isl_basic_set_solve_ilp: avoid double free on error path

[isl.git] / isl_map.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2012-2014 Ecole Normale Superieure
 * Copyright 2014      INRIA Rocquencourt
 * Copyright 2016      INRIA Paris
 * Copyright 2016      Sven Verdoolaege
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
 * B.P. 105 - 78153 Le Chesnay, France
 * and Centre de Recherche Inria de Paris, 2 rue Simone Iff - Voie DQ12,
 * CS 42112, 75589 Paris Cedex 12, France
 */

#include <string.h>
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl_blk.h>
#include <isl/constraint.h>
#include "isl_space_private.h"
#include "isl_equalities.h"
#include <isl_lp_private.h>
#include <isl_seq.h>
#include <isl/set.h>
#include <isl/map.h>
#include <isl_reordering.h>
#include "isl_sample.h"
#include <isl_sort.h>
#include "isl_tab.h"
#include <isl/vec.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl_dim_map.h>
#include <isl_local_space_private.h>
#include <isl_aff_private.h>
#include <isl_options_private.h>
#include <isl_morph.h>
#include <isl_val_private.h>
#include <isl/deprecated/map_int.h>
#include <isl/deprecated/set_int.h>

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

static unsigned n(__isl_keep isl_space *dim, enum isl_dim_type type)
{
        switch (type) {
        case isl_dim_param:     return dim->nparam;
        case isl_dim_in:        return dim->n_in;
        case isl_dim_out:       return dim->n_out;
        case isl_dim_all:       return dim->nparam + dim->n_in + dim->n_out;
        default:                return 0;
        }
}

static unsigned pos(__isl_keep isl_space *dim, enum isl_dim_type type)
{
        switch (type) {
        case isl_dim_param:     return 1;
        case isl_dim_in:        return 1 + dim->nparam;
        case isl_dim_out:       return 1 + dim->nparam + dim->n_in;
        default:                return 0;
        }
}

unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
                                enum isl_dim_type type)
{
        if (!bmap)
                return 0;
        switch (type) {
        case isl_dim_cst:       return 1;
        case isl_dim_param:
        case isl_dim_in:
        case isl_dim_out:       return isl_space_dim(bmap->dim, type);
        case isl_dim_div:       return bmap->n_div;
        case isl_dim_all:       return isl_basic_map_total_dim(bmap);
        default:                return 0;
        }
}

unsigned isl_map_dim(__isl_keep isl_map *map, enum isl_dim_type type)
{
        return map ? n(map->dim, type) : 0;
}

unsigned isl_set_dim(__isl_keep isl_set *set, enum isl_dim_type type)
{
        return set ? n(set->dim, type) : 0;
}

unsigned isl_basic_map_offset(struct isl_basic_map *bmap,
                                        enum isl_dim_type type)
{
        isl_space *space;

        if (!bmap)
                return 0;

        space = bmap->dim;
        switch (type) {
        case isl_dim_cst:       return 0;
        case isl_dim_param:     return 1;
        case isl_dim_in:        return 1 + space->nparam;
        case isl_dim_out:       return 1 + space->nparam + space->n_in;
        case isl_dim_div:       return 1 + space->nparam + space->n_in +
                                                                space->n_out;
        default:                return 0;
        }
}

unsigned isl_basic_set_offset(struct isl_basic_set *bset,
                                        enum isl_dim_type type)
{
        return isl_basic_map_offset(bset, type);
}

static unsigned map_offset(struct isl_map *map, enum isl_dim_type type)
{
        return pos(map->dim, type);
}

unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
                                enum isl_dim_type type)
{
        return isl_basic_map_dim(bset, type);
}

unsigned isl_basic_set_n_dim(__isl_keep isl_basic_set *bset)
{
        return isl_basic_set_dim(bset, isl_dim_set);
}

unsigned isl_basic_set_n_param(__isl_keep isl_basic_set *bset)
{
        return isl_basic_set_dim(bset, isl_dim_param);
}

unsigned isl_basic_set_total_dim(const struct isl_basic_set *bset)
{
        if (!bset)
                return 0;
        return isl_space_dim(bset->dim, isl_dim_all) + bset->n_div;
}

unsigned isl_set_n_dim(__isl_keep isl_set *set)
{
        return isl_set_dim(set, isl_dim_set);
}

unsigned isl_set_n_param(__isl_keep isl_set *set)
{
        return isl_set_dim(set, isl_dim_param);
}

unsigned isl_basic_map_n_in(const struct isl_basic_map *bmap)
{
        return bmap ? bmap->dim->n_in : 0;
}

unsigned isl_basic_map_n_out(const struct isl_basic_map *bmap)
{
        return bmap ? bmap->dim->n_out : 0;
}

unsigned isl_basic_map_n_param(const struct isl_basic_map *bmap)
{
        return bmap ? bmap->dim->nparam : 0;
}

unsigned isl_basic_map_n_div(const struct isl_basic_map *bmap)
{
        return bmap ? bmap->n_div : 0;
}

unsigned isl_basic_map_total_dim(const struct isl_basic_map *bmap)
{
        return bmap ? isl_space_dim(bmap->dim, isl_dim_all) + bmap->n_div : 0;
}

unsigned isl_map_n_in(const struct isl_map *map)
{
        return map ? map->dim->n_in : 0;
}

unsigned isl_map_n_out(const struct isl_map *map)
{
        return map ? map->dim->n_out : 0;
}

unsigned isl_map_n_param(const struct isl_map *map)
{
        return map ? map->dim->nparam : 0;
}

int isl_map_compatible_domain(struct isl_map *map, struct isl_set *set)
{
        int m;
        if (!map || !set)
                return -1;
        m = isl_space_match(map->dim, isl_dim_param, set->dim, isl_dim_param);
        if (m < 0 || !m)
                return m;
        return isl_space_tuple_is_equal(map->dim, isl_dim_in,
                                        set->dim, isl_dim_set);
}

isl_bool isl_basic_map_compatible_domain(__isl_keep isl_basic_map *bmap,
        __isl_keep isl_basic_set *bset)
{
        isl_bool m;
        if (!bmap || !bset)
                return isl_bool_error;
        m = isl_space_match(bmap->dim, isl_dim_param, bset->dim, isl_dim_param);
        if (m < 0 || !m)
                return m;
        return isl_space_tuple_is_equal(bmap->dim, isl_dim_in,
                                        bset->dim, isl_dim_set);
}

int isl_map_compatible_range(__isl_keep isl_map *map, __isl_keep isl_set *set)
{
        int m;
        if (!map || !set)
                return -1;
        m = isl_space_match(map->dim, isl_dim_param, set->dim, isl_dim_param);
        if (m < 0 || !m)
                return m;
        return isl_space_tuple_is_equal(map->dim, isl_dim_out,
                                        set->dim, isl_dim_set);
}

int isl_basic_map_compatible_range(struct isl_basic_map *bmap,
                struct isl_basic_set *bset)
{
        int m;
        if (!bmap || !bset)
                return -1;
        m = isl_space_match(bmap->dim, isl_dim_param, bset->dim, isl_dim_param);
        if (m < 0 || !m)
                return m;
        return isl_space_tuple_is_equal(bmap->dim, isl_dim_out,
                                        bset->dim, isl_dim_set);
}

isl_ctx *isl_basic_map_get_ctx(__isl_keep isl_basic_map *bmap)
{
        return bmap ? bmap->ctx : NULL;
}

isl_ctx *isl_basic_set_get_ctx(__isl_keep isl_basic_set *bset)
{
        return bset ? bset->ctx : NULL;
}

isl_ctx *isl_map_get_ctx(__isl_keep isl_map *map)
{
        return map ? map->ctx : NULL;
}

isl_ctx *isl_set_get_ctx(__isl_keep isl_set *set)
{
        return set ? set->ctx : NULL;
}

__isl_give isl_space *isl_basic_map_get_space(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;
        return isl_space_copy(bmap->dim);
}

__isl_give isl_space *isl_basic_set_get_space(__isl_keep isl_basic_set *bset)
{
        if (!bset)
                return NULL;
        return isl_space_copy(bset->dim);
}

/* Extract the divs in "bmap" as a matrix.
 */
__isl_give isl_mat *isl_basic_map_get_divs(__isl_keep isl_basic_map *bmap)
{
        int i;
        isl_ctx *ctx;
        isl_mat *div;
        unsigned total;
        unsigned cols;

        if (!bmap)
                return NULL;

        ctx = isl_basic_map_get_ctx(bmap);
        total = isl_space_dim(bmap->dim, isl_dim_all);
        cols = 1 + 1 + total + bmap->n_div;
        div = isl_mat_alloc(ctx, bmap->n_div, cols);
        if (!div)
                return NULL;

        for (i = 0; i < bmap->n_div; ++i)
                isl_seq_cpy(div->row[i], bmap->div[i], cols);

        return div;
}

/* Extract the divs in "bset" as a matrix.
 */
__isl_give isl_mat *isl_basic_set_get_divs(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_get_divs(bset);
}

__isl_give isl_local_space *isl_basic_map_get_local_space(
        __isl_keep isl_basic_map *bmap)
{
        isl_mat *div;

        if (!bmap)
                return NULL;

        div = isl_basic_map_get_divs(bmap);
        return isl_local_space_alloc_div(isl_space_copy(bmap->dim), div);
}

__isl_give isl_local_space *isl_basic_set_get_local_space(
        __isl_keep isl_basic_set *bset)
{
        return isl_basic_map_get_local_space(bset);
}

/* For each known div d = floor(f/m), add the constraints
 *
 *              f - m d >= 0
 *              -(f-(m-1)) + m d >= 0
 *
 * Do not finalize the result.
 */
static __isl_give isl_basic_map *add_known_div_constraints(
        __isl_take isl_basic_map *bmap)
{
        int i;
        unsigned n_div;

        if (!bmap)
                return NULL;
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        if (n_div == 0)
                return bmap;
        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_extend_constraints(bmap, 0, 2 * n_div);
        if (!bmap)
                return NULL;
        for (i = 0; i < n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (isl_basic_map_add_div_constraints(bmap, i) < 0)
                        return isl_basic_map_free(bmap);
        }

        return bmap;
}

__isl_give isl_basic_map *isl_basic_map_from_local_space(
        __isl_take isl_local_space *ls)
{
        int i;
        int n_div;
        isl_basic_map *bmap;

        if (!ls)
                return NULL;

        n_div = isl_local_space_dim(ls, isl_dim_div);
        bmap = isl_basic_map_alloc_space(isl_local_space_get_space(ls),
                                        n_div, 0, 2 * n_div);

        for (i = 0; i < n_div; ++i)
                if (isl_basic_map_alloc_div(bmap) < 0)
                        goto error;

        for (i = 0; i < n_div; ++i)
                isl_seq_cpy(bmap->div[i], ls->div->row[i], ls->div->n_col);
        bmap = add_known_div_constraints(bmap);
                                        
        isl_local_space_free(ls);
        return bmap;
error:
        isl_local_space_free(ls);
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_from_local_space(
        __isl_take isl_local_space *ls)
{
        return isl_basic_map_from_local_space(ls);
}

__isl_give isl_space *isl_map_get_space(__isl_keep isl_map *map)
{
        if (!map)
                return NULL;
        return isl_space_copy(map->dim);
}

__isl_give isl_space *isl_set_get_space(__isl_keep isl_set *set)
{
        if (!set)
                return NULL;
        return isl_space_copy(set->dim);
}

__isl_give isl_basic_map *isl_basic_map_set_tuple_name(
        __isl_take isl_basic_map *bmap, enum isl_dim_type type, const char *s)
{
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;
        bmap->dim = isl_space_set_tuple_name(bmap->dim, type, s);
        if (!bmap->dim)
                goto error;
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_set_tuple_name(
        __isl_take isl_basic_set *bset, const char *s)
{
        return isl_basic_map_set_tuple_name(bset, isl_dim_set, s);
}

const char *isl_basic_map_get_tuple_name(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type)
{
        return bmap ? isl_space_get_tuple_name(bmap->dim, type) : NULL;
}

__isl_give isl_map *isl_map_set_tuple_name(__isl_take isl_map *map,
        enum isl_dim_type type, const char *s)
{
        int i;

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

        map->dim = isl_space_set_tuple_name(map->dim, type, s);
        if (!map->dim)
                goto error;

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

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

/* Replace the identifier of the tuple of type "type" by "id".
 */
__isl_give isl_basic_map *isl_basic_map_set_tuple_id(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, __isl_take isl_id *id)
{
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                goto error;
        bmap->dim = isl_space_set_tuple_id(bmap->dim, type, id);
        if (!bmap->dim)
                return isl_basic_map_free(bmap);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_id_free(id);
        return NULL;
}

/* Replace the identifier of the tuple by "id".
 */
__isl_give isl_basic_set *isl_basic_set_set_tuple_id(
        __isl_take isl_basic_set *bset, __isl_take isl_id *id)
{
        return isl_basic_map_set_tuple_id(bset, isl_dim_set, id);
}

/* Does the input or output tuple have a name?
 */
isl_bool isl_map_has_tuple_name(__isl_keep isl_map *map, enum isl_dim_type type)
{
        return map ? isl_space_has_tuple_name(map->dim, type) : isl_bool_error;
}

const char *isl_map_get_tuple_name(__isl_keep isl_map *map,
        enum isl_dim_type type)
{
        return map ? isl_space_get_tuple_name(map->dim, type) : NULL;
}

__isl_give isl_set *isl_set_set_tuple_name(__isl_take isl_set *set,
        const char *s)
{
        return set_from_map(isl_map_set_tuple_name(set_to_map(set),
                                                isl_dim_set, s));
}

__isl_give isl_map *isl_map_set_tuple_id(__isl_take isl_map *map,
        enum isl_dim_type type, __isl_take isl_id *id)
{
        map = isl_map_cow(map);
        if (!map)
                goto error;

        map->dim = isl_space_set_tuple_id(map->dim, type, id);

        return isl_map_reset_space(map, isl_space_copy(map->dim));
error:
        isl_id_free(id);
        return NULL;
}

__isl_give isl_set *isl_set_set_tuple_id(__isl_take isl_set *set,
        __isl_take isl_id *id)
{
        return isl_map_set_tuple_id(set, isl_dim_set, id);
}

__isl_give isl_map *isl_map_reset_tuple_id(__isl_take isl_map *map,
        enum isl_dim_type type)
{
        map = isl_map_cow(map);
        if (!map)
                return NULL;

        map->dim = isl_space_reset_tuple_id(map->dim, type);

        return isl_map_reset_space(map, isl_space_copy(map->dim));
}

__isl_give isl_set *isl_set_reset_tuple_id(__isl_take isl_set *set)
{
        return isl_map_reset_tuple_id(set, isl_dim_set);
}

isl_bool isl_map_has_tuple_id(__isl_keep isl_map *map, enum isl_dim_type type)
{
        return map ? isl_space_has_tuple_id(map->dim, type) : isl_bool_error;
}

__isl_give isl_id *isl_map_get_tuple_id(__isl_keep isl_map *map,
        enum isl_dim_type type)
{
        return map ? isl_space_get_tuple_id(map->dim, type) : NULL;
}

isl_bool isl_set_has_tuple_id(__isl_keep isl_set *set)
{
        return isl_map_has_tuple_id(set, isl_dim_set);
}

__isl_give isl_id *isl_set_get_tuple_id(__isl_keep isl_set *set)
{
        return isl_map_get_tuple_id(set, isl_dim_set);
}

/* Does the set tuple have a name?
 */
isl_bool isl_set_has_tuple_name(__isl_keep isl_set *set)
{
        if (!set)
                return isl_bool_error;
        return isl_space_has_tuple_name(set->dim, isl_dim_set);
}


const char *isl_basic_set_get_tuple_name(__isl_keep isl_basic_set *bset)
{
        return bset ? isl_space_get_tuple_name(bset->dim, isl_dim_set) : NULL;
}

const char *isl_set_get_tuple_name(__isl_keep isl_set *set)
{
        return set ? isl_space_get_tuple_name(set->dim, isl_dim_set) : NULL;
}

const char *isl_basic_map_get_dim_name(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos)
{
        return bmap ? isl_space_get_dim_name(bmap->dim, type, pos) : NULL;
}

const char *isl_basic_set_get_dim_name(__isl_keep isl_basic_set *bset,
        enum isl_dim_type type, unsigned pos)
{
        return bset ? isl_space_get_dim_name(bset->dim, type, pos) : NULL;
}

/* Does the given dimension have a name?
 */
isl_bool isl_map_has_dim_name(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos)
{
        if (!map)
                return isl_bool_error;
        return isl_space_has_dim_name(map->dim, type, pos);
}

const char *isl_map_get_dim_name(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos)
{
        return map ? isl_space_get_dim_name(map->dim, type, pos) : NULL;
}

const char *isl_set_get_dim_name(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return set ? isl_space_get_dim_name(set->dim, type, pos) : NULL;
}

/* Does the given dimension have a name?
 */
isl_bool isl_set_has_dim_name(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        if (!set)
                return isl_bool_error;
        return isl_space_has_dim_name(set->dim, type, pos);
}

__isl_give isl_basic_map *isl_basic_map_set_dim_name(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, const char *s)
{
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;
        bmap->dim = isl_space_set_dim_name(bmap->dim, type, pos, s);
        if (!bmap->dim)
                goto error;
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_map *isl_map_set_dim_name(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, const char *s)
{
        int i;

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

        map->dim = isl_space_set_dim_name(map->dim, type, pos, s);
        if (!map->dim)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_set_dim_name(map->p[i], type, pos, s);
                if (!map->p[i])
                        goto error;
        }

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

__isl_give isl_basic_set *isl_basic_set_set_dim_name(
        __isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned pos, const char *s)
{
        return bset_from_bmap(isl_basic_map_set_dim_name(bset_to_bmap(bset),
                                                        type, pos, s));
}

__isl_give isl_set *isl_set_set_dim_name(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, const char *s)
{
        return set_from_map(isl_map_set_dim_name(set_to_map(set),
                                                        type, pos, s));
}

isl_bool isl_basic_map_has_dim_id(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos)
{
        if (!bmap)
                return isl_bool_error;
        return isl_space_has_dim_id(bmap->dim, type, pos);
}

__isl_give isl_id *isl_basic_set_get_dim_id(__isl_keep isl_basic_set *bset,
        enum isl_dim_type type, unsigned pos)
{
        return bset ? isl_space_get_dim_id(bset->dim, type, pos) : NULL;
}

isl_bool isl_map_has_dim_id(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos)
{
        return map ? isl_space_has_dim_id(map->dim, type, pos) : isl_bool_error;
}

__isl_give isl_id *isl_map_get_dim_id(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos)
{
        return map ? isl_space_get_dim_id(map->dim, type, pos) : NULL;
}

isl_bool isl_set_has_dim_id(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return isl_map_has_dim_id(set, type, pos);
}

__isl_give isl_id *isl_set_get_dim_id(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return isl_map_get_dim_id(set, type, pos);
}

__isl_give isl_map *isl_map_set_dim_id(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
{
        map = isl_map_cow(map);
        if (!map)
                goto error;

        map->dim = isl_space_set_dim_id(map->dim, type, pos, id);

        return isl_map_reset_space(map, isl_space_copy(map->dim));
error:
        isl_id_free(id);
        return NULL;
}

__isl_give isl_set *isl_set_set_dim_id(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
{
        return isl_map_set_dim_id(set, type, pos, id);
}

int isl_map_find_dim_by_id(__isl_keep isl_map *map, enum isl_dim_type type,
        __isl_keep isl_id *id)
{
        if (!map)
                return -1;
        return isl_space_find_dim_by_id(map->dim, type, id);
}

int isl_set_find_dim_by_id(__isl_keep isl_set *set, enum isl_dim_type type,
        __isl_keep isl_id *id)
{
        return isl_map_find_dim_by_id(set, type, id);
}

/* Return the position of the dimension of the given type and name
 * in "bmap".
 * Return -1 if no such dimension can be found.
 */
int isl_basic_map_find_dim_by_name(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, const char *name)
{
        if (!bmap)
                return -1;
        return isl_space_find_dim_by_name(bmap->dim, type, name);
}

int isl_map_find_dim_by_name(__isl_keep isl_map *map, enum isl_dim_type type,
        const char *name)
{
        if (!map)
                return -1;
        return isl_space_find_dim_by_name(map->dim, type, name);
}

int isl_set_find_dim_by_name(__isl_keep isl_set *set, enum isl_dim_type type,
        const char *name)
{
        return isl_map_find_dim_by_name(set, type, name);
}

/* Reset the user pointer on all identifiers of parameters and tuples
 * of the space of "map".
 */
__isl_give isl_map *isl_map_reset_user(__isl_take isl_map *map)
{
        isl_space *space;

        space = isl_map_get_space(map);
        space = isl_space_reset_user(space);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Reset the user pointer on all identifiers of parameters and tuples
 * of the space of "set".
 */
__isl_give isl_set *isl_set_reset_user(__isl_take isl_set *set)
{
        return isl_map_reset_user(set);
}

int isl_basic_map_is_rational(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return -1;
        return ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
}

/* Has "map" been marked as a rational map?
 * In particular, have all basic maps in "map" been marked this way?
 * An empty map is not considered to be rational.
 * Maps where only some of the basic maps are marked rational
 * are not allowed.
 */
isl_bool isl_map_is_rational(__isl_keep isl_map *map)
{
        int i;
        isl_bool rational;

        if (!map)
                return isl_bool_error;
        if (map->n == 0)
                return isl_bool_false;
        rational = isl_basic_map_is_rational(map->p[0]);
        if (rational < 0)
                return rational;
        for (i = 1; i < map->n; ++i) {
                isl_bool rational_i;

                rational_i = isl_basic_map_is_rational(map->p[i]);
                if (rational_i < 0)
                        return rational_i;
                if (rational != rational_i)
                        isl_die(isl_map_get_ctx(map), isl_error_unsupported,
                                "mixed rational and integer basic maps "
                                "not supported", return isl_bool_error);
        }

        return rational;
}

/* Has "set" been marked as a rational set?
 * In particular, have all basic set in "set" been marked this way?
 * An empty set is not considered to be rational.
 * Sets where only some of the basic sets are marked rational
 * are not allowed.
 */
isl_bool isl_set_is_rational(__isl_keep isl_set *set)
{
        return isl_map_is_rational(set);
}

int isl_basic_set_is_rational(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_is_rational(bset);
}

/* Does "bmap" contain any rational points?
 *
 * If "bmap" has an equality for each dimension, equating the dimension
 * to an integer constant, then it has no rational points, even if it
 * is marked as rational.
 */
int isl_basic_map_has_rational(__isl_keep isl_basic_map *bmap)
{
        int has_rational = 1;
        unsigned total;

        if (!bmap)
                return -1;
        if (isl_basic_map_plain_is_empty(bmap))
                return 0;
        if (!isl_basic_map_is_rational(bmap))
                return 0;
        bmap = isl_basic_map_copy(bmap);
        bmap = isl_basic_map_implicit_equalities(bmap);
        if (!bmap)
                return -1;
        total = isl_basic_map_total_dim(bmap);
        if (bmap->n_eq == total) {
                int i, j;
                for (i = 0; i < bmap->n_eq; ++i) {
                        j = isl_seq_first_non_zero(bmap->eq[i] + 1, total);
                        if (j < 0)
                                break;
                        if (!isl_int_is_one(bmap->eq[i][1 + j]) &&
                            !isl_int_is_negone(bmap->eq[i][1 + j]))
                                break;
                        j = isl_seq_first_non_zero(bmap->eq[i] + 1 + j + 1,
                                                    total - j - 1);
                        if (j >= 0)
                                break;
                }
                if (i == bmap->n_eq)
                        has_rational = 0;
        }
        isl_basic_map_free(bmap);

        return has_rational;
}

/* Does "map" contain any rational points?
 */
int isl_map_has_rational(__isl_keep isl_map *map)
{
        int i;
        int has_rational;

        if (!map)
                return -1;
        for (i = 0; i < map->n; ++i) {
                has_rational = isl_basic_map_has_rational(map->p[i]);
                if (has_rational < 0)
                        return -1;
                if (has_rational)
                        return 1;
        }
        return 0;
}

/* Does "set" contain any rational points?
 */
int isl_set_has_rational(__isl_keep isl_set *set)
{
        return isl_map_has_rational(set);
}

/* Is this basic set a parameter domain?
 */
int isl_basic_set_is_params(__isl_keep isl_basic_set *bset)
{
        if (!bset)
                return -1;
        return isl_space_is_params(bset->dim);
}

/* Is this set a parameter domain?
 */
isl_bool isl_set_is_params(__isl_keep isl_set *set)
{
        if (!set)
                return isl_bool_error;
        return isl_space_is_params(set->dim);
}

/* Is this map actually a parameter domain?
 * Users should never call this function.  Outside of isl,
 * a map can never be a parameter domain.
 */
int isl_map_is_params(__isl_keep isl_map *map)
{
        if (!map)
                return -1;
        return isl_space_is_params(map->dim);
}

static struct isl_basic_map *basic_map_init(struct isl_ctx *ctx,
                struct isl_basic_map *bmap, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        int i;
        size_t row_size = 1 + isl_space_dim(bmap->dim, isl_dim_all) + extra;

        bmap->ctx = ctx;
        isl_ctx_ref(ctx);

        bmap->block = isl_blk_alloc(ctx, (n_ineq + n_eq) * row_size);
        if (isl_blk_is_error(bmap->block))
                goto error;

        bmap->ineq = isl_alloc_array(ctx, isl_int *, n_ineq + n_eq);
        if ((n_ineq + n_eq) && !bmap->ineq)
                goto error;

        if (extra == 0) {
                bmap->block2 = isl_blk_empty();
                bmap->div = NULL;
        } else {
                bmap->block2 = isl_blk_alloc(ctx, extra * (1 + row_size));
                if (isl_blk_is_error(bmap->block2))
                        goto error;

                bmap->div = isl_alloc_array(ctx, isl_int *, extra);
                if (!bmap->div)
                        goto error;
        }

        for (i = 0; i < n_ineq + n_eq; ++i)
                bmap->ineq[i] = bmap->block.data + i * row_size;

        for (i = 0; i < extra; ++i)
                bmap->div[i] = bmap->block2.data + i * (1 + row_size);

        bmap->ref = 1;
        bmap->flags = 0;
        bmap->c_size = n_eq + n_ineq;
        bmap->eq = bmap->ineq + n_ineq;
        bmap->extra = extra;
        bmap->n_eq = 0;
        bmap->n_ineq = 0;
        bmap->n_div = 0;
        bmap->sample = NULL;

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

struct isl_basic_set *isl_basic_set_alloc(struct isl_ctx *ctx,
                unsigned nparam, unsigned dim, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        struct isl_basic_map *bmap;
        isl_space *space;

        space = isl_space_set_alloc(ctx, nparam, dim);
        if (!space)
                return NULL;

        bmap = isl_basic_map_alloc_space(space, extra, n_eq, n_ineq);
        return bset_from_bmap(bmap);
}

struct isl_basic_set *isl_basic_set_alloc_space(__isl_take isl_space *dim,
                unsigned extra, unsigned n_eq, unsigned n_ineq)
{
        struct isl_basic_map *bmap;
        if (!dim)
                return NULL;
        isl_assert(dim->ctx, dim->n_in == 0, goto error);
        bmap = isl_basic_map_alloc_space(dim, extra, n_eq, n_ineq);
        return bset_from_bmap(bmap);
error:
        isl_space_free(dim);
        return NULL;
}

struct isl_basic_map *isl_basic_map_alloc_space(__isl_take isl_space *dim,
                unsigned extra, unsigned n_eq, unsigned n_ineq)
{
        struct isl_basic_map *bmap;

        if (!dim)
                return NULL;
        bmap = isl_calloc_type(dim->ctx, struct isl_basic_map);
        if (!bmap)
                goto error;
        bmap->dim = dim;

        return basic_map_init(dim->ctx, bmap, extra, n_eq, n_ineq);
error:
        isl_space_free(dim);
        return NULL;
}

struct isl_basic_map *isl_basic_map_alloc(struct isl_ctx *ctx,
                unsigned nparam, unsigned in, unsigned out, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        struct isl_basic_map *bmap;
        isl_space *dim;

        dim = isl_space_alloc(ctx, nparam, in, out);
        if (!dim)
                return NULL;

        bmap = isl_basic_map_alloc_space(dim, extra, n_eq, n_ineq);
        return bmap;
}

static void dup_constraints(
                struct isl_basic_map *dst, struct isl_basic_map *src)
{
        int i;
        unsigned total = isl_basic_map_total_dim(src);

        for (i = 0; i < src->n_eq; ++i) {
                int j = isl_basic_map_alloc_equality(dst);
                isl_seq_cpy(dst->eq[j], src->eq[i], 1+total);
        }

        for (i = 0; i < src->n_ineq; ++i) {
                int j = isl_basic_map_alloc_inequality(dst);
                isl_seq_cpy(dst->ineq[j], src->ineq[i], 1+total);
        }

        for (i = 0; i < src->n_div; ++i) {
                int j = isl_basic_map_alloc_div(dst);
                isl_seq_cpy(dst->div[j], src->div[i], 1+1+total);
        }
        ISL_F_SET(dst, ISL_BASIC_SET_FINAL);
}

struct isl_basic_map *isl_basic_map_dup(struct isl_basic_map *bmap)
{
        struct isl_basic_map *dup;

        if (!bmap)
                return NULL;
        dup = isl_basic_map_alloc_space(isl_space_copy(bmap->dim),
                        bmap->n_div, bmap->n_eq, bmap->n_ineq);
        if (!dup)
                return NULL;
        dup_constraints(dup, bmap);
        dup->flags = bmap->flags;
        dup->sample = isl_vec_copy(bmap->sample);
        return dup;
}

struct isl_basic_set *isl_basic_set_dup(struct isl_basic_set *bset)
{
        struct isl_basic_map *dup;

        dup = isl_basic_map_dup(bset_to_bmap(bset));
        return bset_from_bmap(dup);
}

struct isl_basic_set *isl_basic_set_copy(struct isl_basic_set *bset)
{
        if (!bset)
                return NULL;

        if (ISL_F_ISSET(bset, ISL_BASIC_SET_FINAL)) {
                bset->ref++;
                return bset;
        }
        return isl_basic_set_dup(bset);
}

struct isl_set *isl_set_copy(struct isl_set *set)
{
        if (!set)
                return NULL;

        set->ref++;
        return set;
}

struct isl_basic_map *isl_basic_map_copy(struct isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;

        if (ISL_F_ISSET(bmap, ISL_BASIC_SET_FINAL)) {
                bmap->ref++;
                return bmap;
        }
        bmap = isl_basic_map_dup(bmap);
        if (bmap)
                ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
        return bmap;
}

struct isl_map *isl_map_copy(struct isl_map *map)
{
        if (!map)
                return NULL;

        map->ref++;
        return map;
}

__isl_null isl_basic_map *isl_basic_map_free(__isl_take isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;

        if (--bmap->ref > 0)
                return NULL;

        isl_ctx_deref(bmap->ctx);
        free(bmap->div);
        isl_blk_free(bmap->ctx, bmap->block2);
        free(bmap->ineq);
        isl_blk_free(bmap->ctx, bmap->block);
        isl_vec_free(bmap->sample);
        isl_space_free(bmap->dim);
        free(bmap);

        return NULL;
}

__isl_null isl_basic_set *isl_basic_set_free(__isl_take isl_basic_set *bset)
{
        return isl_basic_map_free(bset_to_bmap(bset));
}

static int room_for_con(struct isl_basic_map *bmap, unsigned n)
{
        return bmap->n_eq + bmap->n_ineq + n <= bmap->c_size;
}

__isl_give isl_map *isl_map_align_params_map_map_and(
        __isl_take isl_map *map1, __isl_take isl_map *map2,
        __isl_give isl_map *(*fn)(__isl_take isl_map *map1,
                                    __isl_take isl_map *map2))
{
        if (!map1 || !map2)
                goto error;
        if (isl_space_match(map1->dim, isl_dim_param, map2->dim, isl_dim_param))
                return fn(map1, map2);
        if (!isl_space_has_named_params(map1->dim) ||
            !isl_space_has_named_params(map2->dim))
                isl_die(map1->ctx, isl_error_invalid,
                        "unaligned unnamed parameters", goto error);
        map1 = isl_map_align_params(map1, isl_map_get_space(map2));
        map2 = isl_map_align_params(map2, isl_map_get_space(map1));
        return fn(map1, map2);
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

isl_bool isl_map_align_params_map_map_and_test(__isl_keep isl_map *map1,
        __isl_keep isl_map *map2,
        isl_bool (*fn)(__isl_keep isl_map *map1, __isl_keep isl_map *map2))
{
        isl_bool r;

        if (!map1 || !map2)
                return isl_bool_error;
        if (isl_space_match(map1->dim, isl_dim_param, map2->dim, isl_dim_param))
                return fn(map1, map2);
        if (!isl_space_has_named_params(map1->dim) ||
            !isl_space_has_named_params(map2->dim))
                isl_die(map1->ctx, isl_error_invalid,
                        "unaligned unnamed parameters", return isl_bool_error);
        map1 = isl_map_copy(map1);
        map2 = isl_map_copy(map2);
        map1 = isl_map_align_params(map1, isl_map_get_space(map2));
        map2 = isl_map_align_params(map2, isl_map_get_space(map1));
        r = fn(map1, map2);
        isl_map_free(map1);
        isl_map_free(map2);
        return r;
}

int isl_basic_map_alloc_equality(struct isl_basic_map *bmap)
{
        struct isl_ctx *ctx;
        if (!bmap)
                return -1;
        ctx = bmap->ctx;
        isl_assert(ctx, room_for_con(bmap, 1), return -1);
        isl_assert(ctx, (bmap->eq - bmap->ineq) + bmap->n_eq <= bmap->c_size,
                        return -1);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_ALL_EQUALITIES);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
        if ((bmap->eq - bmap->ineq) + bmap->n_eq == bmap->c_size) {
                isl_int *t;
                int j = isl_basic_map_alloc_inequality(bmap);
                if (j < 0)
                        return -1;
                t = bmap->ineq[j];
                bmap->ineq[j] = bmap->ineq[bmap->n_ineq - 1];
                bmap->ineq[bmap->n_ineq - 1] = bmap->eq[-1];
                bmap->eq[-1] = t;
                bmap->n_eq++;
                bmap->n_ineq--;
                bmap->eq--;
                return 0;
        }
        isl_seq_clr(bmap->eq[bmap->n_eq] + 1 + isl_basic_map_total_dim(bmap),
                      bmap->extra - bmap->n_div);
        return bmap->n_eq++;
}

int isl_basic_set_alloc_equality(struct isl_basic_set *bset)
{
        return isl_basic_map_alloc_equality(bset_to_bmap(bset));
}

int isl_basic_map_free_equality(struct isl_basic_map *bmap, unsigned n)
{
        if (!bmap)
                return -1;
        isl_assert(bmap->ctx, n <= bmap->n_eq, return -1);
        bmap->n_eq -= n;
        return 0;
}

int isl_basic_set_free_equality(struct isl_basic_set *bset, unsigned n)
{
        return isl_basic_map_free_equality(bset_to_bmap(bset), n);
}

int isl_basic_map_drop_equality(struct isl_basic_map *bmap, unsigned pos)
{
        isl_int *t;
        if (!bmap)
                return -1;
        isl_assert(bmap->ctx, pos < bmap->n_eq, return -1);

        if (pos != bmap->n_eq - 1) {
                t = bmap->eq[pos];
                bmap->eq[pos] = bmap->eq[bmap->n_eq - 1];
                bmap->eq[bmap->n_eq - 1] = t;
        }
        bmap->n_eq--;
        return 0;
}

int isl_basic_set_drop_equality(struct isl_basic_set *bset, unsigned pos)
{
        return isl_basic_map_drop_equality(bset_to_bmap(bset), pos);
}

/* Turn inequality "pos" of "bmap" into an equality.
 *
 * In particular, we move the inequality in front of the equalities
 * and move the last inequality in the position of the moved inequality.
 * Note that isl_tab_make_equalities_explicit depends on this particular
 * change in the ordering of the constraints.
 */
void isl_basic_map_inequality_to_equality(
                struct isl_basic_map *bmap, unsigned pos)
{
        isl_int *t;

        t = bmap->ineq[pos];
        bmap->ineq[pos] = bmap->ineq[bmap->n_ineq - 1];
        bmap->ineq[bmap->n_ineq - 1] = bmap->eq[-1];
        bmap->eq[-1] = t;
        bmap->n_eq++;
        bmap->n_ineq--;
        bmap->eq--;
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_ALL_EQUALITIES);
}

static int room_for_ineq(struct isl_basic_map *bmap, unsigned n)
{
        return bmap->n_ineq + n <= bmap->eq - bmap->ineq;
}

int isl_basic_map_alloc_inequality(struct isl_basic_map *bmap)
{
        struct isl_ctx *ctx;
        if (!bmap)
                return -1;
        ctx = bmap->ctx;
        isl_assert(ctx, room_for_ineq(bmap, 1), return -1);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_ALL_EQUALITIES);
        isl_seq_clr(bmap->ineq[bmap->n_ineq] +
                      1 + isl_basic_map_total_dim(bmap),
                      bmap->extra - bmap->n_div);
        return bmap->n_ineq++;
}

int isl_basic_set_alloc_inequality(struct isl_basic_set *bset)
{
        return isl_basic_map_alloc_inequality(bset_to_bmap(bset));
}

int isl_basic_map_free_inequality(struct isl_basic_map *bmap, unsigned n)
{
        if (!bmap)
                return -1;
        isl_assert(bmap->ctx, n <= bmap->n_ineq, return -1);
        bmap->n_ineq -= n;
        return 0;
}

int isl_basic_set_free_inequality(struct isl_basic_set *bset, unsigned n)
{
        return isl_basic_map_free_inequality(bset_to_bmap(bset), n);
}

int isl_basic_map_drop_inequality(struct isl_basic_map *bmap, unsigned pos)
{
        isl_int *t;
        if (!bmap)
                return -1;
        isl_assert(bmap->ctx, pos < bmap->n_ineq, return -1);

        if (pos != bmap->n_ineq - 1) {
                t = bmap->ineq[pos];
                bmap->ineq[pos] = bmap->ineq[bmap->n_ineq - 1];
                bmap->ineq[bmap->n_ineq - 1] = t;
                ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        }
        bmap->n_ineq--;
        return 0;
}

int isl_basic_set_drop_inequality(struct isl_basic_set *bset, unsigned pos)
{
        return isl_basic_map_drop_inequality(bset_to_bmap(bset), pos);
}

__isl_give isl_basic_map *isl_basic_map_add_eq(__isl_take isl_basic_map *bmap,
        isl_int *eq)
{
        int k;

        bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
        if (!bmap)
                return NULL;
        k = isl_basic_map_alloc_equality(bmap);
        if (k < 0)
                goto error;
        isl_seq_cpy(bmap->eq[k], eq, 1 + isl_basic_map_total_dim(bmap));
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_add_eq(__isl_take isl_basic_set *bset,
        isl_int *eq)
{
        return bset_from_bmap(isl_basic_map_add_eq(bset_to_bmap(bset), eq));
}

__isl_give isl_basic_map *isl_basic_map_add_ineq(__isl_take isl_basic_map *bmap,
        isl_int *ineq)
{
        int k;

        bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
        if (!bmap)
                return NULL;
        k = isl_basic_map_alloc_inequality(bmap);
        if (k < 0)
                goto error;
        isl_seq_cpy(bmap->ineq[k], ineq, 1 + isl_basic_map_total_dim(bmap));
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_add_ineq(__isl_take isl_basic_set *bset,
        isl_int *ineq)
{
        return bset_from_bmap(isl_basic_map_add_ineq(bset_to_bmap(bset), ineq));
}

int isl_basic_map_alloc_div(struct isl_basic_map *bmap)
{
        if (!bmap)
                return -1;
        isl_assert(bmap->ctx, bmap->n_div < bmap->extra, return -1);
        isl_seq_clr(bmap->div[bmap->n_div] +
                      1 + 1 + isl_basic_map_total_dim(bmap),
                      bmap->extra - bmap->n_div);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
        return bmap->n_div++;
}

int isl_basic_set_alloc_div(struct isl_basic_set *bset)
{
        return isl_basic_map_alloc_div(bset_to_bmap(bset));
}

/* Insert an extra integer division, prescribed by "div", to "bmap"
 * at (integer division) position "pos".
 *
 * The integer division is first added at the end and then moved
 * into the right position.
 */
__isl_give isl_basic_map *isl_basic_map_insert_div(
        __isl_take isl_basic_map *bmap, int pos, __isl_keep isl_vec *div)
{
        int i, k, n_div;

        bmap = isl_basic_map_cow(bmap);
        if (!bmap || !div)
                return isl_basic_map_free(bmap);

        if (div->size != 1 + 1 + isl_basic_map_dim(bmap, isl_dim_all))
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "unexpected size", return isl_basic_map_free(bmap));
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        if (pos < 0 || pos > n_div)
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "invalid position", return isl_basic_map_free(bmap));

        bmap = isl_basic_map_extend_space(bmap,
                                        isl_basic_map_get_space(bmap), 1, 0, 2);
        k = isl_basic_map_alloc_div(bmap);
        if (k < 0)
                return isl_basic_map_free(bmap);
        isl_seq_cpy(bmap->div[k], div->el, div->size);
        isl_int_set_si(bmap->div[k][div->size], 0);

        for (i = k; i > pos; --i)
                isl_basic_map_swap_div(bmap, i, i - 1);

        return bmap;
}

int isl_basic_map_free_div(struct isl_basic_map *bmap, unsigned n)
{
        if (!bmap)
                return -1;
        isl_assert(bmap->ctx, n <= bmap->n_div, return -1);
        bmap->n_div -= n;
        return 0;
}

int isl_basic_set_free_div(struct isl_basic_set *bset, unsigned n)
{
        return isl_basic_map_free_div(bset_to_bmap(bset), n);
}

