drop *_fast_* functions

[isl.git] / isl_map_simplify.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2012-2013 Ecole Normale Superieure
 * Copyright 2014      INRIA Rocquencourt
 *
 * 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 <strings.h>
#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>

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;
                isl_basic_map_free_div(bmap, n);
        } 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 (isl_basic_set *)isl_basic_map_drop((isl_basic_map *)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 (isl_set *)isl_map_drop((isl_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.
 */
static struct isl_basic_map *isl_basic_map_drop_div(
                struct 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);
        isl_basic_map_free_div(bmap, 1);

        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)
{
        return (struct isl_basic_set *)isl_basic_map_normalize_constraints(
                (struct isl_basic_map *)bset);
}

/* 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 void eliminate_div(struct 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);

        isl_basic_map_drop_div(bmap, div);
}

/* Check if elimination of div "div" using equality "eq" would not
 * result in a div depending on a later div.
 */
static int 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 1;

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

        return 1;
}

/* Elimininate 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) {
                        if (!isl_int_is_one(bmap->eq[i][off + d]) &&
                            !isl_int_is_negone(bmap->eq[i][off + d]))
                                continue;
                        if (!ok_to_eliminate_div(bmap, bmap->eq[i], d))
                                continue;
                        modified = 1;
                        *progress = 1;
                        eliminate_div(bmap, bmap->eq[i], d, 1);
                        isl_basic_map_drop_equality(bmap, i);
                        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;
}

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 &&
                    isl_int_is_zero(bmap->div[last_var - total_var][0])) {
                        unsigned div = last_var - total_var;
                        isl_seq_neg(bmap->div[div]+1, bmap->eq[done], 1+total);
                        isl_int_set_si(bmap->div[div][1+1+last_var], 0);
                        isl_int_set(bmap->div[div][0],
                                    bmap->eq[done][1+last_var]);
                        if (progress)
                                *progress = 1;
                        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
                }
        }
        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 (struct isl_basic_set*)isl_basic_map_gauss(
                        (struct isl_basic_map *)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;
}

static int hash_index(isl_int ***index, unsigned int size, int bits,
                        struct isl_basic_map *bmap, int k)
{
        int h;
        unsigned total = isl_basic_map_total_dim(bmap);
        uint32_t hash = isl_seq_get_hash_bits(bmap->ineq[k]+1, total, bits);
        for (h = hash; index[h]; h = (h+1) % size)
                if (&bmap->ineq[k] != index[h] &&
                    isl_seq_eq(bmap->ineq[k]+1, index[h][0]+1, total))
                        break;
        return h;
}

static int set_hash_index(isl_int ***index, unsigned int size, int bits,
                          struct isl_basic_set *bset, int k)
{
        return hash_index(index, size, bits, (struct isl_basic_map *)bset, k);
}

/* 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);
                eliminate_div(bmap, eq.data, l, 1);
                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;
        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:
        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 int 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 0;
        }

        /* 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 0;
        }

        return 1;
}

/* 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 int 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 1;

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

        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) {
                if (isl_int_is_zero(bmap->ineq[k][total + i]))
                        continue;
                if (isl_int_abs_ge(sum, bmap->ineq[k][total + i]))
                        continue;
                if (!better_div_constraint(bmap, i, k))
                        continue;
                if (!ok_to_set_div_from_bound(bmap, i, k))
                        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)
{
        unsigned int size;
        isl_int ***index;
        int k, l, h;
        int bits;
        unsigned total = isl_basic_map_total_dim(bmap);
        isl_int sum;
        isl_ctx *ctx;

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

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

        index[isl_seq_get_hash_bits(bmap->ineq[0]+1, total, bits)] = &bmap->ineq[0];
        for (k = 1; k < bmap->n_ineq; ++k) {
                h = hash_index(index, size, bits, bmap, k);
                if (!index[h]) {
                        index[h] = &bmap->ineq[k];
                        continue;
                }
                if (progress)
                        *progress = 1;
                l = 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(index, size, bits, bmap, k);
                isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total);
                if (!index[h])
                        continue;
                l = 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);

        free(index);
        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 use 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) {
                progress = 0;
                if (!bmap)
                        break;
                if (isl_basic_map_plain_is_empty(bmap))
                        break;
                bmap = isl_basic_map_normalize_constraints(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 (struct isl_basic_set *)
                isl_basic_map_simplify((struct isl_basic_map *)bset);
}


