || !bmap2)
                return isl_bool_error;
        isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
                        return isl_bool_error);
        if (bmap1->n_div || bmap2->n_div)
                return isl_bool_false;
        if (!bmap1->n_eq && !bmap2->n_eq)
                return isl_bool_false;

        total = isl_space_dim(bmap1->dim, isl_dim_all);
        if (total == 0)
                return isl_bool_false;
        v = isl_vec_alloc(bmap1->ctx, 1 + total);
        if (!v)
                goto error;
        elim = isl_alloc_array(bmap1->ctx, int, total);
        if (!elim)
                goto error;
        compute_elimination_index(bmap1, elim);
        for (i = 0; i < bmap2->n_eq; ++i) {
                int reduced;
                reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
                                                        bmap1, elim);
                if (reduced && !isl_int_is_zero(v->block.data[0]) &&
                    isl_seq_first_non_zero(v->block.data + 1, total) == -1)
                        goto disjoint;
        }
        for (i = 0; i < bmap2->n_ineq; ++i) {
                int reduced;
                reduced = reduced_using_equalities(v->block.data,
                                                bmap2->ineq[i], bmap1, elim);
                if (reduced && isl_int_is_neg(v->block.data[0]) &&
                    isl_seq_first_non_zero(v->block.data + 1, total) == -1)
                        goto disjoint;
        }
        compute_elimination_index(bmap2, elim);
        for (i = 0; i < bmap1->n_ineq; ++i) {
                int reduced;
                reduced = reduced_using_equalities(v->block.data,
                                                bmap1->ineq[i], bmap2, elim);
                if (reduced && isl_int_is_neg(v->block.data[0]) &&
                    isl_seq_first_non_zero(v->block.data + 1, total) == -1)
                        goto disjoint;
        }
        isl_vec_free(v);
        free(elim);
        return isl_bool_false;
disjoint:
        isl_vec_free(v);
        free(elim);
        return isl_bool_true;
error:
        isl_vec_free(v);
        free(elim);
        return isl_bool_error;
}

int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1,
        __isl_keep isl_basic_set *bset2)
{
        return isl_basic_map_plain_is_disjoint(bset_to_bmap(bset1),
                                              bset_to_bmap(bset2));
}

/* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
 */
static isl_bool all_pairs(__isl_keep isl_map *map1, __isl_keep isl_map *map2,
        isl_bool (*test)(__isl_keep isl_basic_map *bmap1,
                __isl_keep isl_basic_map *bmap2))
{
        int i, j;

        if (!map1 || !map2)
                return isl_bool_error;

        for (i = 0; i < map1->n; ++i) {
                for (j = 0; j < map2->n; ++j) {
                        isl_bool d = test(map1->p[i], map2->p[j]);
                        if (d != isl_bool_true)
                                return d;
                }
        }

        return isl_bool_true;
}

/* Are "map1" and "map2" obviously disjoint, based on information
 * that can be derived without looking at the individual basic maps?
 *
 * In particular, if one of them is empty or if they live in different spaces
 * (ignoring parameters), then they are clearly disjoint.
 */
static isl_bool isl_map_plain_is_disjoint_global(__isl_keep isl_map *map1,
        __isl_keep isl_map *map2)
{
        isl_bool disjoint;
        isl_bool match;

        if (!map1 || !map2)
                return isl_bool_error;

        disjoint = isl_map_plain_is_empty(map1);
        if (disjoint < 0 || disjoint)
                return disjoint;

        disjoint = isl_map_plain_is_empty(map2);
        if (disjoint < 0 || disjoint)
                return disjoint;

        match = isl_space_tuple_is_equal(map1->dim, isl_dim_in,
                                map2->dim, isl_dim_in);
        if (match < 0 || !match)
                return match < 0 ? isl_bool_error : isl_bool_true;

        match = isl_space_tuple_is_equal(map1->dim, isl_dim_out,
                                map2->dim, isl_dim_out);
        if (match < 0 || !match)
                return match < 0 ? isl_bool_error : isl_bool_true;

        return isl_bool_false;
}

/* Are "map1" and "map2" obviously disjoint?
 *
 * If one of them is empty or if they live in different spaces (ignoring
 * parameters), then they are clearly disjoint.
 * This is checked by isl_map_plain_is_disjoint_global.
 *
 * If they have different parameters, then we skip any further tests.
 *
 * If they are obviously equal, but not obviously empty, then we will
 * not be able to detect if they are disjoint.
 *
 * Otherwise we check if each basic map in "map1" is obviously disjoint
 * from each basic map in "map2".
 */
isl_bool isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
        __isl_keep isl_map *map2)
{
        isl_bool disjoint;
        isl_bool intersect;
        isl_bool match;

        disjoint = isl_map_plain_is_disjoint_global(map1, map2);
        if (disjoint < 0 || disjoint)
                return disjoint;

        match = isl_space_match(map1->dim, isl_dim_param,
                                map2->dim, isl_dim_param);
        if (match < 0 || !match)
                return match < 0 ? isl_bool_error : isl_bool_false;

        intersect = isl_map_plain_is_equal(map1, map2);
        if (intersect < 0 || intersect)
                return intersect < 0 ? isl_bool_error : isl_bool_false;

        return all_pairs(map1, map2, &isl_basic_map_plain_is_disjoint);
}

/* Are "map1" and "map2" disjoint?
 *
 * They are disjoint if they are "obviously disjoint" or if one of them
 * is empty.  Otherwise, they are not disjoint if one of them is universal.
 * If the two inputs are (obviously) equal and not empty, then they are
 * not disjoint.
 * If none of these cases apply, then check if all pairs of basic maps
 * are disjoint.
 */
isl_bool isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{
        isl_bool disjoint;
        isl_bool intersect;

        disjoint = isl_map_plain_is_disjoint_global(map1, map2);
        if (disjoint < 0 || disjoint)
                return disjoint;

        disjoint = isl_map_is_empty(map1);
        if (disjoint < 0 || disjoint)
                return disjoint;

        disjoint = isl_map_is_empty(map2);
        if (disjoint < 0 || disjoint)
                return disjoint;

        intersect = isl_map_plain_is_universe(map1);
        if (intersect < 0 || intersect)
                return intersect < 0 ? isl_bool_error : isl_bool_false;

        intersect = isl_map_plain_is_universe(map2);
        if (intersect < 0 || intersect)
                return intersect < 0 ? isl_bool_error : isl_bool_false;

        intersect = isl_map_plain_is_equal(map1, map2);
        if (intersect < 0 || intersect)
                return isl_bool_not(intersect);