/* Copy constraint from src to dst, putting the vars of src at offset
 * dim_off in dst and the divs of src at offset div_off in dst.
 * If both sets are actually map, then dim_off applies to the input
 * variables.
 */
static void copy_constraint(struct isl_basic_map *dst_map, isl_int *dst,
                            struct isl_basic_map *src_map, isl_int *src,
                            unsigned in_off, unsigned out_off, unsigned div_off)
{
        unsigned src_nparam = isl_basic_map_n_param(src_map);
        unsigned dst_nparam = isl_basic_map_n_param(dst_map);
        unsigned src_in = isl_basic_map_n_in(src_map);
        unsigned dst_in = isl_basic_map_n_in(dst_map);
        unsigned src_out = isl_basic_map_n_out(src_map);
        unsigned dst_out = isl_basic_map_n_out(dst_map);
        isl_int_set(dst[0], src[0]);
        isl_seq_cpy(dst+1, src+1, isl_min(dst_nparam, src_nparam));
        if (dst_nparam > src_nparam)
                isl_seq_clr(dst+1+src_nparam,
                                dst_nparam - src_nparam);
        isl_seq_clr(dst+1+dst_nparam, in_off);
        isl_seq_cpy(dst+1+dst_nparam+in_off,
                    src+1+src_nparam,
                    isl_min(dst_in-in_off, src_in));
        if (dst_in-in_off > src_in)
                isl_seq_clr(dst+1+dst_nparam+in_off+src_in,
                                dst_in - in_off - src_in);
        isl_seq_clr(dst+1+dst_nparam+dst_in, out_off);
        isl_seq_cpy(dst+1+dst_nparam+dst_in+out_off,
                    src+1+src_nparam+src_in,
                    isl_min(dst_out-out_off, src_out));
        if (dst_out-out_off > src_out)
                isl_seq_clr(dst+1+dst_nparam+dst_in+out_off+src_out,
                                dst_out - out_off - src_out);
        isl_seq_clr(dst+1+dst_nparam+dst_in+dst_out, div_off);
        isl_seq_cpy(dst+1+dst_nparam+dst_in+dst_out+div_off,
                    src+1+src_nparam+src_in+src_out,
                    isl_min(dst_map->extra-div_off, src_map->n_div));
        if (dst_map->n_div-div_off > src_map->n_div)
                isl_seq_clr(dst+1+dst_nparam+dst_in+dst_out+
                                div_off+src_map->n_div,
                                dst_map->n_div - div_off - src_map->n_div);
}

static void copy_div(struct isl_basic_map *dst_map, isl_int *dst,
                     struct isl_basic_map *src_map, isl_int *src,
                     unsigned in_off, unsigned out_off, unsigned div_off)
{
        isl_int_set(dst[0], src[0]);
        copy_constraint(dst_map, dst+1, src_map, src+1, in_off, out_off, div_off);
}

static struct isl_basic_map *add_constraints(struct isl_basic_map *bmap1,
                struct isl_basic_map *bmap2, unsigned i_pos, unsigned o_pos)
{
        int i;
        unsigned div_off;

        if (!bmap1 || !bmap2)
                goto error;

        div_off = bmap1->n_div;

        for (i = 0; i < bmap2->n_eq; ++i) {
                int i1 = isl_basic_map_alloc_equality(bmap1);
                if (i1 < 0)
                        goto error;
                copy_constraint(bmap1, bmap1->eq[i1], bmap2, bmap2->eq[i],
                                i_pos, o_pos, div_off);
        }

        for (i = 0; i < bmap2->n_ineq; ++i) {
                int i1 = isl_basic_map_alloc_inequality(bmap1);
                if (i1 < 0)
                        goto error;
                copy_constraint(bmap1, bmap1->ineq[i1], bmap2, bmap2->ineq[i],
                                i_pos, o_pos, div_off);
        }

        for (i = 0; i < bmap2->n_div; ++i) {
                int i1 = isl_basic_map_alloc_div(bmap1);
                if (i1 < 0)
                        goto error;
                copy_div(bmap1, bmap1->div[i1], bmap2, bmap2->div[i],
                         i_pos, o_pos, div_off);
        }

        isl_basic_map_free(bmap2);

        return bmap1;

error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

struct isl_basic_set *isl_basic_set_add_constraints(struct isl_basic_set *bset1,
                struct isl_basic_set *bset2, unsigned pos)
{
        return bset_from_bmap(add_constraints(bset_to_bmap(bset1),
                                                bset_to_bmap(bset2), 0, pos));
}

struct isl_basic_map *isl_basic_map_extend_space(struct isl_basic_map *base,
                __isl_take isl_space *dim, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        struct isl_basic_map *ext;
        unsigned flags;
        int dims_ok;

        if (!dim)
                goto error;

        if (!base)
                goto error;

        dims_ok = isl_space_is_equal(base->dim, dim) &&
                  base->extra >= base->n_div + extra;

        if (dims_ok && room_for_con(base, n_eq + n_ineq) &&
                       room_for_ineq(base, n_ineq)) {
                isl_space_free(dim);
                return base;
        }

        isl_assert(base->ctx, base->dim->nparam <= dim->nparam, goto error);
        isl_assert(base->ctx, base->dim->n_in <= dim->n_in, goto error);
        isl_assert(base->ctx, base->dim->n_out <= dim->n_out, goto error);
        extra += base->extra;
        n_eq += base->n_eq;
        n_ineq += base->n_ineq;

        ext = isl_basic_map_alloc_space(dim, extra, n_eq, n_ineq);
        dim = NULL;
        if (!ext)
                goto error;

        if (dims_ok)
                ext->sample = isl_vec_copy(base->sample);
        flags = base->flags;
        ext = add_constraints(ext, base, 0, 0);
        if (ext) {
                ext->flags = flags;
                ISL_F_CLR(ext, ISL_BASIC_SET_FINAL);
        }

        return ext;

error:
        isl_space_free(dim);
        isl_basic_map_free(base);
        return NULL;
}

struct isl_basic_set *isl_basic_set_extend_space(struct isl_basic_set *base,
                __isl_take isl_space *dim, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        return bset_from_bmap(isl_basic_map_extend_space(bset_to_bmap(base),
                                                    dim, extra, n_eq, n_ineq));
}

struct isl_basic_map *isl_basic_map_extend_constraints(
                struct isl_basic_map *base, unsigned n_eq, unsigned n_ineq)
{
        if (!base)
                return NULL;
        return isl_basic_map_extend_space(base, isl_space_copy(base->dim),
                                        0, n_eq, n_ineq);
}

struct isl_basic_map *isl_basic_map_extend(struct isl_basic_map *base,
                unsigned nparam, unsigned n_in, unsigned n_out, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        struct isl_basic_map *bmap;
        isl_space *dim;

        if (!base)
                return NULL;
        dim = isl_space_alloc(base->ctx, nparam, n_in, n_out);
        if (!dim)
                goto error;

        bmap = isl_basic_map_extend_space(base, dim, extra, n_eq, n_ineq);
        return bmap;
error:
        isl_basic_map_free(base);
        return NULL;
}

struct isl_basic_set *isl_basic_set_extend(struct isl_basic_set *base,
                unsigned nparam, unsigned dim, unsigned extra,
                unsigned n_eq, unsigned n_ineq)
{
        return bset_from_bmap(isl_basic_map_extend(bset_to_bmap(base),
                                        nparam, 0, dim, extra, n_eq, n_ineq));
}

struct isl_basic_set *isl_basic_set_extend_constraints(
                struct isl_basic_set *base, unsigned n_eq, unsigned n_ineq)
{
        isl_basic_map *bmap = bset_to_bmap(base);
        bmap = isl_basic_map_extend_constraints(bmap, n_eq, n_ineq);
        return bset_from_bmap(bmap);
}

struct isl_basic_set *isl_basic_set_cow(struct isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_cow(bset_to_bmap(bset)));
}

struct isl_basic_map *isl_basic_map_cow(struct isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;

        if (bmap->ref > 1) {
                bmap->ref--;
                bmap = isl_basic_map_dup(bmap);
        }
        if (bmap) {
                ISL_F_CLR(bmap, ISL_BASIC_SET_FINAL);
                ISL_F_CLR(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS);
        }
        return bmap;
}

/* Clear all cached information in "map", either because it is about
 * to be modified or because it is being freed.
 * Always return the same pointer that is passed in.
 * This is needed for the use in isl_map_free.
 */
static __isl_give isl_map *clear_caches(__isl_take isl_map *map)
{
        isl_basic_map_free(map->cached_simple_hull[0]);
        isl_basic_map_free(map->cached_simple_hull[1]);
        map->cached_simple_hull[0] = NULL;
        map->cached_simple_hull[1] = NULL;
        return map;
}

struct isl_set *isl_set_cow(struct isl_set *set)
{
        return isl_map_cow(set);
}

/* Return an isl_map that is equal to "map" and that has only
 * a single reference.
 *
 * If the original input already has only one reference, then
 * simply return it, but clear all cached information, since
 * it may be rendered invalid by the operations that will be
 * performed on the result.
 *
 * Otherwise, create a duplicate (without any cached information).
 */
struct isl_map *isl_map_cow(struct isl_map *map)
{
        if (!map)
                return NULL;

        if (map->ref == 1)
                return clear_caches(map);
        map->ref--;
        return isl_map_dup(map);
}

static void swap_vars(struct isl_blk blk, isl_int *a,
                        unsigned a_len, unsigned b_len)
{
        isl_seq_cpy(blk.data, a+a_len, b_len);
        isl_seq_cpy(blk.data+b_len, a, a_len);
        isl_seq_cpy(a, blk.data, b_len+a_len);
}

static __isl_give isl_basic_map *isl_basic_map_swap_vars(
        __isl_take isl_basic_map *bmap, unsigned pos, unsigned n1, unsigned n2)
{
        int i;
        struct isl_blk blk;

        if (!bmap)
                goto error;

        isl_assert(bmap->ctx,
                pos + n1 + n2 <= 1 + isl_basic_map_total_dim(bmap), goto error);

        if (n1 == 0 || n2 == 0)
                return bmap;

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

        blk = isl_blk_alloc(bmap->ctx, n1 + n2);
        if (isl_blk_is_error(blk))
                goto error;

        for (i = 0; i < bmap->n_eq; ++i)
                swap_vars(blk,
                          bmap->eq[i] + pos, n1, n2);

        for (i = 0; i < bmap->n_ineq; ++i)
                swap_vars(blk,
                          bmap->ineq[i] + pos, n1, n2);

        for (i = 0; i < bmap->n_div; ++i)
                swap_vars(blk,
                          bmap->div[i]+1 + pos, n1, n2);

        isl_blk_free(bmap->ctx, blk);

        ISL_F_CLR(bmap, ISL_BASIC_SET_NORMALIZED);
        bmap = isl_basic_map_gauss(bmap, NULL);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_map *isl_basic_map_set_to_empty(struct isl_basic_map *bmap)
{
        int i = 0;
        unsigned total;
        if (!bmap)
                goto error;
        total = isl_basic_map_total_dim(bmap);
        if (isl_basic_map_free_div(bmap, bmap->n_div) < 0)
                return isl_basic_map_free(bmap);
        isl_basic_map_free_inequality(bmap, bmap->n_ineq);
        if (bmap->n_eq > 0)
                isl_basic_map_free_equality(bmap, bmap->n_eq-1);
        else {
                i = isl_basic_map_alloc_equality(bmap);
                if (i < 0)
                        goto error;
        }
        isl_int_set_si(bmap->eq[i][0], 1);
        isl_seq_clr(bmap->eq[i]+1, total);
        ISL_F_SET(bmap, ISL_BASIC_MAP_EMPTY);
        isl_vec_free(bmap->sample);
        bmap->sample = NULL;
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_set *isl_basic_set_set_to_empty(struct isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_set_to_empty(bset_to_bmap(bset)));
}

/* Swap divs "a" and "b" in "bmap" (without modifying any of the constraints
 * of "bmap").
 */
static void swap_div(__isl_keep isl_basic_map *bmap, int a, int b)
{
        isl_int *t = bmap->div[a];
        bmap->div[a] = bmap->div[b];
        bmap->div[b] = t;
}

/* Swap divs "a" and "b" in "bmap" and adjust the constraints and
 * div definitions accordingly.
 */
void isl_basic_map_swap_div(struct isl_basic_map *bmap, int a, int b)
{
        int i;
        unsigned off = isl_space_dim(bmap->dim, isl_dim_all);

        swap_div(bmap, a, b);

        for (i = 0; i < bmap->n_eq; ++i)
                isl_int_swap(bmap->eq[i][1+off+a], bmap->eq[i][1+off+b]);

        for (i = 0; i < bmap->n_ineq; ++i)
                isl_int_swap(bmap->ineq[i][1+off+a], bmap->ineq[i][1+off+b]);

        for (i = 0; i < bmap->n_div; ++i)
                isl_int_swap(bmap->div[i][1+1+off+a], bmap->div[i][1+1+off+b]);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
}

/* Swap divs "a" and "b" in "bset" and adjust the constraints and
 * div definitions accordingly.
 */
void isl_basic_set_swap_div(__isl_keep isl_basic_set *bset, int a, int b)
{
        isl_basic_map_swap_div(bset, a, b);
}

/* 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.
 */
__isl_give isl_map *isl_map_eliminate(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

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

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

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

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

/* 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.
 */
__isl_give isl_set *isl_set_eliminate(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return set_from_map(isl_map_eliminate(set_to_map(set), type, first, 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.
 */
__isl_give isl_set *isl_set_eliminate_dims(__isl_take isl_set *set,
        unsigned first, unsigned n)
{
        return isl_set_eliminate(set, isl_dim_set, first, n);
}

__isl_give isl_basic_map *isl_basic_map_remove_divs(
        __isl_take isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;
        bmap = isl_basic_map_eliminate_vars(bmap,
                            isl_space_dim(bmap->dim, isl_dim_all), bmap->n_div);
        if (!bmap)
                return NULL;
        bmap->n_div = 0;
        return isl_basic_map_finalize(bmap);
}

__isl_give isl_basic_set *isl_basic_set_remove_divs(
        __isl_take isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_remove_divs(bset_to_bmap(bset)));
}

__isl_give isl_map *isl_map_remove_divs(__isl_take isl_map *map)
{
        int i;

        if (!map)
                return NULL;
        if (map->n == 0)
                return map;

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

__isl_give isl_set *isl_set_remove_divs(__isl_take isl_set *set)
{
        return isl_map_remove_divs(set);
}

struct isl_basic_map *isl_basic_map_remove_dims(struct isl_basic_map *bmap,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        if (!bmap)
                return NULL;
        isl_assert(bmap->ctx, first + n <= isl_basic_map_dim(bmap, type),
                        goto error);
        if (n == 0 && !isl_space_is_named_or_nested(bmap->dim, type))
                return bmap;
        bmap = isl_basic_map_eliminate_vars(bmap,
                        isl_basic_map_offset(bmap, type) - 1 + first, n);
        if (!bmap)
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY) && type == isl_dim_div)
                return bmap;
        bmap = isl_basic_map_drop(bmap, type, first, n);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Return true if the definition of the given div (recursively) involves
 * any of the given variables.
 */
static int div_involves_vars(__isl_keep isl_basic_map *bmap, int div,
        unsigned first, unsigned n)
{
        int i;
        unsigned div_offset = isl_basic_map_offset(bmap, isl_dim_div);

        if (isl_int_is_zero(bmap->div[div][0]))
                return 0;
        if (isl_seq_first_non_zero(bmap->div[div] + 1 + first, n) >= 0)
                return 1;

        for (i = bmap->n_div - 1; i >= 0; --i) {
                if (isl_int_is_zero(bmap->div[div][1 + div_offset + i]))
                        continue;
                if (div_involves_vars(bmap, i, first, n))
                        return 1;
        }

        return 0;
}

/* Try and add a lower and/or upper bound on "div" to "bmap"
 * based on inequality "i".
 * "total" is the total number of variables (excluding the divs).
 * "v" is a temporary object that can be used during the calculations.
 * If "lb" is set, then a lower bound should be constructed.
 * If "ub" is set, then an upper bound should be constructed.
 *
 * The calling function has already checked that the inequality does not
 * reference "div", but we still need to check that the inequality is
 * of the right form.  We'll consider the case where we want to construct
 * a lower bound.  The construction of upper bounds is similar.
 *
 * Let "div" be of the form
 *
 *      q = floor((a + f(x))/d)
 *
 * We essentially check if constraint "i" is of the form
 *
 *      b + f(x) >= 0
 *
 * so that we can use it to derive a lower bound on "div".
 * However, we allow a slightly more general form
 *
 *      b + g(x) >= 0
 *
 * with the condition that the coefficients of g(x) - f(x) are all
 * divisible by d.
 * Rewriting this constraint as
 *
 *      0 >= -b - g(x)
 *
 * adding a + f(x) to both sides and dividing by d, we obtain
 *
 *      (a + f(x))/d >= (a-b)/d + (f(x)-g(x))/d
 *
 * Taking the floor on both sides, we obtain
 *
 *      q >= floor((a-b)/d) + (f(x)-g(x))/d
 *
 * or
 *
 *      (g(x)-f(x))/d + ceil((b-a)/d) + q >= 0
 *
 * In the case of an upper bound, we construct the constraint
 *
 *      (g(x)+f(x))/d + floor((b+a)/d) - q >= 0
 *
 */
static __isl_give isl_basic_map *insert_bounds_on_div_from_ineq(
        __isl_take isl_basic_map *bmap, int div, int i,
        unsigned total, isl_int v, int lb, int ub)
{
        int j;

        for (j = 0; (lb || ub) && j < total + bmap->n_div; ++j) {
                if (lb) {
                        isl_int_sub(v, bmap->ineq[i][1 + j],
                                        bmap->div[div][1 + 1 + j]);
                        lb = isl_int_is_divisible_by(v, bmap->div[div][0]);
                }
                if (ub) {
                        isl_int_add(v, bmap->ineq[i][1 + j],
                                        bmap->div[div][1 + 1 + j]);
                        ub = isl_int_is_divisible_by(v, bmap->div[div][0]);
                }
        }
        if (!lb && !ub)
                return bmap;

        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_extend_constraints(bmap, 0, lb + ub);
        if (lb) {
                int k = isl_basic_map_alloc_inequality(bmap);
                if (k < 0)
                        goto error;
                for (j = 0; j < 1 + total + bmap->n_div; ++j) {
                        isl_int_sub(bmap->ineq[k][j], bmap->ineq[i][j],
                                        bmap->div[div][1 + j]);
                        isl_int_cdiv_q(bmap->ineq[k][j],
                                        bmap->ineq[k][j], bmap->div[div][0]);
                }
                isl_int_set_si(bmap->ineq[k][1 + total + div], 1);
        }
        if (ub) {
                int k = isl_basic_map_alloc_inequality(bmap);
                if (k < 0)
                        goto error;
                for (j = 0; j < 1 + total + bmap->n_div; ++j) {
                        isl_int_add(bmap->ineq[k][j], bmap->ineq[i][j],
                                        bmap->div[div][1 + j]);
                        isl_int_fdiv_q(bmap->ineq[k][j],
                                        bmap->ineq[k][j], bmap->div[div][0]);
                }
                isl_int_set_si(bmap->ineq[k][1 + total + div], -1);
        }

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

/* This function is called right before "div" is eliminated from "bmap"
 * using Fourier-Motzkin.
 * Look through the constraints of "bmap" for constraints on the argument
 * of the integer division and use them to construct constraints on the
 * integer division itself.  These constraints can then be combined
 * during the Fourier-Motzkin elimination.
 * Note that it is only useful to introduce lower bounds on "div"
 * if "bmap" already contains upper bounds on "div" as the newly
 * introduce lower bounds can then be combined with the pre-existing
 * upper bounds.  Similarly for upper bounds.
 * We therefore first check if "bmap" contains any lower and/or upper bounds
 * on "div".
 *
 * It is interesting to note that the introduction of these constraints
 * can indeed lead to more accurate results, even when compared to
 * deriving constraints on the argument of "div" from constraints on "div".
 * Consider, for example, the set
 *
 *      { [i,j,k] : 3 + i + 2j >= 0 and 2 * [(i+2j)/4] <= k }
 *
 * The second constraint can be rewritten as
 *
 *      2 * [(-i-2j+3)/4] + k >= 0
 *
 * from which we can derive
 *
 *      -i - 2j + 3 >= -2k
 *
 * or
 *
 *      i + 2j <= 3 + 2k
 *
 * Combined with the first constraint, we obtain
 *
 *      -3 <= 3 + 2k    or      k >= -3
 *
 * If, on the other hand we derive a constraint on [(i+2j)/4] from
 * the first constraint, we obtain
 *
 *      [(i + 2j)/4] >= [-3/4] = -1
 *
 * Combining this constraint with the second constraint, we obtain
 *
 *      k >= -2
 */
static __isl_give isl_basic_map *insert_bounds_on_div(
        __isl_take isl_basic_map *bmap, int div)
{
        int i;
        int check_lb, check_ub;
        isl_int v;
        unsigned total;

        if (!bmap)
                return NULL;

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

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

        check_lb = 0;
        check_ub = 0;
        for (i = 0; (!check_lb || !check_ub) && i < bmap->n_ineq; ++i) {
                int s = isl_int_sgn(bmap->ineq[i][1 + total + div]);
                if (s > 0)
                        check_ub = 1;
                if (s < 0)
                        check_lb = 1;
        }

        if (!check_lb && !check_ub)
                return bmap;

        isl_int_init(v);

        for (i = 0; bmap && i < bmap->n_ineq; ++i) {
                if (!isl_int_is_zero(bmap->ineq[i][1 + total + div]))
                        continue;

                bmap = insert_bounds_on_div_from_ineq(bmap, div, i, total, v,
                                                        check_lb, check_ub);
        }

        isl_int_clear(v);

        return bmap;
}

/* Remove all divs (recursively) involving any of the given dimensions
 * in their definitions.
 */
__isl_give isl_basic_map *isl_basic_map_remove_divs_involving_dims(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

        if (!bmap)
                return NULL;
        isl_assert(bmap->ctx, first + n <= isl_basic_map_dim(bmap, type),
                        goto error);
        first += isl_basic_map_offset(bmap, type);

        for (i = bmap->n_div - 1; i >= 0; --i) {
                if (!div_involves_vars(bmap, i, first, n))
                        continue;
                bmap = insert_bounds_on_div(bmap, i);
                bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, i, 1);
                if (!bmap)
                        return NULL;
                i = bmap->n_div;
        }

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

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

__isl_give isl_map *isl_map_remove_divs_involving_dims(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

        if (!map)
                return NULL;
        if (map->n == 0)
                return map;

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

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

__isl_give isl_set *isl_set_remove_divs_involving_dims(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return set_from_map(isl_map_remove_divs_involving_dims(set_to_map(set),
                                                              type, first, n));
}

/* Does the description of "bmap" depend on the specified dimensions?
 * We also check whether the dimensions appear in any of the div definitions.
 * In principle there is no need for this check.  If the dimensions appear
 * in a div definition, they also appear in the defining constraints of that
 * div.
 */
isl_bool isl_basic_map_involves_dims(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

        if (!bmap)
                return isl_bool_error;

        if (first + n > isl_basic_map_dim(bmap, type))
                isl_die(bmap->ctx, isl_error_invalid,
                        "index out of bounds", return isl_bool_error);

        first += isl_basic_map_offset(bmap, type);
        for (i = 0; i < bmap->n_eq; ++i)
                if (isl_seq_first_non_zero(bmap->eq[i] + first, n) >= 0)
                        return isl_bool_true;
        for (i = 0; i < bmap->n_ineq; ++i)
                if (isl_seq_first_non_zero(bmap->ineq[i] + first, n) >= 0)
                        return isl_bool_true;
        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (isl_seq_first_non_zero(bmap->div[i] + 1 + first, n) >= 0)
                        return isl_bool_true;
        }

        return isl_bool_false;
}

isl_bool isl_map_involves_dims(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

        if (!map)
                return isl_bool_error;

        if (first + n > isl_map_dim(map, type))
                isl_die(map->ctx, isl_error_invalid,
                        "index out of bounds", return isl_bool_error);

        for (i = 0; i < map->n; ++i) {
                isl_bool involves = isl_basic_map_involves_dims(map->p[i],
                                                            type, first, n);
                if (involves < 0 || involves)
                        return involves;
        }

        return isl_bool_false;
}

isl_bool isl_basic_set_involves_dims(__isl_keep isl_basic_set *bset,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return isl_basic_map_involves_dims(bset, type, first, n);
}

isl_bool isl_set_involves_dims(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return isl_map_involves_dims(set, type, first, n);
}

/* Does local variable "div" of "bmap" have a complete explicit representation?
 * Having a complete explicit representation requires not only
 * an explicit representation, but also that all local variables
 * that appear in this explicit representation in turn have
 * a complete explicit representation.
 */
isl_bool isl_basic_map_div_is_known(__isl_keep isl_basic_map *bmap, int div)
{
        int i;
        unsigned div_offset = isl_basic_map_offset(bmap, isl_dim_div);
        isl_bool marked;

        marked = isl_basic_map_div_is_marked_unknown(bmap, div);
        if (marked < 0 || marked)
                return isl_bool_not(marked);

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

                if (isl_int_is_zero(bmap->div[div][1 + div_offset + i]))
                        continue;
                known = isl_basic_map_div_is_known(bmap, i);
                if (known < 0 || !known)
                        return known;
        }

        return isl_bool_true;
}

/* Does local variable "div" of "bset" have a complete explicit representation?
 */
isl_bool isl_basic_set_div_is_known(__isl_keep isl_basic_set *bset, int div)
{
        return isl_basic_map_div_is_known(bset, div);
}

/* Remove all divs that are unknown or defined in terms of unknown divs.
 */
__isl_give isl_basic_map *isl_basic_map_remove_unknown_divs(
        __isl_take isl_basic_map *bmap)
{
        int i;

        if (!bmap)
                return NULL;

        for (i = bmap->n_div - 1; i >= 0; --i) {
                if (isl_basic_map_div_is_known(bmap, i))
                        continue;
                bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, i, 1);
                if (!bmap)
                        return NULL;
                i = bmap->n_div;
        }

        return bmap;
}

/* Remove all divs that are unknown or defined in terms of unknown divs.
 */
__isl_give isl_basic_set *isl_basic_set_remove_unknown_divs(
        __isl_take isl_basic_set *bset)
{
        return isl_basic_map_remove_unknown_divs(bset);
}

__isl_give isl_map *isl_map_remove_unknown_divs(__isl_take isl_map *map)
{
        int i;

        if (!map)
                return NULL;
        if (map->n == 0)
                return map;

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

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

__isl_give isl_set *isl_set_remove_unknown_divs(__isl_take isl_set *set)
{
        return set_from_map(isl_map_remove_unknown_divs(set_to_map(set)));
}

__isl_give isl_basic_set *isl_basic_set_remove_dims(
        __isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        isl_basic_map *bmap = bset_to_bmap(bset);
        bmap = isl_basic_map_remove_dims(bmap, type, first, n);
        return bset_from_bmap(bmap);
}

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

        if (n == 0)
                return map;

        map = isl_map_cow(map);
        if (!map)
                return NULL;
        isl_assert(map->ctx, first + n <= isl_map_dim(map, type), goto error);
        
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_eliminate_vars(map->p[i],
                        isl_basic_map_offset(map->p[i], type) - 1 + first, n);
                if (!map->p[i])
                        goto error;
        }
        map = isl_map_drop(map, type, first, n);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_set *isl_set_remove_dims(__isl_take isl_set *bset,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        return set_from_map(isl_map_remove_dims(set_to_map(bset),
                                                type, first, n));
}

/* Project out n inputs starting at first using Fourier-Motzkin */
struct isl_map *isl_map_remove_inputs(struct isl_map *map,
        unsigned first, unsigned n)
{
        return isl_map_remove_dims(map, isl_dim_in, first, n);
}

static void dump_term(struct isl_basic_map *bmap,
                        isl_int c, int pos, FILE *out)
{
        const char *name;
        unsigned in = isl_basic_map_n_in(bmap);
        unsigned dim = in + isl_basic_map_n_out(bmap);
        unsigned nparam = isl_basic_map_n_param(bmap);
        if (!pos)
                isl_int_print(out, c, 0);
        else {
                if (!isl_int_is_one(c))
                        isl_int_print(out, c, 0);
                if (pos < 1 + nparam) {
                        name = isl_space_get_dim_name(bmap->dim,
                                                isl_dim_param, pos - 1);
                        if (name)
                                fprintf(out, "%s", name);
                        else
                                fprintf(out, "p%d", pos - 1);
                } else if (pos < 1 + nparam + in)
                        fprintf(out, "i%d", pos - 1 - nparam);
                else if (pos < 1 + nparam + dim)
                        fprintf(out, "o%d", pos - 1 - nparam - in);
                else
                        fprintf(out, "e%d", pos - 1 - nparam - dim);
        }
}

static void dump_constraint_sign(struct isl_basic_map *bmap, isl_int *c,
                                int sign, FILE *out)
{
        int i;
        int first;
        unsigned len = 1 + isl_basic_map_total_dim(bmap);
        isl_int v;

        isl_int_init(v);
        for (i = 0, first = 1; i < len; ++i) {
                if (isl_int_sgn(c[i]) * sign <= 0)
                        continue;
                if (!first)
                        fprintf(out, " + ");
                first = 0;
                isl_int_abs(v, c[i]);
                dump_term(bmap, v, i, out);
        }
        isl_int_clear(v);
        if (first)
                fprintf(out, "0");
}

static void dump_constraint(struct isl_basic_map *bmap, isl_int *c,
                                const char *op, FILE *out, int indent)
{
        int i;

        fprintf(out, "%*s", indent, "");

        dump_constraint_sign(bmap, c, 1, out);
        fprintf(out, " %s ", op);
        dump_constraint_sign(bmap, c, -1, out);

        fprintf(out, "\n");

        for (i = bmap->n_div; i < bmap->extra; ++i) {
                if (isl_int_is_zero(c[1+isl_space_dim(bmap->dim, isl_dim_all)+i]))
                        continue;
                fprintf(out, "%*s", indent, "");
                fprintf(out, "ERROR: unused div coefficient not zero\n");
                abort();
        }
}

static void dump_constraints(struct isl_basic_map *bmap,
                                isl_int **c, unsigned n,
                                const char *op, FILE *out, int indent)
{
        int i;

        for (i = 0; i < n; ++i)
                dump_constraint(bmap, c[i], op, out, indent);
}

static void dump_affine(struct isl_basic_map *bmap, isl_int *exp, FILE *out)
{
        int j;
        int first = 1;
        unsigned total = isl_basic_map_total_dim(bmap);

        for (j = 0; j < 1 + total; ++j) {
                if (isl_int_is_zero(exp[j]))
                        continue;
                if (!first && isl_int_is_pos(exp[j]))
                        fprintf(out, "+");
                dump_term(bmap, exp[j], j, out);
                first = 0;
        }
}

static void dump(struct isl_basic_map *bmap, FILE *out, int indent)
{
        int i;

        dump_constraints(bmap, bmap->eq, bmap->n_eq, "=", out, indent);
        dump_constraints(bmap, bmap->ineq, bmap->n_ineq, ">=", out, indent);

        for (i = 0; i < bmap->n_div; ++i) {
                fprintf(out, "%*s", indent, "");
                fprintf(out, "e%d = [(", i);
                dump_affine(bmap, bmap->div[i]+1, out);
                fprintf(out, ")/");
                isl_int_print(out, bmap->div[i][0], 0);
                fprintf(out, "]\n");
        }
}

void isl_basic_set_print_internal(struct isl_basic_set *bset,
        FILE *out, int indent)
{
        if (!bset) {
                fprintf(out, "null basic set\n");
                return;
        }

        fprintf(out, "%*s", indent, "");
        fprintf(out, "ref: %d, nparam: %d, dim: %d, extra: %d, flags: %x\n",
                        bset->ref, bset->dim->nparam, bset->dim->n_out,
                        bset->extra, bset->flags);
        dump(bset_to_bmap(bset), out, indent);
}

void isl_basic_map_print_internal(struct isl_basic_map *bmap,
        FILE *out, int indent)
{
        if (!bmap) {
                fprintf(out, "null basic map\n");
                return;
        }

        fprintf(out, "%*s", indent, "");
        fprintf(out, "ref: %d, nparam: %d, in: %d, out: %d, extra: %d, "
                        "flags: %x, n_name: %d\n",
                bmap->ref,
                bmap->dim->nparam, bmap->dim->n_in, bmap->dim->n_out,
                bmap->extra, bmap->flags, bmap->dim->n_id);
        dump(bmap, out, indent);
}

int isl_inequality_negate(struct isl_basic_map *bmap, unsigned pos)
{
        unsigned total;
        if (!bmap)
                return -1;
        total = isl_basic_map_total_dim(bmap);
        isl_assert(bmap->ctx, pos < bmap->n_ineq, return -1);
        isl_seq_neg(bmap->ineq[pos], bmap->ineq[pos], 1 + total);
        isl_int_sub_ui(bmap->ineq[pos][0], bmap->ineq[pos][0], 1);
        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        return 0;
}

__isl_give isl_set *isl_set_alloc_space(__isl_take isl_space *space, int n,
        unsigned flags)
{
        if (!space)
                return NULL;
        if (isl_space_dim(space, isl_dim_in) != 0)
                isl_die(isl_space_get_ctx(space), isl_error_invalid,
                        "set cannot have input dimensions", goto error);
        return isl_map_alloc_space(space, n, flags);
error:
        isl_space_free(space);
        return NULL;
}

struct isl_set *isl_set_alloc(struct isl_ctx *ctx,
                unsigned nparam, unsigned dim, int n, unsigned flags)
{
        struct isl_set *set;
        isl_space *dims;

        dims = isl_space_alloc(ctx, nparam, 0, dim);
        if (!dims)
                return NULL;

        set = isl_set_alloc_space(dims, n, flags);
        return set;
}

/* Make sure "map" has room for at least "n" more basic maps.
 */
struct isl_map *isl_map_grow(struct isl_map *map, int n)
{
        int i;
        struct isl_map *grown = NULL;

        if (!map)
                return NULL;
        isl_assert(map->ctx, n >= 0, goto error);
        if (map->n + n <= map->size)
                return map;
        grown = isl_map_alloc_space(isl_map_get_space(map), map->n + n, map->flags);
        if (!grown)
                goto error;
        for (i = 0; i < map->n; ++i) {
                grown->p[i] = isl_basic_map_copy(map->p[i]);
                if (!grown->p[i])
                        goto error;
                grown->n++;
        }
        isl_map_free(map);
        return grown;
error:
        isl_map_free(grown);
        isl_map_free(map);
        return NULL;
}

/* Make sure "set" has room for at least "n" more basic sets.
 */
struct isl_set *isl_set_grow(struct isl_set *set, int n)
{
        return set_from_map(isl_map_grow(set_to_map(set), n));
}

struct isl_set *isl_set_dup(struct isl_set *set)
{
        int i;
        struct isl_set *dup;

        if (!set)
                return NULL;

        dup = isl_set_alloc_space(isl_space_copy(set->dim), set->n, set->flags);
        if (!dup)
                return NULL;
        for (i = 0; i < set->n; ++i)
                dup = isl_set_add_basic_set(dup, isl_basic_set_copy(set->p[i]));
        return dup;
}

struct isl_set *isl_set_from_basic_set(struct isl_basic_set *bset)
{
        return isl_map_from_basic_map(bset);
}

struct isl_map *isl_map_from_basic_map(struct isl_basic_map *bmap)
{
        struct isl_map *map;

        if (!bmap)
                return NULL;

        map = isl_map_alloc_space(isl_space_copy(bmap->dim), 1, ISL_MAP_DISJOINT);
        return isl_map_add_basic_map(map, bmap);
}

__isl_give isl_set *isl_set_add_basic_set(__isl_take isl_set *set,
                                                __isl_take isl_basic_set *bset)
{
        return set_from_map(isl_map_add_basic_map(set_to_map(set),
                                                bset_to_bmap(bset)));
}

__isl_null isl_set *isl_set_free(__isl_take isl_set *set)
{
        return isl_map_free(set);
}

void isl_set_print_internal(struct isl_set *set, FILE *out, int indent)
{
        int i;

        if (!set) {
                fprintf(out, "null set\n");
                return;
        }

        fprintf(out, "%*s", indent, "");
        fprintf(out, "ref: %d, n: %d, nparam: %d, dim: %d, flags: %x\n",
                        set->ref, set->n, set->dim->nparam, set->dim->n_out,
                        set->flags);
        for (i = 0; i < set->n; ++i) {
                fprintf(out, "%*s", indent, "");
                fprintf(out, "basic set %d:\n", i);
                isl_basic_set_print_internal(set->p[i], out, indent+4);
        }
}

void isl_map_print_internal(struct isl_map *map, FILE *out, int indent)
{
        int i;

        if (!map) {
                fprintf(out, "null map\n");
                return;
        }

        fprintf(out, "%*s", indent, "");
        fprintf(out, "ref: %d, n: %d, nparam: %d, in: %d, out: %d, "
                     "flags: %x, n_name: %d\n",
                        map->ref, map->n, map->dim->nparam, map->dim->n_in,
                        map->dim->n_out, map->flags, map->dim->n_id);
        for (i = 0; i < map->n; ++i) {
                fprintf(out, "%*s", indent, "");
                fprintf(out, "basic map %d:\n", i);
                isl_basic_map_print_internal(map->p[i], out, indent+4);
        }
}

struct isl_basic_map *isl_basic_map_intersect_domain(
                struct isl_basic_map *bmap, struct isl_basic_set *bset)
{
        struct isl_basic_map *bmap_domain;

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

        isl_assert(bset->ctx, isl_space_match(bmap->dim, isl_dim_param,
                                        bset->dim, isl_dim_param), goto error);

        if (isl_space_dim(bset->dim, isl_dim_set) != 0)
                isl_assert(bset->ctx,
                    isl_basic_map_compatible_domain(bmap, bset), goto error);

        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                goto error;
        bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
                        bset->n_div, bset->n_eq, bset->n_ineq);
        bmap_domain = isl_basic_map_from_domain(bset);
        bmap = add_constraints(bmap, bmap_domain, 0, 0);

        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        isl_basic_set_free(bset);
        return NULL;
}

struct isl_basic_map *isl_basic_map_intersect_range(
                struct isl_basic_map *bmap, struct isl_basic_set *bset)
{
        struct isl_basic_map *bmap_range;

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

        isl_assert(bset->ctx, isl_space_match(bmap->dim, isl_dim_param,
                                        bset->dim, isl_dim_param), goto error);

        if (isl_space_dim(bset->dim, isl_dim_set) != 0)
                isl_assert(bset->ctx,
                    isl_basic_map_compatible_range(bmap, bset), goto error);

        if (isl_basic_set_plain_is_universe(bset)) {
                isl_basic_set_free(bset);
                return bmap;
        }

        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                goto error;
        bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
                        bset->n_div, bset->n_eq, bset->n_ineq);
        bmap_range = bset_to_bmap(bset);
        bmap = add_constraints(bmap, bmap_range, 0, 0);

        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        isl_basic_set_free(bset);
        return NULL;
}

isl_bool isl_basic_map_contains(__isl_keep isl_basic_map *bmap,
        __isl_keep isl_vec *vec)
{
        int i;
        unsigned total;
        isl_int s;

        if (!bmap || !vec)
                return isl_bool_error;

        total = 1 + isl_basic_map_total_dim(bmap);
        if (total != vec->size)
                return isl_bool_error;

        isl_int_init(s);

        for (i = 0; i < bmap->n_eq; ++i) {
                isl_seq_inner_product(vec->el, bmap->eq[i], total, &s);
                if (!isl_int_is_zero(s)) {
                        isl_int_clear(s);
                        return isl_bool_false;
                }
        }

        for (i = 0; i < bmap->n_ineq; ++i) {
                isl_seq_inner_product(vec->el, bmap->ineq[i], total, &s);
                if (isl_int_is_neg(s)) {
                        isl_int_clear(s);
                        return isl_bool_false;
                }
        }

        isl_int_clear(s);

        return isl_bool_true;
}

isl_bool isl_basic_set_contains(__isl_keep isl_basic_set *bset,
        __isl_keep isl_vec *vec)
{
        return isl_basic_map_contains(bset_to_bmap(bset), vec);
}

struct isl_basic_map *isl_basic_map_intersect(
                struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        struct isl_vec *sample = NULL;

        if (!bmap1 || !bmap2)
                goto error;

        isl_assert(bmap1->ctx, isl_space_match(bmap1->dim, isl_dim_param,
                                     bmap2->dim, isl_dim_param), goto error);
        if (isl_space_dim(bmap1->dim, isl_dim_all) ==
                                isl_space_dim(bmap1->dim, isl_dim_param) &&
            isl_space_dim(bmap2->dim, isl_dim_all) !=
                                isl_space_dim(bmap2->dim, isl_dim_param))
                return isl_basic_map_intersect(bmap2, bmap1);

        if (isl_space_dim(bmap2->dim, isl_dim_all) !=
                                        isl_space_dim(bmap2->dim, isl_dim_param))
                isl_assert(bmap1->ctx,
                            isl_space_is_equal(bmap1->dim, bmap2->dim), goto error);

        if (isl_basic_map_plain_is_empty(bmap1)) {
                isl_basic_map_free(bmap2);
                return bmap1;
        }
        if (isl_basic_map_plain_is_empty(bmap2)) {
                isl_basic_map_free(bmap1);
                return bmap2;
        }

        if (bmap1->sample &&
            isl_basic_map_contains(bmap1, bmap1->sample) > 0 &&
            isl_basic_map_contains(bmap2, bmap1->sample) > 0)
                sample = isl_vec_copy(bmap1->sample);
        else if (bmap2->sample &&
            isl_basic_map_contains(bmap1, bmap2->sample) > 0 &&
            isl_basic_map_contains(bmap2, bmap2->sample) > 0)
                sample = isl_vec_copy(bmap2->sample);

        bmap1 = isl_basic_map_cow(bmap1);
        if (!bmap1)
                goto error;
        bmap1 = isl_basic_map_extend_space(bmap1, isl_space_copy(bmap1->dim),
                        bmap2->n_div, bmap2->n_eq, bmap2->n_ineq);
        bmap1 = add_constraints(bmap1, bmap2, 0, 0);

        if (!bmap1)
                isl_vec_free(sample);
        else if (sample) {
                isl_vec_free(bmap1->sample);
                bmap1->sample = sample;
        }

        bmap1 = isl_basic_map_simplify(bmap1);
        return isl_basic_map_finalize(bmap1);
error:
        if (sample)
                isl_vec_free(sample);
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

struct isl_basic_set *isl_basic_set_intersect(
                struct isl_basic_set *bset1, struct isl_basic_set *bset2)
{
        return bset_from_bmap(isl_basic_map_intersect(bset_to_bmap(bset1),
                                                        bset_to_bmap(bset2)));
}