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

        if (!bmap)
                return -1;

        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 0;
                if (isl_seq_first_non_zero(constraint+pos+1,
                                            bmap->n_div-div-1) != -1)
                        return 0;
        } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
                if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
                        return 0;
                if (isl_seq_first_non_zero(constraint+pos+1,
                                            bmap->n_div-div-1) != -1)
                        return 0;
        } else
                return 0;

        return 1;
}

int 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 int 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 0;

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

        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 0;
        }

        return 1;
}

/*
 * 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) {
                if (!div_is_redundant(bmap, i))
                        continue;
                bmap = isl_basic_map_drop_div(bmap, i);
        }
        return bmap;
}

struct isl_basic_map *isl_basic_map_finalize(struct isl_basic_map *bmap)
{
        bmap = remove_redundant_divs(bmap);
        if (!bmap)
                return NULL;
        ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
        return bmap;
}

struct isl_basic_set *isl_basic_set_finalize(struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
                isl_basic_map_finalize((struct isl_basic_map *)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;
                isl_int_set_si(bmap->div[i][0], 0);
        }
        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 (struct isl_basic_set *)isl_basic_map_eliminate_vars(
                        (struct isl_basic_map *)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);
}

/* 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((struct isl_basic_map *)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,
                                        (struct isl_basic_map *)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;
}

static struct isl_basic_set *remove_shifted_constraints(
        struct isl_basic_set *bset, struct isl_basic_set *context)
{
        unsigned int size;
        isl_int ***index;
        int bits;
        int k, h, l;
        isl_ctx *ctx;

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

        size = round_up(4 * (context->n_ineq+1) / 3 - 1);
        if (size == 0)
                return bset;
        bits = ffs(size) - 1;
        ctx = isl_basic_set_get_ctx(bset);
        index = isl_calloc_array(ctx, isl_int **, size);
        if (!index)
                return bset;

        for (k = 0; k < context->n_ineq; ++k) {
                h = set_hash_index(index, size, bits, context, k);
                index[h] = &context->ineq[k];
        }
        for (k = 0; k < bset->n_ineq; ++k) {
                h = set_hash_index(index, size, bits, bset, k);
                if (!index[h])
                        continue;
                l = index[h] - &context->ineq[0];
                if (isl_int_lt(bset->ineq[k][0], context->ineq[l][0]))
                        continue;
                bset = isl_basic_set_cow(bset);
                if (!bset)
                        goto error;
                isl_basic_set_drop_inequality(bset, k);
                --k;
        }
        free(index);
        return bset;
error:
        free(index);
        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 "bset" that do not involve any of
 * the dimensions marked "relevant".
 */
static __isl_give isl_basic_set *drop_unrelated_constraints(
        __isl_take isl_basic_set *bset, int *relevant)
{
        int i, dim;

        dim = isl_basic_set_dim(bset, isl_dim_set);
        for (i = 0; i < dim; ++i)
                if (!relevant[i])
                        break;
        if (i >= dim)
                return bset;

        for (i = bset->n_eq - 1; i >= 0; --i)
                if (!is_related(bset->eq[i] + 1, dim, relevant))
                        isl_basic_set_drop_equality(bset, i);

        for (i = bset->n_ineq - 1; i >= 0; --i)
                if (!is_related(bset->ineq[i] + 1, dim, relevant))
                        isl_basic_set_drop_inequality(bset, i);

        return bset;
}

/* 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;
        }
}

/* 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 construct groups of variables that collect variables that
 * (indirectly) appear in some common constraint of "context".
 * Each group is identified by the first variable in the group,
 * except for the special group of variables that appear in "bset"
 * (or are related to those variables), which is identified by -1.
 * 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 the -1 group with the variables that appear in "bset".
 * Then we initialize groups for the remaining variables.
 * Then we iterate over the constraints of "context" 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.
 */
static __isl_give isl_basic_set *drop_irrelevant_constraints(
        __isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset)
{
        isl_ctx *ctx;
        int *group;
        int dim;
        int i, j;
        int last;

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

        dim = isl_basic_set_dim(bset, isl_dim_set);
        ctx = isl_basic_set_get_ctx(bset);
        group = isl_calloc_array(ctx, int, dim);

        if (!group)
                goto error;