        return all_pairs(map1, map2, &isl_basic_map_is_disjoint);
}

/* Are "bmap1" and "bmap2" disjoint?
 *
 * They are disjoint if they are "obviously disjoint" or if one of them
 * is empty.  Otherwise, they are not disjoint if one of them is universal.
 * If none of these cases apply, we compute the intersection and see if
 * the result is empty.
 */
isl_bool isl_basic_map_is_disjoint(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        isl_bool disjoint;
        isl_bool intersect;
        isl_basic_map *test;

        disjoint = isl_basic_map_plain_is_disjoint(bmap1, bmap2);
        if (disjoint < 0 || disjoint)
                return disjoint;

        disjoint = isl_basic_map_is_empty(bmap1);
        if (disjoint < 0 || disjoint)
                return disjoint;

        disjoint = isl_basic_map_is_empty(bmap2);
        if (disjoint < 0 || disjoint)
                return disjoint;

        intersect = isl_basic_map_plain_is_universe(bmap1);
        if (intersect < 0 || intersect)
                return intersect < 0 ? isl_bool_error : isl_bool_false;

        intersect = isl_basic_map_plain_is_universe(bmap2);
        if (intersect < 0 || intersect)
                return intersect < 0 ? isl_bool_error : isl_bool_false;

        test = isl_basic_map_intersect(isl_basic_map_copy(bmap1),
                isl_basic_map_copy(bmap2));
        disjoint = isl_basic_map_is_empty(test);
        isl_basic_map_free(test);

        return disjoint;
}

/* Are "bset1" and "bset2" disjoint?
 */
isl_bool isl_basic_set_is_disjoint(__isl_keep isl_basic_set *bset1,
        __isl_keep isl_basic_set *bset2)
{
        return isl_basic_map_is_disjoint(bset1, bset2);
}

isl_bool isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
        __isl_keep isl_set *set2)
{
        return isl_map_plain_is_disjoint(set_to_map(set1), set_to_map(set2));
}

/* Are "set1" and "set2" disjoint?
 */
isl_bool isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
{
        return isl_map_is_disjoint(set1, set2);
}

/* Is "v" equal to 0, 1 or -1?
 */
static int is_zero_or_one(isl_int v)
{
        return isl_int_is_zero(v) || isl_int_is_one(v) || isl_int_is_negone(v);
}

/* Check if we can combine a given div with lower bound l and upper
 * bound u with some other div and if so return that other div.
 * Otherwise return -1.
 *
 * We first check that
 *      - the bounds are opposites of each other (except for the constant
 *        term)
 *      - the bounds do not reference any other div
 *      - no div is defined in terms of this div
 *
 * Let m be the size of the range allowed on the div by the bounds.
 * That is, the bounds are of the form
 *
 *      e <= a <= e + m - 1
 *
 * with e some expression in the other variables.
 * We look for another div b such that no third div is defined in terms
 * of this second div b and such that in any constraint that contains
 * a (except for the given lower and upper bound), also contains b
 * with a coefficient that is m times that of b.
 * That is, all constraints (execpt for the lower and upper bound)
 * are of the form
 *
 *      e + f (a + m b) >= 0
 *
 * Furthermore, in the constraints that only contain b, the coefficient
 * of b should be equal to 1 or -1.
 * If so, we return b so that "a + m b" can be replaced by
 * a single div "c = a + m b".
 */
static int div_find_coalesce(struct isl_basic_map *bmap, int *pairs,
        unsigned div, unsigned l, unsigned u)
{
        int i, j;
        unsigned dim;
        int coalesce = -1;

        if (bmap->n_div <= 1)
                return -1;
        dim = isl_space_dim(bmap->dim, isl_dim_all);
        if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim, div) != -1)
                return -1;
        if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + dim + div + 1,
                                   bmap->n_div - div - 1) != -1)
                return -1;
        if (!isl_seq_is_neg(bmap->ineq[l] + 1, bmap->ineq[u] + 1,
                            dim + bmap->n_div))
                return -1;

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (!isl_int_is_zero(bmap->div[i][1 + 1 + dim + div]))
                        return -1;
        }

        isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
        if (isl_int_is_neg(bmap->ineq[l][0])) {
                isl_int_sub(bmap->ineq[l][0],
                            bmap->ineq[l][0], bmap->ineq[u][0]);
                bmap = isl_basic_map_copy(bmap);
                bmap = isl_basic_map_set_to_empty(bmap);
                isl_basic_map_free(bmap);
                return -1;
        }
        isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
        for (i = 0; i < bmap->n_div; ++i) {
                if (i == div)
                        continue;
                if (!pairs[i])
                        continue;
                for (j = 0; j < bmap->n_div; ++j) {
                        if (isl_int_is_zero(bmap->div[j][0]))
                                continue;
                        if (!isl_int_is_zero(bmap->div[j][1 + 1 + dim + i]))
                                break;
                }
                if (j < bmap->n_div)
                        continue;
                for (j = 0; j < bmap->n_ineq; ++j) {
                        int valid;
                        if (j == l || j == u)
                                continue;
                        if (isl_int_is_zero(bmap->ineq[j][1 + dim + div])) {
                                if (is_zero_or_one(bmap->ineq[j][1 + dim + i]))
                                        continue;
                                break;
                        }
                        if (isl_int_is_zero(bmap->ineq[j][1 + dim + i]))
                                break;
                        isl_int_mul(bmap->ineq[j][1 + dim + div],
                                    bmap->ineq[j][1 + dim + div],
                                    bmap->ineq[l][0]);
                        valid = isl_int_eq(bmap->ineq[j][1 + dim + div],
                                           bmap->ineq[j][1 + dim + i]);
                        isl_int_divexact(bmap->ineq[j][1 + dim + div],
                                         bmap->ineq[j][1 + dim + div],
                                         bmap->ineq[l][0]);
                        if (!valid)
                                break;
                }
                if (j < bmap->n_ineq)
                        continue;
                coalesce = i;
                break;
        }
        isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
        isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
        return coalesce;
}

/* Internal data structure used during the construction and/or evaluation of
 * an inequality that ensures that a pair of bounds always allows
 * for an integer value.
 *
 * "tab" is the tableau in which the inequality is evaluated.  It may
 * be NULL until it is actually needed.
 * "v" contains the inequality coefficients.
 * "g", "fl" and "fu" are temporary scalars used during the construction and
 * evaluation.
 */
struct test_ineq_data {
        struct isl_tab *tab;
        isl_vec *v;
        isl_int g;
        isl_int fl;
        isl_int fu;
};

/* Free all the memory allocated by the fields of "data".
 */
static void test_ineq_data_clear(struct test_ineq_data *data)
{
        isl_tab_free(data->tab);
        isl_vec_free(data->v);
        isl_int_clear(data->g);
        isl_int_clear(data->fl);
        isl_int_clear(data->fu);
}