__isl_give isl_basic_set *isl_basic_set_intersect_params(
        __isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2)
{
        return isl_basic_set_intersect(bset1, bset2);
}

/* Special case of isl_map_intersect, where both map1 and map2
 * are convex, without any divs and such that either map1 or map2
 * contains a single constraint.  This constraint is then simply
 * added to the other map.
 */
static __isl_give isl_map *map_intersect_add_constraint(
        __isl_take isl_map *map1, __isl_take isl_map *map2)
{
        isl_assert(map1->ctx, map1->n == 1, goto error);
        isl_assert(map2->ctx, map1->n == 1, goto error);
        isl_assert(map1->ctx, map1->p[0]->n_div == 0, goto error);
        isl_assert(map2->ctx, map1->p[0]->n_div == 0, goto error);

        if (map2->p[0]->n_eq + map2->p[0]->n_ineq != 1)
                return isl_map_intersect(map2, map1);

        isl_assert(map2->ctx,
                    map2->p[0]->n_eq + map2->p[0]->n_ineq == 1, goto error);

        map1 = isl_map_cow(map1);
        if (!map1)
                goto error;
        if (isl_map_plain_is_empty(map1)) {
                isl_map_free(map2);
                return map1;
        }
        map1->p[0] = isl_basic_map_cow(map1->p[0]);
        if (map2->p[0]->n_eq == 1)
                map1->p[0] = isl_basic_map_add_eq(map1->p[0], map2->p[0]->eq[0]);
        else
                map1->p[0] = isl_basic_map_add_ineq(map1->p[0],
                                                        map2->p[0]->ineq[0]);

        map1->p[0] = isl_basic_map_simplify(map1->p[0]);
        map1->p[0] = isl_basic_map_finalize(map1->p[0]);
        if (!map1->p[0])
                goto error;

        if (isl_basic_map_plain_is_empty(map1->p[0])) {
                isl_basic_map_free(map1->p[0]);
                map1->n = 0;
        }

        isl_map_free(map2);

        return map1;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

/* map2 may be either a parameter domain or a map living in the same
 * space as map1.
 */
static __isl_give isl_map *map_intersect_internal(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        unsigned flags = 0;
        isl_map *result;
        int i, j;

        if (!map1 || !map2)
                goto error;

        if ((isl_map_plain_is_empty(map1) ||
             isl_map_plain_is_universe(map2)) &&
            isl_space_is_equal(map1->dim, map2->dim)) {
                isl_map_free(map2);
                return map1;
        }
        if ((isl_map_plain_is_empty(map2) ||
             isl_map_plain_is_universe(map1)) &&
            isl_space_is_equal(map1->dim, map2->dim)) {
                isl_map_free(map1);
                return map2;
        }

        if (map1->n == 1 && map2->n == 1 &&
            map1->p[0]->n_div == 0 && map2->p[0]->n_div == 0 &&
            isl_space_is_equal(map1->dim, map2->dim) &&
            (map1->p[0]->n_eq + map1->p[0]->n_ineq == 1 ||
             map2->p[0]->n_eq + map2->p[0]->n_ineq == 1))
                return map_intersect_add_constraint(map1, map2);

        if (isl_space_dim(map2->dim, isl_dim_all) !=
                                isl_space_dim(map2->dim, isl_dim_param))
                isl_assert(map1->ctx,
                            isl_space_is_equal(map1->dim, map2->dim), goto error);

        if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT) &&
            ISL_F_ISSET(map2, ISL_MAP_DISJOINT))
                ISL_FL_SET(flags, ISL_MAP_DISJOINT);

        result = isl_map_alloc_space(isl_space_copy(map1->dim),
                                map1->n * map2->n, flags);
        if (!result)
                goto error;
        for (i = 0; i < map1->n; ++i)
                for (j = 0; j < map2->n; ++j) {
                        struct isl_basic_map *part;
                        part = isl_basic_map_intersect(
                                    isl_basic_map_copy(map1->p[i]),
                                    isl_basic_map_copy(map2->p[j]));
                        if (isl_basic_map_is_empty(part) < 0)
                                part = isl_basic_map_free(part);
                        result = isl_map_add_basic_map(result, part);
                        if (!result)
                                goto error;
                }
        isl_map_free(map1);
        isl_map_free(map2);
        return result;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

static __isl_give isl_map *map_intersect(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        if (!map1 || !map2)
                goto error;
        if (!isl_space_is_equal(map1->dim, map2->dim))
                isl_die(isl_map_get_ctx(map1), isl_error_invalid,
                        "spaces don't match", goto error);
        return map_intersect_internal(map1, map2);
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

__isl_give isl_map *isl_map_intersect(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_intersect);
}

struct isl_set *isl_set_intersect(struct isl_set *set1, struct isl_set *set2)
{
        return set_from_map(isl_map_intersect(set_to_map(set1),
                                              set_to_map(set2)));
}

/* map_intersect_internal accepts intersections
 * with parameter domains, so we can just call that function.
 */
static __isl_give isl_map *map_intersect_params(__isl_take isl_map *map,
                __isl_take isl_set *params)
{
        return map_intersect_internal(map, params);
}

__isl_give isl_map *isl_map_intersect_params(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_intersect_params);
}

__isl_give isl_set *isl_set_intersect_params(__isl_take isl_set *set,
                __isl_take isl_set *params)
{
        return isl_map_intersect_params(set, params);
}

struct isl_basic_map *isl_basic_map_reverse(struct isl_basic_map *bmap)
{
        isl_space *space;
        unsigned pos, n1, n2;

        if (!bmap)
                return NULL;
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;
        space = isl_space_reverse(isl_space_copy(bmap->dim));
        pos = isl_basic_map_offset(bmap, isl_dim_in);
        n1 = isl_basic_map_dim(bmap, isl_dim_in);
        n2 = isl_basic_map_dim(bmap, isl_dim_out);
        bmap = isl_basic_map_swap_vars(bmap, pos, n1, n2);
        return isl_basic_map_reset_space(bmap, space);
}

static __isl_give isl_basic_map *basic_map_space_reset(
        __isl_take isl_basic_map *bmap, enum isl_dim_type type)
{
        isl_space *space;

        if (!bmap)
                return NULL;
        if (!isl_space_is_named_or_nested(bmap->dim, type))
                return bmap;

        space = isl_basic_map_get_space(bmap);
        space = isl_space_reset(space, type);
        bmap = isl_basic_map_reset_space(bmap, space);
        return bmap;
}

__isl_give isl_basic_map *isl_basic_map_insert_dims(
        __isl_take isl_basic_map *bmap, enum isl_dim_type type,
        unsigned pos, unsigned n)
{
        isl_space *res_dim;
        struct isl_basic_map *res;
        struct isl_dim_map *dim_map;
        unsigned total, off;
        enum isl_dim_type t;

        if (n == 0)
                return basic_map_space_reset(bmap, type);

        if (!bmap)
                return NULL;

        res_dim = isl_space_insert_dims(isl_basic_map_get_space(bmap), type, pos, n);

        total = isl_basic_map_total_dim(bmap) + n;
        dim_map = isl_dim_map_alloc(bmap->ctx, total);
        off = 0;
        for (t = isl_dim_param; t <= isl_dim_out; ++t) {
                if (t != type) {
                        isl_dim_map_dim(dim_map, bmap->dim, t, off);
                } else {
                        unsigned size = isl_basic_map_dim(bmap, t);
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                                0, pos, off);
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                                pos, size - pos, off + pos + n);
                }
                off += isl_space_dim(res_dim, t);
        }
        isl_dim_map_div(dim_map, bmap, off);

        res = isl_basic_map_alloc_space(res_dim,
                        bmap->n_div, bmap->n_eq, bmap->n_ineq);
        if (isl_basic_map_is_rational(bmap))
                res = isl_basic_map_set_rational(res);
        if (isl_basic_map_plain_is_empty(bmap)) {
                isl_basic_map_free(bmap);
                free(dim_map);
                return isl_basic_map_set_to_empty(res);
        }
        res = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
        return isl_basic_map_finalize(res);
}

__isl_give isl_basic_set *isl_basic_set_insert_dims(
        __isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned pos, unsigned n)
{
        return isl_basic_map_insert_dims(bset, type, pos, n);
}

__isl_give isl_basic_map *isl_basic_map_add_dims(__isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned n)
{
        if (!bmap)
                return NULL;
        return isl_basic_map_insert_dims(bmap, type,
                                        isl_basic_map_dim(bmap, type), n);
}

__isl_give isl_basic_set *isl_basic_set_add_dims(__isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned n)
{
        if (!bset)
                return NULL;
        isl_assert(bset->ctx, type != isl_dim_in, goto error);
        return isl_basic_map_add_dims(bset, type, n);
error:
        isl_basic_set_free(bset);
        return NULL;
}

static __isl_give isl_map *map_space_reset(__isl_take isl_map *map,
        enum isl_dim_type type)
{
        isl_space *space;

        if (!map || !isl_space_is_named_or_nested(map->dim, type))
                return map;

        space = isl_map_get_space(map);
        space = isl_space_reset(space, type);
        map = isl_map_reset_space(map, space);
        return map;
}

__isl_give isl_map *isl_map_insert_dims(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, unsigned n)
{
        int i;

        if (n == 0)
                return map_space_reset(map, type);

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

        map->dim = isl_space_insert_dims(map->dim, type, pos, n);
        if (!map->dim)
                goto error;

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

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

__isl_give isl_set *isl_set_insert_dims(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, unsigned n)
{
        return isl_map_insert_dims(set, type, pos, n);
}

__isl_give isl_map *isl_map_add_dims(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned n)
{
        if (!map)
                return NULL;
        return isl_map_insert_dims(map, type, isl_map_dim(map, type), n);
}

__isl_give isl_set *isl_set_add_dims(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned n)
{
        if (!set)
                return NULL;
        isl_assert(set->ctx, type != isl_dim_in, goto error);
        return set_from_map(isl_map_add_dims(set_to_map(set), type, n));
error:
        isl_set_free(set);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_move_dims(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type dst_type, unsigned dst_pos,
        enum isl_dim_type src_type, unsigned src_pos, unsigned n)
{
        struct isl_dim_map *dim_map;
        struct isl_basic_map *res;
        enum isl_dim_type t;
        unsigned total, off;

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

        isl_assert(bmap->ctx, src_pos + n <= isl_basic_map_dim(bmap, src_type),
                goto error);

        if (dst_type == src_type && dst_pos == src_pos)
                return bmap;

        isl_assert(bmap->ctx, dst_type != src_type, goto error);

        if (pos(bmap->dim, dst_type) + dst_pos ==
            pos(bmap->dim, src_type) + src_pos +
                                            ((src_type < dst_type) ? n : 0)) {
                bmap = isl_basic_map_cow(bmap);
                if (!bmap)
                        return NULL;

                bmap->dim = isl_space_move_dims(bmap->dim, dst_type, dst_pos,
                                                src_type, src_pos, n);
                if (!bmap->dim)
                        goto error;

                bmap = isl_basic_map_finalize(bmap);

                return bmap;
        }

        total = isl_basic_map_total_dim(bmap);
        dim_map = isl_dim_map_alloc(bmap->ctx, total);

        off = 0;
        for (t = isl_dim_param; t <= isl_dim_out; ++t) {
                unsigned size = isl_space_dim(bmap->dim, t);
                if (t == dst_type) {
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                            0, dst_pos, off);
                        off += dst_pos;
                        isl_dim_map_dim_range(dim_map, bmap->dim, src_type,
                                            src_pos, n, off);
                        off += n;
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                            dst_pos, size - dst_pos, off);
                        off += size - dst_pos;
                } else if (t == src_type) {
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                            0, src_pos, off);
                        off += src_pos;
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                        src_pos + n, size - src_pos - n, off);
                        off += size - src_pos - n;
                } else {
                        isl_dim_map_dim(dim_map, bmap->dim, t, off);
                        off += size;
                }
        }
        isl_dim_map_div(dim_map, bmap, off);

        res = isl_basic_map_alloc_space(isl_basic_map_get_space(bmap),
                        bmap->n_div, bmap->n_eq, bmap->n_ineq);
        bmap = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
        if (!bmap)
                goto error;

        bmap->dim = isl_space_move_dims(bmap->dim, dst_type, dst_pos,
                                        src_type, src_pos, n);
        if (!bmap->dim)
                goto error;

        ISL_F_CLR(bmap, ISL_BASIC_MAP_NORMALIZED);
        bmap = isl_basic_map_gauss(bmap, NULL);
        bmap = isl_basic_map_finalize(bmap);

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

__isl_give isl_basic_set *isl_basic_set_move_dims(__isl_take isl_basic_set *bset,
        enum isl_dim_type dst_type, unsigned dst_pos,
        enum isl_dim_type src_type, unsigned src_pos, unsigned n)
{
        isl_basic_map *bmap = bset_to_bmap(bset);
        bmap = isl_basic_map_move_dims(bmap, dst_type, dst_pos,
                                        src_type, src_pos, n);
        return bset_from_bmap(bmap);
}

__isl_give isl_set *isl_set_move_dims(__isl_take isl_set *set,
        enum isl_dim_type dst_type, unsigned dst_pos,
        enum isl_dim_type src_type, unsigned src_pos, unsigned n)
{
        if (!set)
                return NULL;
        isl_assert(set->ctx, dst_type != isl_dim_in, goto error);
        return set_from_map(isl_map_move_dims(set_to_map(set),
                                    dst_type, dst_pos, src_type, src_pos, n));
error:
        isl_set_free(set);
        return NULL;
}

__isl_give isl_map *isl_map_move_dims(__isl_take isl_map *map,
        enum isl_dim_type dst_type, unsigned dst_pos,
        enum isl_dim_type src_type, unsigned src_pos, unsigned n)
{
        int i;

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

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

        if (dst_type == src_type && dst_pos == src_pos)
                return map;

        isl_assert(map->ctx, dst_type != src_type, goto error);

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

        map->dim = isl_space_move_dims(map->dim, dst_type, dst_pos, src_type, src_pos, n);
        if (!map->dim)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_move_dims(map->p[i],
                                                dst_type, dst_pos,
                                                src_type, src_pos, n);
                if (!map->p[i])
                        goto error;
        }

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

/* Move the specified dimensions to the last columns right before
 * the divs.  Don't change the dimension specification of bmap.
 * That's the responsibility of the caller.
 */
static __isl_give isl_basic_map *move_last(__isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        struct isl_dim_map *dim_map;
        struct isl_basic_map *res;
        enum isl_dim_type t;
        unsigned total, off;

        if (!bmap)
                return NULL;
        if (pos(bmap->dim, type) + first + n ==
                                1 + isl_space_dim(bmap->dim, isl_dim_all))
                return bmap;

        total = isl_basic_map_total_dim(bmap);
        dim_map = isl_dim_map_alloc(bmap->ctx, total);

        off = 0;
        for (t = isl_dim_param; t <= isl_dim_out; ++t) {
                unsigned size = isl_space_dim(bmap->dim, t);
                if (t == type) {
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                            0, first, off);
                        off += first;
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                            first, n, total - bmap->n_div - n);
                        isl_dim_map_dim_range(dim_map, bmap->dim, t,
                                            first + n, size - (first + n), off);
                        off += size - (first + n);
                } else {
                        isl_dim_map_dim(dim_map, bmap->dim, t, off);
                        off += size;
                }
        }
        isl_dim_map_div(dim_map, bmap, off + n);

        res = isl_basic_map_alloc_space(isl_basic_map_get_space(bmap),
                        bmap->n_div, bmap->n_eq, bmap->n_ineq);
        res = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
        return res;
}

/* Insert "n" rows in the divs of "bmap".
 *
 * The number of columns is not changed, which means that the last
 * dimensions of "bmap" are being reintepreted as the new divs.
 * The space of "bmap" is not adjusted, however, which means
 * that "bmap" is left in an inconsistent state.  Removing "n" dimensions
 * from the space of "bmap" is the responsibility of the caller.
 */
static __isl_give isl_basic_map *insert_div_rows(__isl_take isl_basic_map *bmap,
        int n)
{
        int i;
        size_t row_size;
        isl_int **new_div;
        isl_int *old;

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

        row_size = 1 + isl_space_dim(bmap->dim, isl_dim_all) + bmap->extra;
        old = bmap->block2.data;
        bmap->block2 = isl_blk_extend(bmap->ctx, bmap->block2,
                                        (bmap->extra + n) * (1 + row_size));
        if (!bmap->block2.data)
                return isl_basic_map_free(bmap);
        new_div = isl_alloc_array(bmap->ctx, isl_int *, bmap->extra + n);
        if (!new_div)
                return isl_basic_map_free(bmap);
        for (i = 0; i < n; ++i) {
                new_div[i] = bmap->block2.data +
                                (bmap->extra + i) * (1 + row_size);
                isl_seq_clr(new_div[i], 1 + row_size);
        }
        for (i = 0; i < bmap->extra; ++i)
                new_div[n + i] = bmap->block2.data + (bmap->div[i] - old);
        free(bmap->div);
        bmap->div = new_div;
        bmap->n_div += n;
        bmap->extra += n;

        return bmap;
}

/* Drop constraints from "bmap" that only involve the variables
 * of "type" in the range [first, first + n] that are not related
 * to any of the variables outside that interval.
 * These constraints cannot influence the values for the variables
 * outside the interval, except in case they cause "bmap" to be empty.
 * Only drop the constraints if "bmap" is known to be non-empty.
 */
static __isl_give isl_basic_map *drop_irrelevant_constraints(
        __isl_take isl_basic_map *bmap, enum isl_dim_type type,
        unsigned first, unsigned n)
{
        int i;
        int *groups;
        unsigned dim, n_div;
        isl_bool non_empty;

        non_empty = isl_basic_map_plain_is_non_empty(bmap);
        if (non_empty < 0)
                return isl_basic_map_free(bmap);
        if (!non_empty)
                return bmap;

        dim = isl_basic_map_dim(bmap, isl_dim_all);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        groups = isl_calloc_array(isl_basic_map_get_ctx(bmap), int, dim);
        if (!groups)
                return isl_basic_map_free(bmap);
        first += isl_basic_map_offset(bmap, type) - 1;
        for (i = 0; i < first; ++i)
                groups[i] = -1;
        for (i = first + n; i < dim - n_div; ++i)
                groups[i] = -1;

        bmap = isl_basic_map_drop_unrelated_constraints(bmap, groups);

        return bmap;
}

/* Turn the n dimensions of type type, starting at first
 * into existentially quantified variables.
 *
 * If a subset of the projected out variables are unrelated
 * to any of the variables that remain, then the constraints
 * involving this subset are simply dropped first.
 */
__isl_give isl_basic_map *isl_basic_map_project_out(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned first, unsigned n)
{
        if (n == 0)
                return basic_map_space_reset(bmap, type);
        if (type == isl_dim_div)
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "cannot project out existentially quantified variables",
                        return isl_basic_map_free(bmap));

        bmap = drop_irrelevant_constraints(bmap, type, first, n);
        if (!bmap)
                return NULL;

        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
                return isl_basic_map_remove_dims(bmap, type, first, n);

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

        bmap = move_last(bmap, type, first, n);
        bmap = isl_basic_map_cow(bmap);
        bmap = insert_div_rows(bmap, n);
        if (!bmap)
                return NULL;

        bmap->dim = isl_space_drop_dims(bmap->dim, type, first, n);
        if (!bmap->dim)
                goto error;
        bmap = isl_basic_map_simplify(bmap);
        bmap = isl_basic_map_drop_redundant_divs(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Turn the n dimensions of type type, starting at first
 * into existentially quantified variables.
 */
struct isl_basic_set *isl_basic_set_project_out(struct isl_basic_set *bset,
                enum isl_dim_type type, unsigned first, unsigned n)
{
        return bset_from_bmap(isl_basic_map_project_out(bset_to_bmap(bset),
                                                        type, first, n));
}

/* Turn the n dimensions of type type, starting at first
 * into existentially quantified variables.
 */
__isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;

        if (!map)
                return NULL;

        if (n == 0)
                return map_space_reset(map, type);

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

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

        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_project_out(map->p[i], type, first, n);
                if (!map->p[i])
                        goto error;
        }

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

/* Turn the n dimensions of type type, starting at first
 * into existentially quantified variables.
 */
__isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned first, unsigned n)
{
        return set_from_map(isl_map_project_out(set_to_map(set),
                                                type, first, n));
}

/* Return a map that projects the elements in "set" onto their
 * "n" set dimensions starting at "first".
 * "type" should be equal to isl_dim_set.
 */
__isl_give isl_map *isl_set_project_onto_map(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;
        int dim;
        isl_map *map;

        if (!set)
                return NULL;
        if (type != isl_dim_set)
                isl_die(isl_set_get_ctx(set), isl_error_invalid,
                        "only set dimensions can be projected out", goto error);
        dim = isl_set_dim(set, isl_dim_set);
        if (first + n > dim || first + n < first)
                isl_die(isl_set_get_ctx(set), isl_error_invalid,
                        "index out of bounds", goto error);

        map = isl_map_from_domain(set);
        map = isl_map_add_dims(map, isl_dim_out, n);
        for (i = 0; i < n; ++i)
                map = isl_map_equate(map, isl_dim_in, first + i,
                                        isl_dim_out, i);
        return map;
error:
        isl_set_free(set);
        return NULL;
}

static struct isl_basic_map *add_divs(struct isl_basic_map *bmap, unsigned n)
{
        int i, j;

        for (i = 0; i < n; ++i) {
                j = isl_basic_map_alloc_div(bmap);
                if (j < 0)
                        goto error;
                isl_seq_clr(bmap->div[j], 1+1+isl_basic_map_total_dim(bmap));
        }
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_map *isl_basic_map_apply_range(
                struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        isl_space *dim_result = NULL;
        struct isl_basic_map *bmap;
        unsigned n_in, n_out, n, nparam, total, pos;
        struct isl_dim_map *dim_map1, *dim_map2;

        if (!bmap1 || !bmap2)
                goto error;
        if (!isl_space_match(bmap1->dim, isl_dim_param,
                                bmap2->dim, isl_dim_param))
                isl_die(isl_basic_map_get_ctx(bmap1), isl_error_invalid,
                        "parameters don't match", goto error);
        if (!isl_space_tuple_is_equal(bmap1->dim, isl_dim_out,
                                    bmap2->dim, isl_dim_in))
                isl_die(isl_basic_map_get_ctx(bmap1), isl_error_invalid,
                        "spaces don't match", goto error);

        dim_result = isl_space_join(isl_space_copy(bmap1->dim),
                                  isl_space_copy(bmap2->dim));

        n_in = isl_basic_map_n_in(bmap1);
        n_out = isl_basic_map_n_out(bmap2);
        n = isl_basic_map_n_out(bmap1);
        nparam = isl_basic_map_n_param(bmap1);

        total = nparam + n_in + n_out + bmap1->n_div + bmap2->n_div + n;
        dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
        dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += n_in);
        isl_dim_map_div(dim_map1, bmap1, pos += n_out);
        isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += bmap2->n_div);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos);

        bmap = isl_basic_map_alloc_space(dim_result,
                        bmap1->n_div + bmap2->n_div + n,
                        bmap1->n_eq + bmap2->n_eq,
                        bmap1->n_ineq + bmap2->n_ineq);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
        bmap = add_divs(bmap, n);
        bmap = isl_basic_map_simplify(bmap);
        bmap = isl_basic_map_drop_redundant_divs(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

struct isl_basic_set *isl_basic_set_apply(
                struct isl_basic_set *bset, struct isl_basic_map *bmap)
{
        if (!bset || !bmap)
                goto error;

        isl_assert(bset->ctx, isl_basic_map_compatible_domain(bmap, bset),
                    goto error);

        return bset_from_bmap(isl_basic_map_apply_range(bset_to_bmap(bset),
                                                        bmap));
error:
        isl_basic_set_free(bset);
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_map *isl_basic_map_apply_domain(
                struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        if (!bmap1 || !bmap2)
                goto error;

        isl_assert(bmap1->ctx,
            isl_basic_map_n_in(bmap1) == isl_basic_map_n_in(bmap2), goto error);
        isl_assert(bmap1->ctx,
            isl_basic_map_n_param(bmap1) == isl_basic_map_n_param(bmap2),
            goto error);

        bmap1 = isl_basic_map_reverse(bmap1);
        bmap1 = isl_basic_map_apply_range(bmap1, bmap2);
        return isl_basic_map_reverse(bmap1);
error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

/* Given two basic maps A -> f(A) and B -> g(B), construct a basic map
 * A \cap B -> f(A) + f(B)
 */
struct isl_basic_map *isl_basic_map_sum(
                struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        unsigned n_in, n_out, nparam, total, pos;
        struct isl_basic_map *bmap = NULL;
        struct isl_dim_map *dim_map1, *dim_map2;
        int i;

        if (!bmap1 || !bmap2)
                goto error;

        isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim),
                goto error);

        nparam = isl_basic_map_n_param(bmap1);
        n_in = isl_basic_map_n_in(bmap1);
        n_out = isl_basic_map_n_out(bmap1);

        total = nparam + n_in + n_out + bmap1->n_div + bmap2->n_div + 2 * n_out;
        dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
        dim_map2 = isl_dim_map_alloc(bmap2->ctx, total);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos);
        isl_dim_map_div(dim_map1, bmap1, pos += n_in + n_out);
        isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += bmap2->n_div);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += n_out);

        bmap = isl_basic_map_alloc_space(isl_space_copy(bmap1->dim),
                        bmap1->n_div + bmap2->n_div + 2 * n_out,
                        bmap1->n_eq + bmap2->n_eq + n_out,
                        bmap1->n_ineq + bmap2->n_ineq);
        for (i = 0; i < n_out; ++i) {
                int j = isl_basic_map_alloc_equality(bmap);
                if (j < 0)
                        goto error;
                isl_seq_clr(bmap->eq[j], 1+total);
                isl_int_set_si(bmap->eq[j][1+nparam+n_in+i], -1);
                isl_int_set_si(bmap->eq[j][1+pos+i], 1);
                isl_int_set_si(bmap->eq[j][1+pos-n_out+i], 1);
        }
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
        bmap = add_divs(bmap, 2 * n_out);

        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

/* Given two maps A -> f(A) and B -> g(B), construct a map
 * A \cap B -> f(A) + f(B)
 */
struct isl_map *isl_map_sum(struct isl_map *map1, struct isl_map *map2)
{
        struct isl_map *result;
        int i, j;

        if (!map1 || !map2)
                goto error;

        isl_assert(map1->ctx, isl_space_is_equal(map1->dim, map2->dim), goto error);

        result = isl_map_alloc_space(isl_space_copy(map1->dim),
                                map1->n * map2->n, 0);
        if (!result)
                goto error;
        for (i = 0; i < map1->n; ++i)
                for (j = 0; j < map2->n; ++j) {
                        struct isl_basic_map *part;
                        part = isl_basic_map_sum(
                                    isl_basic_map_copy(map1->p[i]),
                                    isl_basic_map_copy(map2->p[j]));
                        if (isl_basic_map_is_empty(part))
                                isl_basic_map_free(part);
                        else
                                result = isl_map_add_basic_map(result, part);
                        if (!result)
                                goto error;
                }
        isl_map_free(map1);
        isl_map_free(map2);
        return result;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

__isl_give isl_set *isl_set_sum(__isl_take isl_set *set1,
        __isl_take isl_set *set2)
{
        return set_from_map(isl_map_sum(set_to_map(set1), set_to_map(set2)));
}

/* Given a basic map A -> f(A), construct A -> -f(A).
 */
struct isl_basic_map *isl_basic_map_neg(struct isl_basic_map *bmap)
{
        int i, j;
        unsigned off, n;

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

        n = isl_basic_map_dim(bmap, isl_dim_out);
        off = isl_basic_map_offset(bmap, isl_dim_out);
        for (i = 0; i < bmap->n_eq; ++i)
                for (j = 0; j < n; ++j)
                        isl_int_neg(bmap->eq[i][off+j], bmap->eq[i][off+j]);
        for (i = 0; i < bmap->n_ineq; ++i)
                for (j = 0; j < n; ++j)
                        isl_int_neg(bmap->ineq[i][off+j], bmap->ineq[i][off+j]);
        for (i = 0; i < bmap->n_div; ++i)
                for (j = 0; j < n; ++j)
                        isl_int_neg(bmap->div[i][1+off+j], bmap->div[i][1+off+j]);
        bmap = isl_basic_map_gauss(bmap, NULL);
        return isl_basic_map_finalize(bmap);
}

__isl_give isl_basic_set *isl_basic_set_neg(__isl_take isl_basic_set *bset)
{
        return isl_basic_map_neg(bset);
}

/* Given a map A -> f(A), construct A -> -f(A).
 */
struct isl_map *isl_map_neg(struct isl_map *map)
{
        int i;

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

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_neg(map->p[i]);
                if (!map->p[i])
                        goto error;
        }

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

__isl_give isl_set *isl_set_neg(__isl_take isl_set *set)
{
        return set_from_map(isl_map_neg(set_to_map(set)));
}

/* Given a basic map A -> f(A) and an integer d, construct a basic map
 * A -> floor(f(A)/d).
 */
struct isl_basic_map *isl_basic_map_floordiv(struct isl_basic_map *bmap,
                isl_int d)
{
        unsigned n_in, n_out, nparam, total, pos;
        struct isl_basic_map *result = NULL;
        struct isl_dim_map *dim_map;
        int i;

        if (!bmap)
                return NULL;

        nparam = isl_basic_map_n_param(bmap);
        n_in = isl_basic_map_n_in(bmap);
        n_out = isl_basic_map_n_out(bmap);

        total = nparam + n_in + n_out + bmap->n_div + n_out;
        dim_map = isl_dim_map_alloc(bmap->ctx, total);
        isl_dim_map_dim(dim_map, bmap->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map, bmap->dim, isl_dim_in, pos += nparam);
        isl_dim_map_div(dim_map, bmap, pos += n_in + n_out);
        isl_dim_map_dim(dim_map, bmap->dim, isl_dim_out, pos += bmap->n_div);

        result = isl_basic_map_alloc_space(isl_space_copy(bmap->dim),
                        bmap->n_div + n_out,
                        bmap->n_eq, bmap->n_ineq + 2 * n_out);
        result = isl_basic_map_add_constraints_dim_map(result, bmap, dim_map);
        result = add_divs(result, n_out);
        for (i = 0; i < n_out; ++i) {
                int j;
                j = isl_basic_map_alloc_inequality(result);
                if (j < 0)
                        goto error;
                isl_seq_clr(result->ineq[j], 1+total);
                isl_int_neg(result->ineq[j][1+nparam+n_in+i], d);
                isl_int_set_si(result->ineq[j][1+pos+i], 1);
                j = isl_basic_map_alloc_inequality(result);
                if (j < 0)
                        goto error;
                isl_seq_clr(result->ineq[j], 1+total);
                isl_int_set(result->ineq[j][1+nparam+n_in+i], d);
                isl_int_set_si(result->ineq[j][1+pos+i], -1);
                isl_int_sub_ui(result->ineq[j][0], d, 1);
        }

        result = isl_basic_map_simplify(result);
        return isl_basic_map_finalize(result);
error:
        isl_basic_map_free(result);
        return NULL;
}

/* Given a map A -> f(A) and an integer d, construct a map
 * A -> floor(f(A)/d).
 */
struct isl_map *isl_map_floordiv(struct isl_map *map, isl_int d)
{
        int i;

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

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

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

/* Given a map A -> f(A) and an integer d, construct a map
 * A -> floor(f(A)/d).
 */
__isl_give isl_map *isl_map_floordiv_val(__isl_take isl_map *map,
        __isl_take isl_val *d)
{
        if (!map || !d)
                goto error;
        if (!isl_val_is_int(d))
                isl_die(isl_val_get_ctx(d), isl_error_invalid,
                        "expecting integer denominator", goto error);
        map = isl_map_floordiv(map, d->n);
        isl_val_free(d);
        return map;
error:
        isl_map_free(map);
        isl_val_free(d);
        return NULL;
}

static struct isl_basic_map *var_equal(struct isl_basic_map *bmap, unsigned pos)
{
        int i;
        unsigned nparam;
        unsigned n_in;

        i = isl_basic_map_alloc_equality(bmap);
        if (i < 0)
                goto error;
        nparam = isl_basic_map_n_param(bmap);
        n_in = isl_basic_map_n_in(bmap);
        isl_seq_clr(bmap->eq[i], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->eq[i][1+nparam+pos], -1);
        isl_int_set_si(bmap->eq[i][1+nparam+n_in+pos], 1);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Add a constraint to "bmap" expressing i_pos < o_pos
 */
static struct isl_basic_map *var_less(struct isl_basic_map *bmap, unsigned pos)
{
        int i;
        unsigned nparam;
        unsigned n_in;

        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                goto error;
        nparam = isl_basic_map_n_param(bmap);
        n_in = isl_basic_map_n_in(bmap);
        isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->ineq[i][0], -1);
        isl_int_set_si(bmap->ineq[i][1+nparam+pos], -1);
        isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], 1);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Add a constraint to "bmap" expressing i_pos <= o_pos
 */
static __isl_give isl_basic_map *var_less_or_equal(
        __isl_take isl_basic_map *bmap, unsigned pos)
{
        int i;
        unsigned nparam;
        unsigned n_in;

        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                goto error;
        nparam = isl_basic_map_n_param(bmap);
        n_in = isl_basic_map_n_in(bmap);
        isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->ineq[i][1+nparam+pos], -1);
        isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], 1);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Add a constraint to "bmap" expressing i_pos > o_pos
 */
static struct isl_basic_map *var_more(struct isl_basic_map *bmap, unsigned pos)
{
        int i;
        unsigned nparam;
        unsigned n_in;

        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                goto error;
        nparam = isl_basic_map_n_param(bmap);
        n_in = isl_basic_map_n_in(bmap);
        isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->ineq[i][0], -1);
        isl_int_set_si(bmap->ineq[i][1+nparam+pos], 1);
        isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], -1);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Add a constraint to "bmap" expressing i_pos >= o_pos
 */
static __isl_give isl_basic_map *var_more_or_equal(
        __isl_take isl_basic_map *bmap, unsigned pos)
{
        int i;
        unsigned nparam;
        unsigned n_in;

        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                goto error;
        nparam = isl_basic_map_n_param(bmap);
        n_in = isl_basic_map_n_in(bmap);
        isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->ineq[i][1+nparam+pos], 1);
        isl_int_set_si(bmap->ineq[i][1+nparam+n_in+pos], -1);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_equal(
        __isl_take isl_space *dim, unsigned n_equal)
{
        int i;
        struct isl_basic_map *bmap;
        bmap = isl_basic_map_alloc_space(dim, 0, n_equal, 0);
        if (!bmap)
                return NULL;
        for (i = 0; i < n_equal && bmap; ++i)
                bmap = var_equal(bmap, i);
        return isl_basic_map_finalize(bmap);
}

/* Return a relation on of dimension "dim" expressing i_[0..pos] << o_[0..pos]
 */
__isl_give isl_basic_map *isl_basic_map_less_at(__isl_take isl_space *dim,
        unsigned pos)
{
        int i;
        struct isl_basic_map *bmap;
        bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
        if (!bmap)
                return NULL;
        for (i = 0; i < pos && bmap; ++i)
                bmap = var_equal(bmap, i);
        if (bmap)
                bmap = var_less(bmap, pos);
        return isl_basic_map_finalize(bmap);
}

/* Return a relation on "dim" expressing i_[0..pos] <<= o_[0..pos]
 */
__isl_give isl_basic_map *isl_basic_map_less_or_equal_at(
        __isl_take isl_space *dim, unsigned pos)
{
        int i;
        isl_basic_map *bmap;

        bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
        for (i = 0; i < pos; ++i)
                bmap = var_equal(bmap, i);
        bmap = var_less_or_equal(bmap, pos);
        return isl_basic_map_finalize(bmap);
}

/* Return a relation on "dim" expressing i_pos > o_pos
 */
__isl_give isl_basic_map *isl_basic_map_more_at(__isl_take isl_space *dim,
        unsigned pos)
{
        int i;
        struct isl_basic_map *bmap;
        bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
        if (!bmap)
                return NULL;
        for (i = 0; i < pos && bmap; ++i)
                bmap = var_equal(bmap, i);
        if (bmap)
                bmap = var_more(bmap, pos);
        return isl_basic_map_finalize(bmap);
}

/* Return a relation on "dim" expressing i_[0..pos] >>= o_[0..pos]
 */
__isl_give isl_basic_map *isl_basic_map_more_or_equal_at(
        __isl_take isl_space *dim, unsigned pos)
{
        int i;
        isl_basic_map *bmap;

        bmap = isl_basic_map_alloc_space(dim, 0, pos, 1);
        for (i = 0; i < pos; ++i)
                bmap = var_equal(bmap, i);
        bmap = var_more_or_equal(bmap, pos);
        return isl_basic_map_finalize(bmap);
}

static __isl_give isl_map *map_lex_lte_first(__isl_take isl_space *dims,
        unsigned n, int equal)
{
        struct isl_map *map;
        int i;

        if (n == 0 && equal)
                return isl_map_universe(dims);

        map = isl_map_alloc_space(isl_space_copy(dims), n, ISL_MAP_DISJOINT);

        for (i = 0; i + 1 < n; ++i)
                map = isl_map_add_basic_map(map,
                                  isl_basic_map_less_at(isl_space_copy(dims), i));
        if (n > 0) {
                if (equal)
                        map = isl_map_add_basic_map(map,
                              isl_basic_map_less_or_equal_at(dims, n - 1));
                else
                        map = isl_map_add_basic_map(map,
                              isl_basic_map_less_at(dims, n - 1));
        } else
                isl_space_free(dims);

        return map;
}

static __isl_give isl_map *map_lex_lte(__isl_take isl_space *dims, int equal)
{
        if (!dims)
                return NULL;
        return map_lex_lte_first(dims, dims->n_out, equal);
}

__isl_give isl_map *isl_map_lex_lt_first(__isl_take isl_space *dim, unsigned n)
{
        return map_lex_lte_first(dim, n, 0);
}

__isl_give isl_map *isl_map_lex_le_first(__isl_take isl_space *dim, unsigned n)
{
        return map_lex_lte_first(dim, n, 1);
}

__isl_give isl_map *isl_map_lex_lt(__isl_take isl_space *set_dim)
{
        return map_lex_lte(isl_space_map_from_set(set_dim), 0);
}

__isl_give isl_map *isl_map_lex_le(__isl_take isl_space *set_dim)
{
        return map_lex_lte(isl_space_map_from_set(set_dim), 1);
}

static __isl_give isl_map *map_lex_gte_first(__isl_take isl_space *dims,
        unsigned n, int equal)
{
        struct isl_map *map;
        int i;

        if (n == 0 && equal)
                return isl_map_universe(dims);

        map = isl_map_alloc_space(isl_space_copy(dims), n, ISL_MAP_DISJOINT);

        for (i = 0; i + 1 < n; ++i)
                map = isl_map_add_basic_map(map,
                                  isl_basic_map_more_at(isl_space_copy(dims), i));
        if (n > 0) {
                if (equal)
                        map = isl_map_add_basic_map(map,
                              isl_basic_map_more_or_equal_at(dims, n - 1));
                else
                        map = isl_map_add_basic_map(map,
                              isl_basic_map_more_at(dims, n - 1));
        } else
                isl_space_free(dims);

        return map;
}

static __isl_give isl_map *map_lex_gte(__isl_take isl_space *dims, int equal)
{
        if (!dims)
                return NULL;
        return map_lex_gte_first(dims, dims->n_out, equal);
}

__isl_give isl_map *isl_map_lex_gt_first(__isl_take isl_space *dim, unsigned n)
{
        return map_lex_gte_first(dim, n, 0);
}

__isl_give isl_map *isl_map_lex_ge_first(__isl_take isl_space *dim, unsigned n)
{
        return map_lex_gte_first(dim, n, 1);
}

__isl_give isl_map *isl_map_lex_gt(__isl_take isl_space *set_dim)
{
        return map_lex_gte(isl_space_map_from_set(set_dim), 0);
}

__isl_give isl_map *isl_map_lex_ge(__isl_take isl_space *set_dim)
{
        return map_lex_gte(isl_space_map_from_set(set_dim), 1);
}

__isl_give isl_map *isl_set_lex_le_set(__isl_take isl_set *set1,
        __isl_take isl_set *set2)
{
        isl_map *map;
        map = isl_map_lex_le(isl_set_get_space(set1));
        map = isl_map_intersect_domain(map, set1);
        map = isl_map_intersect_range(map, set2);
        return map;
}

__isl_give isl_map *isl_set_lex_lt_set(__isl_take isl_set *set1,
        __isl_take isl_set *set2)
{
        isl_map *map;
        map = isl_map_lex_lt(isl_set_get_space(set1));
        map = isl_map_intersect_domain(map, set1);
        map = isl_map_intersect_range(map, set2);
        return map;
}

__isl_give isl_map *isl_set_lex_ge_set(__isl_take isl_set *set1,
        __isl_take isl_set *set2)
{
        isl_map *map;
        map = isl_map_lex_ge(isl_set_get_space(set1));
        map = isl_map_intersect_domain(map, set1);
        map = isl_map_intersect_range(map, set2);
        return map;
}

__isl_give isl_map *isl_set_lex_gt_set(__isl_take isl_set *set1,
        __isl_take isl_set *set2)
{
        isl_map *map;
        map = isl_map_lex_gt(isl_set_get_space(set1));
        map = isl_map_intersect_domain(map, set1);
        map = isl_map_intersect_range(map, set2);
        return map;
}

__isl_give isl_map *isl_map_lex_le_map(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *map;
        map = isl_map_lex_le(isl_space_range(isl_map_get_space(map1)));
        map = isl_map_apply_domain(map, isl_map_reverse(map1));
        map = isl_map_apply_range(map, isl_map_reverse(map2));
        return map;
}

__isl_give isl_map *isl_map_lex_lt_map(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *map;
        map = isl_map_lex_lt(isl_space_range(isl_map_get_space(map1)));
        map = isl_map_apply_domain(map, isl_map_reverse(map1));
        map = isl_map_apply_range(map, isl_map_reverse(map2));
        return map;
}

__isl_give isl_map *isl_map_lex_ge_map(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *map;
        map = isl_map_lex_ge(isl_space_range(isl_map_get_space(map1)));
        map = isl_map_apply_domain(map, isl_map_reverse(map1));
        map = isl_map_apply_range(map, isl_map_reverse(map2));
        return map;
}