        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;
        }

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

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

        context = drop_unrelated_constraints(context, group);

        free(group);
        return context;
error:
        free(group);
        return isl_basic_set_free(context);
}

/* Remove all information from bset that is redundant in the context
 * of context.  Both bset and context are assumed to be full-dimensional.
 *
 * We first remove the inequalities from "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_basic_set *context)
{
        int i, k;
        isl_basic_set *combined = NULL;
        struct isl_tab *tab = NULL;
        unsigned context_ineq;
        unsigned total;

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

        if (isl_basic_set_is_universe(bset)) {
                isl_basic_set_free(context);
                return bset;
        }

        if (isl_basic_set_is_universe(context)) {
                isl_basic_set_free(context);
                return bset;
        }

        bset = remove_shifted_constraints(bset, context);
        if (!bset)
                goto error;
        if (bset->n_ineq == 0)
                goto done;

        context = drop_irrelevant_constraints(context, bset);
        if (!context)
                goto error;
        if (isl_basic_set_is_universe(context)) {
                isl_basic_set_free(context);
                return bset;
        }

        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, i) < 0)
                        goto error;
        if (isl_tab_extend_cons(tab, bset->n_ineq) < 0)
                goto error;
        for (i = 0; i < bset->n_ineq; ++i)
                if (isl_tab_add_ineq(tab, bset->ineq[i]) < 0)
                        goto error;
        bset = isl_basic_set_add_constraints(combined, bset, 0);
        combined = NULL;
        if (!bset)
                goto error;
        if (isl_tab_detect_redundant(tab) < 0)
                goto error;
        total = isl_basic_set_total_dim(bset);
        for (i = context_ineq; i < bset->n_ineq; ++i) {
                int is_empty;
                if (tab->con[i].is_redundant)
                        continue;
                tab->con[i].is_redundant = 1;
                combined = isl_basic_set_dup(bset);
                combined = isl_basic_set_update_from_tab(combined, tab);
                combined = isl_basic_set_extend_constraints(combined, 0, 1);
                k = isl_basic_set_alloc_inequality(combined);
                if (k < 0)
                        goto error;
                isl_seq_neg(combined->ineq[k], bset->ineq[i], 1 + total);
                isl_int_sub_ui(combined->ineq[k][0], combined->ineq[k][0], 1);
                is_empty = isl_basic_set_is_empty(combined);
                if (is_empty < 0)
                        goto error;
                isl_basic_set_free(combined);
                combined = NULL;
                if (!is_empty)
                        tab->con[i].is_redundant = 0;
        }
        for (i = 0; i < context_ineq; ++i)
                tab->con[i].is_redundant = 1;
        bset = isl_basic_set_update_from_tab(bset, tab);
        if (bset) {
                ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
                ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
        }

        isl_tab_free(tab);
done:
        bset = isl_basic_set_simplify(bset);
        bset = isl_basic_set_finalize(bset);
        isl_basic_set_free(context);
        return bset;
error:
        isl_tab_free(tab);
        isl_basic_set_free(combined);
        isl_basic_set_free(context);
        isl_basic_set_free(bset);
        return NULL;
}

/* 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 integer affine hull of the intersection,
 * compute the gist inside this affine hull and then add back
 * those equalities that are not implied by the context.
 *
 * 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, *T2;
        isl_basic_set *aff;
        isl_basic_set *aff_context;
        unsigned total;

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

        context = drop_irrelevant_constraints(context, bset);

        aff = isl_basic_set_copy(bset);
        aff = isl_basic_set_intersect(aff, isl_basic_set_copy(context));
        aff = isl_basic_set_affine_hull(aff);
        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_full(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, &T2);
        if (T && T->n_col == 0) {
                isl_mat_free(T);
                isl_mat_free(T2);
                isl_basic_set_free(context);
                isl_basic_set_free(aff);
                return isl_basic_set_set_to_empty(bset);
        }

        aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));

        bset = isl_basic_set_preimage(bset, isl_mat_copy(T));
        context = isl_basic_set_preimage(context, T);

        bset = uset_gist_full(bset, context);
        bset = isl_basic_set_preimage(bset, T2);
        bset = isl_basic_set_intersect(bset, aff);
        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;
}