/* Is the inequality stored in data->v satisfied by "bmap"?
 * That is, does it only attain non-negative values?
 * data->tab is a tableau corresponding to "bmap".
 */
static isl_bool test_ineq_is_satisfied(__isl_keep isl_basic_map *bmap,
        struct test_ineq_data *data)
{
        isl_ctx *ctx;
        enum isl_lp_result res;

        ctx = isl_basic_map_get_ctx(bmap);
        if (!data->tab)
                data->tab = isl_tab_from_basic_map(bmap, 0);
        res = isl_tab_min(data->tab, data->v->el, ctx->one, &data->g, NULL, 0);
        if (res == isl_lp_error)
                return isl_bool_error;
        return res == isl_lp_ok && isl_int_is_nonneg(data->g);
}

/* Given a lower and an upper bound on div i, do they always allow
 * for an integer value of the given div?
 * Determine this property by constructing an inequality
 * such that the property is guaranteed when the inequality is nonnegative.
 * The lower bound is inequality l, while the upper bound is inequality u.
 * The constructed inequality is stored in data->v.
 *
 * Let the upper bound be
 *
 *      -n_u a + e_u >= 0
 *
 * and the lower bound
 *
 *      n_l a + e_l >= 0
 *
 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
 * We have
 *
 *      - f_u e_l <= f_u f_l g a <= f_l e_u
 *
 * Since all variables are integer valued, this is equivalent to
 *
 *      - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
 *
 * If this interval is at least f_u f_l g, then it contains at least
 * one integer value for a.
 * That is, the test constraint is
 *
 *      f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
 *
 * or
 *
 *      f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
 *
 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
 * then the constraint can be scaled down by a factor g',
 * with the constant term replaced by
 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
 * Note that the result of applying Fourier-Motzkin to this pair
 * of constraints is
 *
 *      f_l e_u + f_u e_l >= 0
 *
 * If the constant term of the scaled down version of this constraint,
 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
 * term of the scaled down test constraint, then the test constraint
 * is known to hold and no explicit evaluation is required.
 * This is essentially the Omega test.
 *
 * If the test constraint consists of only a constant term, then
 * it is sufficient to look at the sign of this constant term.
 */
static isl_bool int_between_bounds(__isl_keep isl_basic_map *bmap, int i,
        int l, int u, struct test_ineq_data *data)
{
        unsigned offset, n_div;
        offset = isl_basic_map_offset(bmap, isl_dim_div);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);

        isl_int_gcd(data->g,
                    bmap->ineq[l][offset + i], bmap->ineq[u][offset + i]);
        isl_int_divexact(data->fl, bmap->ineq[l][offset + i], data->g);
        isl_int_divexact(data->fu, bmap->ineq[u][offset + i], data->g);
        isl_int_neg(data->fu, data->fu);
        isl_seq_combine(data->v->el, data->fl, bmap->ineq[u],
                        data->fu, bmap->ineq[l], offset + n_div);
        isl_int_mul(data->g, data->g, data->fl);
        isl_int_mul(data->g, data->g, data->fu);
        isl_int_sub(data->g, data->g, data->fl);
        isl_int_sub(data->g, data->g, data->fu);
        isl_int_add_ui(data->g, data->g, 1);
        isl_int_sub(data->fl, data->v->el[0], data->g);

        isl_seq_gcd(data->v->el + 1, offset - 1 + n_div, &data->g);
        if (isl_int_is_zero(data->g))
                return isl_int_is_nonneg(data->fl);
        if (isl_int_is_one(data->g)) {
                isl_int_set(data->v->el[0], data->fl);
                return test_ineq_is_satisfied(bmap, data);
        }
        isl_int_fdiv_q(data->fl, data->fl, data->g);
        isl_int_fdiv_q(data->v->el[0], data->v->el[0], data->g);
        if (isl_int_eq(data->fl, data->v->el[0]))
                return isl_bool_true;
        isl_int_set(data->v->el[0], data->fl);
        isl_seq_scale_down(data->v->el + 1, data->v->el + 1, data->g,
                            offset - 1 + n_div);

        return test_ineq_is_satisfied(bmap, data);
}

/* Remove more kinds of divs that are not strictly needed.
 * In particular, if all pairs of lower and upper bounds on a div
 * are such that they allow at least one integer value of the div,
 * then we can eliminate the div using Fourier-Motzkin without
 * introducing any spurious solutions.
 *
 * If at least one of the two constraints has a unit coefficient for the div,
 * then the presence of such a value is guaranteed so there is no need to check.
 * In particular, the value attained by the bound with unit coefficient
 * can serve as this intermediate value.
 */
static struct isl_basic_map *drop_more_redundant_divs(
        struct isl_basic_map *bmap, int *pairs, int n)
{
        isl_ctx *ctx;
        struct test_ineq_data data = { NULL, NULL };
        unsigned off, n_div;
        int remove = -1;

        isl_int_init(data.g);
        isl_int_init(data.fl);
        isl_int_init(data.fu);

        if (!bmap)
                goto error;

        ctx = isl_basic_map_get_ctx(bmap);
        off = isl_basic_map_offset(bmap, isl_dim_div);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        data.v = isl_vec_alloc(ctx, off + n_div);
        if (!data.v)
                goto error;

        while (n > 0) {
                int i, l, u;
                int best = -1;
                isl_bool has_int;

                for (i = 0; i < n_div; ++i) {
                        if (!pairs[i])
                                continue;
                        if (best >= 0 && pairs[best] <= pairs[i])
                                continue;
                        best = i;
                }

                i = best;
                for (l = 0; l < bmap->n_ineq; ++l) {
                        if (!isl_int_is_pos(bmap->ineq[l][off + i]))
                                continue;
                        if (isl_int_is_one(bmap->ineq[l][off + i]))
                                continue;
                        for (u = 0; u < bmap->n_ineq; ++u) {
                                if (!isl_int_is_neg(bmap->ineq[u][off + i]))
                                        continue;
                                if (isl_int_is_negone(bmap->ineq[u][off + i]))
                                        continue;
                                has_int = int_between_bounds(bmap, i, l, u,
                                                                &data);
                                if (has_int < 0)
                                        goto error;
                                if (data.tab && data.tab->empty)
                                        break;
                                if (!has_int)
                                        break;
                        }
                        if (u < bmap->n_ineq)
                                break;
                }
                if (data.tab && data.tab->empty) {
                        bmap = isl_basic_map_set_to_empty(bmap);
                        break;
                }
                if (l == bmap->n_ineq) {
                        remove = i;
                        break;
                }
                pairs[i] = 0;
                --n;
        }

        test_ineq_data_clear(&data);

        free(pairs);

        if (remove < 0)
                return bmap;

        bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1);
        return isl_basic_map_drop_redundant_divs(bmap);
error:
        free(pairs);
        isl_basic_map_free(bmap);
        test_ineq_data_clear(&data);
        return NULL;
}