__isl_give isl_map *isl_map_lex_gt_map(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *map;
        map = isl_map_lex_gt(isl_space_range(isl_map_get_space(map1)));
        map = isl_map_apply_domain(map, isl_map_reverse(map1));
        map = isl_map_apply_range(map, isl_map_reverse(map2));
        return map;
}

static __isl_give isl_basic_map *basic_map_from_basic_set(
        __isl_take isl_basic_set *bset, __isl_take isl_space *dim)
{
        struct isl_basic_map *bmap;

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

        isl_assert(bset->ctx, isl_space_compatible_internal(bset->dim, dim),
                goto error);
        isl_space_free(bset->dim);
        bmap = bset_to_bmap(bset);
        bmap->dim = dim;
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_set_free(bset);
        isl_space_free(dim);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_from_basic_set(
        __isl_take isl_basic_set *bset, __isl_take isl_space *space)
{
        return basic_map_from_basic_set(bset, space);
}

/* For a div d = floor(f/m), add the constraint
 *
 *              f - m d >= 0
 */
static int add_upper_div_constraint(__isl_keep isl_basic_map *bmap,
        unsigned pos, isl_int *div)
{
        int i;
        unsigned total = isl_basic_map_total_dim(bmap);

        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                return -1;
        isl_seq_cpy(bmap->ineq[i], div + 1, 1 + total);
        isl_int_neg(bmap->ineq[i][1 + pos], div[0]);

        return 0;
}

/* For a div d = floor(f/m), add the constraint
 *
 *              -(f-(m-1)) + m d >= 0
 */
static int add_lower_div_constraint(__isl_keep isl_basic_map *bmap,
        unsigned pos, isl_int *div)
{
        int i;
        unsigned total = isl_basic_map_total_dim(bmap);

        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                return -1;
        isl_seq_neg(bmap->ineq[i], div + 1, 1 + total);
        isl_int_set(bmap->ineq[i][1 + pos], div[0]);
        isl_int_add(bmap->ineq[i][0], bmap->ineq[i][0], bmap->ineq[i][1 + pos]);
        isl_int_sub_ui(bmap->ineq[i][0], bmap->ineq[i][0], 1);

        return 0;
}

/* For a div d = floor(f/m), add the constraints
 *
 *              f - m d >= 0
 *              -(f-(m-1)) + m d >= 0
 *
 * Note that the second constraint is the negation of
 *
 *              f - m d >= m
 */
int isl_basic_map_add_div_constraints_var(__isl_keep isl_basic_map *bmap,
        unsigned pos, isl_int *div)
{
        if (add_upper_div_constraint(bmap, pos, div) < 0)
                return -1;
        if (add_lower_div_constraint(bmap, pos, div) < 0)
                return -1;
        return 0;
}

int isl_basic_set_add_div_constraints_var(__isl_keep isl_basic_set *bset,
        unsigned pos, isl_int *div)
{
        return isl_basic_map_add_div_constraints_var(bset_to_bmap(bset),
                                                        pos, div);
}

int isl_basic_map_add_div_constraints(struct isl_basic_map *bmap, unsigned div)
{
        unsigned total = isl_basic_map_total_dim(bmap);
        unsigned div_pos = total - bmap->n_div + div;

        return isl_basic_map_add_div_constraints_var(bmap, div_pos,
                                                        bmap->div[div]);
}

/* For each known div d = floor(f/m), add the constraints
 *
 *              f - m d >= 0
 *              -(f-(m-1)) + m d >= 0
 *
 * Remove duplicate constraints in case of some these div constraints
 * already appear in "bmap".
 */
__isl_give isl_basic_map *isl_basic_map_add_known_div_constraints(
        __isl_take isl_basic_map *bmap)
{
        unsigned n_div;

        if (!bmap)
                return NULL;
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        if (n_div == 0)
                return bmap;

        bmap = add_known_div_constraints(bmap);
        bmap = isl_basic_map_remove_duplicate_constraints(bmap, NULL, 0);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
}

/* Add the div constraint of sign "sign" for div "div" of "bmap".
 *
 * In particular, if this div is of the form d = floor(f/m),
 * then add the constraint
 *
 *              f - m d >= 0
 *
 * if sign < 0 or the constraint
 *
 *              -(f-(m-1)) + m d >= 0
 *
 * if sign > 0.
 */
int isl_basic_map_add_div_constraint(__isl_keep isl_basic_map *bmap,
        unsigned div, int sign)
{
        unsigned total;
        unsigned div_pos;

        if (!bmap)
                return -1;

        total = isl_basic_map_total_dim(bmap);
        div_pos = total - bmap->n_div + div;

        if (sign < 0)
                return add_upper_div_constraint(bmap, div_pos, bmap->div[div]);
        else
                return add_lower_div_constraint(bmap, div_pos, bmap->div[div]);
}

int isl_basic_set_add_div_constraints(struct isl_basic_set *bset, unsigned div)
{
        return isl_basic_map_add_div_constraints(bset, div);
}

struct isl_basic_set *isl_basic_map_underlying_set(
                struct isl_basic_map *bmap)
{
        if (!bmap)
                goto error;
        if (bmap->dim->nparam == 0 && bmap->dim->n_in == 0 &&
            bmap->n_div == 0 &&
            !isl_space_is_named_or_nested(bmap->dim, isl_dim_in) &&
            !isl_space_is_named_or_nested(bmap->dim, isl_dim_out))
                return bset_from_bmap(bmap);
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                goto error;
        bmap->dim = isl_space_underlying(bmap->dim, bmap->n_div);
        if (!bmap->dim)
                goto error;
        bmap->extra -= bmap->n_div;
        bmap->n_div = 0;
        bmap = isl_basic_map_finalize(bmap);
        return bset_from_bmap(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_underlying_set(
                __isl_take isl_basic_set *bset)
{
        return isl_basic_map_underlying_set(bset_to_bmap(bset));
}

/* Replace each element in "list" by the result of applying
 * isl_basic_map_underlying_set to the element.
 */
__isl_give isl_basic_set_list *isl_basic_map_list_underlying_set(
        __isl_take isl_basic_map_list *list)
{
        int i, n;

        if (!list)
                return NULL;

        n = isl_basic_map_list_n_basic_map(list);
        for (i = 0; i < n; ++i) {
                isl_basic_map *bmap;
                isl_basic_set *bset;

                bmap = isl_basic_map_list_get_basic_map(list, i);
                bset = isl_basic_set_underlying_set(bmap);
                list = isl_basic_set_list_set_basic_set(list, i, bset);
        }

        return list;
}

struct isl_basic_map *isl_basic_map_overlying_set(
        struct isl_basic_set *bset, struct isl_basic_map *like)
{
        struct isl_basic_map *bmap;
        struct isl_ctx *ctx;
        unsigned total;
        int i;

        if (!bset || !like)
                goto error;
        ctx = bset->ctx;
        isl_assert(ctx, bset->n_div == 0, goto error);
        isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
        isl_assert(ctx, bset->dim->n_out == isl_basic_map_total_dim(like),
                        goto error);
        if (like->n_div == 0) {
                isl_space *space = isl_basic_map_get_space(like);
                isl_basic_map_free(like);
                return isl_basic_map_reset_space(bset, space);
        }
        bset = isl_basic_set_cow(bset);
        if (!bset)
                goto error;
        total = bset->dim->n_out + bset->extra;
        bmap = bset_to_bmap(bset);
        isl_space_free(bmap->dim);
        bmap->dim = isl_space_copy(like->dim);
        if (!bmap->dim)
                goto error;
        bmap->n_div = like->n_div;
        bmap->extra += like->n_div;
        if (bmap->extra) {
                unsigned ltotal;
                isl_int **div;
                ltotal = total - bmap->extra + like->extra;
                if (ltotal > total)
                        ltotal = total;
                bmap->block2 = isl_blk_extend(ctx, bmap->block2,
                                        bmap->extra * (1 + 1 + total));
                if (isl_blk_is_error(bmap->block2))
                        goto error;
                div = isl_realloc_array(ctx, bmap->div, isl_int *, bmap->extra);
                if (!div)
                        goto error;
                bmap->div = div;
                for (i = 0; i < bmap->extra; ++i)
                        bmap->div[i] = bmap->block2.data + i * (1 + 1 + total);
                for (i = 0; i < like->n_div; ++i) {
                        isl_seq_cpy(bmap->div[i], like->div[i], 1 + 1 + ltotal);
                        isl_seq_clr(bmap->div[i]+1+1+ltotal, total - ltotal);
                }
                bmap = isl_basic_map_add_known_div_constraints(bmap);
        }
        isl_basic_map_free(like);
        bmap = isl_basic_map_simplify(bmap);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_basic_map_free(like);
        isl_basic_set_free(bset);
        return NULL;
}

struct isl_basic_set *isl_basic_set_from_underlying_set(
        struct isl_basic_set *bset, struct isl_basic_set *like)
{
        return bset_from_bmap(isl_basic_map_overlying_set(bset,
                                                        bset_to_bmap(like)));
}

struct isl_set *isl_set_from_underlying_set(
        struct isl_set *set, struct isl_basic_set *like)
{
        int i;

        if (!set || !like)
                goto error;
        isl_assert(set->ctx, set->dim->n_out == isl_basic_set_total_dim(like),
                    goto error);
        if (isl_space_is_equal(set->dim, like->dim) && like->n_div == 0) {
                isl_basic_set_free(like);
                return set;
        }
        set = isl_set_cow(set);
        if (!set)
                goto error;
        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_from_underlying_set(set->p[i],
                                                      isl_basic_set_copy(like));
                if (!set->p[i])
                        goto error;
        }
        isl_space_free(set->dim);
        set->dim = isl_space_copy(like->dim);
        if (!set->dim)
                goto error;
        isl_basic_set_free(like);
        return set;
error:
        isl_basic_set_free(like);
        isl_set_free(set);
        return NULL;
}

struct isl_set *isl_map_underlying_set(struct isl_map *map)
{
        int i;

        map = isl_map_cow(map);
        if (!map)
                return NULL;
        map->dim = isl_space_cow(map->dim);
        if (!map->dim)
                goto error;

        for (i = 1; i < map->n; ++i)
                isl_assert(map->ctx, map->p[0]->n_div == map->p[i]->n_div,
                                goto error);
        for (i = 0; i < map->n; ++i) {
                map->p[i] = bset_to_bmap(
                                isl_basic_map_underlying_set(map->p[i]));
                if (!map->p[i])
                        goto error;
        }
        if (map->n == 0)
                map->dim = isl_space_underlying(map->dim, 0);
        else {
                isl_space_free(map->dim);
                map->dim = isl_space_copy(map->p[0]->dim);
        }
        if (!map->dim)
                goto error;
        return set_from_map(map);
error:
        isl_map_free(map);
        return NULL;
}

struct isl_set *isl_set_to_underlying_set(struct isl_set *set)
{
        return set_from_map(isl_map_underlying_set(set_to_map(set)));
}

/* Replace the space of "bmap" by "space".
 *
 * If the space of "bmap" is identical to "space" (including the identifiers
 * of the input and output dimensions), then simply return the original input.
 */
__isl_give isl_basic_map *isl_basic_map_reset_space(
        __isl_take isl_basic_map *bmap, __isl_take isl_space *space)
{
        isl_bool equal;

        if (!bmap)
                goto error;
        equal = isl_space_is_equal(bmap->dim, space);
        if (equal >= 0 && equal)
                equal = isl_space_match(bmap->dim, isl_dim_in,
                                        space, isl_dim_in);
        if (equal >= 0 && equal)
                equal = isl_space_match(bmap->dim, isl_dim_out,
                                        space, isl_dim_out);
        if (equal < 0)
                goto error;
        if (equal) {
                isl_space_free(space);
                return bmap;
        }
        bmap = isl_basic_map_cow(bmap);
        if (!bmap || !space)
                goto error;

        isl_space_free(bmap->dim);
        bmap->dim = space;

        bmap = isl_basic_map_finalize(bmap);

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

__isl_give isl_basic_set *isl_basic_set_reset_space(
        __isl_take isl_basic_set *bset, __isl_take isl_space *dim)
{
        return bset_from_bmap(isl_basic_map_reset_space(bset_to_bmap(bset),
                                                        dim));
}

__isl_give isl_map *isl_map_reset_space(__isl_take isl_map *map,
        __isl_take isl_space *dim)
{
        int i;

        map = isl_map_cow(map);
        if (!map || !dim)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_reset_space(map->p[i],
                                                    isl_space_copy(dim));
                if (!map->p[i])
                        goto error;
        }
        isl_space_free(map->dim);
        map->dim = dim;

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

__isl_give isl_set *isl_set_reset_space(__isl_take isl_set *set,
        __isl_take isl_space *dim)
{
        return set_from_map(isl_map_reset_space(set_to_map(set), dim));
}

/* Compute the parameter domain of the given basic set.
 */
__isl_give isl_basic_set *isl_basic_set_params(__isl_take isl_basic_set *bset)
{
        isl_space *space;
        unsigned n;

        if (isl_basic_set_is_params(bset))
                return bset;

        n = isl_basic_set_dim(bset, isl_dim_set);
        bset = isl_basic_set_project_out(bset, isl_dim_set, 0, n);
        space = isl_basic_set_get_space(bset);
        space = isl_space_params(space);
        bset = isl_basic_set_reset_space(bset, space);
        return bset;
}

/* Construct a zero-dimensional basic set with the given parameter domain.
 */
__isl_give isl_basic_set *isl_basic_set_from_params(
        __isl_take isl_basic_set *bset)
{
        isl_space *space;
        space = isl_basic_set_get_space(bset);
        space = isl_space_set_from_params(space);
        bset = isl_basic_set_reset_space(bset, space);
        return bset;
}

/* Compute the parameter domain of the given set.
 */
__isl_give isl_set *isl_set_params(__isl_take isl_set *set)
{
        isl_space *space;
        unsigned n;

        if (isl_set_is_params(set))
                return set;

        n = isl_set_dim(set, isl_dim_set);
        set = isl_set_project_out(set, isl_dim_set, 0, n);
        space = isl_set_get_space(set);
        space = isl_space_params(space);
        set = isl_set_reset_space(set, space);
        return set;
}

/* Construct a zero-dimensional set with the given parameter domain.
 */
__isl_give isl_set *isl_set_from_params(__isl_take isl_set *set)
{
        isl_space *space;
        space = isl_set_get_space(set);
        space = isl_space_set_from_params(space);
        set = isl_set_reset_space(set, space);
        return set;
}

/* Compute the parameter domain of the given map.
 */
__isl_give isl_set *isl_map_params(__isl_take isl_map *map)
{
        isl_space *space;
        unsigned n;

        n = isl_map_dim(map, isl_dim_in);
        map = isl_map_project_out(map, isl_dim_in, 0, n);
        n = isl_map_dim(map, isl_dim_out);
        map = isl_map_project_out(map, isl_dim_out, 0, n);
        space = isl_map_get_space(map);
        space = isl_space_params(space);
        map = isl_map_reset_space(map, space);
        return map;
}

struct isl_basic_set *isl_basic_map_domain(struct isl_basic_map *bmap)
{
        isl_space *space;
        unsigned n_out;

        if (!bmap)
                return NULL;
        space = isl_space_domain(isl_basic_map_get_space(bmap));

        n_out = isl_basic_map_n_out(bmap);
        bmap = isl_basic_map_project_out(bmap, isl_dim_out, 0, n_out);

        return isl_basic_map_reset_space(bmap, space);
}

int isl_basic_map_may_be_set(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return -1;
        return isl_space_may_be_set(bmap->dim);
}

/* Is this basic map actually a set?
 * Users should never call this function.  Outside of isl,
 * the type should indicate whether something is a set or a map.
 */
int isl_basic_map_is_set(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return -1;
        return isl_space_is_set(bmap->dim);
}

struct isl_basic_set *isl_basic_map_range(struct isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;
        if (isl_basic_map_is_set(bmap))
                return bmap;
        return isl_basic_map_domain(isl_basic_map_reverse(bmap));
}

__isl_give isl_basic_map *isl_basic_map_domain_map(
        __isl_take isl_basic_map *bmap)
{
        int i, k;
        isl_space *dim;
        isl_basic_map *domain;
        int nparam, n_in, n_out;
        unsigned total;

        nparam = isl_basic_map_dim(bmap, isl_dim_param);
        n_in = isl_basic_map_dim(bmap, isl_dim_in);
        n_out = isl_basic_map_dim(bmap, isl_dim_out);

        dim = isl_space_from_range(isl_space_domain(isl_basic_map_get_space(bmap)));
        domain = isl_basic_map_universe(dim);

        bmap = isl_basic_map_from_domain(isl_basic_map_wrap(bmap));
        bmap = isl_basic_map_apply_range(bmap, domain);
        bmap = isl_basic_map_extend_constraints(bmap, n_in, 0);

        total = isl_basic_map_total_dim(bmap);

        for (i = 0; i < n_in; ++i) {
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_clr(bmap->eq[k], 1 + total);
                isl_int_set_si(bmap->eq[k][1 + nparam + i], -1);
                isl_int_set_si(bmap->eq[k][1 + nparam + n_in + n_out + i], 1);
        }

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

__isl_give isl_basic_map *isl_basic_map_range_map(
        __isl_take isl_basic_map *bmap)
{
        int i, k;
        isl_space *dim;
        isl_basic_map *range;
        int nparam, n_in, n_out;
        unsigned total;

        nparam = isl_basic_map_dim(bmap, isl_dim_param);
        n_in = isl_basic_map_dim(bmap, isl_dim_in);
        n_out = isl_basic_map_dim(bmap, isl_dim_out);

        dim = isl_space_from_range(isl_space_range(isl_basic_map_get_space(bmap)));
        range = isl_basic_map_universe(dim);

        bmap = isl_basic_map_from_domain(isl_basic_map_wrap(bmap));
        bmap = isl_basic_map_apply_range(bmap, range);
        bmap = isl_basic_map_extend_constraints(bmap, n_out, 0);

        total = isl_basic_map_total_dim(bmap);

        for (i = 0; i < n_out; ++i) {
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_clr(bmap->eq[k], 1 + total);
                isl_int_set_si(bmap->eq[k][1 + nparam + n_in + i], -1);
                isl_int_set_si(bmap->eq[k][1 + nparam + n_in + n_out + i], 1);
        }

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

int isl_map_may_be_set(__isl_keep isl_map *map)
{
        if (!map)
                return -1;
        return isl_space_may_be_set(map->dim);
}

/* Is this map actually a set?
 * Users should never call this function.  Outside of isl,
 * the type should indicate whether something is a set or a map.
 */
int isl_map_is_set(__isl_keep isl_map *map)
{
        if (!map)
                return -1;
        return isl_space_is_set(map->dim);
}

struct isl_set *isl_map_range(struct isl_map *map)
{
        int i;
        struct isl_set *set;

        if (!map)
                goto error;
        if (isl_map_is_set(map))
                return set_from_map(map);

        map = isl_map_cow(map);
        if (!map)
                goto error;

        set = set_from_map(map);
        set->dim = isl_space_range(set->dim);
        if (!set->dim)
                goto error;
        for (i = 0; i < map->n; ++i) {
                set->p[i] = isl_basic_map_range(map->p[i]);
                if (!set->p[i])
                        goto error;
        }
        ISL_F_CLR(set, ISL_MAP_DISJOINT);
        ISL_F_CLR(set, ISL_SET_NORMALIZED);
        return set;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map)
{
        int i;

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

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

__isl_give isl_map *isl_map_range_map(__isl_take isl_map *map)
{
        int i;
        isl_space *range_dim;

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

        range_dim = isl_space_range(isl_map_get_space(map));
        range_dim = isl_space_from_range(range_dim);
        map->dim = isl_space_from_domain(isl_space_wrap(map->dim));
        map->dim = isl_space_join(map->dim, range_dim);
        if (!map->dim)
                goto error;
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_range_map(map->p[i]);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_DISJOINT);
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

/* Given a wrapped map of the form A[B -> C],
 * return the map A[B -> C] -> B.
 */
__isl_give isl_map *isl_set_wrapped_domain_map(__isl_take isl_set *set)
{
        isl_id *id;
        isl_map *map;

        if (!set)
                return NULL;
        if (!isl_set_has_tuple_id(set))
                return isl_map_domain_map(isl_set_unwrap(set));

        id = isl_set_get_tuple_id(set);
        map = isl_map_domain_map(isl_set_unwrap(set));
        map = isl_map_set_tuple_id(map, isl_dim_in, id);

        return map;
}

__isl_give isl_map *isl_map_from_set(__isl_take isl_set *set,
        __isl_take isl_space *dim)
{
        int i;
        struct isl_map *map = NULL;

        set = isl_set_cow(set);
        if (!set || !dim)
                goto error;
        isl_assert(set->ctx, isl_space_compatible_internal(set->dim, dim),
                goto error);
        map = set_to_map(set);
        for (i = 0; i < set->n; ++i) {
                map->p[i] = basic_map_from_basic_set(
                                set->p[i], isl_space_copy(dim));
                if (!map->p[i])
                        goto error;
        }
        isl_space_free(map->dim);
        map->dim = dim;
        return map;
error:
        isl_space_free(dim);
        isl_set_free(set);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_from_domain(
        __isl_take isl_basic_set *bset)
{
        return isl_basic_map_reverse(isl_basic_map_from_range(bset));
}

__isl_give isl_basic_map *isl_basic_map_from_range(
        __isl_take isl_basic_set *bset)
{
        isl_space *space;
        space = isl_basic_set_get_space(bset);
        space = isl_space_from_range(space);
        bset = isl_basic_set_reset_space(bset, space);
        return bset_to_bmap(bset);
}

/* Create a relation with the given set as range.
 * The domain of the created relation is a zero-dimensional
 * flat anonymous space.
 */
__isl_give isl_map *isl_map_from_range(__isl_take isl_set *set)
{
        isl_space *space;
        space = isl_set_get_space(set);
        space = isl_space_from_range(space);
        set = isl_set_reset_space(set, space);
        return set_to_map(set);
}

/* Create a relation with the given set as domain.
 * The range of the created relation is a zero-dimensional
 * flat anonymous space.
 */
__isl_give isl_map *isl_map_from_domain(__isl_take isl_set *set)
{
        return isl_map_reverse(isl_map_from_range(set));
}

__isl_give isl_basic_map *isl_basic_map_from_domain_and_range(
        __isl_take isl_basic_set *domain, __isl_take isl_basic_set *range)
{
        return isl_basic_map_apply_range(isl_basic_map_reverse(domain), range);
}

__isl_give isl_map *isl_map_from_domain_and_range(__isl_take isl_set *domain,
        __isl_take isl_set *range)
{
        return isl_map_apply_range(isl_map_reverse(domain), range);
}

/* Return a newly allocated isl_map with given space and flags and
 * room for "n" basic maps.
 * Make sure that all cached information is cleared.
 */
__isl_give isl_map *isl_map_alloc_space(__isl_take isl_space *space, int n,
        unsigned flags)
{
        struct isl_map *map;

        if (!space)
                return NULL;
        if (n < 0)
                isl_die(space->ctx, isl_error_internal,
                        "negative number of basic maps", goto error);
        map = isl_calloc(space->ctx, struct isl_map,
                        sizeof(struct isl_map) +
                        (n - 1) * sizeof(struct isl_basic_map *));
        if (!map)
                goto error;

        map->ctx = space->ctx;
        isl_ctx_ref(map->ctx);
        map->ref = 1;
        map->size = n;
        map->n = 0;
        map->dim = space;
        map->flags = flags;
        return map;
error:
        isl_space_free(space);
        return NULL;
}

struct isl_map *isl_map_alloc(struct isl_ctx *ctx,
                unsigned nparam, unsigned in, unsigned out, int n,
                unsigned flags)
{
        struct isl_map *map;
        isl_space *dims;

        dims = isl_space_alloc(ctx, nparam, in, out);
        if (!dims)
                return NULL;

        map = isl_map_alloc_space(dims, n, flags);
        return map;
}

__isl_give isl_basic_map *isl_basic_map_empty(__isl_take isl_space *dim)
{
        struct isl_basic_map *bmap;
        bmap = isl_basic_map_alloc_space(dim, 0, 1, 0);
        bmap = isl_basic_map_set_to_empty(bmap);
        return bmap;
}

__isl_give isl_basic_set *isl_basic_set_empty(__isl_take isl_space *dim)
{
        struct isl_basic_set *bset;
        bset = isl_basic_set_alloc_space(dim, 0, 1, 0);
        bset = isl_basic_set_set_to_empty(bset);
        return bset;
}

__isl_give isl_basic_map *isl_basic_map_universe(__isl_take isl_space *dim)
{
        struct isl_basic_map *bmap;
        bmap = isl_basic_map_alloc_space(dim, 0, 0, 0);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
}

__isl_give isl_basic_set *isl_basic_set_universe(__isl_take isl_space *dim)
{
        struct isl_basic_set *bset;
        bset = isl_basic_set_alloc_space(dim, 0, 0, 0);
        bset = isl_basic_set_finalize(bset);
        return bset;
}

__isl_give isl_basic_map *isl_basic_map_nat_universe(__isl_take isl_space *dim)
{
        int i;
        unsigned total = isl_space_dim(dim, isl_dim_all);
        isl_basic_map *bmap;

        bmap= isl_basic_map_alloc_space(dim, 0, 0, total);
        for (i = 0; i < total; ++i) {
                int k = isl_basic_map_alloc_inequality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_clr(bmap->ineq[k], 1 + total);
                isl_int_set_si(bmap->ineq[k][1 + i], 1);
        }
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_nat_universe(__isl_take isl_space *dim)
{
        return isl_basic_map_nat_universe(dim);
}

__isl_give isl_map *isl_map_nat_universe(__isl_take isl_space *dim)
{
        return isl_map_from_basic_map(isl_basic_map_nat_universe(dim));
}

__isl_give isl_set *isl_set_nat_universe(__isl_take isl_space *dim)
{
        return isl_map_nat_universe(dim);
}

__isl_give isl_map *isl_map_empty(__isl_take isl_space *dim)
{
        return isl_map_alloc_space(dim, 0, ISL_MAP_DISJOINT);
}

__isl_give isl_set *isl_set_empty(__isl_take isl_space *dim)
{
        return isl_set_alloc_space(dim, 0, ISL_MAP_DISJOINT);
}

__isl_give isl_map *isl_map_universe(__isl_take isl_space *dim)
{
        struct isl_map *map;
        if (!dim)
                return NULL;
        map = isl_map_alloc_space(isl_space_copy(dim), 1, ISL_MAP_DISJOINT);
        map = isl_map_add_basic_map(map, isl_basic_map_universe(dim));
        return map;
}

__isl_give isl_set *isl_set_universe(__isl_take isl_space *dim)
{
        struct isl_set *set;
        if (!dim)
                return NULL;
        set = isl_set_alloc_space(isl_space_copy(dim), 1, ISL_MAP_DISJOINT);
        set = isl_set_add_basic_set(set, isl_basic_set_universe(dim));
        return set;
}

struct isl_map *isl_map_dup(struct isl_map *map)
{
        int i;
        struct isl_map *dup;

        if (!map)
                return NULL;
        dup = isl_map_alloc_space(isl_space_copy(map->dim), map->n, map->flags);
        for (i = 0; i < map->n; ++i)
                dup = isl_map_add_basic_map(dup, isl_basic_map_copy(map->p[i]));
        return dup;
}

__isl_give isl_map *isl_map_add_basic_map(__isl_take isl_map *map,
                                                __isl_take isl_basic_map *bmap)
{
        if (!bmap || !map)
                goto error;
        if (isl_basic_map_plain_is_empty(bmap)) {
                isl_basic_map_free(bmap);
                return map;
        }
        isl_assert(map->ctx, isl_space_is_equal(map->dim, bmap->dim), goto error);
        isl_assert(map->ctx, map->n < map->size, goto error);
        map->p[map->n] = bmap;
        map->n++;
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        if (map)
                isl_map_free(map);
        if (bmap)
                isl_basic_map_free(bmap);
        return NULL;
}

__isl_null isl_map *isl_map_free(__isl_take isl_map *map)
{
        int i;

        if (!map)
                return NULL;

        if (--map->ref > 0)
                return NULL;

        clear_caches(map);
        isl_ctx_deref(map->ctx);
        for (i = 0; i < map->n; ++i)
                isl_basic_map_free(map->p[i]);
        isl_space_free(map->dim);
        free(map);

        return NULL;
}

static struct isl_basic_map *isl_basic_map_fix_pos_si(
        struct isl_basic_map *bmap, unsigned pos, int value)
{
        int j;

        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
        j = isl_basic_map_alloc_equality(bmap);
        if (j < 0)
                goto error;
        isl_seq_clr(bmap->eq[j] + 1, isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->eq[j][pos], -1);
        isl_int_set_si(bmap->eq[j][0], value);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

static __isl_give isl_basic_map *isl_basic_map_fix_pos(
        __isl_take isl_basic_map *bmap, unsigned pos, isl_int value)
{
        int j;

        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
        j = isl_basic_map_alloc_equality(bmap);
        if (j < 0)
                goto error;
        isl_seq_clr(bmap->eq[j] + 1, isl_basic_map_total_dim(bmap));
        isl_int_set_si(bmap->eq[j][pos], -1);
        isl_int_set(bmap->eq[j][0], value);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_basic_map *isl_basic_map_fix_si(struct isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos, int value)
{
        if (!bmap)
                return NULL;
        isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), goto error);
        return isl_basic_map_fix_pos_si(bmap,
                isl_basic_map_offset(bmap, type) + pos, value);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_fix(__isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos, isl_int value)
{
        if (!bmap)
                return NULL;
        isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), goto error);
        return isl_basic_map_fix_pos(bmap,
                isl_basic_map_offset(bmap, type) + pos, value);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Fix the value of the variable at position "pos" of type "type" of "bmap"
 * to be equal to "v".
 */
__isl_give isl_basic_map *isl_basic_map_fix_val(__isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
{
        if (!bmap || !v)
                goto error;
        if (!isl_val_is_int(v))
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "expecting integer value", goto error);
        if (pos >= isl_basic_map_dim(bmap, type))
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "index out of bounds", goto error);
        pos += isl_basic_map_offset(bmap, type);
        bmap = isl_basic_map_fix_pos(bmap, pos, v->n);
        isl_val_free(v);
        return bmap;
error:
        isl_basic_map_free(bmap);
        isl_val_free(v);
        return NULL;
}

/* Fix the value of the variable at position "pos" of type "type" of "bset"
 * to be equal to "v".
 */
__isl_give isl_basic_set *isl_basic_set_fix_val(__isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
{
        return isl_basic_map_fix_val(bset, type, pos, v);
}

struct isl_basic_set *isl_basic_set_fix_si(struct isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos, int value)
{
        return bset_from_bmap(isl_basic_map_fix_si(bset_to_bmap(bset),
                                                    type, pos, value));
}

__isl_give isl_basic_set *isl_basic_set_fix(__isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos, isl_int value)
{
        return bset_from_bmap(isl_basic_map_fix(bset_to_bmap(bset),
                                                    type, pos, value));
}

struct isl_basic_map *isl_basic_map_fix_input_si(struct isl_basic_map *bmap,
                unsigned input, int value)
{
        return isl_basic_map_fix_si(bmap, isl_dim_in, input, value);
}

struct isl_basic_set *isl_basic_set_fix_dim_si(struct isl_basic_set *bset,
                unsigned dim, int value)
{
        return bset_from_bmap(isl_basic_map_fix_si(bset_to_bmap(bset),
                                        isl_dim_set, dim, value));
}

static int remove_if_empty(__isl_keep isl_map *map, int i)
{
        int empty = isl_basic_map_plain_is_empty(map->p[i]);

        if (empty < 0)
                return -1;
        if (!empty)
                return 0;

        isl_basic_map_free(map->p[i]);
        if (i != map->n - 1) {
                ISL_F_CLR(map, ISL_MAP_NORMALIZED);
                map->p[i] = map->p[map->n - 1];
        }
        map->n--;

        return 0;
}

/* Perform "fn" on each basic map of "map", where we may not be holding
 * the only reference to "map".
 * In particular, "fn" should be a semantics preserving operation
 * that we want to apply to all copies of "map".  We therefore need
 * to be careful not to modify "map" in a way that breaks "map"
 * in case anything goes wrong.
 */
__isl_give isl_map *isl_map_inline_foreach_basic_map(__isl_take isl_map *map,
        __isl_give isl_basic_map *(*fn)(__isl_take isl_basic_map *bmap))
{
        struct isl_basic_map *bmap;
        int i;

        if (!map)
                return NULL;

        for (i = map->n - 1; i >= 0; --i) {
                bmap = isl_basic_map_copy(map->p[i]);
                bmap = fn(bmap);
                if (!bmap)
                        goto error;
                isl_basic_map_free(map->p[i]);
                map->p[i] = bmap;
                if (remove_if_empty(map, i) < 0)
                        goto error;
        }

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

struct isl_map *isl_map_fix_si(struct isl_map *map,
                enum isl_dim_type type, unsigned pos, int value)
{
        int i;

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

        isl_assert(map->ctx, pos < isl_map_dim(map, type), goto error);
        for (i = map->n - 1; i >= 0; --i) {
                map->p[i] = isl_basic_map_fix_si(map->p[i], type, pos, value);
                if (remove_if_empty(map, i) < 0)
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_set *isl_set_fix_si(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value)
{
        return set_from_map(isl_map_fix_si(set_to_map(set), type, pos, value));
}

__isl_give isl_map *isl_map_fix(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, isl_int value)
{
        int i;

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

        isl_assert(map->ctx, pos < isl_map_dim(map, type), goto error);
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_fix(map->p[i], type, pos, value);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_set *isl_set_fix(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, isl_int value)
{
        return set_from_map(isl_map_fix(set_to_map(set), type, pos, value));
}

/* Fix the value of the variable at position "pos" of type "type" of "map"
 * to be equal to "v".
 */
__isl_give isl_map *isl_map_fix_val(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
{
        int i;

        map = isl_map_cow(map);
        if (!map || !v)
                goto error;

        if (!isl_val_is_int(v))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "expecting integer value", goto error);
        if (pos >= isl_map_dim(map, type))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "index out of bounds", goto error);
        for (i = map->n - 1; i >= 0; --i) {
                map->p[i] = isl_basic_map_fix_val(map->p[i], type, pos,
                                                        isl_val_copy(v));
                if (remove_if_empty(map, i) < 0)
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        isl_val_free(v);
        return map;
error:
        isl_map_free(map);
        isl_val_free(v);
        return NULL;
}

/* Fix the value of the variable at position "pos" of type "type" of "set"
 * to be equal to "v".
 */
__isl_give isl_set *isl_set_fix_val(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, __isl_take isl_val *v)
{
        return isl_map_fix_val(set, type, pos, v);
}

struct isl_map *isl_map_fix_input_si(struct isl_map *map,
                unsigned input, int value)
{
        return isl_map_fix_si(map, isl_dim_in, input, value);
}

struct isl_set *isl_set_fix_dim_si(struct isl_set *set, unsigned dim, int value)
{
        return set_from_map(isl_map_fix_si(set_to_map(set),
                                                isl_dim_set, dim, value));
}

static __isl_give isl_basic_map *basic_map_bound_si(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, int value, int upper)
{
        int j;

        if (!bmap)
                return NULL;
        isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type), goto error);
        pos += isl_basic_map_offset(bmap, type);
        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
        j = isl_basic_map_alloc_inequality(bmap);
        if (j < 0)
                goto error;
        isl_seq_clr(bmap->ineq[j], 1 + isl_basic_map_total_dim(bmap));
        if (upper) {
                isl_int_set_si(bmap->ineq[j][pos], -1);
                isl_int_set_si(bmap->ineq[j][0], value);
        } else {
                isl_int_set_si(bmap->ineq[j][pos], 1);
                isl_int_set_si(bmap->ineq[j][0], -value);
        }
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_lower_bound_si(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, int value)
{
        return basic_map_bound_si(bmap, type, pos, value, 0);
}

/* Constrain the values of the given dimension to be no greater than "value".
 */
__isl_give isl_basic_map *isl_basic_map_upper_bound_si(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, int value)
{
        return basic_map_bound_si(bmap, type, pos, value, 1);
}

struct isl_basic_set *isl_basic_set_lower_bound_dim(struct isl_basic_set *bset,
        unsigned dim, isl_int value)
{
        int j;

        bset = isl_basic_set_cow(bset);
        bset = isl_basic_set_extend_constraints(bset, 0, 1);
        j = isl_basic_set_alloc_inequality(bset);
        if (j < 0)
                goto error;
        isl_seq_clr(bset->ineq[j], 1 + isl_basic_set_total_dim(bset));
        isl_int_set_si(bset->ineq[j][1 + isl_basic_set_n_param(bset) + dim], 1);
        isl_int_neg(bset->ineq[j][0], value);
        bset = isl_basic_set_simplify(bset);
        return isl_basic_set_finalize(bset);
error:
        isl_basic_set_free(bset);
        return NULL;
}

static __isl_give isl_map *map_bound_si(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, int value, int upper)
{
        int i;

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

        isl_assert(map->ctx, pos < isl_map_dim(map, type), goto error);
        for (i = 0; i < map->n; ++i) {
                map->p[i] = basic_map_bound_si(map->p[i],
                                                 type, pos, value, upper);
                if (!map->p[i])
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_map *isl_map_lower_bound_si(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, int value)
{
        return map_bound_si(map, type, pos, value, 0);
}

__isl_give isl_map *isl_map_upper_bound_si(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, int value)
{
        return map_bound_si(map, type, pos, value, 1);
}

__isl_give isl_set *isl_set_lower_bound_si(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value)
{
        return set_from_map(isl_map_lower_bound_si(set_to_map(set),
                                                        type, pos, value));
}

__isl_give isl_set *isl_set_upper_bound_si(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, int value)
{
        return isl_map_upper_bound_si(set, type, pos, value);
}

/* Bound the given variable of "bmap" from below (or above is "upper"
 * is set) to "value".
 */
static __isl_give isl_basic_map *basic_map_bound(
        __isl_take isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, isl_int value, int upper)
{
        int j;

        if (!bmap)
                return NULL;
        if (pos >= isl_basic_map_dim(bmap, type))
                isl_die(bmap->ctx, isl_error_invalid,
                        "index out of bounds", goto error);
        pos += isl_basic_map_offset(bmap, type);
        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
        j = isl_basic_map_alloc_inequality(bmap);
        if (j < 0)
                goto error;
        isl_seq_clr(bmap->ineq[j], 1 + isl_basic_map_total_dim(bmap));
        if (upper) {
                isl_int_set_si(bmap->ineq[j][pos], -1);
                isl_int_set(bmap->ineq[j][0], value);
        } else {
                isl_int_set_si(bmap->ineq[j][pos], 1);
                isl_int_neg(bmap->ineq[j][0], value);
        }
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Bound the given variable of "map" from below (or above is "upper"
 * is set) to "value".
 */
static __isl_give isl_map *map_bound(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, isl_int value, int upper)
{
        int i;

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

        if (pos >= isl_map_dim(map, type))
                isl_die(map->ctx, isl_error_invalid,
                        "index out of bounds", goto error);
        for (i = map->n - 1; i >= 0; --i) {
                map->p[i] = basic_map_bound(map->p[i], type, pos, value, upper);
                if (remove_if_empty(map, i) < 0)
                        goto error;
        }
        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_map *isl_map_lower_bound(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, isl_int value)
{
        return map_bound(map, type, pos, value, 0);
}

__isl_give isl_map *isl_map_upper_bound(__isl_take isl_map *map,
        enum isl_dim_type type, unsigned pos, isl_int value)
{
        return map_bound(map, type, pos, value, 1);
}

__isl_give isl_set *isl_set_lower_bound(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, isl_int value)
{
        return isl_map_lower_bound(set, type, pos, value);
}

__isl_give isl_set *isl_set_upper_bound(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, isl_int value)
{
        return isl_map_upper_bound(set, type, pos, value);
}

/* Force the values of the variable at position "pos" of type "type" of "set"
 * to be no smaller than "value".
 */
__isl_give isl_set *isl_set_lower_bound_val(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, __isl_take isl_val *value)
{
        if (!value)
                goto error;
        if (!isl_val_is_int(value))
                isl_die(isl_set_get_ctx(set), isl_error_invalid,
                        "expecting integer value", goto error);
        set = isl_set_lower_bound(set, type, pos, value->n);
        isl_val_free(value);
        return set;
error:
        isl_val_free(value);
        isl_set_free(set);
        return NULL;
}

/* Force the values of the variable at position "pos" of type "type" of "set"
 * to be no greater than "value".
 */
__isl_give isl_set *isl_set_upper_bound_val(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, __isl_take isl_val *value)
{
        if (!value)
                goto error;
        if (!isl_val_is_int(value))
                isl_die(isl_set_get_ctx(set), isl_error_invalid,
                        "expecting integer value", goto error);
        set = isl_set_upper_bound(set, type, pos, value->n);
        isl_val_free(value);
        return set;
error:
        isl_val_free(value);
        isl_set_free(set);
        return NULL;
}

struct isl_set *isl_set_lower_bound_dim(struct isl_set *set, unsigned dim,
                                        isl_int value)
{
        int i;

        set = isl_set_cow(set);
        if (!set)
                return NULL;

        isl_assert(set->ctx, dim < isl_set_n_dim(set), goto error);
        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_lower_bound_dim(set->p[i], dim, value);
                if (!set->p[i])
                        goto error;
        }
        return set;
error:
        isl_set_free(set);
        return NULL;
}

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

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

        map->dim = isl_space_reverse(map->dim);
        if (!map->dim)
                goto error;
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_reverse(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;
}

#undef TYPE
#define TYPE    isl_pw_multi_aff
#undef SUFFIX
#define SUFFIX  _pw_multi_aff
#undef EMPTY
#define EMPTY   isl_pw_multi_aff_empty
#undef ADD
#define ADD     isl_pw_multi_aff_union_add
#include "isl_map_lexopt_templ.c"