/* Normalize the divs in "bmap" in the context of the equalities in "context".
 * We simply add the equalities in context to bmap and then do a regular
 * div normalizations.  Better results can be obtained by normalizing
 * only the divs in bmap than do not also appear in context.
 * We need to be careful to reduce the divs using the equalities
 * so that later calls to isl_basic_map_overlying_set wouldn't introduce
 * spurious constraints.
 */
static struct isl_basic_map *normalize_divs_in_context(
        struct isl_basic_map *bmap, struct isl_basic_map *context)
{
        int i;
        unsigned total_context;
        int div_eq;

        div_eq = n_pure_div_eq(bmap);
        if (div_eq == 0)
                return bmap;

        bmap = isl_basic_map_cow(bmap);
        if (context->n_div > 0)
                bmap = isl_basic_map_align_divs(bmap, context);

        total_context = isl_basic_map_total_dim(context);
        bmap = isl_basic_map_extend_constraints(bmap, context->n_eq, 0);
        for (i = 0; i < context->n_eq; ++i) {
                int k;
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        return isl_basic_map_free(bmap);
                isl_seq_cpy(bmap->eq[k], context->eq[i], 1 + total_context);
                isl_seq_clr(bmap->eq[k] + 1 + total_context,
                                isl_basic_map_total_dim(bmap) - total_context);
        }
        bmap = isl_basic_map_gauss(bmap, NULL);
        bmap = normalize_divs(bmap, NULL);
        bmap = isl_basic_map_gauss(bmap, NULL);
        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.
 */
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 n_div, n_eq, n_ineq;

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

        if (isl_basic_map_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;

        if (context->n_eq)
                bmap = normalize_divs_in_context(bmap, context);

        context = isl_basic_map_align_divs(context, bmap);
        bmap = isl_basic_map_align_divs(bmap, context);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);

        bset = uset_gist(isl_basic_map_underlying_set(isl_basic_map_copy(bmap)),
                    isl_basic_map_underlying_set(isl_basic_map_copy(context)));

        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 = 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;
}

/* 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 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;
        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) {
                isl_map *res = isl_map_universe(isl_map_get_space(map));
                isl_map_free(map);
                isl_map_free(context);
                return res;
        }

        context = isl_map_compute_divs(context);
        if (!context)
                goto error;
        if (isl_map_n_basic_map(context) == 1) {
                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 (struct isl_basic_set *)isl_basic_map_gist(
                (struct isl_basic_map *)bset, (struct isl_basic_map *)context);
}

__isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
        __isl_take isl_basic_set *context)
{
        return (struct isl_set *)isl_map_gist_basic_map((struct isl_map *)set,
                                        (struct isl_basic_map *)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 (struct isl_set *)isl_map_gist((struct isl_map *)set,
                                        (struct isl_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.
 */
int 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 -1;
        isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
                        return -1);
        if (bmap1->n_div || bmap2->n_div)
                return 0;
        if (!bmap1->n_eq && !bmap2->n_eq)
                return 0;

        total = isl_space_dim(bmap1->dim, isl_dim_all);
        if (total == 0)
                return 0;
        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 0;
disjoint:
        isl_vec_free(v);
        free(elim);
        return 1;
error:
        isl_vec_free(v);
        free(elim);
        return -1;
}

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((struct isl_basic_map *)bset1,
                                              (struct isl_basic_map *)bset2);
}

/* 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.
 *
 * 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".
 */
int isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
        __isl_keep isl_map *map2)
{
        int i, j;
        int disjoint;
        int intersect;
        int match;

        if (!map1 || !map2)
                return -1;

        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 ? -1 : 1;

        match = isl_space_tuple_is_equal(map1->dim, isl_dim_out,
                                map2->dim, isl_dim_out);
        if (match < 0 || !match)
                return match < 0 ? -1 : 1;

        match = isl_space_match(map1->dim, isl_dim_param,
                                map2->dim, isl_dim_param);
        if (match < 0 || !match)
                return match < 0 ? -1 : 0;

        intersect = isl_map_plain_is_equal(map1, map2);
        if (intersect < 0 || intersect)
                return intersect < 0 ? -1 : 0;

        for (i = 0; i < map1->n; ++i) {
                for (j = 0; j < map2->n; ++j) {
                        int d = isl_basic_map_plain_is_disjoint(map1->p[i],
                                                               map2->p[j]);
                        if (d != 1)
                                return d;
                }
        }
        return 1;
}