/* Given a pair of divs div1 and div2 such that, except for the lower bound l
 * and the upper bound u, div1 always occurs together with div2 in the form
 * (div1 + m div2), where m is the constant range on the variable div1
 * allowed by l and u, replace the pair div1 and div2 by a single
 * div that is equal to div1 + m div2.
 *
 * The new div will appear in the location that contains div2.
 * We need to modify all constraints that contain
 * div2 = (div - div1) / m
 * The coefficient of div2 is known to be equal to 1 or -1.
 * (If a constraint does not contain div2, it will also not contain div1.)
 * If the constraint also contains div1, then we know they appear
 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
 * i.e., the coefficient of div is f.
 *
 * Otherwise, we first need to introduce div1 into the constraint.
 * Let the l be
 *
 *      div1 + f >=0
 *
 * and u
 *
 *      -div1 + f' >= 0
 *
 * A lower bound on div2
 *
 *      div2 + t >= 0
 *
 * can be replaced by
 *
 *      m div2 + div1 + m t + f >= 0
 *
 * An upper bound
 *
 *      -div2 + t >= 0
 *
 * can be replaced by
 *
 *      -(m div2 + div1) + m t + f' >= 0
 *
 * These constraint are those that we would obtain from eliminating
 * div1 using Fourier-Motzkin.
 *
 * After all constraints have been modified, we drop the lower and upper
 * bound and then drop div1.
 */
static struct isl_basic_map *coalesce_divs(struct isl_basic_map *bmap,
        unsigned div1, unsigned div2, unsigned l, unsigned u)
{
        isl_ctx *ctx;
        isl_int m;
        unsigned dim, total;
        int i;

        ctx = isl_basic_map_get_ctx(bmap);

        dim = isl_space_dim(bmap->dim, isl_dim_all);
        total = 1 + dim + bmap->n_div;

        isl_int_init(m);
        isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
        isl_int_add_ui(m, m, 1);

        for (i = 0; i < bmap->n_ineq; ++i) {
                if (i == l || i == u)
                        continue;
                if (isl_int_is_zero(bmap->ineq[i][1 + dim + div2]))
                        continue;
                if (isl_int_is_zero(bmap->ineq[i][1 + dim + div1])) {
                        if (isl_int_is_pos(bmap->ineq[i][1 + dim + div2]))
                                isl_seq_combine(bmap->ineq[i], m, bmap->ineq[i],
                                                ctx->one, bmap->ineq[l], total);
                        else
                                isl_seq_combine(bmap->ineq[i], m, bmap->ineq[i],
                                                ctx->one, bmap->ineq[u], total);
                }
                isl_int_set(bmap->ineq[i][1 + dim + div2],
                            bmap->ineq[i][1 + dim + div1]);
                isl_int_set_si(bmap->ineq[i][1 + dim + div1], 0);
        }

        isl_int_clear(m);
        if (l > u) {
                isl_basic_map_drop_inequality(bmap, l);
                isl_basic_map_drop_inequality(bmap, u);
        } else {
                isl_basic_map_drop_inequality(bmap, u);
                isl_basic_map_drop_inequality(bmap, l);
        }
        bmap = isl_basic_map_drop_div(bmap, div1);
        return bmap;
}

/* First check if we can coalesce any pair of divs and
 * then continue with dropping more redundant divs.
 *
 * We loop over all pairs of lower and upper bounds on a div
 * with coefficient 1 and -1, respectively, check if there
 * is any other div "c" with which we can coalesce the div
 * and if so, perform the coalescing.
 */
static struct isl_basic_map *coalesce_or_drop_more_redundant_divs(
        struct isl_basic_map *bmap, int *pairs, int n)
{
        int i, l, u;
        unsigned dim;

        dim = isl_space_dim(bmap->dim, isl_dim_all);

        for (i = 0; i < bmap->n_div; ++i) {
                if (!pairs[i])
                        continue;
                for (l = 0; l < bmap->n_ineq; ++l) {
                        if (!isl_int_is_one(bmap->ineq[l][1 + dim + i]))
                                continue;
                        for (u = 0; u < bmap->n_ineq; ++u) {
                                int c;

                                if (!isl_int_is_negone(bmap->ineq[u][1+dim+i]))
                                        continue;
                                c = div_find_coalesce(bmap, pairs, i, l, u);
                                if (c < 0)
                                        continue;
                                free(pairs);
                                bmap = coalesce_divs(bmap, i, c, l, u);
                                return isl_basic_map_drop_redundant_divs(bmap);
                        }
                }
        }

        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
                free(pairs);
                return bmap;
        }

        return drop_more_redundant_divs(bmap, pairs, n);
}

/* Are the "n" coefficients starting at "first" of inequality constraints
 * "i" and "j" of "bmap" equal to each other?
 */
static int is_parallel_part(__isl_keep isl_basic_map *bmap, int i, int j,
        int first, int n)
{
        return isl_seq_eq(bmap->ineq[i] + first, bmap->ineq[j] + first, n);
}

/* Are the "n" coefficients starting at "first" of inequality constraints
 * "i" and "j" of "bmap" opposite to each other?
 */
static int is_opposite_part(__isl_keep isl_basic_map *bmap, int i, int j,
        int first, int n)
{
        return isl_seq_is_neg(bmap->ineq[i] + first, bmap->ineq[j] + first, n);
}

/* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
 * apart from the constant term?
 */
static isl_bool is_opposite(__isl_keep isl_basic_map *bmap, int i, int j)
{
        unsigned total;

        total = isl_basic_map_dim(bmap, isl_dim_all);
        return is_opposite_part(bmap, i, j, 1, total);
}

/* Are inequality constraints "i" and "j" of "bmap" equal to each other,
 * apart from the constant term and the coefficient at position "pos"?
 */
static int is_parallel_except(__isl_keep isl_basic_map *bmap, int i, int j,
        int pos)
{
        unsigned total;

        total = isl_basic_map_dim(bmap, isl_dim_all);
        return is_parallel_part(bmap, i, j, 1, pos - 1) &&
                is_parallel_part(bmap, i, j, pos + 1, total - pos);
}

/* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
 * apart from the constant term and the coefficient at position "pos"?
 */
static int is_opposite_except(__isl_keep isl_basic_map *bmap, int i, int j,
        int pos)
{
        unsigned total;

        total = isl_basic_map_dim(bmap, isl_dim_all);
        return is_opposite_part(bmap, i, j, 1, pos - 1) &&
                is_opposite_part(bmap, i, j, pos + 1, total - pos);
}

/* Restart isl_basic_map_drop_redundant_divs after "bmap" has
 * been modified, simplying it if "simplify" is set.
 * Free the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 */
static __isl_give isl_basic_map *drop_redundant_divs_again(
        __isl_take isl_basic_map *bmap, __isl_take int *pairs, int simplify)
{
        if (simplify)
                bmap = isl_basic_map_simplify(bmap);
        free(pairs);
        return isl_basic_map_drop_redundant_divs(bmap);
}

/* Is "div" the single unknown existentially quantified variable
 * in inequality constraint "ineq" of "bmap"?
 * "div" is known to have a non-zero coefficient in "ineq".
 */
static isl_bool single_unknown(__isl_keep isl_basic_map *bmap, int ineq,
        int div)
{
        int i;
        unsigned n_div, o_div;
        isl_bool known;

        known = isl_basic_map_div_is_known(bmap, div);
        if (known < 0 || known)
                return isl_bool_not(known);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        if (n_div == 1)
                return isl_bool_true;
        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        for (i = 0; i < n_div; ++i) {
                isl_bool known;

                if (i == div)
                        continue;
                if (isl_int_is_zero(bmap->ineq[ineq][o_div + i]))
                        continue;
                known = isl_basic_map_div_is_known(bmap, i);
                if (known < 0 || !known)
                        return known;
        }

        return isl_bool_true;
}

/* Does integer division "div" have coefficient 1 in inequality constraint
 * "ineq" of "map"?
 */
static isl_bool has_coef_one(__isl_keep isl_basic_map *bmap, int div, int ineq)
{
        unsigned o_div;

        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        if (isl_int_is_one(bmap->ineq[ineq][o_div + div]))
                return isl_bool_true;

        return isl_bool_false;
}

/* Turn inequality constraint "ineq" of "bmap" into an equality and
 * then try and drop redundant divs again,
 * freeing the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 */
static __isl_give isl_basic_map *set_eq_and_try_again(
        __isl_take isl_basic_map *bmap, int ineq, __isl_take int *pairs)
{
        bmap = isl_basic_map_cow(bmap);
        isl_basic_map_inequality_to_equality(bmap, ineq);
        return drop_redundant_divs_again(bmap, pairs, 1);
}

/* Drop the integer division at position "div", along with the two
 * inequality constraints "ineq1" and "ineq2" in which it appears
 * from "bmap" and then try and drop redundant divs again,
 * freeing the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 */
static __isl_give isl_basic_map *drop_div_and_try_again(
        __isl_take isl_basic_map *bmap, int div, int ineq1, int ineq2,
        __isl_take int *pairs)
{
        if (ineq1 > ineq2) {
                isl_basic_map_drop_inequality(bmap, ineq1);
                isl_basic_map_drop_inequality(bmap, ineq2);
        } else {
                isl_basic_map_drop_inequality(bmap, ineq2);
                isl_basic_map_drop_inequality(bmap, ineq1);
        }
        bmap = isl_basic_map_drop_div(bmap, div);
        return drop_redundant_divs_again(bmap, pairs, 0);
}

/* Given two inequality constraints
 *
 *      f(x) + n d + c >= 0,            (ineq)
 *
 * with d the variable at position "pos", and
 *
 *      f(x) + c0 >= 0,                 (lower)
 *
 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
 * determined by the first constraint.
 * That is, store
 *
 *      ceil((c0 - c)/n)
 *
 * in *l.
 */
static void lower_bound_from_parallel(__isl_keep isl_basic_map *bmap,
        int ineq, int lower, int pos, isl_int *l)
{
        isl_int_neg(*l, bmap->ineq[ineq][0]);
        isl_int_add(*l, *l, bmap->ineq[lower][0]);
        isl_int_cdiv_q(*l, *l, bmap->ineq[ineq][pos]);
}

/* Given two inequality constraints
 *
 *      f(x) + n d + c >= 0,            (ineq)
 *
 * with d the variable at position "pos", and
 *
 *      -f(x) - c0 >= 0,                (upper)
 *
 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
 * determined by the first constraint.
 * That is, store
 *
 *      ceil((-c1 - c)/n)
 *
 * in *u.
 */
static void lower_bound_from_opposite(__isl_keep isl_basic_map *bmap,
        int ineq, int upper, int pos, isl_int *u)
{
        isl_int_neg(*u, bmap->ineq[ineq][0]);
        isl_int_sub(*u, *u, bmap->ineq[upper][0]);
        isl_int_cdiv_q(*u, *u, bmap->ineq[ineq][pos]);
}

/* Given a lower bound constraint "ineq" on "div" in "bmap",
 * does the corresponding lower bound have a fixed value in "bmap"?
 *
 * In particular, "ineq" is of the form
 *
 *      f(x) + n d + c >= 0
 *
 * with n > 0, c the constant term and
 * d the existentially quantified variable "div".
 * That is, the lower bound is
 *
 *      ceil((-f(x) - c)/n)
 *
 * Look for a pair of constraints
 *
 *      f(x) + c0 >= 0
 *      -f(x) + c1 >= 0
 *
 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
 * That is, check that
 *
 *      ceil((-c1 - c)/n) = ceil((c0 - c)/n)
 *
 * If so, return the index of inequality f(x) + c0 >= 0.
 * Otherwise, return -1.
 */
static int lower_bound_is_cst(__isl_keep isl_basic_map *bmap, int div, int ineq)
{
        int i;
        int lower = -1, upper = -1;
        unsigned o_div, n_div;
        isl_int l, u;
        int equal;