/* Given a map "map", compute the lexicographically minimal
 * (or maximal) image element for each domain element in dom,
 * in the form of an isl_pw_multi_aff.
 * If "empty" is not NULL, then set *empty to those elements in dom that
 * do not have an image element.
 * If "flags" includes ISL_OPT_FULL, then "dom" is NULL and the optimum
 * should be computed over the domain of "map".  "empty" is also NULL
 * in this case.
 *
 * We first compute the lexicographically minimal or maximal element
 * in the first basic map.  This results in a partial solution "res"
 * and a subset "todo" of dom that still need to be handled.
 * We then consider each of the remaining maps in "map" and successively
 * update both "res" and "todo".
 * If "empty" is NULL, then the todo sets are not needed and therefore
 * also not computed.
 */
static __isl_give isl_pw_multi_aff *isl_map_partial_lexopt_aligned_pw_multi_aff(
        __isl_take isl_map *map, __isl_take isl_set *dom,
        __isl_give isl_set **empty, unsigned flags)
{
        int i;
        int full;
        isl_pw_multi_aff *res;
        isl_set *todo;

        full = ISL_FL_ISSET(flags, ISL_OPT_FULL);
        if (!map || (!full && !dom))
                goto error;

        if (isl_map_plain_is_empty(map)) {
                if (empty)
                        *empty = dom;
                else
                        isl_set_free(dom);
                return isl_pw_multi_aff_from_map(map);
        }

        res = basic_map_partial_lexopt_pw_multi_aff(
                                            isl_basic_map_copy(map->p[0]),
                                            isl_set_copy(dom), empty, flags);

        if (empty)
                todo = *empty;
        for (i = 1; i < map->n; ++i) {
                isl_pw_multi_aff *res_i;

                res_i = basic_map_partial_lexopt_pw_multi_aff(
                                            isl_basic_map_copy(map->p[i]),
                                            isl_set_copy(dom), empty, flags);

                if (ISL_FL_ISSET(flags, ISL_OPT_MAX))
                        res = isl_pw_multi_aff_union_lexmax(res, res_i);
                else
                        res = isl_pw_multi_aff_union_lexmin(res, res_i);

                if (empty)
                        todo = isl_set_intersect(todo, *empty);
        }

        isl_set_free(dom);
        isl_map_free(map);

        if (empty)
                *empty = todo;

        return res;
error:
        if (empty)
                *empty = NULL;
        isl_set_free(dom);
        isl_map_free(map);
        return NULL;
}

#undef TYPE
#define TYPE    isl_map
#undef SUFFIX
#define SUFFIX
#undef EMPTY
#define EMPTY   isl_map_empty
#undef ADD
#define ADD     isl_map_union_disjoint
#include "isl_map_lexopt_templ.c"

/* Given a map "map", compute the lexicographically minimal
 * (or maximal) image element for each domain element in "dom",
 * in the form of an isl_map.
 * If "empty" is not NULL, then set *empty to those elements in "dom" that
 * do not have an image element.
 * If "flags" includes ISL_OPT_FULL, then "dom" is NULL and the optimum
 * should be computed over the domain of "map".  "empty" is also NULL
 * in this case.
 *
 * If the input consists of more than one disjunct, then first
 * compute the desired result in the form of an isl_pw_multi_aff and
 * then convert that into an isl_map.
 *
 * This function used to have an explicit implementation in terms
 * of isl_maps, but it would continually intersect the domains of
 * partial results with the complement of the domain of the next
 * partial solution, potentially leading to an explosion in the number
 * of disjuncts if there are several disjuncts in the input.
 * An even earlier implementation of this function would look for
 * better results in the domain of the partial result and for extra
 * results in the complement of this domain, which would lead to
 * even more splintering.
 */
static __isl_give isl_map *isl_map_partial_lexopt_aligned(
        __isl_take isl_map *map, __isl_take isl_set *dom,
        __isl_give isl_set **empty, unsigned flags)
{
        int full;
        struct isl_map *res;
        isl_pw_multi_aff *pma;

        full = ISL_FL_ISSET(flags, ISL_OPT_FULL);
        if (!map || (!full && !dom))
                goto error;

        if (isl_map_plain_is_empty(map)) {
                if (empty)
                        *empty = dom;
                else
                        isl_set_free(dom);
                return map;
        }

        if (map->n == 1) {
                res = basic_map_partial_lexopt(isl_basic_map_copy(map->p[0]),
                                                dom, empty, flags);
                isl_map_free(map);
                return res;
        }

        pma = isl_map_partial_lexopt_aligned_pw_multi_aff(map, dom, empty,
                                                        flags);
        return isl_map_from_pw_multi_aff(pma);
error:
        if (empty)
                *empty = NULL;
        isl_set_free(dom);
        isl_map_free(map);
        return NULL;
}

__isl_give isl_map *isl_map_partial_lexmax(
                __isl_take isl_map *map, __isl_take isl_set *dom,
                __isl_give isl_set **empty)
{
        return isl_map_partial_lexopt(map, dom, empty, ISL_OPT_MAX);
}

__isl_give isl_map *isl_map_partial_lexmin(
                __isl_take isl_map *map, __isl_take isl_set *dom,
                __isl_give isl_set **empty)
{
        return isl_map_partial_lexopt(map, dom, empty, 0);
}

__isl_give isl_set *isl_set_partial_lexmin(
                __isl_take isl_set *set, __isl_take isl_set *dom,
                __isl_give isl_set **empty)
{
        return set_from_map(isl_map_partial_lexmin(set_to_map(set),
                                                    dom, empty));
}

__isl_give isl_set *isl_set_partial_lexmax(
                __isl_take isl_set *set, __isl_take isl_set *dom,
                __isl_give isl_set **empty)
{
        return set_from_map(isl_map_partial_lexmax(set_to_map(set),
                                                    dom, empty));
}

/* Compute the lexicographic minimum (or maximum if "flags" includes
 * ISL_OPT_MAX) of "bset" over its parametric domain.
 */
__isl_give isl_set *isl_basic_set_lexopt(__isl_take isl_basic_set *bset,
        unsigned flags)
{
        return isl_basic_map_lexopt(bset, flags);
}

__isl_give isl_map *isl_basic_map_lexmax(__isl_take isl_basic_map *bmap)
{
        return isl_basic_map_lexopt(bmap, ISL_OPT_MAX);
}

__isl_give isl_set *isl_basic_set_lexmin(__isl_take isl_basic_set *bset)
{
        return set_from_map(isl_basic_map_lexmin(bset_to_bmap(bset)));
}

__isl_give isl_set *isl_basic_set_lexmax(__isl_take isl_basic_set *bset)
{
        return set_from_map(isl_basic_map_lexmax(bset_to_bmap(bset)));
}

/* Compute the lexicographic minimum of "bset" over its parametric domain
 * for the purpose of quantifier elimination.
 * That is, find an explicit representation for all the existentially
 * quantified variables in "bset" by computing their lexicographic
 * minimum.
 */
static __isl_give isl_set *isl_basic_set_lexmin_compute_divs(
        __isl_take isl_basic_set *bset)
{
        return isl_basic_set_lexopt(bset, ISL_OPT_QE);
}

/* Extract the first and only affine expression from list
 * and then add it to *pwaff with the given dom.
 * This domain is known to be disjoint from other domains
 * because of the way isl_basic_map_foreach_lexmax works.
 */
static int update_dim_opt(__isl_take isl_basic_set *dom,
        __isl_take isl_aff_list *list, void *user)
{
        isl_ctx *ctx = isl_basic_set_get_ctx(dom);
        isl_aff *aff;
        isl_pw_aff **pwaff = user;
        isl_pw_aff *pwaff_i;

        if (!list)
                goto error;
        if (isl_aff_list_n_aff(list) != 1)
                isl_die(ctx, isl_error_internal,
                        "expecting single element list", goto error);

        aff = isl_aff_list_get_aff(list, 0);
        pwaff_i = isl_pw_aff_alloc(isl_set_from_basic_set(dom), aff);

        *pwaff = isl_pw_aff_add_disjoint(*pwaff, pwaff_i);

        isl_aff_list_free(list);

        return 0;
error:
        isl_basic_set_free(dom);
        isl_aff_list_free(list);
        return -1;
}

/* Given a basic map with one output dimension, compute the minimum or
 * maximum of that dimension as an isl_pw_aff.
 *
 * The isl_pw_aff is constructed by having isl_basic_map_foreach_lexopt
 * call update_dim_opt on each leaf of the result.
 */
static __isl_give isl_pw_aff *basic_map_dim_opt(__isl_keep isl_basic_map *bmap,
        int max)
{
        isl_space *dim = isl_basic_map_get_space(bmap);
        isl_pw_aff *pwaff;
        int r;

        dim = isl_space_from_domain(isl_space_domain(dim));
        dim = isl_space_add_dims(dim, isl_dim_out, 1);
        pwaff = isl_pw_aff_empty(dim);

        r = isl_basic_map_foreach_lexopt(bmap, max, &update_dim_opt, &pwaff);
        if (r < 0)
                return isl_pw_aff_free(pwaff);

        return pwaff;
}

/* Compute the minimum or maximum of the given output dimension
 * as a function of the parameters and the input dimensions,
 * but independently of the other output dimensions.
 *
 * We first project out the other output dimension and then compute
 * the "lexicographic" maximum in each basic map, combining the results
 * using isl_pw_aff_union_max.
 */
static __isl_give isl_pw_aff *map_dim_opt(__isl_take isl_map *map, int pos,
        int max)
{
        int i;
        isl_pw_aff *pwaff;
        unsigned n_out;

        n_out = isl_map_dim(map, isl_dim_out);
        map = isl_map_project_out(map, isl_dim_out, pos + 1, n_out - (pos + 1));
        map = isl_map_project_out(map, isl_dim_out, 0, pos);
        if (!map)
                return NULL;

        if (map->n == 0) {
                isl_space *dim = isl_map_get_space(map);
                isl_map_free(map);
                return isl_pw_aff_empty(dim);
        }

        pwaff = basic_map_dim_opt(map->p[0], max);
        for (i = 1; i < map->n; ++i) {
                isl_pw_aff *pwaff_i;

                pwaff_i = basic_map_dim_opt(map->p[i], max);
                pwaff = isl_pw_aff_union_opt(pwaff, pwaff_i, max);
        }

        isl_map_free(map);

        return pwaff;
}

/* Compute the minimum of the given output dimension as a function of the
 * parameters and input dimensions, but independently of
 * the other output dimensions.
 */
__isl_give isl_pw_aff *isl_map_dim_min(__isl_take isl_map *map, int pos)
{
        return map_dim_opt(map, pos, 0);
}

/* Compute the maximum of the given output dimension as a function of the
 * parameters and input dimensions, but independently of
 * the other output dimensions.
 */
__isl_give isl_pw_aff *isl_map_dim_max(__isl_take isl_map *map, int pos)
{
        return map_dim_opt(map, pos, 1);
}

/* Compute the minimum or maximum of the given set dimension
 * as a function of the parameters,
 * but independently of the other set dimensions.
 */
static __isl_give isl_pw_aff *set_dim_opt(__isl_take isl_set *set, int pos,
        int max)
{
        return map_dim_opt(set, pos, max);
}

/* Compute the maximum of the given set dimension as a function of the
 * parameters, but independently of the other set dimensions.
 */
__isl_give isl_pw_aff *isl_set_dim_max(__isl_take isl_set *set, int pos)
{
        return set_dim_opt(set, pos, 1);
}

/* Compute the minimum of the given set dimension as a function of the
 * parameters, but independently of the other set dimensions.
 */
__isl_give isl_pw_aff *isl_set_dim_min(__isl_take isl_set *set, int pos)
{
        return set_dim_opt(set, pos, 0);
}

/* Apply a preimage specified by "mat" on the parameters of "bset".
 * bset is assumed to have only parameters and divs.
 */
static struct isl_basic_set *basic_set_parameter_preimage(
        struct isl_basic_set *bset, struct isl_mat *mat)
{
        unsigned nparam;

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

        bset->dim = isl_space_cow(bset->dim);
        if (!bset->dim)
                goto error;

        nparam = isl_basic_set_dim(bset, isl_dim_param);

        isl_assert(bset->ctx, mat->n_row == 1 + nparam, goto error);

        bset->dim->nparam = 0;
        bset->dim->n_out = nparam;
        bset = isl_basic_set_preimage(bset, mat);
        if (bset) {
                bset->dim->nparam = bset->dim->n_out;
                bset->dim->n_out = 0;
        }
        return bset;
error:
        isl_mat_free(mat);
        isl_basic_set_free(bset);
        return NULL;
}

/* Apply a preimage specified by "mat" on the parameters of "set".
 * set is assumed to have only parameters and divs.
 */
static __isl_give isl_set *set_parameter_preimage(__isl_take isl_set *set,
        __isl_take isl_mat *mat)
{
        isl_space *space;
        unsigned nparam;

        if (!set || !mat)
                goto error;

        nparam = isl_set_dim(set, isl_dim_param);

        if (mat->n_row != 1 + nparam)
                isl_die(isl_set_get_ctx(set), isl_error_internal,
                        "unexpected number of rows", goto error);

        space = isl_set_get_space(set);
        space = isl_space_move_dims(space, isl_dim_set, 0,
                                    isl_dim_param, 0, nparam);
        set = isl_set_reset_space(set, space);
        set = isl_set_preimage(set, mat);
        nparam = isl_set_dim(set, isl_dim_out);
        space = isl_set_get_space(set);
        space = isl_space_move_dims(space, isl_dim_param, 0,
                                    isl_dim_out, 0, nparam);
        set = isl_set_reset_space(set, space);
        return set;
error:
        isl_mat_free(mat);
        isl_set_free(set);
        return NULL;
}

/* Intersect the basic set "bset" with the affine space specified by the
 * equalities in "eq".
 */
static struct isl_basic_set *basic_set_append_equalities(
        struct isl_basic_set *bset, struct isl_mat *eq)
{
        int i, k;
        unsigned len;

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

        bset = isl_basic_set_extend_space(bset, isl_space_copy(bset->dim), 0,
                                        eq->n_row, 0);
        if (!bset)
                goto error;

        len = 1 + isl_space_dim(bset->dim, isl_dim_all) + bset->extra;
        for (i = 0; i < eq->n_row; ++i) {
                k = isl_basic_set_alloc_equality(bset);
                if (k < 0)
                        goto error;
                isl_seq_cpy(bset->eq[k], eq->row[i], eq->n_col);
                isl_seq_clr(bset->eq[k] + eq->n_col, len - eq->n_col);
        }
        isl_mat_free(eq);

        bset = isl_basic_set_gauss(bset, NULL);
        bset = isl_basic_set_finalize(bset);

        return bset;
error:
        isl_mat_free(eq);
        isl_basic_set_free(bset);
        return NULL;
}

/* Intersect the set "set" with the affine space specified by the
 * equalities in "eq".
 */
static struct isl_set *set_append_equalities(struct isl_set *set,
        struct isl_mat *eq)
{
        int i;

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

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

/* Given a basic set "bset" that only involves parameters and existentially
 * quantified variables, return the index of the first equality
 * that only involves parameters.  If there is no such equality then
 * return bset->n_eq.
 *
 * This function assumes that isl_basic_set_gauss has been called on "bset".
 */
static int first_parameter_equality(__isl_keep isl_basic_set *bset)
{
        int i, j;
        unsigned nparam, n_div;

        if (!bset)
                return -1;

        nparam = isl_basic_set_dim(bset, isl_dim_param);
        n_div = isl_basic_set_dim(bset, isl_dim_div);

        for (i = 0, j = n_div - 1; i < bset->n_eq && j >= 0; --j) {
                if (!isl_int_is_zero(bset->eq[i][1 + nparam + j]))
                        ++i;
        }

        return i;
}

/* Compute an explicit representation for the existentially quantified
 * variables in "bset" by computing the "minimal value" of the set
 * variables.  Since there are no set variables, the computation of
 * the minimal value essentially computes an explicit representation
 * of the non-empty part(s) of "bset".
 *
 * The input only involves parameters and existentially quantified variables.
 * All equalities among parameters have been removed.
 *
 * Since the existentially quantified variables in the result are in general
 * going to be different from those in the input, we first replace
 * them by the minimal number of variables based on their equalities.
 * This should simplify the parametric integer programming.
 */
static __isl_give isl_set *base_compute_divs(__isl_take isl_basic_set *bset)
{
        isl_morph *morph1, *morph2;
        isl_set *set;
        unsigned n;

        if (!bset)
                return NULL;
        if (bset->n_eq == 0)
                return isl_basic_set_lexmin_compute_divs(bset);

        morph1 = isl_basic_set_parameter_compression(bset);
        bset = isl_morph_basic_set(isl_morph_copy(morph1), bset);
        bset = isl_basic_set_lift(bset);
        morph2 = isl_basic_set_variable_compression(bset, isl_dim_set);
        bset = isl_morph_basic_set(morph2, bset);
        n = isl_basic_set_dim(bset, isl_dim_set);
        bset = isl_basic_set_project_out(bset, isl_dim_set, 0, n);

        set = isl_basic_set_lexmin_compute_divs(bset);

        set = isl_morph_set(isl_morph_inverse(morph1), set);

        return set;
}

/* Project the given basic set onto its parameter domain, possibly introducing
 * new, explicit, existential variables in the constraints.
 * The input has parameters and (possibly implicit) existential variables.
 * The output has the same parameters, but only
 * explicit existentially quantified variables.
 *
 * The actual projection is performed by pip, but pip doesn't seem
 * to like equalities very much, so we first remove the equalities
 * among the parameters by performing a variable compression on
 * the parameters.  Afterward, an inverse transformation is performed
 * and the equalities among the parameters are inserted back in.
 *
 * The variable compression on the parameters may uncover additional
 * equalities that were only implicit before.  We therefore check
 * if there are any new parameter equalities in the result and
 * if so recurse.  The removal of parameter equalities is required
 * for the parameter compression performed by base_compute_divs.
 */
static struct isl_set *parameter_compute_divs(struct isl_basic_set *bset)
{
        int i;
        struct isl_mat *eq;
        struct isl_mat *T, *T2;
        struct isl_set *set;
        unsigned nparam;

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

        if (bset->n_eq == 0)
                return base_compute_divs(bset);

        bset = isl_basic_set_gauss(bset, NULL);
        if (!bset)
                return NULL;
        if (isl_basic_set_plain_is_empty(bset))
                return isl_set_from_basic_set(bset);

        i = first_parameter_equality(bset);
        if (i == bset->n_eq)
                return base_compute_divs(bset);

        nparam = isl_basic_set_dim(bset, isl_dim_param);
        eq = isl_mat_sub_alloc6(bset->ctx, bset->eq, i, bset->n_eq - i,
                0, 1 + nparam);
        eq = isl_mat_cow(eq);
        T = isl_mat_variable_compression(isl_mat_copy(eq), &T2);
        if (T && T->n_col == 0) {
                isl_mat_free(T);
                isl_mat_free(T2);
                isl_mat_free(eq);
                bset = isl_basic_set_set_to_empty(bset);
                return isl_set_from_basic_set(bset);
        }
        bset = basic_set_parameter_preimage(bset, T);

        i = first_parameter_equality(bset);
        if (!bset)
                set = NULL;
        else if (i == bset->n_eq)
                set = base_compute_divs(bset);
        else
                set = parameter_compute_divs(bset);
        set = set_parameter_preimage(set, T2);
        set = set_append_equalities(set, eq);
        return set;
}

/* Insert the divs from "ls" before those of "bmap".
 *
 * The number of columns is not changed, which means that the last
 * dimensions of "bmap" are being reintepreted as the divs from "ls".
 * The caller is responsible for removing the same number of dimensions
 * from the space of "bmap".
 */
static __isl_give isl_basic_map *insert_divs_from_local_space(
        __isl_take isl_basic_map *bmap, __isl_keep isl_local_space *ls)
{
        int i;
        int n_div;
        int old_n_div;

        n_div = isl_local_space_dim(ls, isl_dim_div);
        if (n_div == 0)
                return bmap;

        old_n_div = bmap->n_div;
        bmap = insert_div_rows(bmap, n_div);
        if (!bmap)
                return NULL;

        for (i = 0; i < n_div; ++i) {
                isl_seq_cpy(bmap->div[i], ls->div->row[i], ls->div->n_col);
                isl_seq_clr(bmap->div[i] + ls->div->n_col, old_n_div);
        }

        return bmap;
}

/* Replace the space of "bmap" by the space and divs of "ls".
 *
 * If "ls" has any divs, then we simplify the result since we may
 * have discovered some additional equalities that could simplify
 * the div expressions.
 */
static __isl_give isl_basic_map *basic_replace_space_by_local_space(
        __isl_take isl_basic_map *bmap, __isl_take isl_local_space *ls)
{
        int n_div;

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

        n_div = isl_local_space_dim(ls, isl_dim_div);
        bmap = insert_divs_from_local_space(bmap, ls);
        if (!bmap)
                goto error;

        isl_space_free(bmap->dim);
        bmap->dim = isl_local_space_get_space(ls);
        if (!bmap->dim)
                goto error;

        isl_local_space_free(ls);
        if (n_div > 0)
                bmap = isl_basic_map_simplify(bmap);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_basic_map_free(bmap);
        isl_local_space_free(ls);
        return NULL;
}

/* Replace the space of "map" by the space and divs of "ls".
 */
static __isl_give isl_map *replace_space_by_local_space(__isl_take isl_map *map,
        __isl_take isl_local_space *ls)
{
        int i;

        map = isl_map_cow(map);
        if (!map || !ls)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = basic_replace_space_by_local_space(map->p[i],
                                                    isl_local_space_copy(ls));
                if (!map->p[i])
                        goto error;
        }
        isl_space_free(map->dim);
        map->dim = isl_local_space_get_space(ls);
        if (!map->dim)
                goto error;

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

/* Compute an explicit representation for the existentially
 * quantified variables for which do not know any explicit representation yet.
 *
 * We first sort the existentially quantified variables so that the
 * existentially quantified variables for which we already have an explicit
 * representation are placed before those for which we do not.
 * The input dimensions, the output dimensions and the existentially
 * quantified variables for which we already have an explicit
 * representation are then turned into parameters.
 * compute_divs returns a map with the same parameters and
 * no input or output dimensions and the dimension specification
 * is reset to that of the input, including the existentially quantified
 * variables for which we already had an explicit representation.
 */
static struct isl_map *compute_divs(struct isl_basic_map *bmap)
{
        struct isl_basic_set *bset;
        struct isl_set *set;
        struct isl_map *map;
        isl_space *dim;
        isl_local_space *ls;
        unsigned         nparam;
        unsigned         n_in;
        unsigned         n_out;
        int n_known;
        int i;

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

        n_known = isl_basic_map_first_unknown_div(bmap);
        if (n_known < 0)
                return isl_map_from_basic_map(isl_basic_map_free(bmap));

        nparam = isl_basic_map_dim(bmap, isl_dim_param);
        n_in = isl_basic_map_dim(bmap, isl_dim_in);
        n_out = isl_basic_map_dim(bmap, isl_dim_out);
        dim = isl_space_set_alloc(bmap->ctx,
                                    nparam + n_in + n_out + n_known, 0);
        if (!dim)
                goto error;

        ls = isl_basic_map_get_local_space(bmap);
        ls = isl_local_space_drop_dims(ls, isl_dim_div,
                                        n_known, bmap->n_div - n_known);
        if (n_known > 0) {
                for (i = n_known; i < bmap->n_div; ++i)
                        swap_div(bmap, i - n_known, i);
                bmap->n_div -= n_known;
                bmap->extra -= n_known;
        }
        bmap = isl_basic_map_reset_space(bmap, dim);
        bset = bset_from_bmap(bmap);

        set = parameter_compute_divs(bset);
        map = set_to_map(set);
        map = replace_space_by_local_space(map, ls);

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

/* Remove the explicit representation of local variable "div",
 * if there is any.
 */
__isl_give isl_basic_map *isl_basic_map_mark_div_unknown(
        __isl_take isl_basic_map *bmap, int div)
{
        isl_bool unknown;

        unknown = isl_basic_map_div_is_marked_unknown(bmap, div);
        if (unknown < 0)
                return isl_basic_map_free(bmap);
        if (unknown)
                return bmap;

        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;
        isl_int_set_si(bmap->div[div][0], 0);
        return bmap;
}

/* Is local variable "div" of "bmap" marked as not having an explicit
 * representation?
 * Note that even if "div" is not marked in this way and therefore
 * has an explicit representation, this representation may still
 * depend (indirectly) on other local variables that do not
 * have an explicit representation.
 */
isl_bool isl_basic_map_div_is_marked_unknown(__isl_keep isl_basic_map *bmap,
        int div)
{
        if (!bmap)
                return isl_bool_error;
        if (div < 0 || div >= isl_basic_map_dim(bmap, isl_dim_div))
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "position out of bounds", return isl_bool_error);
        return isl_int_is_zero(bmap->div[div][0]);
}

/* Return the position of the first local variable that does not
 * have an explicit representation.
 * Return the total number of local variables if they all have
 * an explicit representation.
 * Return -1 on error.
 */
int isl_basic_map_first_unknown_div(__isl_keep isl_basic_map *bmap)
{
        int i;

        if (!bmap)
                return -1;

        for (i = 0; i < bmap->n_div; ++i) {
                if (!isl_basic_map_div_is_known(bmap, i))
                        return i;
        }
        return bmap->n_div;
}

/* Return the position of the first local variable that does not
 * have an explicit representation.
 * Return the total number of local variables if they all have
 * an explicit representation.
 * Return -1 on error.
 */
int isl_basic_set_first_unknown_div(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_first_unknown_div(bset);
}

/* Does "bmap" have an explicit representation for all local variables?
 */
isl_bool isl_basic_map_divs_known(__isl_keep isl_basic_map *bmap)
{
        int first, n;

        n = isl_basic_map_dim(bmap, isl_dim_div);
        first = isl_basic_map_first_unknown_div(bmap);
        if (first < 0)
                return isl_bool_error;
        return first == n;
}

/* Do all basic maps in "map" have an explicit representation
 * for all local variables?
 */
isl_bool isl_map_divs_known(__isl_keep isl_map *map)
{
        int i;

        if (!map)
                return isl_bool_error;

        for (i = 0; i < map->n; ++i) {
                int known = isl_basic_map_divs_known(map->p[i]);
                if (known <= 0)
                        return known;
        }

        return isl_bool_true;
}

/* If bmap contains any unknown divs, then compute explicit
 * expressions for them.  However, this computation may be
 * quite expensive, so first try to remove divs that aren't
 * strictly needed.
 */
struct isl_map *isl_basic_map_compute_divs(struct isl_basic_map *bmap)
{
        int known;
        struct isl_map *map;

        known = isl_basic_map_divs_known(bmap);
        if (known < 0)
                goto error;
        if (known)
                return isl_map_from_basic_map(bmap);

        bmap = isl_basic_map_drop_redundant_divs(bmap);

        known = isl_basic_map_divs_known(bmap);
        if (known < 0)
                goto error;
        if (known)
                return isl_map_from_basic_map(bmap);

        map = compute_divs(bmap);
        return map;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

struct isl_map *isl_map_compute_divs(struct isl_map *map)
{
        int i;
        int known;
        struct isl_map *res;

        if (!map)
                return NULL;
        if (map->n == 0)
                return map;

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

        res = isl_basic_map_compute_divs(isl_basic_map_copy(map->p[0]));
        for (i = 1 ; i < map->n; ++i) {
                struct isl_map *r2;
                r2 = isl_basic_map_compute_divs(isl_basic_map_copy(map->p[i]));
                if (ISL_F_ISSET(map, ISL_MAP_DISJOINT))
                        res = isl_map_union_disjoint(res, r2);
                else
                        res = isl_map_union(res, r2);
        }
        isl_map_free(map);

        return res;
}

struct isl_set *isl_basic_set_compute_divs(struct isl_basic_set *bset)
{
        return set_from_map(isl_basic_map_compute_divs(bset_to_bmap(bset)));
}

struct isl_set *isl_set_compute_divs(struct isl_set *set)
{
        return set_from_map(isl_map_compute_divs(set_to_map(set)));
}

struct isl_set *isl_map_domain(struct isl_map *map)
{
        int i;
        struct isl_set *set;

        if (!map)
                goto error;

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

        set = set_from_map(map);
        set->dim = isl_space_domain(set->dim);
        if (!set->dim)
                goto error;
        for (i = 0; i < map->n; ++i) {
                set->p[i] = isl_basic_map_domain(map->p[i]);
                if (!set->p[i])
                        goto error;
        }
        ISL_F_CLR(set, ISL_MAP_DISJOINT);
        ISL_F_CLR(set, ISL_SET_NORMALIZED);
        return set;
error:
        isl_map_free(map);
        return NULL;
}

/* Return the union of "map1" and "map2", where we assume for now that
 * "map1" and "map2" are disjoint.  Note that the basic maps inside
 * "map1" or "map2" may not be disjoint from each other.
 * Also note that this function is also called from isl_map_union,
 * which takes care of handling the situation where "map1" and "map2"
 * may not be disjoint.
 *
 * If one of the inputs is empty, we can simply return the other input.
 * Similarly, if one of the inputs is universal, then it is equal to the union.
 */
static __isl_give isl_map *map_union_disjoint(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        int i;
        unsigned flags = 0;
        struct isl_map *map = NULL;
        int is_universe;

        if (!map1 || !map2)
                goto error;

        if (!isl_space_is_equal(map1->dim, map2->dim))
                isl_die(isl_map_get_ctx(map1), isl_error_invalid,
                        "spaces don't match", goto error);

        if (map1->n == 0) {
                isl_map_free(map1);
                return map2;
        }
        if (map2->n == 0) {
                isl_map_free(map2);
                return map1;
        }

        is_universe = isl_map_plain_is_universe(map1);
        if (is_universe < 0)
                goto error;
        if (is_universe) {
                isl_map_free(map2);
                return map1;
        }

        is_universe = isl_map_plain_is_universe(map2);
        if (is_universe < 0)
                goto error;
        if (is_universe) {
                isl_map_free(map1);
                return map2;
        }

        if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT) &&
            ISL_F_ISSET(map2, ISL_MAP_DISJOINT))
                ISL_FL_SET(flags, ISL_MAP_DISJOINT);

        map = isl_map_alloc_space(isl_space_copy(map1->dim),
                                map1->n + map2->n, flags);
        if (!map)
                goto error;
        for (i = 0; i < map1->n; ++i) {
                map = isl_map_add_basic_map(map,
                                  isl_basic_map_copy(map1->p[i]));
                if (!map)
                        goto error;
        }
        for (i = 0; i < map2->n; ++i) {
                map = isl_map_add_basic_map(map,
                                  isl_basic_map_copy(map2->p[i]));
                if (!map)
                        goto error;
        }
        isl_map_free(map1);
        isl_map_free(map2);
        return map;
error:
        isl_map_free(map);
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

/* Return the union of "map1" and "map2", where "map1" and "map2" are
 * guaranteed to be disjoint by the caller.
 *
 * Note that this functions is called from within isl_map_make_disjoint,
 * so we have to be careful not to touch the constraints of the inputs
 * in any way.
 */
__isl_give isl_map *isl_map_union_disjoint(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_union_disjoint);
}

/* Return the union of "map1" and "map2", where "map1" and "map2" may
 * not be disjoint.  The parameters are assumed to have been aligned.
 *
 * We currently simply call map_union_disjoint, the internal operation
 * of which does not really depend on the inputs being disjoint.
 * If the result contains more than one basic map, then we clear
 * the disjoint flag since the result may contain basic maps from
 * both inputs and these are not guaranteed to be disjoint.
 *
 * As a special case, if "map1" and "map2" are obviously equal,
 * then we simply return "map1".
 */
static __isl_give isl_map *map_union_aligned(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        int equal;

        if (!map1 || !map2)
                goto error;

        equal = isl_map_plain_is_equal(map1, map2);
        if (equal < 0)
                goto error;
        if (equal) {
                isl_map_free(map2);
                return map1;
        }

        map1 = map_union_disjoint(map1, map2);
        if (!map1)
                return NULL;
        if (map1->n > 1)
                ISL_F_CLR(map1, ISL_MAP_DISJOINT);
        return map1;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

/* Return the union of "map1" and "map2", where "map1" and "map2" may
 * not be disjoint.
 */
__isl_give isl_map *isl_map_union(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_union_aligned);
}

struct isl_set *isl_set_union_disjoint(
                        struct isl_set *set1, struct isl_set *set2)
{
        return set_from_map(isl_map_union_disjoint(set_to_map(set1),
                                                    set_to_map(set2)));
}

struct isl_set *isl_set_union(struct isl_set *set1, struct isl_set *set2)
{
        return set_from_map(isl_map_union(set_to_map(set1), set_to_map(set2)));
}

/* Apply "fn" to pairs of elements from "map" and "set" and collect
 * the results.
 *
 * "map" and "set" are assumed to be compatible and non-NULL.
 */
static __isl_give isl_map *map_intersect_set(__isl_take isl_map *map,
        __isl_take isl_set *set,
        __isl_give isl_basic_map *fn(__isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *bset))
{
        unsigned flags = 0;
        struct isl_map *result;
        int i, j;

        if (isl_set_plain_is_universe(set)) {
                isl_set_free(set);
                return map;
        }

        if (ISL_F_ISSET(map, ISL_MAP_DISJOINT) &&
            ISL_F_ISSET(set, ISL_MAP_DISJOINT))
                ISL_FL_SET(flags, ISL_MAP_DISJOINT);

        result = isl_map_alloc_space(isl_space_copy(map->dim),
                                        map->n * set->n, flags);
        for (i = 0; result && i < map->n; ++i)
                for (j = 0; j < set->n; ++j) {
                        result = isl_map_add_basic_map(result,
                                        fn(isl_basic_map_copy(map->p[i]),
                                            isl_basic_set_copy(set->p[j])));
                        if (!result)
                                break;
                }

        isl_map_free(map);
        isl_set_free(set);
        return result;
}

static __isl_give isl_map *map_intersect_range(__isl_take isl_map *map,
        __isl_take isl_set *set)
{
        if (!map || !set)
                goto error;

        if (!isl_map_compatible_range(map, set))
                isl_die(set->ctx, isl_error_invalid,
                        "incompatible spaces", goto error);

        return map_intersect_set(map, set, &isl_basic_map_intersect_range);
error:
        isl_map_free(map);
        isl_set_free(set);
        return NULL;
}

__isl_give isl_map *isl_map_intersect_range(__isl_take isl_map *map,
        __isl_take isl_set *set)
{
        return isl_map_align_params_map_map_and(map, set, &map_intersect_range);
}

static __isl_give isl_map *map_intersect_domain(__isl_take isl_map *map,
        __isl_take isl_set *set)
{
        if (!map || !set)
                goto error;

        if (!isl_map_compatible_domain(map, set))
                isl_die(set->ctx, isl_error_invalid,
                        "incompatible spaces", goto error);

        return map_intersect_set(map, set, &isl_basic_map_intersect_domain);
error:
        isl_map_free(map);
        isl_set_free(set);
        return NULL;
}

__isl_give isl_map *isl_map_intersect_domain(__isl_take isl_map *map,
        __isl_take isl_set *set)
{
        return isl_map_align_params_map_map_and(map, set,
                                                &map_intersect_domain);
}

static __isl_give isl_map *map_apply_domain(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        if (!map1 || !map2)
                goto error;
        map1 = isl_map_reverse(map1);
        map1 = isl_map_apply_range(map1, map2);
        return isl_map_reverse(map1);
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

__isl_give isl_map *isl_map_apply_domain(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_apply_domain);
}

static __isl_give isl_map *map_apply_range(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_space *dim_result;
        struct isl_map *result;
        int i, j;

        if (!map1 || !map2)
                goto error;

        dim_result = isl_space_join(isl_space_copy(map1->dim),
                                  isl_space_copy(map2->dim));

        result = isl_map_alloc_space(dim_result, map1->n * map2->n, 0);
        if (!result)
                goto error;
        for (i = 0; i < map1->n; ++i)
                for (j = 0; j < map2->n; ++j) {
                        result = isl_map_add_basic_map(result,
                            isl_basic_map_apply_range(
                                isl_basic_map_copy(map1->p[i]),
                                isl_basic_map_copy(map2->p[j])));
                        if (!result)
                                goto error;
                }
        isl_map_free(map1);
        isl_map_free(map2);
        if (result && result->n <= 1)
                ISL_F_SET(result, ISL_MAP_DISJOINT);
        return result;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

__isl_give isl_map *isl_map_apply_range(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_apply_range);
}

/*
 * returns range - domain
 */
struct isl_basic_set *isl_basic_map_deltas(struct isl_basic_map *bmap)
{
        isl_space *target_space;
        struct isl_basic_set *bset;
        unsigned dim;
        unsigned nparam;
        int i;

        if (!bmap)
                goto error;
        isl_assert(bmap->ctx, isl_space_tuple_is_equal(bmap->dim, isl_dim_in,
                                                  bmap->dim, isl_dim_out),
                   goto error);
        target_space = isl_space_domain(isl_basic_map_get_space(bmap));
        dim = isl_basic_map_n_in(bmap);
        nparam = isl_basic_map_n_param(bmap);
        bmap = isl_basic_map_from_range(isl_basic_map_wrap(bmap));
        bmap = isl_basic_map_add_dims(bmap, isl_dim_in, dim);
        bmap = isl_basic_map_extend_constraints(bmap, dim, 0);
        for (i = 0; i < dim; ++i) {
                int j = isl_basic_map_alloc_equality(bmap);
                if (j < 0) {
                        bmap = isl_basic_map_free(bmap);
                        break;
                }
                isl_seq_clr(bmap->eq[j], 1 + isl_basic_map_total_dim(bmap));
                isl_int_set_si(bmap->eq[j][1+nparam+i], 1);
                isl_int_set_si(bmap->eq[j][1+nparam+dim+i], 1);
                isl_int_set_si(bmap->eq[j][1+nparam+2*dim+i], -1);
        }
        bset = isl_basic_map_domain(bmap);
        bset = isl_basic_set_reset_space(bset, target_space);
        return bset;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/*
 * returns range - domain
 */
__isl_give isl_set *isl_map_deltas(__isl_take isl_map *map)
{
        int i;
        isl_space *dim;
        struct isl_set *result;

        if (!map)
                return NULL;

        isl_assert(map->ctx, isl_space_tuple_is_equal(map->dim, isl_dim_in,
                                                 map->dim, isl_dim_out),
                   goto error);
        dim = isl_map_get_space(map);
        dim = isl_space_domain(dim);
        result = isl_set_alloc_space(dim, map->n, 0);
        if (!result)
                goto error;
        for (i = 0; i < map->n; ++i)
                result = isl_set_add_basic_set(result,
                          isl_basic_map_deltas(isl_basic_map_copy(map->p[i])));
        isl_map_free(map);
        return result;
error:
        isl_map_free(map);
        return NULL;
}

/*
 * returns [domain -> range] -> range - domain
 */
__isl_give isl_basic_map *isl_basic_map_deltas_map(
        __isl_take isl_basic_map *bmap)
{
        int i, k;
        isl_space *dim;
        isl_basic_map *domain;
        int nparam, n;
        unsigned total;

        if (!isl_space_tuple_is_equal(bmap->dim, isl_dim_in,
                                        bmap->dim, isl_dim_out))
                isl_die(bmap->ctx, isl_error_invalid,
                        "domain and range don't match", goto error);

        nparam = isl_basic_map_dim(bmap, isl_dim_param);
        n = isl_basic_map_dim(bmap, isl_dim_in);

        dim = isl_space_from_range(isl_space_domain(isl_basic_map_get_space(bmap)));
        domain = isl_basic_map_universe(dim);

        bmap = isl_basic_map_from_domain(isl_basic_map_wrap(bmap));
        bmap = isl_basic_map_apply_range(bmap, domain);
        bmap = isl_basic_map_extend_constraints(bmap, n, 0);

        total = isl_basic_map_total_dim(bmap);

        for (i = 0; i < n; ++i) {
                k = isl_basic_map_alloc_equality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_clr(bmap->eq[k], 1 + total);
                isl_int_set_si(bmap->eq[k][1 + nparam + i], 1);
                isl_int_set_si(bmap->eq[k][1 + nparam + n + i], -1);
                isl_int_set_si(bmap->eq[k][1 + nparam + n + n + i], 1);
        }

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

/*
 * returns [domain -> range] -> range - domain
 */
__isl_give isl_map *isl_map_deltas_map(__isl_take isl_map *map)
{
        int i;
        isl_space *domain_dim;

        if (!map)
                return NULL;

        if (!isl_space_tuple_is_equal(map->dim, isl_dim_in,
                                        map->dim, isl_dim_out))
                isl_die(map->ctx, isl_error_invalid,
                        "domain and range don't match", goto error);

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

        domain_dim = isl_space_from_range(isl_space_domain(isl_map_get_space(map)));
        map->dim = isl_space_from_domain(isl_space_wrap(map->dim));
        map->dim = isl_space_join(map->dim, domain_dim);
        if (!map->dim)
                goto error;
        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_deltas_map(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;
}

static __isl_give isl_basic_map *basic_map_identity(__isl_take isl_space *dims)
{
        struct isl_basic_map *bmap;
        unsigned nparam;
        unsigned dim;
        int i;

        if (!dims)
                return NULL;

        nparam = dims->nparam;
        dim = dims->n_out;
        bmap = isl_basic_map_alloc_space(dims, 0, dim, 0);
        if (!bmap)
                goto error;

        for (i = 0; i < dim; ++i) {
                int j = isl_basic_map_alloc_equality(bmap);
                if (j < 0)
                        goto error;
                isl_seq_clr(bmap->eq[j], 1 + isl_basic_map_total_dim(bmap));
                isl_int_set_si(bmap->eq[j][1+nparam+i], 1);
                isl_int_set_si(bmap->eq[j][1+nparam+dim+i], -1);
        }
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_identity(__isl_take isl_space *dim)
{
        if (!dim)
                return NULL;
        if (dim->n_in != dim->n_out)
                isl_die(dim->ctx, isl_error_invalid,
                        "number of input and output dimensions needs to be "
                        "the same", goto error);
        return basic_map_identity(dim);
error:
        isl_space_free(dim);
        return NULL;
}

__isl_give isl_map *isl_map_identity(__isl_take isl_space *dim)
{
        return isl_map_from_basic_map(isl_basic_map_identity(dim));
}

__isl_give isl_map *isl_set_identity(__isl_take isl_set *set)
{
        isl_space *dim = isl_set_get_space(set);
        isl_map *id;
        id = isl_map_identity(isl_space_map_from_set(dim));
        return isl_map_intersect_range(id, set);
}