/* 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 none of these cases apply, we compute the intersection and see if
 * the result is empty.
 */
int isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{
        int disjoint;
        int intersect;
        isl_map *test;

        disjoint = isl_map_plain_is_disjoint(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 ? -1 : 0;

        intersect = isl_map_plain_is_universe(map2);
        if (intersect < 0 || intersect)
                return intersect < 0 ? -1 : 0;

        test = isl_map_intersect(isl_map_copy(map1), isl_map_copy(map2));
        disjoint = isl_map_is_empty(test);
        isl_map_free(test);

        return 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.
 */
int isl_basic_map_is_disjoint(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        int disjoint;
        int 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_is_universe(bmap1);
        if (intersect < 0 || intersect)
                return intersect < 0 ? -1 : 0;

        intersect = isl_basic_map_is_universe(bmap2);
        if (intersect < 0 || intersect)
                return intersect < 0 ? -1 : 0;

        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?
 */
int 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);
}

int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
        __isl_keep isl_set *set2)
{
        return isl_map_plain_is_disjoint((struct isl_map *)set1,
                                        (struct isl_map *)set2);
}

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

/* 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
 *
 * 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]))
                                continue;
                        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;
}

/* Given a lower and an upper bound on div i, construct an inequality
 * that when nonnegative ensures that this pair of bounds always allows
 * for an integer value of the given div.
 * The lower bound is inequality l, while the upper bound is inequality u.
 * The constructed inequality is stored in ineq.
 * g, fl, fu are temporary scalars.
 *
 * 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
 */
static void construct_test_ineq(struct isl_basic_map *bmap, int i,
        int l, int u, isl_int *ineq, isl_int g, isl_int fl, isl_int fu)
{
        unsigned dim;
        dim = isl_space_dim(bmap->dim, isl_dim_all);

        isl_int_gcd(g, bmap->ineq[l][1 + dim + i], bmap->ineq[u][1 + dim + i]);
        isl_int_divexact(fl, bmap->ineq[l][1 + dim + i], g);
        isl_int_divexact(fu, bmap->ineq[u][1 + dim + i], g);
        isl_int_neg(fu, fu);
        isl_seq_combine(ineq, fl, bmap->ineq[u], fu, bmap->ineq[l],
                        1 + dim + bmap->n_div);
        isl_int_add(ineq[0], ineq[0], fl);
        isl_int_add(ineq[0], ineq[0], fu);
        isl_int_sub_ui(ineq[0], ineq[0], 1);
        isl_int_mul(g, g, fl);
        isl_int_mul(g, g, fu);
        isl_int_sub(ineq[0], ineq[0], g);
}

/* 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,
 * the we can eliminate the div using Fourier-Motzkin without
 * introducing any spurious solutions.
 */
static struct isl_basic_map *drop_more_redundant_divs(
        struct isl_basic_map *bmap, int *pairs, int n)
{
        struct isl_tab *tab = NULL;
        struct isl_vec *vec = NULL;
        unsigned dim;
        int remove = -1;
        isl_int g, fl, fu;

        isl_int_init(g);
        isl_int_init(fl);
        isl_int_init(fu);

        if (!bmap)
                goto error;

        dim = isl_space_dim(bmap->dim, isl_dim_all);
        vec = isl_vec_alloc(bmap->ctx, 1 + dim + bmap->n_div);
        if (!vec)
                goto error;

        tab = isl_tab_from_basic_map(bmap, 0);

        while (n > 0) {
                int i, l, u;
                int best = -1;
                enum isl_lp_result res;

                for (i = 0; i < bmap->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][1 + dim + i]))
                                continue;
                        for (u = 0; u < bmap->n_ineq; ++u) {
                                if (!isl_int_is_neg(bmap->ineq[u][1 + dim + i]))
                                        continue;
                                construct_test_ineq(bmap, i, l, u,
                                                    vec->el, g, fl, fu);
                                res = isl_tab_min(tab, vec->el,
                                                  bmap->ctx->one, &g, NULL, 0);
                                if (res == isl_lp_error)
                                        goto error;
                                if (res == isl_lp_empty) {
                                        bmap = isl_basic_map_set_to_empty(bmap);
                                        break;
                                }
                                if (res != isl_lp_ok || isl_int_is_neg(g))
                                        break;
                        }
                        if (u < bmap->n_ineq)
                                break;
                }
                if (l == bmap->n_ineq) {
                        remove = i;
                        break;
                }
                pairs[i] = 0;
                --n;
        }

        isl_tab_free(tab);
        isl_vec_free(vec);

        isl_int_clear(g);
        isl_int_clear(fl);
        isl_int_clear(fu);