        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        for (i = 0; i < bmap->n_ineq && (lower < 0 || upper < 0); ++i) {
                if (i == ineq)
                        continue;
                if (!isl_int_is_zero(bmap->ineq[i][o_div + div]))
                        continue;
                if (lower < 0 &&
                    is_parallel_except(bmap, ineq, i, o_div + div)) {
                        lower = i;
                        continue;
                }
                if (upper < 0 &&
                    is_opposite_except(bmap, ineq, i, o_div + div)) {
                        upper = i;
                }
        }

        if (lower < 0 || upper < 0)
                return -1;

        isl_int_init(l);
        isl_int_init(u);

        lower_bound_from_parallel(bmap, ineq, lower, o_div + div, &l);
        lower_bound_from_opposite(bmap, ineq, upper, o_div + div, &u);

        equal = isl_int_eq(l, u);

        isl_int_clear(l);
        isl_int_clear(u);

        return equal ? lower : -1;
}

/* Given a lower bound constraint "ineq" on the existentially quantified
 * variable "div", such that the corresponding lower bound has
 * a fixed value in "bmap", assign this fixed value to the variable and
 * then try and drop redundant divs again,
 * freeing the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 * "lower" determines the constant value for the lower bound.
 *
 * In particular, "ineq" is of the form
 *
 *      f(x) + n d + c >= 0,
 *
 * while "lower" is of the form
 *
 *      f(x) + c0 >= 0
 *
 * The lower bound is ceil((-f(x) - c)/n) and its constant value
 * is ceil((c0 - c)/n).
 */
static __isl_give isl_basic_map *fix_cst_lower(__isl_take isl_basic_map *bmap,
        int div, int ineq, int lower, int *pairs)
{
        isl_int c;
        unsigned o_div;

        isl_int_init(c);

        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        lower_bound_from_parallel(bmap, ineq, lower, o_div + div, &c);
        bmap = isl_basic_map_fix(bmap, isl_dim_div, div, c);
        free(pairs);

        isl_int_clear(c);

        return isl_basic_map_drop_redundant_divs(bmap);
}

/* Remove divs that are not strictly needed based on the inequality
 * constraints.
 * In particular, if a div only occurs positively (or negatively)
 * in constraints, then it can simply be dropped.
 * Also, if a div occurs in only two constraints and if moreover
 * those two constraints are opposite to each other, except for the constant
 * term and if the sum of the constant terms is such that for any value
 * of the other values, there is always at least one integer value of the
 * div, i.e., if one plus this sum is greater than or equal to
 * the (absolute value) of the coefficient of the div in the constraints,
 * then we can also simply drop the div.
 *
 * If an existentially quantified variable does not have an explicit
 * representation, appears in only a single lower bound that does not
 * involve any other such existentially quantified variables and appears
 * in this lower bound with coefficient 1,
 * then fix the variable to the value of the lower bound.  That is,
 * turn the inequality into an equality.
 * If for any value of the other variables, there is any value
 * for the existentially quantified variable satisfying the constraints,
 * then this lower bound also satisfies the constraints.
 * It is therefore safe to pick this lower bound.
 *
 * The same reasoning holds even if the coefficient is not one.
 * However, fixing the variable to the value of the lower bound may
 * in general introduce an extra integer division, in which case
 * it may be better to pick another value.
 * If this integer division has a known constant value, then plugging
 * in this constant value removes the existentially quantified variable
 * completely.  In particular, if the lower bound is of the form
 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
 * then the existentially quantified variable can be assigned this
 * shared value.
 *
 * We skip divs that appear in equalities or in the definition of other divs.
 * Divs that appear in the definition of other divs usually occur in at least
 * 4 constraints, but the constraints may have been simplified.
 *
 * If any divs are left after these simple checks then we move on
 * to more complicated cases in drop_more_redundant_divs.
 */
static __isl_give isl_basic_map *isl_basic_map_drop_redundant_divs_ineq(
        __isl_take isl_basic_map *bmap)
{
        int i, j;
        unsigned off;
        int *pairs = NULL;
        int n = 0;

        if (!bmap)
                goto error;
        if (bmap->n_div == 0)
                return bmap;

        off = isl_space_dim(bmap->dim, isl_dim_all);
        pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
        if (!pairs)
                goto error;

        for (i = 0; i < bmap->n_div; ++i) {
                int pos, neg;
                int last_pos, last_neg;
                int redundant;
                int defined;
                isl_bool opp, set_div;

                defined = !isl_int_is_zero(bmap->div[i][0]);
                for (j = i; j < bmap->n_div; ++j)
                        if (!isl_int_is_zero(bmap->div[j][1 + 1 + off + i]))
                                break;
                if (j < bmap->n_div)
                        continue;
                for (j = 0; j < bmap->n_eq; ++j)
                        if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
                                break;
                if (j < bmap->n_eq)
                        continue;
                ++n;
                pos = neg = 0;
                for (j = 0; j < bmap->n_ineq; ++j) {
                        if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
                                last_pos = j;
                                ++pos;
                        }
                        if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
                                last_neg = j;
                                ++neg;
                        }
                }
                pairs[i] = pos * neg;
                if (pairs[i] == 0) {
                        for (j = bmap->n_ineq - 1; j >= 0; --j)
                                if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
                                        isl_basic_map_drop_inequality(bmap, j);
                        bmap = isl_basic_map_drop_div(bmap, i);
                        return drop_redundant_divs_again(bmap, pairs, 0);
                }
                if (pairs[i] != 1)
                        opp = isl_bool_false;
                else
                        opp = is_opposite(bmap, last_pos, last_neg);
                if (opp < 0)
                        goto error;
                if (!opp) {
                        int lower;
                        isl_bool single, one;

                        if (pos != 1)
                                continue;
                        single = single_unknown(bmap, last_pos, i);
                        if (single < 0)
                                goto error;
                        if (!single)
                                continue;
                        one = has_coef_one(bmap, i, last_pos);
                        if (one < 0)
                                goto error;
                        if (one)
                                return set_eq_and_try_again(bmap, last_pos,
                                                            pairs);
                        lower = lower_bound_is_cst(bmap, i, last_pos);
                        if (lower >= 0)
                                return fix_cst_lower(bmap, i, last_pos, lower,
                                                pairs);
                        continue;
                }

                isl_int_add(bmap->ineq[last_pos][0],
                            bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
                isl_int_add_ui(bmap->ineq[last_pos][0],
                               bmap->ineq[last_pos][0], 1);
                redundant = isl_int_ge(bmap->ineq[last_pos][0],
                                bmap->ineq[last_pos][1+off+i]);
                isl_int_sub_ui(bmap->ineq[last_pos][0],
                               bmap->ineq[last_pos][0], 1);
                isl_int_sub(bmap->ineq[last_pos][0],
                            bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
                if (redundant)
                        return drop_div_and_try_again(bmap, i,
                                                    last_pos, last_neg, pairs);
                if (defined)
                        set_div = isl_bool_false;
                else
                        set_div = ok_to_set_div_from_bound(bmap, i, last_pos);
                if (set_div < 0)
                        return isl_basic_map_free(bmap);
                if (set_div) {
                        bmap = set_div_from_lower_bound(bmap, i, last_pos);
                        return drop_redundant_divs_again(bmap, pairs, 1);
                }
                pairs[i] = 0;
                --n;
        }

        if (n > 0)
                return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);

        free(pairs);
        return bmap;
error:
        free(pairs);
        isl_basic_map_free(bmap);
        return NULL;
}