/* Construct a basic set with all set dimensions having only non-negative
 * values.
 */
__isl_give isl_basic_set *isl_basic_set_positive_orthant(
        __isl_take isl_space *space)
{
        int i;
        unsigned nparam;
        unsigned dim;
        struct isl_basic_set *bset;

        if (!space)
                return NULL;
        nparam = space->nparam;
        dim = space->n_out;
        bset = isl_basic_set_alloc_space(space, 0, 0, dim);
        if (!bset)
                return NULL;
        for (i = 0; i < dim; ++i) {
                int k = isl_basic_set_alloc_inequality(bset);
                if (k < 0)
                        goto error;
                isl_seq_clr(bset->ineq[k], 1 + isl_basic_set_total_dim(bset));
                isl_int_set_si(bset->ineq[k][1 + nparam + i], 1);
        }
        return bset;
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Construct the half-space x_pos >= 0.
 */
static __isl_give isl_basic_set *nonneg_halfspace(__isl_take isl_space *dim,
        int pos)
{
        int k;
        isl_basic_set *nonneg;

        nonneg = isl_basic_set_alloc_space(dim, 0, 0, 1);
        k = isl_basic_set_alloc_inequality(nonneg);
        if (k < 0)
                goto error;
        isl_seq_clr(nonneg->ineq[k], 1 + isl_basic_set_total_dim(nonneg));
        isl_int_set_si(nonneg->ineq[k][pos], 1);

        return isl_basic_set_finalize(nonneg);
error:
        isl_basic_set_free(nonneg);
        return NULL;
}

/* Construct the half-space x_pos <= -1.
 */
static __isl_give isl_basic_set *neg_halfspace(__isl_take isl_space *dim, int pos)
{
        int k;
        isl_basic_set *neg;

        neg = isl_basic_set_alloc_space(dim, 0, 0, 1);
        k = isl_basic_set_alloc_inequality(neg);
        if (k < 0)
                goto error;
        isl_seq_clr(neg->ineq[k], 1 + isl_basic_set_total_dim(neg));
        isl_int_set_si(neg->ineq[k][0], -1);
        isl_int_set_si(neg->ineq[k][pos], -1);

        return isl_basic_set_finalize(neg);
error:
        isl_basic_set_free(neg);
        return NULL;
}

__isl_give isl_set *isl_set_split_dims(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;
        unsigned offset;
        isl_basic_set *nonneg;
        isl_basic_set *neg;

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

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

        offset = pos(set->dim, type);
        for (i = 0; i < n; ++i) {
                nonneg = nonneg_halfspace(isl_set_get_space(set),
                                          offset + first + i);
                neg = neg_halfspace(isl_set_get_space(set), offset + first + i);

                set = isl_set_intersect(set, isl_basic_set_union(nonneg, neg));
        }

        return set;
error:
        isl_set_free(set);
        return NULL;
}

static int foreach_orthant(__isl_take isl_set *set, int *signs, int first,
        int len, int (*fn)(__isl_take isl_set *orthant, int *signs, void *user),
        void *user)
{
        isl_set *half;

        if (!set)
                return -1;
        if (isl_set_plain_is_empty(set)) {
                isl_set_free(set);
                return 0;
        }
        if (first == len)
                return fn(set, signs, user);

        signs[first] = 1;
        half = isl_set_from_basic_set(nonneg_halfspace(isl_set_get_space(set),
                                                        1 + first));
        half = isl_set_intersect(half, isl_set_copy(set));
        if (foreach_orthant(half, signs, first + 1, len, fn, user) < 0)
                goto error;

        signs[first] = -1;
        half = isl_set_from_basic_set(neg_halfspace(isl_set_get_space(set),
                                                        1 + first));
        half = isl_set_intersect(half, set);
        return foreach_orthant(half, signs, first + 1, len, fn, user);
error:
        isl_set_free(set);
        return -1;
}

/* Call "fn" on the intersections of "set" with each of the orthants
 * (except for obviously empty intersections).  The orthant is identified
 * by the signs array, with each entry having value 1 or -1 according
 * to the sign of the corresponding variable.
 */
int isl_set_foreach_orthant(__isl_keep isl_set *set,
        int (*fn)(__isl_take isl_set *orthant, int *signs, void *user),
        void *user)
{
        unsigned nparam;
        unsigned nvar;
        int *signs;
        int r;

        if (!set)
                return -1;
        if (isl_set_plain_is_empty(set))
                return 0;

        nparam = isl_set_dim(set, isl_dim_param);
        nvar = isl_set_dim(set, isl_dim_set);

        signs = isl_alloc_array(set->ctx, int, nparam + nvar);

        r = foreach_orthant(isl_set_copy(set), signs, 0, nparam + nvar,
                            fn, user);

        free(signs);

        return r;
}

isl_bool isl_set_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
{
        return isl_map_is_equal(set_to_map(set1), set_to_map(set2));
}

isl_bool isl_basic_map_is_subset(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        int is_subset;
        struct isl_map *map1;
        struct isl_map *map2;

        if (!bmap1 || !bmap2)
                return isl_bool_error;

        map1 = isl_map_from_basic_map(isl_basic_map_copy(bmap1));
        map2 = isl_map_from_basic_map(isl_basic_map_copy(bmap2));

        is_subset = isl_map_is_subset(map1, map2);

        isl_map_free(map1);
        isl_map_free(map2);

        return is_subset;
}

isl_bool isl_basic_set_is_subset(__isl_keep isl_basic_set *bset1,
        __isl_keep isl_basic_set *bset2)
{
        return isl_basic_map_is_subset(bset1, bset2);
}

isl_bool isl_basic_map_is_equal(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        isl_bool is_subset;

        if (!bmap1 || !bmap2)
                return isl_bool_error;
        is_subset = isl_basic_map_is_subset(bmap1, bmap2);
        if (is_subset != isl_bool_true)
                return is_subset;
        is_subset = isl_basic_map_is_subset(bmap2, bmap1);
        return is_subset;
}

isl_bool isl_basic_set_is_equal(__isl_keep isl_basic_set *bset1,
        __isl_keep isl_basic_set *bset2)
{
        return isl_basic_map_is_equal(
                bset_to_bmap(bset1), bset_to_bmap(bset2));
}

isl_bool isl_map_is_empty(__isl_keep isl_map *map)
{
        int i;
        int is_empty;

        if (!map)
                return isl_bool_error;
        for (i = 0; i < map->n; ++i) {
                is_empty = isl_basic_map_is_empty(map->p[i]);
                if (is_empty < 0)
                        return isl_bool_error;
                if (!is_empty)
                        return isl_bool_false;
        }
        return isl_bool_true;
}

isl_bool isl_map_plain_is_empty(__isl_keep isl_map *map)
{
        return map ? map->n == 0 : isl_bool_error;
}

isl_bool isl_set_plain_is_empty(__isl_keep isl_set *set)
{
        return set ? set->n == 0 : isl_bool_error;
}

isl_bool isl_set_is_empty(__isl_keep isl_set *set)
{
        return isl_map_is_empty(set_to_map(set));
}

int isl_map_has_equal_space(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{
        if (!map1 || !map2)
                return -1;

        return isl_space_is_equal(map1->dim, map2->dim);
}

int isl_set_has_equal_space(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
{
        if (!set1 || !set2)
                return -1;

        return isl_space_is_equal(set1->dim, set2->dim);
}

static isl_bool map_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{
        isl_bool is_subset;

        if (!map1 || !map2)
                return isl_bool_error;
        is_subset = isl_map_is_subset(map1, map2);
        if (is_subset != isl_bool_true)
                return is_subset;
        is_subset = isl_map_is_subset(map2, map1);
        return is_subset;
}

/* Is "map1" equal to "map2"?
 *
 * First check if they are obviously equal.
 * If not, then perform a more detailed analysis.
 */
isl_bool isl_map_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{
        isl_bool equal;

        equal = isl_map_plain_is_equal(map1, map2);
        if (equal < 0 || equal)
                return equal;
        return isl_map_align_params_map_map_and_test(map1, map2, &map_is_equal);
}

isl_bool isl_basic_map_is_strict_subset(
                struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        isl_bool is_subset;

        if (!bmap1 || !bmap2)
                return isl_bool_error;
        is_subset = isl_basic_map_is_subset(bmap1, bmap2);
        if (is_subset != isl_bool_true)
                return is_subset;
        is_subset = isl_basic_map_is_subset(bmap2, bmap1);
        if (is_subset == isl_bool_error)
                return is_subset;
        return !is_subset;
}

isl_bool isl_map_is_strict_subset(__isl_keep isl_map *map1,
        __isl_keep isl_map *map2)
{
        isl_bool is_subset;

        if (!map1 || !map2)
                return isl_bool_error;
        is_subset = isl_map_is_subset(map1, map2);
        if (is_subset != isl_bool_true)
                return is_subset;
        is_subset = isl_map_is_subset(map2, map1);
        if (is_subset == isl_bool_error)
                return is_subset;
        return !is_subset;
}

isl_bool isl_set_is_strict_subset(__isl_keep isl_set *set1,
        __isl_keep isl_set *set2)
{
        return isl_map_is_strict_subset(set_to_map(set1), set_to_map(set2));
}

/* Is "bmap" obviously equal to the universe with the same space?
 *
 * That is, does it not have any constraints?
 */
isl_bool isl_basic_map_plain_is_universe(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return isl_bool_error;
        return bmap->n_eq == 0 && bmap->n_ineq == 0;
}

/* Is "bset" obviously equal to the universe with the same space?
 */
isl_bool isl_basic_set_plain_is_universe(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_plain_is_universe(bset);
}

/* If "c" does not involve any existentially quantified variables,
 * then set *univ to false and abort
 */
static isl_stat involves_divs(__isl_take isl_constraint *c, void *user)
{
        isl_bool *univ = user;
        unsigned n;

        n = isl_constraint_dim(c, isl_dim_div);
        *univ = isl_constraint_involves_dims(c, isl_dim_div, 0, n);
        isl_constraint_free(c);
        if (*univ < 0 || !*univ)
                return isl_stat_error;
        return isl_stat_ok;
}

/* Is "bmap" equal to the universe with the same space?
 *
 * First check if it is obviously equal to the universe.
 * If not and if there are any constraints not involving
 * existentially quantified variables, then it is certainly
 * not equal to the universe.
 * Otherwise, check if the universe is a subset of "bmap".
 */
isl_bool isl_basic_map_is_universe(__isl_keep isl_basic_map *bmap)
{
        isl_bool univ;
        isl_basic_map *test;

        univ = isl_basic_map_plain_is_universe(bmap);
        if (univ < 0 || univ)
                return univ;
        if (isl_basic_map_dim(bmap, isl_dim_div) == 0)
                return isl_bool_false;
        univ = isl_bool_true;
        if (isl_basic_map_foreach_constraint(bmap, &involves_divs, &univ) < 0 &&
            univ)
                return isl_bool_error;
        if (univ < 0 || !univ)
                return univ;
        test = isl_basic_map_universe(isl_basic_map_get_space(bmap));
        univ = isl_basic_map_is_subset(test, bmap);
        isl_basic_map_free(test);
        return univ;
}

/* Is "bset" equal to the universe with the same space?
 */
isl_bool isl_basic_set_is_universe(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_is_universe(bset);
}

isl_bool isl_map_plain_is_universe(__isl_keep isl_map *map)
{
        int i;

        if (!map)
                return isl_bool_error;

        for (i = 0; i < map->n; ++i) {
                isl_bool r = isl_basic_map_plain_is_universe(map->p[i]);
                if (r < 0 || r)
                        return r;
        }

        return isl_bool_false;
}

isl_bool isl_set_plain_is_universe(__isl_keep isl_set *set)
{
        return isl_map_plain_is_universe(set_to_map(set));
}

isl_bool isl_basic_map_is_empty(__isl_keep isl_basic_map *bmap)
{
        struct isl_basic_set *bset = NULL;
        struct isl_vec *sample = NULL;
        isl_bool empty, non_empty;

        if (!bmap)
                return isl_bool_error;

        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                return isl_bool_true;

        if (isl_basic_map_plain_is_universe(bmap))
                return isl_bool_false;

        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
                struct isl_basic_map *copy = isl_basic_map_copy(bmap);
                copy = isl_basic_map_remove_redundancies(copy);
                empty = isl_basic_map_plain_is_empty(copy);
                isl_basic_map_free(copy);
                return empty;
        }

        non_empty = isl_basic_map_plain_is_non_empty(bmap);
        if (non_empty < 0)
                return isl_bool_error;
        if (non_empty)
                return isl_bool_false;
        isl_vec_free(bmap->sample);
        bmap->sample = NULL;
        bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
        if (!bset)
                return isl_bool_error;
        sample = isl_basic_set_sample_vec(bset);
        if (!sample)
                return isl_bool_error;
        empty = sample->size == 0;
        isl_vec_free(bmap->sample);
        bmap->sample = sample;
        if (empty)
                ISL_F_SET(bmap, ISL_BASIC_MAP_EMPTY);

        return empty;
}

isl_bool isl_basic_map_plain_is_empty(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return isl_bool_error;
        return ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY);
}

isl_bool isl_basic_set_plain_is_empty(__isl_keep isl_basic_set *bset)
{
        if (!bset)
                return isl_bool_error;
        return ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY);
}

/* Is "bmap" known to be non-empty?
 *
 * That is, is the cached sample still valid?
 */
isl_bool isl_basic_map_plain_is_non_empty(__isl_keep isl_basic_map *bmap)
{
        unsigned total;

        if (!bmap)
                return isl_bool_error;
        if (!bmap->sample)
                return isl_bool_false;
        total = 1 + isl_basic_map_total_dim(bmap);
        if (bmap->sample->size != total)
                return isl_bool_false;
        return isl_basic_map_contains(bmap, bmap->sample);
}

isl_bool isl_basic_set_is_empty(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_is_empty(bset_to_bmap(bset));
}

struct isl_map *isl_basic_map_union(
        struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        struct isl_map *map;
        if (!bmap1 || !bmap2)
                goto error;

        isl_assert(bmap1->ctx, isl_space_is_equal(bmap1->dim, bmap2->dim), goto error);

        map = isl_map_alloc_space(isl_space_copy(bmap1->dim), 2, 0);
        if (!map)
                goto error;
        map = isl_map_add_basic_map(map, bmap1);
        map = isl_map_add_basic_map(map, bmap2);
        return map;
error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

struct isl_set *isl_basic_set_union(
                struct isl_basic_set *bset1, struct isl_basic_set *bset2)
{
        return set_from_map(isl_basic_map_union(bset_to_bmap(bset1),
                                                bset_to_bmap(bset2)));
}

/* Order divs such that any div only depends on previous divs */
struct isl_basic_map *isl_basic_map_order_divs(struct isl_basic_map *bmap)
{
        int i;
        unsigned off;

        if (!bmap)
                return NULL;

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

        for (i = 0; i < bmap->n_div; ++i) {
                int pos;
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                pos = isl_seq_first_non_zero(bmap->div[i]+1+1+off+i,
                                                            bmap->n_div-i);
                if (pos == -1)
                        continue;
                if (pos == 0)
                        isl_die(isl_basic_map_get_ctx(bmap), isl_error_internal,
                                "integer division depends on itself",
                                return isl_basic_map_free(bmap));
                isl_basic_map_swap_div(bmap, i, i + pos);
                --i;
        }
        return bmap;
}

struct isl_basic_set *isl_basic_set_order_divs(struct isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_order_divs(bset_to_bmap(bset)));
}

__isl_give isl_map *isl_map_order_divs(__isl_take isl_map *map)
{
        int i;

        if (!map)
                return 0;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_order_divs(map->p[i]);
                if (!map->p[i])
                        goto error;
        }

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

/* Sort the local variables of "bset".
 */
__isl_give isl_basic_set *isl_basic_set_sort_divs(
        __isl_take isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_sort_divs(bset_to_bmap(bset)));
}

/* Apply the expansion computed by isl_merge_divs.
 * The expansion itself is given by "exp" while the resulting
 * list of divs is given by "div".
 *
 * Move the integer divisions of "bmap" into the right position
 * according to "exp" and then introduce the additional integer
 * divisions, adding div constraints.
 * The moving should be done first to avoid moving coefficients
 * in the definitions of the extra integer divisions.
 */
__isl_give isl_basic_map *isl_basic_map_expand_divs(
        __isl_take isl_basic_map *bmap, __isl_take isl_mat *div, int *exp)
{
        int i, j;
        int n_div;

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

        if (div->n_row < bmap->n_div)
                isl_die(isl_mat_get_ctx(div), isl_error_invalid,
                        "not an expansion", goto error);

        n_div = bmap->n_div;
        bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
                                            div->n_row - n_div, 0,
                                            2 * (div->n_row - n_div));

        for (i = n_div; i < div->n_row; ++i)
                if (isl_basic_map_alloc_div(bmap) < 0)
                        goto error;

        for (j = n_div - 1; j >= 0; --j) {
                if (exp[j] == j)
                        break;
                isl_basic_map_swap_div(bmap, j, exp[j]);
        }
        j = 0;
        for (i = 0; i < div->n_row; ++i) {
                if (j < n_div && exp[j] == i) {
                        j++;
                } else {
                        isl_seq_cpy(bmap->div[i], div->row[i], div->n_col);
                        if (isl_basic_map_div_is_marked_unknown(bmap, i))
                                continue;
                        if (isl_basic_map_add_div_constraints(bmap, i) < 0)
                                goto error;
                }
        }

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

/* Apply the expansion computed by isl_merge_divs.
 * The expansion itself is given by "exp" while the resulting
 * list of divs is given by "div".
 */
__isl_give isl_basic_set *isl_basic_set_expand_divs(
        __isl_take isl_basic_set *bset, __isl_take isl_mat *div, int *exp)
{
        return isl_basic_map_expand_divs(bset, div, exp);
}

/* Look for a div in dst that corresponds to the div "div" in src.
 * The divs before "div" in src and dst are assumed to be the same.
 * 
 * Returns -1 if no corresponding div was found and the position
 * of the corresponding div in dst otherwise.
 */
static int find_div(struct isl_basic_map *dst,
                        struct isl_basic_map *src, unsigned div)
{
        int i;

        unsigned total = isl_space_dim(src->dim, isl_dim_all);

        isl_assert(dst->ctx, div <= dst->n_div, return -1);
        for (i = div; i < dst->n_div; ++i)
                if (isl_seq_eq(dst->div[i], src->div[div], 1+1+total+div) &&
                    isl_seq_first_non_zero(dst->div[i]+1+1+total+div,
                                                dst->n_div - div) == -1)
                        return i;
        return -1;
}

/* Align the divs of "dst" to those of "src", adding divs from "src"
 * if needed.  That is, make sure that the first src->n_div divs
 * of the result are equal to those of src.
 *
 * The result is not finalized as by design it will have redundant
 * divs if any divs from "src" were copied.
 */
__isl_give isl_basic_map *isl_basic_map_align_divs(
        __isl_take isl_basic_map *dst, __isl_keep isl_basic_map *src)
{
        int i;
        int known, extended;
        unsigned total;

        if (!dst || !src)
                return isl_basic_map_free(dst);

        if (src->n_div == 0)
                return dst;

        known = isl_basic_map_divs_known(src);
        if (known < 0)
                return isl_basic_map_free(dst);
        if (!known)
                isl_die(isl_basic_map_get_ctx(src), isl_error_invalid,
                        "some src divs are unknown",
                        return isl_basic_map_free(dst));

        src = isl_basic_map_order_divs(src);

        extended = 0;
        total = isl_space_dim(src->dim, isl_dim_all);
        for (i = 0; i < src->n_div; ++i) {
                int j = find_div(dst, src, i);
                if (j < 0) {
                        if (!extended) {
                                int extra = src->n_div - i;
                                dst = isl_basic_map_cow(dst);
                                if (!dst)
                                        return NULL;
                                dst = isl_basic_map_extend_space(dst,
                                                isl_space_copy(dst->dim),
                                                extra, 0, 2 * extra);
                                extended = 1;
                        }
                        j = isl_basic_map_alloc_div(dst);
                        if (j < 0)
                                return isl_basic_map_free(dst);
                        isl_seq_cpy(dst->div[j], src->div[i], 1+1+total+i);
                        isl_seq_clr(dst->div[j]+1+1+total+i, dst->n_div - i);
                        if (isl_basic_map_add_div_constraints(dst, j) < 0)
                                return isl_basic_map_free(dst);
                }
                if (j != i)
                        isl_basic_map_swap_div(dst, i, j);
        }
        return dst;
}

struct isl_basic_set *isl_basic_set_align_divs(
                struct isl_basic_set *dst, struct isl_basic_set *src)
{
        return bset_from_bmap(isl_basic_map_align_divs(bset_to_bmap(dst),
                                                        bset_to_bmap(src)));
}

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

        if (!map)
                return NULL;
        if (map->n == 0)
                return map;
        map = isl_map_compute_divs(map);
        map = isl_map_cow(map);
        if (!map)
                return NULL;

        for (i = 1; i < map->n; ++i)
                map->p[0] = isl_basic_map_align_divs(map->p[0], map->p[i]);
        for (i = 1; i < map->n; ++i) {
                map->p[i] = isl_basic_map_align_divs(map->p[i], map->p[0]);
                if (!map->p[i])
                        return isl_map_free(map);
        }

        ISL_F_CLR(map, ISL_MAP_NORMALIZED);
        return map;
}

struct isl_set *isl_set_align_divs(struct isl_set *set)
{
        return set_from_map(isl_map_align_divs(set_to_map(set)));
}

/* Align the divs of the basic maps in "map" to those
 * of the basic maps in "list", as well as to the other basic maps in "map".
 * The elements in "list" are assumed to have known divs.
 */
__isl_give isl_map *isl_map_align_divs_to_basic_map_list(
        __isl_take isl_map *map, __isl_keep isl_basic_map_list *list)
{
        int i, n;

        map = isl_map_compute_divs(map);
        map = isl_map_cow(map);
        if (!map || !list)
                return isl_map_free(map);
        if (map->n == 0)
                return map;

        n = isl_basic_map_list_n_basic_map(list);
        for (i = 0; i < n; ++i) {
                isl_basic_map *bmap;

                bmap = isl_basic_map_list_get_basic_map(list, i);
                map->p[0] = isl_basic_map_align_divs(map->p[0], bmap);
                isl_basic_map_free(bmap);
        }
        if (!map->p[0])
                return isl_map_free(map);

        return isl_map_align_divs(map);
}

/* Align the divs of each element of "list" to those of "bmap".
 * Both "bmap" and the elements of "list" are assumed to have known divs.
 */
__isl_give isl_basic_map_list *isl_basic_map_list_align_divs_to_basic_map(
        __isl_take isl_basic_map_list *list, __isl_keep isl_basic_map *bmap)
{
        int i, n;

        if (!list || !bmap)
                return isl_basic_map_list_free(list);

        n = isl_basic_map_list_n_basic_map(list);
        for (i = 0; i < n; ++i) {
                isl_basic_map *bmap_i;

                bmap_i = isl_basic_map_list_get_basic_map(list, i);
                bmap_i = isl_basic_map_align_divs(bmap_i, bmap);
                list = isl_basic_map_list_set_basic_map(list, i, bmap_i);
        }

        return list;
}

static __isl_give isl_set *set_apply( __isl_take isl_set *set,
        __isl_take isl_map *map)
{
        if (!set || !map)
                goto error;
        isl_assert(set->ctx, isl_map_compatible_domain(map, set), goto error);
        map = isl_map_intersect_domain(map, set);
        set = isl_map_range(map);
        return set;
error:
        isl_set_free(set);
        isl_map_free(map);
        return NULL;
}

__isl_give isl_set *isl_set_apply( __isl_take isl_set *set,
        __isl_take isl_map *map)
{
        return isl_map_align_params_map_map_and(set, map, &set_apply);
}

/* There is no need to cow as removing empty parts doesn't change
 * the meaning of the set.
 */
struct isl_map *isl_map_remove_empty_parts(struct isl_map *map)
{
        int i;

        if (!map)
                return NULL;

        for (i = map->n - 1; i >= 0; --i)
                remove_if_empty(map, i);

        return map;
}

struct isl_set *isl_set_remove_empty_parts(struct isl_set *set)
{
        return set_from_map(isl_map_remove_empty_parts(set_to_map(set)));
}

static __isl_give isl_basic_map *map_copy_basic_map(__isl_keep isl_map *map)
{
        struct isl_basic_map *bmap;
        if (!map || map->n == 0)
                return NULL;
        bmap = map->p[map->n-1];
        isl_assert(map->ctx, ISL_F_ISSET(bmap, ISL_BASIC_SET_FINAL), return NULL);
        return isl_basic_map_copy(bmap);
}

__isl_give isl_basic_map *isl_map_copy_basic_map(__isl_keep isl_map *map)
{
        return map_copy_basic_map(map);
}

struct isl_basic_set *isl_set_copy_basic_set(struct isl_set *set)
{
        return bset_from_bmap(map_copy_basic_map(set_to_map(set)));
}

static __isl_give isl_map *map_drop_basic_map(__isl_take isl_map *map,
                                                __isl_keep isl_basic_map *bmap)
{
        int i;

        if (!map || !bmap)
                goto error;
        for (i = map->n-1; i >= 0; --i) {
                if (map->p[i] != bmap)
                        continue;
                map = isl_map_cow(map);
                if (!map)
                        goto error;
                isl_basic_map_free(map->p[i]);
                if (i != map->n-1) {
                        ISL_F_CLR(map, ISL_SET_NORMALIZED);
                        map->p[i] = map->p[map->n-1];
                }
                map->n--;
                return map;
        }
        return map;
error:
        isl_map_free(map);
        return NULL;
}

__isl_give isl_map *isl_map_drop_basic_map(__isl_take isl_map *map,
        __isl_keep isl_basic_map *bmap)
{
        return map_drop_basic_map(map, bmap);
}

struct isl_set *isl_set_drop_basic_set(struct isl_set *set,
                                                struct isl_basic_set *bset)
{
        return set_from_map(map_drop_basic_map(set_to_map(set),
                                                bset_to_bmap(bset)));
}

/* Given two basic sets bset1 and bset2, compute the maximal difference
 * between the values of dimension pos in bset1 and those in bset2
 * for any common value of the parameters and dimensions preceding pos.
 */
static enum isl_lp_result basic_set_maximal_difference_at(
        __isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2,
        int pos, isl_int *opt)
{
        isl_basic_map *bmap1;
        isl_basic_map *bmap2;
        struct isl_ctx *ctx;
        struct isl_vec *obj;
        unsigned total;
        unsigned nparam;
        unsigned dim1;
        enum isl_lp_result res;

        if (!bset1 || !bset2)
                return isl_lp_error;

        nparam = isl_basic_set_n_param(bset1);
        dim1 = isl_basic_set_n_dim(bset1);

        bmap1 = isl_basic_map_from_range(isl_basic_set_copy(bset1));
        bmap2 = isl_basic_map_from_range(isl_basic_set_copy(bset2));
        bmap1 = isl_basic_map_move_dims(bmap1, isl_dim_in, 0,
                                        isl_dim_out, 0, pos);
        bmap2 = isl_basic_map_move_dims(bmap2, isl_dim_in, 0,
                                        isl_dim_out, 0, pos);
        bmap1 = isl_basic_map_range_product(bmap1, bmap2);
        if (!bmap1)
                return isl_lp_error;

        total = isl_basic_map_total_dim(bmap1);
        ctx = bmap1->ctx;
        obj = isl_vec_alloc(ctx, 1 + total);
        if (!obj)
                goto error;
        isl_seq_clr(obj->block.data, 1 + total);
        isl_int_set_si(obj->block.data[1+nparam+pos], 1);
        isl_int_set_si(obj->block.data[1+nparam+pos+(dim1-pos)], -1);
        res = isl_basic_map_solve_lp(bmap1, 1, obj->block.data, ctx->one,
                                        opt, NULL, NULL);
        isl_basic_map_free(bmap1);
        isl_vec_free(obj);
        return res;
error:
        isl_basic_map_free(bmap1);
        return isl_lp_error;
}

/* Given two _disjoint_ basic sets bset1 and bset2, check whether
 * for any common value of the parameters and dimensions preceding pos
 * in both basic sets, the values of dimension pos in bset1 are
 * smaller or larger than those in bset2.
 *
 * Returns
 *       1 if bset1 follows bset2
 *      -1 if bset1 precedes bset2
 *       0 if bset1 and bset2 are incomparable
 *      -2 if some error occurred.
 */
int isl_basic_set_compare_at(struct isl_basic_set *bset1,
        struct isl_basic_set *bset2, int pos)
{
        isl_int opt;
        enum isl_lp_result res;
        int cmp;

        isl_int_init(opt);

        res = basic_set_maximal_difference_at(bset1, bset2, pos, &opt);

        if (res == isl_lp_empty)
                cmp = 0;
        else if ((res == isl_lp_ok && isl_int_is_pos(opt)) ||
                  res == isl_lp_unbounded)
                cmp = 1;
        else if (res == isl_lp_ok && isl_int_is_neg(opt))
                cmp = -1;
        else
                cmp = -2;

        isl_int_clear(opt);
        return cmp;
}

/* Given two basic sets bset1 and bset2, check whether
 * for any common value of the parameters and dimensions preceding pos
 * there is a value of dimension pos in bset1 that is larger
 * than a value of the same dimension in bset2.
 *
 * Return
 *       1 if there exists such a pair
 *       0 if there is no such pair, but there is a pair of equal values
 *      -1 otherwise
 *      -2 if some error occurred.
 */
int isl_basic_set_follows_at(__isl_keep isl_basic_set *bset1,
        __isl_keep isl_basic_set *bset2, int pos)
{
        isl_int opt;
        enum isl_lp_result res;
        int cmp;

        isl_int_init(opt);

        res = basic_set_maximal_difference_at(bset1, bset2, pos, &opt);

        if (res == isl_lp_empty)
                cmp = -1;
        else if ((res == isl_lp_ok && isl_int_is_pos(opt)) ||
                  res == isl_lp_unbounded)
                cmp = 1;
        else if (res == isl_lp_ok && isl_int_is_neg(opt))
                cmp = -1;
        else if (res == isl_lp_ok)
                cmp = 0;
        else
                cmp = -2;

        isl_int_clear(opt);
        return cmp;
}

/* Given two sets set1 and set2, check whether
 * for any common value of the parameters and dimensions preceding pos
 * there is a value of dimension pos in set1 that is larger
 * than a value of the same dimension in set2.
 *
 * Return
 *       1 if there exists such a pair
 *       0 if there is no such pair, but there is a pair of equal values
 *      -1 otherwise
 *      -2 if some error occurred.
 */
int isl_set_follows_at(__isl_keep isl_set *set1,
        __isl_keep isl_set *set2, int pos)
{
        int i, j;
        int follows = -1;

        if (!set1 || !set2)
                return -2;

        for (i = 0; i < set1->n; ++i)
                for (j = 0; j < set2->n; ++j) {
                        int f;
                        f = isl_basic_set_follows_at(set1->p[i], set2->p[j], pos);
                        if (f == 1 || f == -2)
                                return f;
                        if (f > follows)
                                follows = f;
                }

        return follows;
}

static int isl_basic_map_plain_has_fixed_var(__isl_keep isl_basic_map *bmap,
        unsigned pos, isl_int *val)
{
        int i;
        int d;
        unsigned total;

        if (!bmap)
                return -1;
        total = isl_basic_map_total_dim(bmap);
        for (i = 0, d = total-1; i < bmap->n_eq && d+1 > pos; ++i) {
                for (; d+1 > pos; --d)
                        if (!isl_int_is_zero(bmap->eq[i][1+d]))
                                break;
                if (d != pos)
                        continue;
                if (isl_seq_first_non_zero(bmap->eq[i]+1, d) != -1)
                        return 0;
                if (isl_seq_first_non_zero(bmap->eq[i]+1+d+1, total-d-1) != -1)
                        return 0;
                if (!isl_int_is_one(bmap->eq[i][1+d]))
                        return 0;
                if (val)
                        isl_int_neg(*val, bmap->eq[i][0]);
                return 1;
        }
        return 0;
}

static int isl_map_plain_has_fixed_var(__isl_keep isl_map *map,
        unsigned pos, isl_int *val)
{
        int i;
        isl_int v;
        isl_int tmp;
        int fixed;

        if (!map)
                return -1;
        if (map->n == 0)
                return 0;
        if (map->n == 1)
                return isl_basic_map_plain_has_fixed_var(map->p[0], pos, val); 
        isl_int_init(v);
        isl_int_init(tmp);
        fixed = isl_basic_map_plain_has_fixed_var(map->p[0], pos, &v); 
        for (i = 1; fixed == 1 && i < map->n; ++i) {
                fixed = isl_basic_map_plain_has_fixed_var(map->p[i], pos, &tmp); 
                if (fixed == 1 && isl_int_ne(tmp, v))
                        fixed = 0;
        }
        if (val)
                isl_int_set(*val, v);
        isl_int_clear(tmp);
        isl_int_clear(v);
        return fixed;
}

static int isl_basic_set_plain_has_fixed_var(__isl_keep isl_basic_set *bset,
        unsigned pos, isl_int *val)
{
        return isl_basic_map_plain_has_fixed_var(bset_to_bmap(bset),
                                                pos, val);
}

static int isl_set_plain_has_fixed_var(__isl_keep isl_set *set, unsigned pos,
        isl_int *val)
{
        return isl_map_plain_has_fixed_var(set_to_map(set), pos, val);
}

int isl_basic_map_plain_is_fixed(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, isl_int *val)
{
        if (pos >= isl_basic_map_dim(bmap, type))
                return -1;
        return isl_basic_map_plain_has_fixed_var(bmap,
                isl_basic_map_offset(bmap, type) - 1 + pos, val);
}

/* If "bmap" obviously lies on a hyperplane where the given dimension
 * has a fixed value, then return that value.
 * Otherwise return NaN.
 */
__isl_give isl_val *isl_basic_map_plain_get_val_if_fixed(
        __isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos)
{
        isl_ctx *ctx;
        isl_val *v;
        int fixed;

        if (!bmap)
                return NULL;
        ctx = isl_basic_map_get_ctx(bmap);
        v = isl_val_alloc(ctx);
        if (!v)
                return NULL;
        fixed = isl_basic_map_plain_is_fixed(bmap, type, pos, &v->n);
        if (fixed < 0)
                return isl_val_free(v);
        if (fixed) {
                isl_int_set_si(v->d, 1);
                return v;
        }
        isl_val_free(v);
        return isl_val_nan(ctx);
}

int isl_map_plain_is_fixed(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos, isl_int *val)
{
        if (pos >= isl_map_dim(map, type))
                return -1;
        return isl_map_plain_has_fixed_var(map,
                map_offset(map, type) - 1 + pos, val);
}

/* If "map" obviously lies on a hyperplane where the given dimension
 * has a fixed value, then return that value.
 * Otherwise return NaN.
 */
__isl_give isl_val *isl_map_plain_get_val_if_fixed(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos)
{
        isl_ctx *ctx;
        isl_val *v;
        int fixed;

        if (!map)
                return NULL;
        ctx = isl_map_get_ctx(map);
        v = isl_val_alloc(ctx);
        if (!v)
                return NULL;
        fixed = isl_map_plain_is_fixed(map, type, pos, &v->n);
        if (fixed < 0)
                return isl_val_free(v);
        if (fixed) {
                isl_int_set_si(v->d, 1);
                return v;
        }
        isl_val_free(v);
        return isl_val_nan(ctx);
}

/* If "set" obviously lies on a hyperplane where the given dimension
 * has a fixed value, then return that value.
 * Otherwise return NaN.
 */
__isl_give isl_val *isl_set_plain_get_val_if_fixed(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return isl_map_plain_get_val_if_fixed(set, type, pos);
}

int isl_set_plain_is_fixed(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos, isl_int *val)
{
        return isl_map_plain_is_fixed(set, type, pos, val);
}

/* Check if dimension dim has fixed value and if so and if val is not NULL,
 * then return this fixed value in *val.
 */
int isl_basic_set_plain_dim_is_fixed(__isl_keep isl_basic_set *bset,
        unsigned dim, isl_int *val)
{
        return isl_basic_set_plain_has_fixed_var(bset,
                                        isl_basic_set_n_param(bset) + dim, val);
}

/* Check if dimension dim has fixed value and if so and if val is not NULL,
 * then return this fixed value in *val.
 */
int isl_set_plain_dim_is_fixed(__isl_keep isl_set *set,
        unsigned dim, isl_int *val)
{
        return isl_set_plain_has_fixed_var(set, isl_set_n_param(set) + dim, val);
}

/* Check if input variable in has fixed value and if so and if val is not NULL,
 * then return this fixed value in *val.
 */
int isl_map_plain_input_is_fixed(__isl_keep isl_map *map,
        unsigned in, isl_int *val)
{
        return isl_map_plain_has_fixed_var(map, isl_map_n_param(map) + in, val);
}

/* Check if dimension dim has an (obvious) fixed lower bound and if so
 * and if val is not NULL, then return this lower bound in *val.
 */
int isl_basic_set_plain_dim_has_fixed_lower_bound(
        __isl_keep isl_basic_set *bset, unsigned dim, isl_int *val)
{
        int i, i_eq = -1, i_ineq = -1;
        isl_int *c;
        unsigned total;
        unsigned nparam;

        if (!bset)
                return -1;
        total = isl_basic_set_total_dim(bset);
        nparam = isl_basic_set_n_param(bset);
        for (i = 0; i < bset->n_eq; ++i) {
                if (isl_int_is_zero(bset->eq[i][1+nparam+dim]))
                        continue;
                if (i_eq != -1)
                        return 0;
                i_eq = i;
        }
        for (i = 0; i < bset->n_ineq; ++i) {
                if (!isl_int_is_pos(bset->ineq[i][1+nparam+dim]))
                        continue;
                if (i_eq != -1 || i_ineq != -1)
                        return 0;
                i_ineq = i;
        }
        if (i_eq == -1 && i_ineq == -1)
                return 0;
        c = i_eq != -1 ? bset->eq[i_eq] : bset->ineq[i_ineq];
        /* The coefficient should always be one due to normalization. */
        if (!isl_int_is_one(c[1+nparam+dim]))
                return 0;
        if (isl_seq_first_non_zero(c+1, nparam+dim) != -1)
                return 0;
        if (isl_seq_first_non_zero(c+1+nparam+dim+1,
                                        total - nparam - dim - 1) != -1)
                return 0;
        if (val)
                isl_int_neg(*val, c[0]);
        return 1;
}

int isl_set_plain_dim_has_fixed_lower_bound(__isl_keep isl_set *set,
        unsigned dim, isl_int *val)
{
        int i;
        isl_int v;
        isl_int tmp;
        int fixed;

        if (!set)
                return -1;
        if (set->n == 0)
                return 0;
        if (set->n == 1)
                return isl_basic_set_plain_dim_has_fixed_lower_bound(set->p[0],
                                                                dim, val);
        isl_int_init(v);
        isl_int_init(tmp);
        fixed = isl_basic_set_plain_dim_has_fixed_lower_bound(set->p[0],
                                                                dim, &v);
        for (i = 1; fixed == 1 && i < set->n; ++i) {
                fixed = isl_basic_set_plain_dim_has_fixed_lower_bound(set->p[i],
                                                                dim, &tmp);
                if (fixed == 1 && isl_int_ne(tmp, v))
                        fixed = 0;
        }
        if (val)
                isl_int_set(*val, v);
        isl_int_clear(tmp);
        isl_int_clear(v);
        return fixed;
}

/* Return -1 if the constraint "c1" should be sorted before "c2"
 * and 1 if it should be sorted after "c2".
 * Return 0 if the two constraints are the same (up to the constant term).
 *
 * In particular, if a constraint involves later variables than another
 * then it is sorted after this other constraint.
 * uset_gist depends on constraints without existentially quantified
 * variables sorting first.
 *
 * For constraints that have the same latest variable, those
 * with the same coefficient for this latest variable (first in absolute value
 * and then in actual value) are grouped together.
 * This is useful for detecting pairs of constraints that can
 * be chained in their printed representation.
 *
 * Finally, within a group, constraints are sorted according to
 * their coefficients (excluding the constant term).
 */
static int sort_constraint_cmp(const void *p1, const void *p2, void *arg)
{
        isl_int **c1 = (isl_int **) p1;
        isl_int **c2 = (isl_int **) p2;
        int l1, l2;
        unsigned size = *(unsigned *) arg;
        int cmp;

        l1 = isl_seq_last_non_zero(*c1 + 1, size);
        l2 = isl_seq_last_non_zero(*c2 + 1, size);

        if (l1 != l2)
                return l1 - l2;

        cmp = isl_int_abs_cmp((*c1)[1 + l1], (*c2)[1 + l1]);
        if (cmp != 0)
                return cmp;
        cmp = isl_int_cmp((*c1)[1 + l1], (*c2)[1 + l1]);
        if (cmp != 0)
                return -cmp;

        return isl_seq_cmp(*c1 + 1, *c2 + 1, size);
}

/* Return -1 if the constraint "c1" of "bmap" is sorted before "c2"
 * by isl_basic_map_sort_constraints, 1 if it is sorted after "c2"
 * and 0 if the two constraints are the same (up to the constant term).
 */
int isl_basic_map_constraint_cmp(__isl_keep isl_basic_map *bmap,
        isl_int *c1, isl_int *c2)
{
        unsigned total;

        if (!bmap)
                return -2;
        total = isl_basic_map_total_dim(bmap);
        return sort_constraint_cmp(&c1, &c2, &total);
}

__isl_give isl_basic_map *isl_basic_map_sort_constraints(
        __isl_take isl_basic_map *bmap)
{
        unsigned total;

        if (!bmap)
                return NULL;
        if (bmap->n_ineq == 0)
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED))
                return bmap;
        total = isl_basic_map_total_dim(bmap);
        if (isl_sort(bmap->ineq, bmap->n_ineq, sizeof(isl_int *),
                    &sort_constraint_cmp, &total) < 0)
                return isl_basic_map_free(bmap);
        return bmap;
}