        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);
        isl_tab_free(tab);
        isl_vec_free(vec);
        isl_int_clear(g);
        isl_int_clear(fl);
        isl_int_clear(fu);
        return NULL;
}

/* Given a pair of divs div1 and div2 such that, expect 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
 * (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
 *
 *      n div2 + t >= 0
 *
 * can be replaced by
 *
 *      (n * (m div 2 + div1) + m t + n f)/g >= 0
 *
 * with g = gcd(m,n).
 * An upper bound
 *
 *      -n div2 + t >= 0
 *
 * can be replaced by
 *
 *      (-n * (m div2 + div1) + m t + n f')/g >= 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_int a;
        isl_int b;
        isl_int m;
        unsigned dim, total;
        int i;

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

        isl_int_init(a);
        isl_int_init(b);
        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])) {
                        isl_int_gcd(b, m, bmap->ineq[i][1 + dim + div2]);
                        isl_int_divexact(a, m, b);
                        isl_int_divexact(b, bmap->ineq[i][1 + dim + div2], b);
                        if (isl_int_is_pos(b)) {
                                isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
                                                b, bmap->ineq[l], total);
                        } else {
                                isl_int_neg(b, b);
                                isl_seq_combine(bmap->ineq[i], a, bmap->ineq[i],
                                                b, 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(a);
        isl_int_clear(b);
        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))
                return bmap;

        return drop_more_redundant_divs(bmap, pairs, n);
}

/* Remove divs that are not strictly needed.
 * 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 coefficent of the div in the constraints,
 * then we can also simply drop the div.
 *
 * 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.
 */
struct isl_basic_map *isl_basic_map_drop_redundant_divs(
        struct 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;

                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);
                        free(pairs);
                        return isl_basic_map_drop_redundant_divs(bmap);
                }
                if (pairs[i] != 1)
                        continue;
                if (!isl_seq_is_neg(bmap->ineq[last_pos] + 1,
                                    bmap->ineq[last_neg] + 1,
                                    off + bmap->n_div))
                        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) {
                        if (defined ||
                            !ok_to_set_div_from_bound(bmap, i, last_pos)) {
                                pairs[i] = 0;
                                --n;
                                continue;
                        }
                        bmap = set_div_from_lower_bound(bmap, i, last_pos);
                        bmap = isl_basic_map_simplify(bmap);
                        free(pairs);
                        return isl_basic_map_drop_redundant_divs(bmap);
                }
                if (last_pos > last_neg) {
                        isl_basic_map_drop_inequality(bmap, last_pos);
                        isl_basic_map_drop_inequality(bmap, last_neg);
                } else {
                        isl_basic_map_drop_inequality(bmap, last_neg);
                        isl_basic_map_drop_inequality(bmap, last_pos);
                }
                bmap = isl_basic_map_drop_div(bmap, i);
                free(pairs);
                return isl_basic_map_drop_redundant_divs(bmap);
        }

        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;
}

struct isl_basic_set *isl_basic_set_drop_redundant_divs(
        struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
            isl_basic_map_drop_redundant_divs((struct isl_basic_map *)bset);
}

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 (struct isl_set *)
            isl_map_drop_redundant_divs((struct isl_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".
 *
 * That is, if the integer division has the form
 *
 *      floor(f(x)/d)
 *
 * then replace it by
 *
 *      floor((f(x) + shift * d)/d) - shift
 */
__isl_give isl_basic_map *isl_basic_map_shift_div(
        __isl_take isl_basic_map *bmap, int div, isl_int shift)
{
        int i;
        unsigned total;

        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], 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][0],
                                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][0],
                                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],
                                shift, bmap->div[i][1 + 1 + total + div]);
        }

        return bmap;
}