/* Consider the coefficients at "c" as a row vector and replace
 * them with their product with "T".  "T" is assumed to be a square matrix.
 */
static isl_stat preimage(isl_int *c, __isl_keep isl_mat *T)
{
        int n;
        isl_ctx *ctx;
        isl_vec *v;

        if (!T)
                return isl_stat_error;
        n = isl_mat_rows(T);
        if (isl_seq_first_non_zero(c, n) == -1)
                return isl_stat_ok;
        ctx = isl_mat_get_ctx(T);
        v = isl_vec_alloc(ctx, n);
        if (!v)
                return isl_stat_error;
        isl_seq_swp_or_cpy(v->el, c, n);
        v = isl_vec_mat_product(v, isl_mat_copy(T));
        if (!v)
                return isl_stat_error;
        isl_seq_swp_or_cpy(c, v->el, n);
        isl_vec_free(v);

        return isl_stat_ok;
}

/* Plug in T for the variables in "bmap" starting at "pos".
 * T is a linear unimodular matrix, i.e., without constant term.
 */
static __isl_give isl_basic_map *isl_basic_map_preimage_vars(
        __isl_take isl_basic_map *bmap, unsigned pos, __isl_take isl_mat *T)
{
        int i;
        unsigned n, total;

        bmap = isl_basic_map_cow(bmap);
        if (!bmap || !T)
                goto error;

        n = isl_mat_cols(T);
        if (n != isl_mat_rows(T))
                isl_die(isl_mat_get_ctx(T), isl_error_invalid,
                        "expecting square matrix", goto error);

        total = isl_basic_map_dim(bmap, isl_dim_all);
        if (pos + n > total || pos + n < pos)
                isl_die(isl_mat_get_ctx(T), isl_error_invalid,
                        "invalid range", goto error);

        for (i = 0; i < bmap->n_eq; ++i)
                if (preimage(bmap->eq[i] + 1 + pos, T) < 0)
                        goto error;
        for (i = 0; i < bmap->n_ineq; ++i)
                if (preimage(bmap->ineq[i] + 1 + pos, T) < 0)
                        goto error;
        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_basic_map_div_is_marked_unknown(bmap, i))
                        continue;
                if (preimage(bmap->div[i] + 1 + 1 + pos, T) < 0)
                        goto error;
        }

        isl_mat_free(T);
        return bmap;
error:
        isl_basic_map_free(bmap);
        isl_mat_free(T);
        return NULL;
}

/* Remove divs that are not strictly needed.
 *
 * First look for an equality constraint involving two or more
 * existentially quantified variables without an explicit
 * representation.  Replace the combination that appears
 * in the equality constraint by a single existentially quantified
 * variable such that the equality can be used to derive
 * an explicit representation for the variable.
 * If there are no more such equality constraints, then continue
 * with isl_basic_map_drop_redundant_divs_ineq.
 *
 * In particular, if the equality constraint is of the form
 *
 *      f(x) + \sum_i c_i a_i = 0
 *
 * with a_i existentially quantified variable without explicit
 * representation, then apply a transformation on the existentially
 * quantified variables to turn the constraint into
 *
 *      f(x) + g a_1' = 0
 *
 * with g the gcd of the c_i.
 * In order to easily identify which existentially quantified variables
 * have a complete explicit representation, i.e., without being defined
 * in terms of other existentially quantified variables without
 * an explicit representation, the existentially quantified variables
 * are first sorted.
 *
 * The variable transformation is computed by extending the row
 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
 *
 *      [a_1']   [c_1/g ... c_n/g]   [ a_1 ]
 *      [a_2']                       [ a_2 ]
 *       ...   =         U             ....
 *      [a_n']                       [ a_n ]
 *
 * with [c_1/g ... c_n/g] representing the first row of U.
 * The inverse of U is then plugged into the original constraints.
 * The call to isl_basic_map_simplify makes sure the explicit
 * representation for a_1' is extracted from the equality constraint.
 */
__isl_give isl_basic_map *isl_basic_map_drop_redundant_divs(
        __isl_take isl_basic_map *bmap)
{
        int first;
        int i;
        unsigned o_div, n_div;
        int l;
        isl_ctx *ctx;
        isl_mat *T;

        if (!bmap)
                return NULL;
        if (isl_basic_map_divs_known(bmap))
                return isl_basic_map_drop_redundant_divs_ineq(bmap);
        if (bmap->n_eq == 0)
                return isl_basic_map_drop_redundant_divs_ineq(bmap);
        bmap = isl_basic_map_sort_divs(bmap);
        if (!bmap)
                return NULL;

        first = isl_basic_map_first_unknown_div(bmap);
        if (first < 0)
                return isl_basic_map_free(bmap);

        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);

        for (i = 0; i < bmap->n_eq; ++i) {
                l = isl_seq_first_non_zero(bmap->eq[i] + o_div + first,
                                            n_div - (first));
                if (l < 0)
                        continue;
                l += first;
                if (isl_seq_first_non_zero(bmap->eq[i] + o_div + l + 1,
                                            n_div - (l + 1)) == -1)
                        continue;
                break;
        }
        if (i >= bmap->n_eq)
                return isl_basic_map_drop_redundant_divs_ineq(bmap);

        ctx = isl_basic_map_get_ctx(bmap);
        T = isl_mat_alloc(ctx, n_div - l, n_div - l);
        if (!T)
                return isl_basic_map_free(bmap);
        isl_seq_cpy(T->row[0], bmap->eq[i] + o_div + l, n_div - l);
        T = isl_mat_normalize_row(T, 0);
        T = isl_mat_unimodular_complete(T, 1);
        T = isl_mat_right_inverse(T);

        for (i = l; i < n_div; ++i)
                bmap = isl_basic_map_mark_div_unknown(bmap, i);
        bmap = isl_basic_map_preimage_vars(bmap, o_div - 1 + l, T);
        bmap = isl_basic_map_simplify(bmap);

        return isl_basic_map_drop_redundant_divs(bmap);
}

struct isl_basic_set *isl_basic_set_drop_redundant_divs(
        struct isl_basic_set *bset)
{
        isl_basic_map *bmap = bset_to_bmap(bset);
        return bset_from_bmap(isl_basic_map_drop_redundant_divs(bmap));
}

struct isl_map *isl_map_drop_redundant_divs(struct isl_map *map)
{
        int i;

        if (!map)
                return NULL;
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_drop_redundant_divs(map->p[i]);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

struct isl_set *isl_set_drop_redundant_divs(struct isl_set *set)
{
        return set_from_map(isl_map_drop_redundant_divs(set_to_map(set)));
}