__isl_give isl_basic_set *isl_basic_set_sort_constraints(
        __isl_take isl_basic_set *bset)
{
        isl_basic_map *bmap = bset_to_bmap(bset);
        return bset_from_bmap(isl_basic_map_sort_constraints(bmap));
}

struct isl_basic_map *isl_basic_map_normalize(struct isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED))
                return bmap;
        bmap = isl_basic_map_remove_redundancies(bmap);
        bmap = isl_basic_map_sort_constraints(bmap);
        if (bmap)
                ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED);
        return bmap;
}

struct isl_basic_set *isl_basic_set_normalize(struct isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_normalize(bset_to_bmap(bset)));
}

int isl_basic_map_plain_cmp(const __isl_keep isl_basic_map *bmap1,
        const __isl_keep isl_basic_map *bmap2)
{
        int i, cmp;
        unsigned total;

        if (!bmap1 || !bmap2)
                return -1;

        if (bmap1 == bmap2)
                return 0;
        if (ISL_F_ISSET(bmap1, ISL_BASIC_MAP_RATIONAL) !=
            ISL_F_ISSET(bmap2, ISL_BASIC_MAP_RATIONAL))
                return ISL_F_ISSET(bmap1, ISL_BASIC_MAP_RATIONAL) ? -1 : 1;
        if (isl_basic_map_n_param(bmap1) != isl_basic_map_n_param(bmap2))
                return isl_basic_map_n_param(bmap1) - isl_basic_map_n_param(bmap2);
        if (isl_basic_map_n_in(bmap1) != isl_basic_map_n_in(bmap2))
                return isl_basic_map_n_out(bmap1) - isl_basic_map_n_out(bmap2);
        if (isl_basic_map_n_out(bmap1) != isl_basic_map_n_out(bmap2))
                return isl_basic_map_n_out(bmap1) - isl_basic_map_n_out(bmap2);
        if (ISL_F_ISSET(bmap1, ISL_BASIC_MAP_EMPTY) &&
            ISL_F_ISSET(bmap2, ISL_BASIC_MAP_EMPTY))
                return 0;
        if (ISL_F_ISSET(bmap1, ISL_BASIC_MAP_EMPTY))
                return 1;
        if (ISL_F_ISSET(bmap2, ISL_BASIC_MAP_EMPTY))
                return -1;
        if (bmap1->n_eq != bmap2->n_eq)
                return bmap1->n_eq - bmap2->n_eq;
        if (bmap1->n_ineq != bmap2->n_ineq)
                return bmap1->n_ineq - bmap2->n_ineq;
        if (bmap1->n_div != bmap2->n_div)
                return bmap1->n_div - bmap2->n_div;
        total = isl_basic_map_total_dim(bmap1);
        for (i = 0; i < bmap1->n_eq; ++i) {
                cmp = isl_seq_cmp(bmap1->eq[i], bmap2->eq[i], 1+total);
                if (cmp)
                        return cmp;
        }
        for (i = 0; i < bmap1->n_ineq; ++i) {
                cmp = isl_seq_cmp(bmap1->ineq[i], bmap2->ineq[i], 1+total);
                if (cmp)
                        return cmp;
        }
        for (i = 0; i < bmap1->n_div; ++i) {
                cmp = isl_seq_cmp(bmap1->div[i], bmap2->div[i], 1+1+total);
                if (cmp)
                        return cmp;
        }
        return 0;
}

int isl_basic_set_plain_cmp(const __isl_keep isl_basic_set *bset1,
        const __isl_keep isl_basic_set *bset2)
{
        return isl_basic_map_plain_cmp(bset1, bset2);
}

int isl_set_plain_cmp(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
{
        int i, cmp;

        if (set1 == set2)
                return 0;
        if (set1->n != set2->n)
                return set1->n - set2->n;

        for (i = 0; i < set1->n; ++i) {
                cmp = isl_basic_set_plain_cmp(set1->p[i], set2->p[i]);
                if (cmp)
                        return cmp;
        }

        return 0;
}

isl_bool isl_basic_map_plain_is_equal(__isl_keep isl_basic_map *bmap1,
        __isl_keep isl_basic_map *bmap2)
{
        if (!bmap1 || !bmap2)
                return isl_bool_error;
        return isl_basic_map_plain_cmp(bmap1, bmap2) == 0;
}

isl_bool isl_basic_set_plain_is_equal(__isl_keep isl_basic_set *bset1,
        __isl_keep isl_basic_set *bset2)
{
        return isl_basic_map_plain_is_equal(bset_to_bmap(bset1),
                                            bset_to_bmap(bset2));
}

static int qsort_bmap_cmp(const void *p1, const void *p2)
{
        const struct isl_basic_map *bmap1 = *(const struct isl_basic_map **)p1;
        const struct isl_basic_map *bmap2 = *(const struct isl_basic_map **)p2;

        return isl_basic_map_plain_cmp(bmap1, bmap2);
}

/* Sort the basic maps of "map" and remove duplicate basic maps.
 *
 * While removing basic maps, we make sure that the basic maps remain
 * sorted because isl_map_normalize expects the basic maps of the result
 * to be sorted.
 */
static __isl_give isl_map *sort_and_remove_duplicates(__isl_take isl_map *map)
{
        int i, j;

        map = isl_map_remove_empty_parts(map);
        if (!map)
                return NULL;
        qsort(map->p, map->n, sizeof(struct isl_basic_map *), qsort_bmap_cmp);
        for (i = map->n - 1; i >= 1; --i) {
                if (!isl_basic_map_plain_is_equal(map->p[i - 1], map->p[i]))
                        continue;
                isl_basic_map_free(map->p[i-1]);
                for (j = i; j < map->n; ++j)
                        map->p[j - 1] = map->p[j];
                map->n--;
        }

        return map;
}

/* Remove obvious duplicates among the basic maps of "map".
 *
 * Unlike isl_map_normalize, this function does not remove redundant
 * constraints and only removes duplicates that have exactly the same
 * constraints in the input.  It does sort the constraints and
 * the basic maps to ease the detection of duplicates.
 *
 * If "map" has already been normalized or if the basic maps are
 * disjoint, then there can be no duplicates.
 */
__isl_give isl_map *isl_map_remove_obvious_duplicates(__isl_take isl_map *map)
{
        int i;
        isl_basic_map *bmap;

        if (!map)
                return NULL;
        if (map->n <= 1)
                return map;
        if (ISL_F_ISSET(map, ISL_MAP_NORMALIZED | ISL_MAP_DISJOINT))
                return map;
        for (i = 0; i < map->n; ++i) {
                bmap = isl_basic_map_copy(map->p[i]);
                bmap = isl_basic_map_sort_constraints(bmap);
                if (!bmap)
                        return isl_map_free(map);
                isl_basic_map_free(map->p[i]);
                map->p[i] = bmap;
        }

        map = sort_and_remove_duplicates(map);
        return map;
}

/* We normalize in place, but if anything goes wrong we need
 * to return NULL, so we need to make sure we don't change the
 * meaning of any possible other copies of map.
 */
__isl_give isl_map *isl_map_normalize(__isl_take isl_map *map)
{
        int i;
        struct isl_basic_map *bmap;

        if (!map)
                return NULL;
        if (ISL_F_ISSET(map, ISL_MAP_NORMALIZED))
                return map;
        for (i = 0; i < map->n; ++i) {
                bmap = isl_basic_map_normalize(isl_basic_map_copy(map->p[i]));
                if (!bmap)
                        goto error;
                isl_basic_map_free(map->p[i]);
                map->p[i] = bmap;
        }

        map = sort_and_remove_duplicates(map);
        if (map)
                ISL_F_SET(map, ISL_MAP_NORMALIZED);
        return map;
error:
        isl_map_free(map);
        return NULL;
}

struct isl_set *isl_set_normalize(struct isl_set *set)
{
        return set_from_map(isl_map_normalize(set_to_map(set)));
}

isl_bool isl_map_plain_is_equal(__isl_keep isl_map *map1,
        __isl_keep isl_map *map2)
{
        int i;
        isl_bool equal;

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

        if (map1 == map2)
                return isl_bool_true;
        if (!isl_space_is_equal(map1->dim, map2->dim))
                return isl_bool_false;

        map1 = isl_map_copy(map1);
        map2 = isl_map_copy(map2);
        map1 = isl_map_normalize(map1);
        map2 = isl_map_normalize(map2);
        if (!map1 || !map2)
                goto error;
        equal = map1->n == map2->n;
        for (i = 0; equal && i < map1->n; ++i) {
                equal = isl_basic_map_plain_is_equal(map1->p[i], map2->p[i]);
                if (equal < 0)
                        goto error;
        }
        isl_map_free(map1);
        isl_map_free(map2);
        return equal;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return isl_bool_error;
}

isl_bool isl_set_plain_is_equal(__isl_keep isl_set *set1,
        __isl_keep isl_set *set2)
{
        return isl_map_plain_is_equal(set_to_map(set1), set_to_map(set2));
}

/* Return an interval that ranges from min to max (inclusive)
 */
struct isl_basic_set *isl_basic_set_interval(struct isl_ctx *ctx,
        isl_int min, isl_int max)
{
        int k;
        struct isl_basic_set *bset = NULL;

        bset = isl_basic_set_alloc(ctx, 0, 1, 0, 0, 2);
        if (!bset)
                goto error;

        k = isl_basic_set_alloc_inequality(bset);
        if (k < 0)
                goto error;
        isl_int_set_si(bset->ineq[k][1], 1);
        isl_int_neg(bset->ineq[k][0], min);

        k = isl_basic_set_alloc_inequality(bset);
        if (k < 0)
                goto error;
        isl_int_set_si(bset->ineq[k][1], -1);
        isl_int_set(bset->ineq[k][0], max);

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

/* Return the basic maps in "map" as a list.
 */
__isl_give isl_basic_map_list *isl_map_get_basic_map_list(
        __isl_keep isl_map *map)
{
        int i;
        isl_ctx *ctx;
        isl_basic_map_list *list;

        if (!map)
                return NULL;
        ctx = isl_map_get_ctx(map);
        list = isl_basic_map_list_alloc(ctx, map->n);

        for (i = 0; i < map->n; ++i) {
                isl_basic_map *bmap;

                bmap = isl_basic_map_copy(map->p[i]);
                list = isl_basic_map_list_add(list, bmap);
        }

        return list;
}

/* Return the intersection of the elements in the non-empty list "list".
 * All elements are assumed to live in the same space.
 */
__isl_give isl_basic_map *isl_basic_map_list_intersect(
        __isl_take isl_basic_map_list *list)
{
        int i, n;
        isl_basic_map *bmap;

        if (!list)
                return NULL;
        n = isl_basic_map_list_n_basic_map(list);
        if (n < 1)
                isl_die(isl_basic_map_list_get_ctx(list), isl_error_invalid,
                        "expecting non-empty list", goto error);

        bmap = isl_basic_map_list_get_basic_map(list, 0);
        for (i = 1; i < n; ++i) {
                isl_basic_map *bmap_i;

                bmap_i = isl_basic_map_list_get_basic_map(list, i);
                bmap = isl_basic_map_intersect(bmap, bmap_i);
        }

        isl_basic_map_list_free(list);
        return bmap;
error:
        isl_basic_map_list_free(list);
        return NULL;
}

/* Return the intersection of the elements in the non-empty list "list".
 * All elements are assumed to live in the same space.
 */
__isl_give isl_basic_set *isl_basic_set_list_intersect(
        __isl_take isl_basic_set_list *list)
{
        return isl_basic_map_list_intersect(list);
}

/* Return the union of the elements of "list".
 * The list is required to have at least one element.
 */
__isl_give isl_set *isl_basic_set_list_union(
        __isl_take isl_basic_set_list *list)
{
        int i, n;
        isl_space *space;
        isl_basic_set *bset;
        isl_set *set;

        if (!list)
                return NULL;
        n = isl_basic_set_list_n_basic_set(list);
        if (n < 1)
                isl_die(isl_basic_set_list_get_ctx(list), isl_error_invalid,
                        "expecting non-empty list", goto error);

        bset = isl_basic_set_list_get_basic_set(list, 0);
        space = isl_basic_set_get_space(bset);
        isl_basic_set_free(bset);

        set = isl_set_alloc_space(space, n, 0);
        for (i = 0; i < n; ++i) {
                bset = isl_basic_set_list_get_basic_set(list, i);
                set = isl_set_add_basic_set(set, bset);
        }

        isl_basic_set_list_free(list);
        return set;
error:
        isl_basic_set_list_free(list);
        return NULL;
}

/* Return the union of the elements in the non-empty list "list".
 * All elements are assumed to live in the same space.
 */
__isl_give isl_set *isl_set_list_union(__isl_take isl_set_list *list)
{
        int i, n;
        isl_set *set;

        if (!list)
                return NULL;
        n = isl_set_list_n_set(list);
        if (n < 1)
                isl_die(isl_set_list_get_ctx(list), isl_error_invalid,
                        "expecting non-empty list", goto error);

        set = isl_set_list_get_set(list, 0);
        for (i = 1; i < n; ++i) {
                isl_set *set_i;

                set_i = isl_set_list_get_set(list, i);
                set = isl_set_union(set, set_i);
        }

        isl_set_list_free(list);
        return set;
error:
        isl_set_list_free(list);
        return NULL;
}

/* Return the Cartesian product of the basic sets in list (in the given order).
 */
__isl_give isl_basic_set *isl_basic_set_list_product(
        __isl_take struct isl_basic_set_list *list)
{
        int i;
        unsigned dim;
        unsigned nparam;
        unsigned extra;
        unsigned n_eq;
        unsigned n_ineq;
        struct isl_basic_set *product = NULL;

        if (!list)
                goto error;
        isl_assert(list->ctx, list->n > 0, goto error);
        isl_assert(list->ctx, list->p[0], goto error);
        nparam = isl_basic_set_n_param(list->p[0]);
        dim = isl_basic_set_n_dim(list->p[0]);
        extra = list->p[0]->n_div;
        n_eq = list->p[0]->n_eq;
        n_ineq = list->p[0]->n_ineq;
        for (i = 1; i < list->n; ++i) {
                isl_assert(list->ctx, list->p[i], goto error);
                isl_assert(list->ctx,
                    nparam == isl_basic_set_n_param(list->p[i]), goto error);
                dim += isl_basic_set_n_dim(list->p[i]);
                extra += list->p[i]->n_div;
                n_eq += list->p[i]->n_eq;
                n_ineq += list->p[i]->n_ineq;
        }
        product = isl_basic_set_alloc(list->ctx, nparam, dim, extra,
                                        n_eq, n_ineq);
        if (!product)
                goto error;
        dim = 0;
        for (i = 0; i < list->n; ++i) {
                isl_basic_set_add_constraints(product,
                                        isl_basic_set_copy(list->p[i]), dim);
                dim += isl_basic_set_n_dim(list->p[i]);
        }
        isl_basic_set_list_free(list);
        return product;
error:
        isl_basic_set_free(product);
        isl_basic_set_list_free(list);
        return NULL;
}

struct isl_basic_map *isl_basic_map_product(
                struct isl_basic_map *bmap1, struct isl_basic_map *bmap2)
{
        isl_space *dim_result = NULL;
        struct isl_basic_map *bmap;
        unsigned in1, in2, out1, out2, nparam, total, pos;
        struct isl_dim_map *dim_map1, *dim_map2;

        if (!bmap1 || !bmap2)
                goto error;

        isl_assert(bmap1->ctx, isl_space_match(bmap1->dim, isl_dim_param,
                                     bmap2->dim, isl_dim_param), goto error);
        dim_result = isl_space_product(isl_space_copy(bmap1->dim),
                                                   isl_space_copy(bmap2->dim));

        in1 = isl_basic_map_n_in(bmap1);
        in2 = isl_basic_map_n_in(bmap2);
        out1 = isl_basic_map_n_out(bmap1);
        out2 = isl_basic_map_n_out(bmap2);
        nparam = isl_basic_map_n_param(bmap1);

        total = nparam + in1 + in2 + out1 + out2 + bmap1->n_div + bmap2->n_div;
        dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
        dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos += in1);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += in2);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += out1);
        isl_dim_map_div(dim_map1, bmap1, pos += out2);
        isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);

        bmap = isl_basic_map_alloc_space(dim_result,
                        bmap1->n_div + bmap2->n_div,
                        bmap1->n_eq + bmap2->n_eq,
                        bmap1->n_ineq + bmap2->n_ineq);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_flat_product(
        __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
{
        isl_basic_map *prod;

        prod = isl_basic_map_product(bmap1, bmap2);
        prod = isl_basic_map_flatten(prod);
        return prod;
}

__isl_give isl_basic_set *isl_basic_set_flat_product(
        __isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2)
{
        return isl_basic_map_flat_range_product(bset1, bset2);
}

__isl_give isl_basic_map *isl_basic_map_domain_product(
        __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
{
        isl_space *space_result = NULL;
        isl_basic_map *bmap;
        unsigned in1, in2, out, nparam, total, pos;
        struct isl_dim_map *dim_map1, *dim_map2;

        if (!bmap1 || !bmap2)
                goto error;

        space_result = isl_space_domain_product(isl_space_copy(bmap1->dim),
                                                isl_space_copy(bmap2->dim));

        in1 = isl_basic_map_dim(bmap1, isl_dim_in);
        in2 = isl_basic_map_dim(bmap2, isl_dim_in);
        out = isl_basic_map_dim(bmap1, isl_dim_out);
        nparam = isl_basic_map_dim(bmap1, isl_dim_param);

        total = nparam + in1 + in2 + out + bmap1->n_div + bmap2->n_div;
        dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
        dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos += in1);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += in2);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos);
        isl_dim_map_div(dim_map1, bmap1, pos += out);
        isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);

        bmap = isl_basic_map_alloc_space(space_result,
                        bmap1->n_div + bmap2->n_div,
                        bmap1->n_eq + bmap2->n_eq,
                        bmap1->n_ineq + bmap2->n_ineq);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_range_product(
        __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
{
        int rational;
        isl_space *dim_result = NULL;
        isl_basic_map *bmap;
        unsigned in, out1, out2, nparam, total, pos;
        struct isl_dim_map *dim_map1, *dim_map2;

        rational = isl_basic_map_is_rational(bmap1);
        if (rational >= 0 && rational)
                rational = isl_basic_map_is_rational(bmap2);
        if (!bmap1 || !bmap2 || rational < 0)
                goto error;

        if (!isl_space_match(bmap1->dim, isl_dim_param,
                            bmap2->dim, isl_dim_param))
                isl_die(isl_basic_map_get_ctx(bmap1), isl_error_invalid,
                        "parameters don't match", goto error);

        dim_result = isl_space_range_product(isl_space_copy(bmap1->dim),
                                           isl_space_copy(bmap2->dim));

        in = isl_basic_map_dim(bmap1, isl_dim_in);
        out1 = isl_basic_map_n_out(bmap1);
        out2 = isl_basic_map_n_out(bmap2);
        nparam = isl_basic_map_n_param(bmap1);

        total = nparam + in + out1 + out2 + bmap1->n_div + bmap2->n_div;
        dim_map1 = isl_dim_map_alloc(bmap1->ctx, total);
        dim_map2 = isl_dim_map_alloc(bmap1->ctx, total);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_param, pos = 0);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_in, pos += nparam);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_in, pos);
        isl_dim_map_dim(dim_map1, bmap1->dim, isl_dim_out, pos += in);
        isl_dim_map_dim(dim_map2, bmap2->dim, isl_dim_out, pos += out1);
        isl_dim_map_div(dim_map1, bmap1, pos += out2);
        isl_dim_map_div(dim_map2, bmap2, pos += bmap1->n_div);

        bmap = isl_basic_map_alloc_space(dim_result,
                        bmap1->n_div + bmap2->n_div,
                        bmap1->n_eq + bmap2->n_eq,
                        bmap1->n_ineq + bmap2->n_ineq);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap1, dim_map1);
        bmap = isl_basic_map_add_constraints_dim_map(bmap, bmap2, dim_map2);
        if (rational)
                bmap = isl_basic_map_set_rational(bmap);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_map_free(bmap1);
        isl_basic_map_free(bmap2);
        return NULL;
}

__isl_give isl_basic_map *isl_basic_map_flat_range_product(
        __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2)
{
        isl_basic_map *prod;

        prod = isl_basic_map_range_product(bmap1, bmap2);
        prod = isl_basic_map_flatten_range(prod);
        return prod;
}

/* Apply "basic_map_product" to each pair of basic maps in "map1" and "map2"
 * and collect the results.
 * The result live in the space obtained by calling "space_product"
 * on the spaces of "map1" and "map2".
 * If "remove_duplicates" is set then the result may contain duplicates
 * (even if the inputs do not) and so we try and remove the obvious
 * duplicates.
 */
static __isl_give isl_map *map_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2,
        __isl_give isl_space *(*space_product)(__isl_take isl_space *left,
                                           __isl_take isl_space *right),
        __isl_give isl_basic_map *(*basic_map_product)(
                __isl_take isl_basic_map *left,
                __isl_take isl_basic_map *right),
        int remove_duplicates)
{
        unsigned flags = 0;
        struct isl_map *result;
        int i, j;

        if (!map1 || !map2)
                goto error;

        isl_assert(map1->ctx, isl_space_match(map1->dim, isl_dim_param,
                                         map2->dim, isl_dim_param), goto error);

        if (ISL_F_ISSET(map1, ISL_MAP_DISJOINT) &&
            ISL_F_ISSET(map2, ISL_MAP_DISJOINT))
                ISL_FL_SET(flags, ISL_MAP_DISJOINT);

        result = isl_map_alloc_space(space_product(isl_space_copy(map1->dim),
                                               isl_space_copy(map2->dim)),
                                map1->n * map2->n, flags);
        if (!result)
                goto error;
        for (i = 0; i < map1->n; ++i)
                for (j = 0; j < map2->n; ++j) {
                        struct isl_basic_map *part;
                        part = basic_map_product(isl_basic_map_copy(map1->p[i]),
                                                 isl_basic_map_copy(map2->p[j]));
                        if (isl_basic_map_is_empty(part))
                                isl_basic_map_free(part);
                        else
                                result = isl_map_add_basic_map(result, part);
                        if (!result)
                                goto error;
                }
        if (remove_duplicates)
                result = isl_map_remove_obvious_duplicates(result);
        isl_map_free(map1);
        isl_map_free(map2);
        return result;
error:
        isl_map_free(map1);
        isl_map_free(map2);
        return NULL;
}

/* Given two maps A -> B and C -> D, construct a map [A -> C] -> [B -> D]
 */
static __isl_give isl_map *map_product_aligned(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return map_product(map1, map2, &isl_space_product,
                        &isl_basic_map_product, 0);
}

__isl_give isl_map *isl_map_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2, &map_product_aligned);
}

/* Given two maps A -> B and C -> D, construct a map (A, C) -> (B, D)
 */
__isl_give isl_map *isl_map_flat_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *prod;

        prod = isl_map_product(map1, map2);
        prod = isl_map_flatten(prod);
        return prod;
}

/* Given two set A and B, construct its Cartesian product A x B.
 */
struct isl_set *isl_set_product(struct isl_set *set1, struct isl_set *set2)
{
        return isl_map_range_product(set1, set2);
}

__isl_give isl_set *isl_set_flat_product(__isl_take isl_set *set1,
        __isl_take isl_set *set2)
{
        return isl_map_flat_range_product(set1, set2);
}

/* Given two maps A -> B and C -> D, construct a map [A -> C] -> (B * D)
 */
static __isl_give isl_map *map_domain_product_aligned(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return map_product(map1, map2, &isl_space_domain_product,
                                &isl_basic_map_domain_product, 1);
}

/* Given two maps A -> B and C -> D, construct a map (A * C) -> [B -> D]
 */
static __isl_give isl_map *map_range_product_aligned(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return map_product(map1, map2, &isl_space_range_product,
                                &isl_basic_map_range_product, 1);
}

__isl_give isl_map *isl_map_domain_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2,
                                                &map_domain_product_aligned);
}

__isl_give isl_map *isl_map_range_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        return isl_map_align_params_map_map_and(map1, map2,
                                                &map_range_product_aligned);
}

/* Given a map of the form [A -> B] -> [C -> D], return the map A -> C.
 */
__isl_give isl_map *isl_map_factor_domain(__isl_take isl_map *map)
{
        isl_space *space;
        int total1, keep1, total2, keep2;

        if (!map)
                return NULL;
        if (!isl_space_domain_is_wrapping(map->dim) ||
            !isl_space_range_is_wrapping(map->dim))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "not a product", return isl_map_free(map));

        space = isl_map_get_space(map);
        total1 = isl_space_dim(space, isl_dim_in);
        total2 = isl_space_dim(space, isl_dim_out);
        space = isl_space_factor_domain(space);
        keep1 = isl_space_dim(space, isl_dim_in);
        keep2 = isl_space_dim(space, isl_dim_out);
        map = isl_map_project_out(map, isl_dim_in, keep1, total1 - keep1);
        map = isl_map_project_out(map, isl_dim_out, keep2, total2 - keep2);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Given a map of the form [A -> B] -> [C -> D], return the map B -> D.
 */
__isl_give isl_map *isl_map_factor_range(__isl_take isl_map *map)
{
        isl_space *space;
        int total1, keep1, total2, keep2;

        if (!map)
                return NULL;
        if (!isl_space_domain_is_wrapping(map->dim) ||
            !isl_space_range_is_wrapping(map->dim))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "not a product", return isl_map_free(map));

        space = isl_map_get_space(map);
        total1 = isl_space_dim(space, isl_dim_in);
        total2 = isl_space_dim(space, isl_dim_out);
        space = isl_space_factor_range(space);
        keep1 = isl_space_dim(space, isl_dim_in);
        keep2 = isl_space_dim(space, isl_dim_out);
        map = isl_map_project_out(map, isl_dim_in, 0, total1 - keep1);
        map = isl_map_project_out(map, isl_dim_out, 0, total2 - keep2);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Given a map of the form [A -> B] -> C, return the map A -> C.
 */
__isl_give isl_map *isl_map_domain_factor_domain(__isl_take isl_map *map)
{
        isl_space *space;
        int total, keep;

        if (!map)
                return NULL;
        if (!isl_space_domain_is_wrapping(map->dim))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "domain is not a product", return isl_map_free(map));

        space = isl_map_get_space(map);
        total = isl_space_dim(space, isl_dim_in);
        space = isl_space_domain_factor_domain(space);
        keep = isl_space_dim(space, isl_dim_in);
        map = isl_map_project_out(map, isl_dim_in, keep, total - keep);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Given a map of the form [A -> B] -> C, return the map B -> C.
 */
__isl_give isl_map *isl_map_domain_factor_range(__isl_take isl_map *map)
{
        isl_space *space;
        int total, keep;

        if (!map)
                return NULL;
        if (!isl_space_domain_is_wrapping(map->dim))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "domain is not a product", return isl_map_free(map));

        space = isl_map_get_space(map);
        total = isl_space_dim(space, isl_dim_in);
        space = isl_space_domain_factor_range(space);
        keep = isl_space_dim(space, isl_dim_in);
        map = isl_map_project_out(map, isl_dim_in, 0, total - keep);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Given a map A -> [B -> C], extract the map A -> B.
 */
__isl_give isl_map *isl_map_range_factor_domain(__isl_take isl_map *map)
{
        isl_space *space;
        int total, keep;

        if (!map)
                return NULL;
        if (!isl_space_range_is_wrapping(map->dim))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "range is not a product", return isl_map_free(map));

        space = isl_map_get_space(map);
        total = isl_space_dim(space, isl_dim_out);
        space = isl_space_range_factor_domain(space);
        keep = isl_space_dim(space, isl_dim_out);
        map = isl_map_project_out(map, isl_dim_out, keep, total - keep);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Given a map A -> [B -> C], extract the map A -> C.
 */
__isl_give isl_map *isl_map_range_factor_range(__isl_take isl_map *map)
{
        isl_space *space;
        int total, keep;

        if (!map)
                return NULL;
        if (!isl_space_range_is_wrapping(map->dim))
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "range is not a product", return isl_map_free(map));

        space = isl_map_get_space(map);
        total = isl_space_dim(space, isl_dim_out);
        space = isl_space_range_factor_range(space);
        keep = isl_space_dim(space, isl_dim_out);
        map = isl_map_project_out(map, isl_dim_out, 0, total - keep);
        map = isl_map_reset_space(map, space);

        return map;
}

/* Given two maps A -> B and C -> D, construct a map (A, C) -> (B * D)
 */
__isl_give isl_map *isl_map_flat_domain_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *prod;

        prod = isl_map_domain_product(map1, map2);
        prod = isl_map_flatten_domain(prod);
        return prod;
}

/* Given two maps A -> B and C -> D, construct a map (A * C) -> (B, D)
 */
__isl_give isl_map *isl_map_flat_range_product(__isl_take isl_map *map1,
        __isl_take isl_map *map2)
{
        isl_map *prod;

        prod = isl_map_range_product(map1, map2);
        prod = isl_map_flatten_range(prod);
        return prod;
}

uint32_t isl_basic_map_get_hash(__isl_keep isl_basic_map *bmap)
{
        int i;
        uint32_t hash = isl_hash_init();
        unsigned total;

        if (!bmap)
                return 0;
        bmap = isl_basic_map_copy(bmap);
        bmap = isl_basic_map_normalize(bmap);
        if (!bmap)
                return 0;
        total = isl_basic_map_total_dim(bmap);
        isl_hash_byte(hash, bmap->n_eq & 0xFF);
        for (i = 0; i < bmap->n_eq; ++i) {
                uint32_t c_hash;
                c_hash = isl_seq_get_hash(bmap->eq[i], 1 + total);
                isl_hash_hash(hash, c_hash);
        }
        isl_hash_byte(hash, bmap->n_ineq & 0xFF);
        for (i = 0; i < bmap->n_ineq; ++i) {
                uint32_t c_hash;
                c_hash = isl_seq_get_hash(bmap->ineq[i], 1 + total);
                isl_hash_hash(hash, c_hash);
        }
        isl_hash_byte(hash, bmap->n_div & 0xFF);
        for (i = 0; i < bmap->n_div; ++i) {
                uint32_t c_hash;
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                isl_hash_byte(hash, i & 0xFF);
                c_hash = isl_seq_get_hash(bmap->div[i], 1 + 1 + total);
                isl_hash_hash(hash, c_hash);
        }
        isl_basic_map_free(bmap);
        return hash;
}

uint32_t isl_basic_set_get_hash(__isl_keep isl_basic_set *bset)
{
        return isl_basic_map_get_hash(bset_to_bmap(bset));
}

uint32_t isl_map_get_hash(__isl_keep isl_map *map)
{
        int i;
        uint32_t hash;

        if (!map)
                return 0;
        map = isl_map_copy(map);
        map = isl_map_normalize(map);
        if (!map)
                return 0;

        hash = isl_hash_init();
        for (i = 0; i < map->n; ++i) {
                uint32_t bmap_hash;
                bmap_hash = isl_basic_map_get_hash(map->p[i]);
                isl_hash_hash(hash, bmap_hash);
        }
                
        isl_map_free(map);

        return hash;
}

uint32_t isl_set_get_hash(__isl_keep isl_set *set)
{
        return isl_map_get_hash(set_to_map(set));
}

/* Check if the value for dimension dim is completely determined
 * by the values of the other parameters and variables.
 * That is, check if dimension dim is involved in an equality.
 */
int isl_basic_set_dim_is_unique(struct isl_basic_set *bset, unsigned dim)
{
        int i;
        unsigned nparam;

        if (!bset)
                return -1;
        nparam = isl_basic_set_n_param(bset);
        for (i = 0; i < bset->n_eq; ++i)
                if (!isl_int_is_zero(bset->eq[i][1 + nparam + dim]))
                        return 1;
        return 0;
}

/* Check if the value for dimension dim is completely determined
 * by the values of the other parameters and variables.
 * That is, check if dimension dim is involved in an equality
 * for each of the subsets.
 */
int isl_set_dim_is_unique(struct isl_set *set, unsigned dim)
{
        int i;

        if (!set)
                return -1;
        for (i = 0; i < set->n; ++i) {
                int unique;
                unique = isl_basic_set_dim_is_unique(set->p[i], dim);
                if (unique != 1)
                        return unique;
        }
        return 1;
}

/* Return the number of basic maps in the (current) representation of "map".
 */
int isl_map_n_basic_map(__isl_keep isl_map *map)
{
        return map ? map->n : 0;
}

int isl_set_n_basic_set(__isl_keep isl_set *set)
{
        return set ? set->n : 0;
}

isl_stat isl_map_foreach_basic_map(__isl_keep isl_map *map,
        isl_stat (*fn)(__isl_take isl_basic_map *bmap, void *user), void *user)
{
        int i;

        if (!map)
                return isl_stat_error;

        for (i = 0; i < map->n; ++i)
                if (fn(isl_basic_map_copy(map->p[i]), user) < 0)
                        return isl_stat_error;

        return isl_stat_ok;
}

isl_stat isl_set_foreach_basic_set(__isl_keep isl_set *set,
        isl_stat (*fn)(__isl_take isl_basic_set *bset, void *user), void *user)
{
        int i;

        if (!set)
                return isl_stat_error;

        for (i = 0; i < set->n; ++i)
                if (fn(isl_basic_set_copy(set->p[i]), user) < 0)
                        return isl_stat_error;

        return isl_stat_ok;
}

/* Return a list of basic sets, the union of which is equal to "set".
 */
__isl_give isl_basic_set_list *isl_set_get_basic_set_list(
        __isl_keep isl_set *set)
{
        int i;
        isl_basic_set_list *list;

        if (!set)
                return NULL;

        list = isl_basic_set_list_alloc(isl_set_get_ctx(set), set->n);
        for (i = 0; i < set->n; ++i) {
                isl_basic_set *bset;

                bset = isl_basic_set_copy(set->p[i]);
                list = isl_basic_set_list_add(list, bset);
        }

        return list;
}

__isl_give isl_basic_set *isl_basic_set_lift(__isl_take isl_basic_set *bset)
{
        isl_space *dim;

        if (!bset)
                return NULL;

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

        dim = isl_basic_set_get_space(bset);
        dim = isl_space_lift(dim, bset->n_div);
        if (!dim)
                goto error;
        isl_space_free(bset->dim);
        bset->dim = dim;
        bset->extra -= bset->n_div;
        bset->n_div = 0;

        bset = isl_basic_set_finalize(bset);

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

__isl_give isl_set *isl_set_lift(__isl_take isl_set *set)
{
        int i;
        isl_space *dim;
        unsigned n_div;

        set = isl_set_align_divs(set);

        if (!set)
                return NULL;

        set = isl_set_cow(set);
        if (!set)
                return NULL;

        n_div = set->p[0]->n_div;
        dim = isl_set_get_space(set);
        dim = isl_space_lift(dim, n_div);
        if (!dim)
                goto error;
        isl_space_free(set->dim);
        set->dim = dim;

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

        return set;
error:
        isl_set_free(set);
        return NULL;
}

__isl_give isl_map *isl_set_lifting(__isl_take isl_set *set)
{
        isl_space *dim;
        struct isl_basic_map *bmap;
        unsigned n_set;
        unsigned n_div;
        unsigned n_param;
        unsigned total;
        int i, k, l;

        set = isl_set_align_divs(set);

        if (!set)
                return NULL;

        dim = isl_set_get_space(set);
        if (set->n == 0 || set->p[0]->n_div == 0) {
                isl_set_free(set);
                return isl_map_identity(isl_space_map_from_set(dim));
        }

        n_div = set->p[0]->n_div;
        dim = isl_space_map_from_set(dim);
        n_param = isl_space_dim(dim, isl_dim_param);
        n_set = isl_space_dim(dim, isl_dim_in);
        dim = isl_space_extend(dim, n_param, n_set, n_set + n_div);
        bmap = isl_basic_map_alloc_space(dim, 0, n_set, 2 * n_div);
        for (i = 0; i < n_set; ++i)
                bmap = var_equal(bmap, i);

        total = n_param + n_set + n_set + n_div;
        for (i = 0; i < n_div; ++i) {
                k = isl_basic_map_alloc_inequality(bmap);
                if (k < 0)
                        goto error;
                isl_seq_cpy(bmap->ineq[k], set->p[0]->div[i]+1, 1+n_param);
                isl_seq_clr(bmap->ineq[k]+1+n_param, n_set);
                isl_seq_cpy(bmap->ineq[k]+1+n_param+n_set,
                            set->p[0]->div[i]+1+1+n_param, n_set + n_div);
                isl_int_neg(bmap->ineq[k][1+n_param+n_set+n_set+i],
                            set->p[0]->div[i][0]);

                l = isl_basic_map_alloc_inequality(bmap);
                if (l < 0)
                        goto error;
                isl_seq_neg(bmap->ineq[l], bmap->ineq[k], 1 + total);
                isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0],
                            set->p[0]->div[i][0]);
                isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
        }

        isl_set_free(set);
        bmap = isl_basic_map_simplify(bmap);
        bmap = isl_basic_map_finalize(bmap);
        return isl_map_from_basic_map(bmap);
error:
        isl_set_free(set);
        isl_basic_map_free(bmap);
        return NULL;
}

int isl_basic_set_size(__isl_keep isl_basic_set *bset)
{
        unsigned dim;
        int size = 0;

        if (!bset)
                return -1;

        dim = isl_basic_set_total_dim(bset);
        size += bset->n_eq * (1 + dim);
        size += bset->n_ineq * (1 + dim);
        size += bset->n_div * (2 + dim);

        return size;
}

int isl_set_size(__isl_keep isl_set *set)
{
        int i;
        int size = 0;

        if (!set)
                return -1;

        for (i = 0; i < set->n; ++i)
                size += isl_basic_set_size(set->p[i]);

        return size;
}

/* Check if there is any lower bound (if lower == 0) and/or upper
 * bound (if upper == 0) on the specified dim.
 */
static isl_bool basic_map_dim_is_bounded(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos, int lower, int upper)
{
        int i;

        if (!bmap)
                return isl_bool_error;

        isl_assert(bmap->ctx, pos < isl_basic_map_dim(bmap, type),
                return isl_bool_error);

        pos += isl_basic_map_offset(bmap, type);

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

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

        for (i = 0; i < bmap->n_ineq; ++i) {
                int sgn = isl_int_sgn(bmap->ineq[i][pos]);
                if (sgn > 0)
                        lower = 1;
                if (sgn < 0)
                        upper = 1;
        }

        return lower && upper;
}

int isl_basic_map_dim_is_bounded(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos)
{
        return basic_map_dim_is_bounded(bmap, type, pos, 0, 0);
}

isl_bool isl_basic_map_dim_has_lower_bound(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos)
{
        return basic_map_dim_is_bounded(bmap, type, pos, 0, 1);
}

isl_bool isl_basic_map_dim_has_upper_bound(__isl_keep isl_basic_map *bmap,
        enum isl_dim_type type, unsigned pos)
{
        return basic_map_dim_is_bounded(bmap, type, pos, 1, 0);
}

int isl_map_dim_is_bounded(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos)
{
        int i;

        if (!map)
                return -1;

        for (i = 0; i < map->n; ++i) {
                int bounded;
                bounded = isl_basic_map_dim_is_bounded(map->p[i], type, pos);
                if (bounded < 0 || !bounded)
                        return bounded;
        }

        return 1;
}

/* Return 1 if the specified dim is involved in both an upper bound
 * and a lower bound.
 */
int isl_set_dim_is_bounded(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return isl_map_dim_is_bounded(set_to_map(set), type, pos);
}

/* Does "map" have a bound (according to "fn") for any of its basic maps?
 */
static isl_bool has_any_bound(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos,
        isl_bool (*fn)(__isl_keep isl_basic_map *bmap,
                  enum isl_dim_type type, unsigned pos))
{
        int i;

        if (!map)
                return isl_bool_error;

        for (i = 0; i < map->n; ++i) {
                isl_bool bounded;
                bounded = fn(map->p[i], type, pos);
                if (bounded < 0 || bounded)
                        return bounded;
        }

        return isl_bool_false;
}

/* Return 1 if the specified dim is involved in any lower bound.
 */
isl_bool isl_set_dim_has_any_lower_bound(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return has_any_bound(set, type, pos,
                                &isl_basic_map_dim_has_lower_bound);
}

/* Return 1 if the specified dim is involved in any upper bound.
 */
isl_bool isl_set_dim_has_any_upper_bound(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return has_any_bound(set, type, pos,
                                &isl_basic_map_dim_has_upper_bound);
}

/* Does "map" have a bound (according to "fn") for all of its basic maps?
 */
static isl_bool has_bound(__isl_keep isl_map *map,
        enum isl_dim_type type, unsigned pos,
        isl_bool (*fn)(__isl_keep isl_basic_map *bmap,
                  enum isl_dim_type type, unsigned pos))
{
        int i;

        if (!map)
                return isl_bool_error;

        for (i = 0; i < map->n; ++i) {
                isl_bool bounded;
                bounded = fn(map->p[i], type, pos);
                if (bounded < 0 || !bounded)
                        return bounded;
        }

        return isl_bool_true;
}

/* Return 1 if the specified dim has a lower bound (in each of its basic sets).
 */
isl_bool isl_set_dim_has_lower_bound(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return has_bound(set, type, pos, &isl_basic_map_dim_has_lower_bound);
}

/* Return 1 if the specified dim has an upper bound (in each of its basic sets).
 */
isl_bool isl_set_dim_has_upper_bound(__isl_keep isl_set *set,
        enum isl_dim_type type, unsigned pos)
{
        return has_bound(set, type, pos, &isl_basic_map_dim_has_upper_bound);
}

/* For each of the "n" variables starting at "first", determine
 * the sign of the variable and put the results in the first "n"
 * elements of the array "signs".
 * Sign
 *      1 means that the variable is non-negative
 *      -1 means that the variable is non-positive
 *      0 means the variable attains both positive and negative values.
 */
int isl_basic_set_vars_get_sign(__isl_keep isl_basic_set *bset,
        unsigned first, unsigned n, int *signs)
{
        isl_vec *bound = NULL;
        struct isl_tab *tab = NULL;
        struct isl_tab_undo *snap;
        int i;

        if (!bset || !signs)
                return -1;

        bound = isl_vec_alloc(bset->ctx, 1 + isl_basic_set_total_dim(bset));
        tab = isl_tab_from_basic_set(bset, 0);
        if (!bound || !tab)
                goto error;

        isl_seq_clr(bound->el, bound->size);
        isl_int_set_si(bound->el[0], -1);

        snap = isl_tab_snap(tab);
        for (i = 0; i < n; ++i) {
                int empty;

                isl_int_set_si(bound->el[1 + first + i], -1);
                if (isl_tab_add_ineq(tab, bound->el) < 0)
                        goto error;
                empty = tab->empty;
                isl_int_set_si(bound->el[1 + first + i], 0);
                if (isl_tab_rollback(tab, snap) < 0)
                        goto error;

                if (empty) {
                        signs[i] = 1;
                        continue;
                }

                isl_int_set_si(bound->el[1 + first + i], 1);
                if (isl_tab_add_ineq(tab, bound->el) < 0)
                        goto error;
                empty = tab->empty;
                isl_int_set_si(bound->el[1 + first + i], 0);
                if (isl_tab_rollback(tab, snap) < 0)
                        goto error;

                signs[i] = empty ? -1 : 0;
        }

        isl_tab_free(tab);
        isl_vec_free(bound);
        return 0;
error:
        isl_tab_free(tab);
        isl_vec_free(bound);
        return -1;
}

int isl_basic_set_dims_get_sign(__isl_keep isl_basic_set *bset,
        enum isl_dim_type type, unsigned first, unsigned n, int *signs)
{
        if (!bset || !signs)
                return -1;
        isl_assert(bset->ctx, first + n <= isl_basic_set_dim(bset, type),
                return -1);

        first += pos(bset->dim, type) - 1;
        return isl_basic_set_vars_get_sign(bset, first, n, signs);
}