/* Does "bmap" satisfy any equality that involves more than 2 variables
 * and/or has coefficients different from -1 and 1?
 */
static int has_multiple_var_equality(__isl_keep isl_basic_map *bmap)
{
        int i;
        unsigned total;

        total = isl_basic_map_dim(bmap, isl_dim_all);

        for (i = 0; i < bmap->n_eq; ++i) {
                int j, k;

                j = isl_seq_first_non_zero(bmap->eq[i] + 1, total);
                if (j < 0)
                        continue;
                if (!isl_int_is_one(bmap->eq[i][1 + j]) &&
                    !isl_int_is_negone(bmap->eq[i][1 + j]))
                        return 1;

                j += 1;
                k = isl_seq_first_non_zero(bmap->eq[i] + 1 + j, total - j);
                if (k < 0)
                        continue;
                j += k;
                if (!isl_int_is_one(bmap->eq[i][1 + j]) &&
                    !isl_int_is_negone(bmap->eq[i][1 + j]))
                        return 1;

                j += 1;
                k = isl_seq_first_non_zero(bmap->eq[i] + 1 + j, total - j);
                if (k >= 0)
                        return 1;
        }

        return 0;
}

/* Remove any common factor g from the constraint coefficients in "v".
 * The constant term is stored in the first position and is replaced
 * by floor(c/g).  If any common factor is removed and if this results
 * in a tightening of the constraint, then set *tightened.
 */
static __isl_give isl_vec *normalize_constraint(__isl_take isl_vec *v,
        int *tightened)
{
        isl_ctx *ctx;

        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))
                return v;
        if (isl_int_is_one(ctx->normalize_gcd))
                return v;
        v = isl_vec_cow(v);
        if (!v)
                return NULL;
        if (tightened && !isl_int_is_divisible_by(v->el[0], ctx->normalize_gcd))
                *tightened = 1;
        isl_int_fdiv_q(v->el[0], v->el[0], ctx->normalize_gcd);
        isl_seq_scale_down(v->el + 1, v->el + 1, ctx->normalize_gcd,
                                v->size - 1);
        return v;
}

/* If "bmap" is an integer set that satisfies any equality involving
 * more than 2 variables and/or has coefficients different from -1 and 1,
 * then use variable compression to reduce the coefficients by removing
 * any (hidden) common factor.
 * In particular, apply the variable compression to each constraint,
 * factor out any common factor in the non-constant coefficients and
 * then apply the inverse of the compression.
 * At the end, we mark the basic map as having reduced constants.
 * If this flag is still set on the next invocation of this function,
 * then we skip the computation.
 *
 * Removing a common factor may result in a tightening of some of
 * the constraints.  If this happens, then we may end up with two
 * opposite inequalities that can be replaced by an equality.
 * We therefore call isl_basic_map_detect_inequality_pairs,
 * which checks for such pairs of inequalities as well as eliminate_divs_eq
 * and isl_basic_map_gauss if such a pair was found.
 */
__isl_give isl_basic_map *isl_basic_map_reduce_coefficients(
        __isl_take isl_basic_map *bmap)
{
        unsigned total;
        isl_ctx *ctx;
        isl_vec *v;
        isl_mat *eq, *T, *T2;
        int i;
        int tightened;

        if (!bmap)
                return NULL;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS))
                return bmap;
        if (isl_basic_map_is_rational(bmap))
                return bmap;
        if (bmap->n_eq == 0)
                return bmap;
        if (!has_multiple_var_equality(bmap))
                return bmap;

        total = isl_basic_map_dim(bmap, isl_dim_all);
        ctx = isl_basic_map_get_ctx(bmap);
        v = isl_vec_alloc(ctx, 1 + total);
        if (!v)
                return isl_basic_map_free(bmap);

        eq = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap->n_eq, 0, 1 + total);
        T = isl_mat_variable_compression(eq, &T2);
        if (!T || !T2)
                goto error;
        if (T->n_col == 0) {
                isl_mat_free(T);
                isl_mat_free(T2);
                isl_vec_free(v);
                return isl_basic_map_set_to_empty(bmap);
        }

        tightened = 0;
        for (i = 0; i < bmap->n_ineq; ++i) {
                isl_seq_cpy(v->el, bmap->ineq[i], 1 + total);
                v = isl_vec_mat_product(v, isl_mat_copy(T));
                v = normalize_constraint(v, &tightened);
                v = isl_vec_mat_product(v, isl_mat_copy(T2));
                if (!v)
                        goto error;
                isl_seq_cpy(bmap->ineq[i], v->el, 1 + total);
        }

        isl_mat_free(T);
        isl_mat_free(T2);
        isl_vec_free(v);

        ISL_F_SET(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS);

        if (tightened) {
                int progress = 0;

                bmap = isl_basic_map_detect_inequality_pairs(bmap, &progress);
                if (progress) {
                        bmap = eliminate_divs_eq(bmap, &progress);
                        bmap = isl_basic_map_gauss(bmap, NULL);
                }
        }

        return bmap;
error:
        isl_mat_free(T);
        isl_mat_free(T2);
        isl_vec_free(v);
        return isl_basic_map_free(bmap);
}

/* Shift the integer division at position "div" of "bmap"
 * by "shift" times the variable at position "pos".
 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
 * corresponds to the constant term.
 *
 * That is, if the integer division has the form
 *
 *      floor(f(x)/d)
 *
 * then replace it by
 *
 *      floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
 */
__isl_give isl_basic_map *isl_basic_map_shift_div(
        __isl_take isl_basic_map *bmap, int div, int pos, isl_int shift)
{
        int i;
        unsigned total;

        if (isl_int_is_zero(shift))
                return bmap;
        if (!bmap)
                return NULL;

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

        isl_int_addmul(bmap->div[div][1 + pos], shift, bmap->div[div][0]);

        for (i = 0; i < bmap->n_eq; ++i) {
                if (isl_int_is_zero(bmap->eq[i][1 + total + div]))
                        continue;
                isl_int_submul(bmap->eq[i][pos],
                                shift, bmap->eq[i][1 + total + div]);
        }
        for (i = 0; i < bmap->n_ineq; ++i) {
                if (isl_int_is_zero(bmap->ineq[i][1 + total + div]))
                        continue;
                isl_int_submul(bmap->ineq[i][pos],
                                shift, bmap->ineq[i][1 + total + div]);
        }
        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (isl_int_is_zero(bmap->div[i][1 + 1 + total + div]))
                        continue;
                isl_int_submul(bmap->div[i][1 + pos],
                                shift, bmap->div[i][1 + 1 + total + div]);
        }

        return bmap;
}