/* Is it possible for the integer division "div" to depend (possibly
 * indirectly) on any output dimensions?
 *
 * If the div is undefined, then we conservatively assume that it
 * may depend on them.
 * Otherwise, we check if it actually depends on them or on any integer
 * divisions that may depend on them.
 */
static int div_may_involve_output(__isl_keep isl_basic_map *bmap, int div)
{
        int i;
        unsigned n_out, o_out;
        unsigned n_div, o_div;

        if (isl_int_is_zero(bmap->div[div][0]))
                return 1;

        n_out = isl_basic_map_dim(bmap, isl_dim_out);
        o_out = isl_basic_map_offset(bmap, isl_dim_out);

        if (isl_seq_first_non_zero(bmap->div[div] + 1 + o_out, n_out) != -1)
                return 1;

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

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

        return 0;
}

/* Return the first integer division of "bmap" in the range
 * [first, first + n[ that may depend on any output dimensions and
 * that has a non-zero coefficient in "c" (where the first coefficient
 * in "c" corresponds to integer division "first").
 */
static int first_div_may_involve_output(__isl_keep isl_basic_map *bmap,
        isl_int *c, int first, int n)
{
        int k;

        if (!bmap)
                return -1;

        for (k = first; k < first + n; ++k) {
                if (isl_int_is_zero(c[k]))
                        continue;
                if (div_may_involve_output(bmap, k))
                        return k;
        }

        return first + n;
}

/* Look for a pair of inequality constraints in "bmap" of the form
 *
 *      -l + i >= 0             or              i >= l
 * and
 *      n + l - i >= 0          or              i <= l + n
 *
 * with n < "m" and i the output dimension at position "pos".
 * (Note that n >= 0 as otherwise the two constraints would conflict.)
 * Furthermore, "l" is only allowed to involve parameters, input dimensions
 * and earlier output dimensions, as well as integer divisions that do
 * not involve any of the output dimensions.
 *
 * Return the index of the first inequality constraint or bmap->n_ineq
 * if no such pair can be found.
 */
static int find_modulo_constraint_pair(__isl_keep isl_basic_map *bmap,
        int pos, isl_int m)
{
        int i, j;
        isl_ctx *ctx;
        unsigned total;
        unsigned n_div, o_div;
        unsigned n_out, o_out;
        int less;

        if (!bmap)
                return -1;

        ctx = isl_basic_map_get_ctx(bmap);
        total = isl_basic_map_total_dim(bmap);
        n_out = isl_basic_map_dim(bmap, isl_dim_out);
        o_out = isl_basic_map_offset(bmap, isl_dim_out);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        o_div = isl_basic_map_offset(bmap, isl_dim_div);
        for (i = 0; i < bmap->n_ineq; ++i) {
                if (!isl_int_abs_eq(bmap->ineq[i][o_out + pos], ctx->one))
                        continue;
                if (isl_seq_first_non_zero(bmap->ineq[i] + o_out + pos + 1,
                                        n_out - (pos + 1)) != -1)
                        continue;
                if (first_div_may_involve_output(bmap, bmap->ineq[i] + o_div,
                                                0, n_div) < n_div)
                        continue;
                for (j = i + 1; j < bmap->n_ineq; ++j) {
                        if (!isl_int_abs_eq(bmap->ineq[j][o_out + pos],
                                            ctx->one))
                                continue;
                        if (!isl_seq_is_neg(bmap->ineq[i] + 1,
                                            bmap->ineq[j] + 1, total))
                                continue;
                        break;
                }
                if (j >= bmap->n_ineq)
                        continue;
                isl_int_add(bmap->ineq[i][0],
                            bmap->ineq[i][0], bmap->ineq[j][0]);
                less = isl_int_abs_lt(bmap->ineq[i][0], m);
                isl_int_sub(bmap->ineq[i][0],
                            bmap->ineq[i][0], bmap->ineq[j][0]);
                if (!less)
                        continue;
                if (isl_int_is_one(bmap->ineq[i][o_out + pos]))
                        return i;
                else
                        return j;
        }

        return bmap->n_ineq;
}

/* Return the index of the equality of "bmap" that defines
 * the output dimension "pos" in terms of earlier dimensions.
 * The equality may also involve integer divisions, as long
 * as those integer divisions are defined in terms of
 * parameters or input dimensions.
 * In this case, *div is set to the number of integer divisions and
 * *ineq is set to the number of inequality constraints (provided
 * div and ineq are not NULL).
 *
 * The equality may also involve a single integer division involving
 * the output dimensions (typically only output dimension "pos") as
 * long as the coefficient of output dimension "pos" is 1 or -1 and
 * there is a pair of constraints i >= l and i <= l + n, with i referring
 * to output dimension "pos", l an expression involving only earlier
 * dimensions and n smaller than the coefficient of the integer division
 * in the equality.  In this case, the output dimension can be defined
 * in terms of a modulo expression that does not involve the integer division.
 * *div is then set to this single integer division and
 * *ineq is set to the index of constraint i >= l.
 *
 * Return bmap->n_eq if there is no such equality.
 * Return -1 on error.
 */
int isl_basic_map_output_defining_equality(__isl_keep isl_basic_map *bmap,
        int pos, int *div, int *ineq)
{
        int j, k, l;
        unsigned n_out, o_out;
        unsigned n_div, o_div;

        if (!bmap)
                return -1;

        n_out = isl_basic_map_dim(bmap, isl_dim_out);
        o_out = isl_basic_map_offset(bmap, isl_dim_out);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        o_div = isl_basic_map_offset(bmap, isl_dim_div);

        if (ineq)
                *ineq = bmap->n_ineq;
        if (div)
                *div = n_div;
        for (j = 0; j < bmap->n_eq; ++j) {
                if (isl_int_is_zero(bmap->eq[j][o_out + pos]))
                        continue;
                if (isl_seq_first_non_zero(bmap->eq[j] + o_out + pos + 1,
                                        n_out - (pos + 1)) != -1)
                        continue;
                k = first_div_may_involve_output(bmap, bmap->eq[j] + o_div,
                                                0, n_div);
                if (k >= n_div)
                        return j;
                if (!isl_int_is_one(bmap->eq[j][o_out + pos]) &&
                    !isl_int_is_negone(bmap->eq[j][o_out + pos]))
                        continue;
                if (first_div_may_involve_output(bmap, bmap->eq[j] + o_div,
                                                k + 1, n_div - (k+1)) < n_div)
                        continue;
                l = find_modulo_constraint_pair(bmap, pos,
                                                bmap->eq[j][o_div + k]);
                if (l < 0)
                        return -1;
                if (l >= bmap->n_ineq)
                        continue;
                if (div)
                        *div = k;
                if (ineq)
                        *ineq = l;
                return j;
        }

        return bmap->n_eq;
}

/* Check if the given basic map is obviously single-valued.
 * In particular, for each output dimension, check that there is
 * an equality that defines the output dimension in terms of
 * earlier dimensions.
 */
isl_bool isl_basic_map_plain_is_single_valued(__isl_keep isl_basic_map *bmap)
{
        int i;
        unsigned n_out;

        if (!bmap)
                return isl_bool_error;

        n_out = isl_basic_map_dim(bmap, isl_dim_out);

        for (i = 0; i < n_out; ++i) {
                int eq;

                eq = isl_basic_map_output_defining_equality(bmap, i,
                                                            NULL, NULL);
                if (eq < 0)
                        return isl_bool_error;
                if (eq >= bmap->n_eq)
                        return isl_bool_false;
        }

        return isl_bool_true;
}

/* Check if the given basic map is single-valued.
 * We simply compute
 *
 *      M \circ M^-1
 *
 * and check if the result is a subset of the identity mapping.
 */
isl_bool isl_basic_map_is_single_valued(__isl_keep isl_basic_map *bmap)
{
        isl_space *space;
        isl_basic_map *test;
        isl_basic_map *id;
        isl_bool sv;

        sv = isl_basic_map_plain_is_single_valued(bmap);
        if (sv < 0 || sv)
                return sv;

        test = isl_basic_map_reverse(isl_basic_map_copy(bmap));
        test = isl_basic_map_apply_range(test, isl_basic_map_copy(bmap));

        space = isl_basic_map_get_space(bmap);
        space = isl_space_map_from_set(isl_space_range(space));
        id = isl_basic_map_identity(space);

        sv = isl_basic_map_is_subset(test, id);

        isl_basic_map_free(test);
        isl_basic_map_free(id);

        return sv;
}

/* Check if the given map is obviously single-valued.
 */
isl_bool isl_map_plain_is_single_valued(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;
        if (map->n == 0)
                return isl_bool_true;
        if (map->n >= 2)
                return isl_bool_false;

        return isl_basic_map_plain_is_single_valued(map->p[0]);
}

/* Check if the given map is single-valued.
 * We simply compute
 *
 *      M \circ M^-1
 *
 * and check if the result is a subset of the identity mapping.
 */
isl_bool isl_map_is_single_valued(__isl_keep isl_map *map)
{
        isl_space *dim;
        isl_map *test;
        isl_map *id;
        isl_bool sv;

        sv = isl_map_plain_is_single_valued(map);
        if (sv < 0 || sv)
                return sv;

        test = isl_map_reverse(isl_map_copy(map));
        test = isl_map_apply_range(test, isl_map_copy(map));

        dim = isl_space_map_from_set(isl_space_range(isl_map_get_space(map)));
        id = isl_map_identity(dim);

        sv = isl_map_is_subset(test, id);

        isl_map_free(test);
        isl_map_free(id);

        return sv;
}

isl_bool isl_map_is_injective(__isl_keep isl_map *map)
{
        isl_bool in;

        map = isl_map_copy(map);
        map = isl_map_reverse(map);
        in = isl_map_is_single_valued(map);
        isl_map_free(map);

        return in;
}

/* Check if the given map is obviously injective.
 */
isl_bool isl_map_plain_is_injective(__isl_keep isl_map *map)
{
        isl_bool in;

        map = isl_map_copy(map);
        map = isl_map_reverse(map);
        in = isl_map_plain_is_single_valued(map);
        isl_map_free(map);

        return in;
}

isl_bool isl_map_is_bijective(__isl_keep isl_map *map)
{
        isl_bool sv;

        sv = isl_map_is_single_valued(map);
        if (sv < 0 || !sv)
                return sv;

        return isl_map_is_injective(map);
}

isl_bool isl_set_is_singleton(__isl_keep isl_set *set)
{
        return isl_map_is_single_valued(set_to_map(set));
}

/* Does "map" only map elements to themselves?
 *
 * If the domain and range spaces are different, then "map"
 * is considered not to be an identity relation, even if it is empty.
 * Otherwise, construct the maximal identity relation and
 * check whether "map" is a subset of this relation.
 */
isl_bool isl_map_is_identity(__isl_keep isl_map *map)
{
        isl_space *space;
        isl_map *id;
        isl_bool equal, is_identity;

        space = isl_map_get_space(map);
        equal = isl_space_tuple_is_equal(space, isl_dim_in, space, isl_dim_out);
        isl_space_free(space);
        if (equal < 0 || !equal)
                return equal;

        id = isl_map_identity(isl_map_get_space(map));
        is_identity = isl_map_is_subset(map, id);
        isl_map_free(id);

        return is_identity;
}

int isl_map_is_translation(__isl_keep isl_map *map)
{
        int ok;
        isl_set *delta;

        delta = isl_map_deltas(isl_map_copy(map));
        ok = isl_set_is_singleton(delta);
        isl_set_free(delta);

        return ok;
}

static int unique(isl_int *p, unsigned pos, unsigned len)
{
        if (isl_seq_first_non_zero(p, pos) != -1)
                return 0;
        if (isl_seq_first_non_zero(p + pos + 1, len - pos - 1) != -1)
                return 0;
        return 1;
}

int isl_basic_set_is_box(__isl_keep isl_basic_set *bset)
{
        int i, j;
        unsigned nvar;
        unsigned ovar;

        if (!bset)
                return -1;

        if (isl_basic_set_dim(bset, isl_dim_div) != 0)
                return 0;

        nvar = isl_basic_set_dim(bset, isl_dim_set);
        ovar = isl_space_offset(bset->dim, isl_dim_set);
        for (j = 0; j < nvar; ++j) {
                int lower = 0, upper = 0;
                for (i = 0; i < bset->n_eq; ++i) {
                        if (isl_int_is_zero(bset->eq[i][1 + ovar + j]))
                                continue;
                        if (!unique(bset->eq[i] + 1 + ovar, j, nvar))
                                return 0;
                        break;
                }
                if (i < bset->n_eq)
                        continue;
                for (i = 0; i < bset->n_ineq; ++i) {
                        if (isl_int_is_zero(bset->ineq[i][1 + ovar + j]))
                                continue;
                        if (!unique(bset->ineq[i] + 1 + ovar, j, nvar))
                                return 0;
                        if (isl_int_is_pos(bset->ineq[i][1 + ovar + j]))
                                lower = 1;
                        else
                                upper = 1;
                }
                if (!lower || !upper)
                        return 0;
        }

        return 1;
}

int isl_set_is_box(__isl_keep isl_set *set)
{
        if (!set)
                return -1;
        if (set->n != 1)
                return 0;

        return isl_basic_set_is_box(set->p[0]);
}

isl_bool isl_basic_set_is_wrapping(__isl_keep isl_basic_set *bset)
{
        if (!bset)
                return isl_bool_error;
        
        return isl_space_is_wrapping(bset->dim);
}

isl_bool isl_set_is_wrapping(__isl_keep isl_set *set)
{
        if (!set)
                return isl_bool_error;
        
        return isl_space_is_wrapping(set->dim);
}

/* Modify the space of "map" through a call to "change".
 * If "can_change" is set (not NULL), then first call it to check
 * if the modification is allowed, printing the error message "cannot_change"
 * if it is not.
 */
static __isl_give isl_map *isl_map_change_space(__isl_take isl_map *map,
        isl_bool (*can_change)(__isl_keep isl_map *map),
        const char *cannot_change,
        __isl_give isl_space *(*change)(__isl_take isl_space *space))
{
        isl_bool ok;
        isl_space *space;

        if (!map)
                return NULL;

        ok = can_change ? can_change(map) : isl_bool_true;
        if (ok < 0)
                return isl_map_free(map);
        if (!ok)
                isl_die(isl_map_get_ctx(map), isl_error_invalid, cannot_change,
                        return isl_map_free(map));

        space = change(isl_map_get_space(map));
        map = isl_map_reset_space(map, space);

        return map;
}

/* Is the domain of "map" a wrapped relation?
 */
isl_bool isl_map_domain_is_wrapping(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;

        return isl_space_domain_is_wrapping(map->dim);
}

/* Is the range of "map" a wrapped relation?
 */
isl_bool isl_map_range_is_wrapping(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;

        return isl_space_range_is_wrapping(map->dim);
}

__isl_give isl_basic_set *isl_basic_map_wrap(__isl_take isl_basic_map *bmap)
{
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;

        bmap->dim = isl_space_wrap(bmap->dim);
        if (!bmap->dim)
                goto error;

        bmap = isl_basic_map_finalize(bmap);

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

/* Given a map A -> B, return the set (A -> B).
 */
__isl_give isl_set *isl_map_wrap(__isl_take isl_map *map)
{
        return isl_map_change_space(map, NULL, NULL, &isl_space_wrap);
}

__isl_give isl_basic_map *isl_basic_set_unwrap(__isl_take isl_basic_set *bset)
{
        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;

        bset->dim = isl_space_unwrap(bset->dim);
        if (!bset->dim)
                goto error;

        bset = isl_basic_set_finalize(bset);

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

/* Given a set (A -> B), return the map A -> B.
 * Error out if "set" is not of the form (A -> B).
 */
__isl_give isl_map *isl_set_unwrap(__isl_take isl_set *set)
{
        return isl_map_change_space(set, &isl_set_is_wrapping,
                                    "not a wrapping set", &isl_space_unwrap);
}

__isl_give isl_basic_map *isl_basic_map_reset(__isl_take isl_basic_map *bmap,
        enum isl_dim_type type)
{
        if (!bmap)
                return NULL;

        if (!isl_space_is_named_or_nested(bmap->dim, type))
                return bmap;

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

        bmap->dim = isl_space_reset(bmap->dim, type);
        if (!bmap->dim)
                goto error;

        bmap = isl_basic_map_finalize(bmap);

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

__isl_give isl_map *isl_map_reset(__isl_take isl_map *map,
        enum isl_dim_type type)
{
        int i;

        if (!map)
                return NULL;

        if (!isl_space_is_named_or_nested(map->dim, type))
                return map;

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

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_reset(map->p[i], type);
                if (!map->p[i])
                        goto error;
        }
        map->dim = isl_space_reset(map->dim, type);
        if (!map->dim)
                goto error;

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

__isl_give isl_basic_map *isl_basic_map_flatten(__isl_take isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;

        if (!bmap->dim->nested[0] && !bmap->dim->nested[1])
                return bmap;

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

        bmap->dim = isl_space_flatten(bmap->dim);
        if (!bmap->dim)
                goto error;

        bmap = isl_basic_map_finalize(bmap);

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

__isl_give isl_basic_set *isl_basic_set_flatten(__isl_take isl_basic_set *bset)
{
        return bset_from_bmap(isl_basic_map_flatten(bset_to_bmap(bset)));
}

__isl_give isl_basic_map *isl_basic_map_flatten_domain(
        __isl_take isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;

        if (!bmap->dim->nested[0])
                return bmap;

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

        bmap->dim = isl_space_flatten_domain(bmap->dim);
        if (!bmap->dim)
                goto error;

        bmap = isl_basic_map_finalize(bmap);

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

__isl_give isl_basic_map *isl_basic_map_flatten_range(
        __isl_take isl_basic_map *bmap)
{
        if (!bmap)
                return NULL;

        if (!bmap->dim->nested[1])
                return bmap;

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

        bmap->dim = isl_space_flatten_range(bmap->dim);
        if (!bmap->dim)
                goto error;

        bmap = isl_basic_map_finalize(bmap);

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

/* Remove any internal structure from the spaces of domain and range of "map".
 */
__isl_give isl_map *isl_map_flatten(__isl_take isl_map *map)
{
        if (!map)
                return NULL;

        if (!map->dim->nested[0] && !map->dim->nested[1])
                return map;

        return isl_map_change_space(map, NULL, NULL, &isl_space_flatten);
}

__isl_give isl_set *isl_set_flatten(__isl_take isl_set *set)
{
        return set_from_map(isl_map_flatten(set_to_map(set)));
}

__isl_give isl_map *isl_set_flatten_map(__isl_take isl_set *set)
{
        isl_space *dim, *flat_dim;
        isl_map *map;

        dim = isl_set_get_space(set);
        flat_dim = isl_space_flatten(isl_space_copy(dim));
        map = isl_map_identity(isl_space_join(isl_space_reverse(dim), flat_dim));
        map = isl_map_intersect_domain(map, set);

        return map;
}

/* Remove any internal structure from the space of the domain of "map".
 */
__isl_give isl_map *isl_map_flatten_domain(__isl_take isl_map *map)
{
        if (!map)
                return NULL;

        if (!map->dim->nested[0])
                return map;

        return isl_map_change_space(map, NULL, NULL, &isl_space_flatten_domain);
}

/* Remove any internal structure from the space of the range of "map".
 */
__isl_give isl_map *isl_map_flatten_range(__isl_take isl_map *map)
{
        if (!map)
                return NULL;

        if (!map->dim->nested[1])
                return map;

        return isl_map_change_space(map, NULL, NULL, &isl_space_flatten_range);
}

/* Reorder the dimensions of "bmap" according to the given dim_map
 * and set the dimension specification to "dim" and
 * perform Gaussian elimination on the result.
 */
__isl_give isl_basic_map *isl_basic_map_realign(__isl_take isl_basic_map *bmap,
        __isl_take isl_space *dim, __isl_take struct isl_dim_map *dim_map)
{
        isl_basic_map *res;
        unsigned flags;

        bmap = isl_basic_map_cow(bmap);
        if (!bmap || !dim || !dim_map)
                goto error;

        flags = bmap->flags;
        ISL_FL_CLR(flags, ISL_BASIC_MAP_FINAL);
        ISL_FL_CLR(flags, ISL_BASIC_MAP_NORMALIZED);
        ISL_FL_CLR(flags, ISL_BASIC_MAP_NORMALIZED_DIVS);
        res = isl_basic_map_alloc_space(dim,
                        bmap->n_div, bmap->n_eq, bmap->n_ineq);
        res = isl_basic_map_add_constraints_dim_map(res, bmap, dim_map);
        if (res)
                res->flags = flags;
        res = isl_basic_map_gauss(res, NULL);
        res = isl_basic_map_finalize(res);
        return res;
error:
        free(dim_map);
        isl_basic_map_free(bmap);
        isl_space_free(dim);
        return NULL;
}

/* Reorder the dimensions of "map" according to given reordering.
 */
__isl_give isl_map *isl_map_realign(__isl_take isl_map *map,
        __isl_take isl_reordering *r)
{
        int i;
        struct isl_dim_map *dim_map;

        map = isl_map_cow(map);
        dim_map = isl_dim_map_from_reordering(r);
        if (!map || !r || !dim_map)
                goto error;

        for (i = 0; i < map->n; ++i) {
                struct isl_dim_map *dim_map_i;

                dim_map_i = isl_dim_map_extend(dim_map, map->p[i]);

                map->p[i] = isl_basic_map_realign(map->p[i],
                                            isl_space_copy(r->dim), dim_map_i);

                if (!map->p[i])
                        goto error;
        }

        map = isl_map_reset_space(map, isl_space_copy(r->dim));

        isl_reordering_free(r);
        free(dim_map);
        return map;
error:
        free(dim_map);
        isl_map_free(map);
        isl_reordering_free(r);
        return NULL;
}

__isl_give isl_set *isl_set_realign(__isl_take isl_set *set,
        __isl_take isl_reordering *r)
{
        return set_from_map(isl_map_realign(set_to_map(set), r));
}

__isl_give isl_map *isl_map_align_params(__isl_take isl_map *map,
        __isl_take isl_space *model)
{
        isl_ctx *ctx;

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

        ctx = isl_space_get_ctx(model);
        if (!isl_space_has_named_params(model))
                isl_die(ctx, isl_error_invalid,
                        "model has unnamed parameters", goto error);
        if (!isl_space_has_named_params(map->dim))
                isl_die(ctx, isl_error_invalid,
                        "relation has unnamed parameters", goto error);
        if (!isl_space_match(map->dim, isl_dim_param, model, isl_dim_param)) {
                isl_reordering *exp;

                model = isl_space_drop_dims(model, isl_dim_in,
                                        0, isl_space_dim(model, isl_dim_in));
                model = isl_space_drop_dims(model, isl_dim_out,
                                        0, isl_space_dim(model, isl_dim_out));
                exp = isl_parameter_alignment_reordering(map->dim, model);
                exp = isl_reordering_extend_space(exp, isl_map_get_space(map));
                map = isl_map_realign(map, exp);
        }

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

__isl_give isl_set *isl_set_align_params(__isl_take isl_set *set,
        __isl_take isl_space *model)
{
        return isl_map_align_params(set, model);
}

/* Align the parameters of "bmap" to those of "model", introducing
 * additional parameters if needed.
 */
__isl_give isl_basic_map *isl_basic_map_align_params(
        __isl_take isl_basic_map *bmap, __isl_take isl_space *model)
{
        isl_ctx *ctx;

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

        ctx = isl_space_get_ctx(model);
        if (!isl_space_has_named_params(model))
                isl_die(ctx, isl_error_invalid,
                        "model has unnamed parameters", goto error);
        if (!isl_space_has_named_params(bmap->dim))
                isl_die(ctx, isl_error_invalid,
                        "relation has unnamed parameters", goto error);
        if (!isl_space_match(bmap->dim, isl_dim_param, model, isl_dim_param)) {
                isl_reordering *exp;
                struct isl_dim_map *dim_map;

                model = isl_space_drop_dims(model, isl_dim_in,
                                        0, isl_space_dim(model, isl_dim_in));
                model = isl_space_drop_dims(model, isl_dim_out,
                                        0, isl_space_dim(model, isl_dim_out));
                exp = isl_parameter_alignment_reordering(bmap->dim, model);
                exp = isl_reordering_extend_space(exp,
                                        isl_basic_map_get_space(bmap));
                dim_map = isl_dim_map_from_reordering(exp);
                bmap = isl_basic_map_realign(bmap,
                                    exp ? isl_space_copy(exp->dim) : NULL,
                                    isl_dim_map_extend(dim_map, bmap));
                isl_reordering_free(exp);
                free(dim_map);
        }

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

/* Align the parameters of "bset" to those of "model", introducing
 * additional parameters if needed.
 */
__isl_give isl_basic_set *isl_basic_set_align_params(
        __isl_take isl_basic_set *bset, __isl_take isl_space *model)
{
        return isl_basic_map_align_params(bset, model);
}

__isl_give isl_mat *isl_basic_map_equalities_matrix(
                __isl_keep isl_basic_map *bmap, enum isl_dim_type c1,
                enum isl_dim_type c2, enum isl_dim_type c3,
                enum isl_dim_type c4, enum isl_dim_type c5)
{
        enum isl_dim_type c[5] = { c1, c2, c3, c4, c5 };
        struct isl_mat *mat;
        int i, j, k;
        int pos;

        if (!bmap)
                return NULL;
        mat = isl_mat_alloc(bmap->ctx, bmap->n_eq,
                                isl_basic_map_total_dim(bmap) + 1);
        if (!mat)
                return NULL;
        for (i = 0; i < bmap->n_eq; ++i)
                for (j = 0, pos = 0; j < 5; ++j) {
                        int off = isl_basic_map_offset(bmap, c[j]);
                        for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
                                isl_int_set(mat->row[i][pos],
                                            bmap->eq[i][off + k]);
                                ++pos;
                        }
                }

        return mat;
}

__isl_give isl_mat *isl_basic_map_inequalities_matrix(
                __isl_keep isl_basic_map *bmap, enum isl_dim_type c1,
                enum isl_dim_type c2, enum isl_dim_type c3,
                enum isl_dim_type c4, enum isl_dim_type c5)
{
        enum isl_dim_type c[5] = { c1, c2, c3, c4, c5 };
        struct isl_mat *mat;
        int i, j, k;
        int pos;

        if (!bmap)
                return NULL;
        mat = isl_mat_alloc(bmap->ctx, bmap->n_ineq,
                                isl_basic_map_total_dim(bmap) + 1);
        if (!mat)
                return NULL;
        for (i = 0; i < bmap->n_ineq; ++i)
                for (j = 0, pos = 0; j < 5; ++j) {
                        int off = isl_basic_map_offset(bmap, c[j]);
                        for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
                                isl_int_set(mat->row[i][pos],
                                            bmap->ineq[i][off + k]);
                                ++pos;
                        }
                }

        return mat;
}

__isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
        __isl_take isl_space *dim,
        __isl_take isl_mat *eq, __isl_take isl_mat *ineq, enum isl_dim_type c1,
        enum isl_dim_type c2, enum isl_dim_type c3,
        enum isl_dim_type c4, enum isl_dim_type c5)
{
        enum isl_dim_type c[5] = { c1, c2, c3, c4, c5 };
        isl_basic_map *bmap;
        unsigned total;
        unsigned extra;
        int i, j, k, l;
        int pos;

        if (!dim || !eq || !ineq)
                goto error;

        if (eq->n_col != ineq->n_col)
                isl_die(dim->ctx, isl_error_invalid,
                        "equalities and inequalities matrices should have "
                        "same number of columns", goto error);

        total = 1 + isl_space_dim(dim, isl_dim_all);

        if (eq->n_col < total)
                isl_die(dim->ctx, isl_error_invalid,
                        "number of columns too small", goto error);

        extra = eq->n_col - total;

        bmap = isl_basic_map_alloc_space(isl_space_copy(dim), extra,
                                       eq->n_row, ineq->n_row);
        if (!bmap)
                goto error;
        for (i = 0; i < extra; ++i) {
                k = isl_basic_map_alloc_div(bmap);
                if (k < 0)
                        goto error;
                isl_int_set_si(bmap->div[k][0], 0);
        }
        for (i = 0; i < eq->n_row; ++i) {
                l = isl_basic_map_alloc_equality(bmap);
                if (l < 0)
                        goto error;
                for (j = 0, pos = 0; j < 5; ++j) {
                        int off = isl_basic_map_offset(bmap, c[j]);
                        for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
                                isl_int_set(bmap->eq[l][off + k], 
                                            eq->row[i][pos]);
                                ++pos;
                        }
                }
        }
        for (i = 0; i < ineq->n_row; ++i) {
                l = isl_basic_map_alloc_inequality(bmap);
                if (l < 0)
                        goto error;
                for (j = 0, pos = 0; j < 5; ++j) {
                        int off = isl_basic_map_offset(bmap, c[j]);
                        for (k = 0; k < isl_basic_map_dim(bmap, c[j]); ++k) {
                                isl_int_set(bmap->ineq[l][off + k], 
                                            ineq->row[i][pos]);
                                ++pos;
                        }
                }
        }

        isl_space_free(dim);
        isl_mat_free(eq);
        isl_mat_free(ineq);

        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_space_free(dim);
        isl_mat_free(eq);
        isl_mat_free(ineq);
        return NULL;
}

__isl_give isl_mat *isl_basic_set_equalities_matrix(
        __isl_keep isl_basic_set *bset, enum isl_dim_type c1,
        enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4)
{
        return isl_basic_map_equalities_matrix(bset_to_bmap(bset),
                                                c1, c2, c3, c4, isl_dim_in);
}

__isl_give isl_mat *isl_basic_set_inequalities_matrix(
        __isl_keep isl_basic_set *bset, enum isl_dim_type c1,
        enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4)
{
        return isl_basic_map_inequalities_matrix(bset_to_bmap(bset),
                                                 c1, c2, c3, c4, isl_dim_in);
}

__isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
        __isl_take isl_space *dim,
        __isl_take isl_mat *eq, __isl_take isl_mat *ineq, enum isl_dim_type c1,
        enum isl_dim_type c2, enum isl_dim_type c3, enum isl_dim_type c4)
{
        isl_basic_map *bmap;
        bmap = isl_basic_map_from_constraint_matrices(dim, eq, ineq,
                                                   c1, c2, c3, c4, isl_dim_in);
        return bset_from_bmap(bmap);
}

isl_bool isl_basic_map_can_zip(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return isl_bool_error;
        
        return isl_space_can_zip(bmap->dim);
}

isl_bool isl_map_can_zip(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;
        
        return isl_space_can_zip(map->dim);
}

/* Given a basic map (A -> B) -> (C -> D), return the corresponding basic map
 * (A -> C) -> (B -> D).
 */
__isl_give isl_basic_map *isl_basic_map_zip(__isl_take isl_basic_map *bmap)
{
        unsigned pos;
        unsigned n1;
        unsigned n2;

        if (!bmap)
                return NULL;

        if (!isl_basic_map_can_zip(bmap))
                isl_die(bmap->ctx, isl_error_invalid,
                        "basic map cannot be zipped", goto error);
        pos = isl_basic_map_offset(bmap, isl_dim_in) +
                isl_space_dim(bmap->dim->nested[0], isl_dim_in);
        n1 = isl_space_dim(bmap->dim->nested[0], isl_dim_out);
        n2 = isl_space_dim(bmap->dim->nested[1], isl_dim_in);
        bmap = isl_basic_map_cow(bmap);
        bmap = isl_basic_map_swap_vars(bmap, pos, n1, n2);
        if (!bmap)
                return NULL;
        bmap->dim = isl_space_zip(bmap->dim);
        if (!bmap->dim)
                goto error;
        bmap = isl_basic_map_mark_final(bmap);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Given a map (A -> B) -> (C -> D), return the corresponding map
 * (A -> C) -> (B -> D).
 */
__isl_give isl_map *isl_map_zip(__isl_take isl_map *map)
{
        int i;

        if (!map)
                return NULL;

        if (!isl_map_can_zip(map))
                isl_die(map->ctx, isl_error_invalid, "map cannot be zipped",
                        goto error);

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

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_zip(map->p[i]);
                if (!map->p[i])
                        goto error;
        }

        map->dim = isl_space_zip(map->dim);
        if (!map->dim)
                goto error;

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

/* Can we apply isl_basic_map_curry to "bmap"?
 * That is, does it have a nested relation in its domain?
 */
isl_bool isl_basic_map_can_curry(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return isl_bool_error;

        return isl_space_can_curry(bmap->dim);
}

/* Can we apply isl_map_curry to "map"?
 * That is, does it have a nested relation in its domain?
 */
isl_bool isl_map_can_curry(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;

        return isl_space_can_curry(map->dim);
}

/* Given a basic map (A -> B) -> C, return the corresponding basic map
 * A -> (B -> C).
 */
__isl_give isl_basic_map *isl_basic_map_curry(__isl_take isl_basic_map *bmap)
{

        if (!bmap)
                return NULL;

        if (!isl_basic_map_can_curry(bmap))
                isl_die(bmap->ctx, isl_error_invalid,
                        "basic map cannot be curried", goto error);
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;
        bmap->dim = isl_space_curry(bmap->dim);
        if (!bmap->dim)
                goto error;
        bmap = isl_basic_map_mark_final(bmap);
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Given a map (A -> B) -> C, return the corresponding map
 * A -> (B -> C).
 */
__isl_give isl_map *isl_map_curry(__isl_take isl_map *map)
{
        return isl_map_change_space(map, &isl_map_can_curry,
                                    "map cannot be curried", &isl_space_curry);
}

/* Can isl_map_range_curry be applied to "map"?
 * That is, does it have a nested relation in its range,
 * the domain of which is itself a nested relation?
 */
isl_bool isl_map_can_range_curry(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;

        return isl_space_can_range_curry(map->dim);
}

/* Given a map A -> ((B -> C) -> D), return the corresponding map
 * A -> (B -> (C -> D)).
 */
__isl_give isl_map *isl_map_range_curry(__isl_take isl_map *map)
{
        return isl_map_change_space(map, &isl_map_can_range_curry,
                                    "map range cannot be curried",
                                    &isl_space_range_curry);
}

/* Can we apply isl_basic_map_uncurry to "bmap"?
 * That is, does it have a nested relation in its domain?
 */
isl_bool isl_basic_map_can_uncurry(__isl_keep isl_basic_map *bmap)
{
        if (!bmap)
                return isl_bool_error;

        return isl_space_can_uncurry(bmap->dim);
}

/* Can we apply isl_map_uncurry to "map"?
 * That is, does it have a nested relation in its domain?
 */
isl_bool isl_map_can_uncurry(__isl_keep isl_map *map)
{
        if (!map)
                return isl_bool_error;

        return isl_space_can_uncurry(map->dim);
}

/* Given a basic map A -> (B -> C), return the corresponding basic map
 * (A -> B) -> C.
 */
__isl_give isl_basic_map *isl_basic_map_uncurry(__isl_take isl_basic_map *bmap)
{

        if (!bmap)
                return NULL;

        if (!isl_basic_map_can_uncurry(bmap))
                isl_die(bmap->ctx, isl_error_invalid,
                        "basic map cannot be uncurried",
                        return isl_basic_map_free(bmap));
        bmap = isl_basic_map_cow(bmap);
        if (!bmap)
                return NULL;
        bmap->dim = isl_space_uncurry(bmap->dim);
        if (!bmap->dim)
                return isl_basic_map_free(bmap);
        bmap = isl_basic_map_mark_final(bmap);
        return bmap;
}

/* Given a map A -> (B -> C), return the corresponding map
 * (A -> B) -> C.
 */
__isl_give isl_map *isl_map_uncurry(__isl_take isl_map *map)
{
        return isl_map_change_space(map, &isl_map_can_uncurry,
                                "map cannot be uncurried", &isl_space_uncurry);
}

/* Construct a basic map mapping the domain of the affine expression
 * to a one-dimensional range prescribed by the affine expression.
 * If "rational" is set, then construct a rational basic map.
 *
 * A NaN affine expression cannot be converted to a basic map.
 */
static __isl_give isl_basic_map *isl_basic_map_from_aff2(
        __isl_take isl_aff *aff, int rational)
{
        int k;
        int pos;
        isl_bool is_nan;
        isl_local_space *ls;
        isl_basic_map *bmap = NULL;

        if (!aff)
                return NULL;
        is_nan = isl_aff_is_nan(aff);
        if (is_nan < 0)
                goto error;
        if (is_nan)
                isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
                        "cannot convert NaN", goto error);

        ls = isl_aff_get_local_space(aff);
        bmap = isl_basic_map_from_local_space(ls);
        bmap = isl_basic_map_extend_constraints(bmap, 1, 0);
        k = isl_basic_map_alloc_equality(bmap);
        if (k < 0)
                goto error;

        pos = isl_basic_map_offset(bmap, isl_dim_out);
        isl_seq_cpy(bmap->eq[k], aff->v->el + 1, pos);
        isl_int_neg(bmap->eq[k][pos], aff->v->el[0]);
        isl_seq_cpy(bmap->eq[k] + pos + 1, aff->v->el + 1 + pos,
                    aff->v->size - (pos + 1));

        isl_aff_free(aff);
        if (rational)
                bmap = isl_basic_map_set_rational(bmap);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_aff_free(aff);
        isl_basic_map_free(bmap);
        return NULL;
}

/* Construct a basic map mapping the domain of the affine expression
 * to a one-dimensional range prescribed by the affine expression.
 */
__isl_give isl_basic_map *isl_basic_map_from_aff(__isl_take isl_aff *aff)
{
        return isl_basic_map_from_aff2(aff, 0);
}

/* Construct a map mapping the domain of the affine expression
 * to a one-dimensional range prescribed by the affine expression.
 */
__isl_give isl_map *isl_map_from_aff(__isl_take isl_aff *aff)
{
        isl_basic_map *bmap;

        bmap = isl_basic_map_from_aff(aff);
        return isl_map_from_basic_map(bmap);
}

/* Construct a basic map mapping the domain the multi-affine expression
 * to its range, with each dimension in the range equated to the
 * corresponding affine expression.
 * If "rational" is set, then construct a rational basic map.
 */
__isl_give isl_basic_map *isl_basic_map_from_multi_aff2(
        __isl_take isl_multi_aff *maff, int rational)
{
        int i;
        isl_space *space;
        isl_basic_map *bmap;

        if (!maff)
                return NULL;

        if (isl_space_dim(maff->space, isl_dim_out) != maff->n)
                isl_die(isl_multi_aff_get_ctx(maff), isl_error_internal,
                        "invalid space", goto error);

        space = isl_space_domain(isl_multi_aff_get_space(maff));
        bmap = isl_basic_map_universe(isl_space_from_domain(space));
        if (rational)
                bmap = isl_basic_map_set_rational(bmap);

        for (i = 0; i < maff->n; ++i) {
                isl_aff *aff;
                isl_basic_map *bmap_i;

                aff = isl_aff_copy(maff->p[i]);
                bmap_i = isl_basic_map_from_aff2(aff, rational);

                bmap = isl_basic_map_flat_range_product(bmap, bmap_i);
        }

        bmap = isl_basic_map_reset_space(bmap, isl_multi_aff_get_space(maff));

        isl_multi_aff_free(maff);
        return bmap;
error:
        isl_multi_aff_free(maff);
        return NULL;
}

/* Construct a basic map mapping the domain the multi-affine expression
 * to its range, with each dimension in the range equated to the
 * corresponding affine expression.
 */
__isl_give isl_basic_map *isl_basic_map_from_multi_aff(
        __isl_take isl_multi_aff *ma)
{
        return isl_basic_map_from_multi_aff2(ma, 0);
}

/* Construct a map mapping the domain the multi-affine expression
 * to its range, with each dimension in the range equated to the
 * corresponding affine expression.
 */
__isl_give isl_map *isl_map_from_multi_aff(__isl_take isl_multi_aff *maff)
{
        isl_basic_map *bmap;

        bmap = isl_basic_map_from_multi_aff(maff);
        return isl_map_from_basic_map(bmap);
}

/* Construct a basic map mapping a domain in the given space to
 * to an n-dimensional range, with n the number of elements in the list,
 * where each coordinate in the range is prescribed by the
 * corresponding affine expression.
 * The domains of all affine expressions in the list are assumed to match
 * domain_dim.
 */
__isl_give isl_basic_map *isl_basic_map_from_aff_list(
        __isl_take isl_space *domain_dim, __isl_take isl_aff_list *list)
{
        int i;
        isl_space *dim;
        isl_basic_map *bmap;

        if (!list)
                return NULL;

        dim = isl_space_from_domain(domain_dim);
        bmap = isl_basic_map_universe(dim);

        for (i = 0; i < list->n; ++i) {
                isl_aff *aff;
                isl_basic_map *bmap_i;

                aff = isl_aff_copy(list->p[i]);
                bmap_i = isl_basic_map_from_aff(aff);

                bmap = isl_basic_map_flat_range_product(bmap, bmap_i);
        }

        isl_aff_list_free(list);
        return bmap;
}

__isl_give isl_set *isl_set_equate(__isl_take isl_set *set,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        return isl_map_equate(set, type1, pos1, type2, pos2);
}

/* Construct a basic map where the given dimensions are equal to each other.
 */
static __isl_give isl_basic_map *equator(__isl_take isl_space *space,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *bmap = NULL;
        int i;

        if (!space)
                return NULL;

        if (pos1 >= isl_space_dim(space, type1))
                isl_die(isl_space_get_ctx(space), isl_error_invalid,
                        "index out of bounds", goto error);
        if (pos2 >= isl_space_dim(space, type2))
                isl_die(isl_space_get_ctx(space), isl_error_invalid,
                        "index out of bounds", goto error);

        if (type1 == type2 && pos1 == pos2)
                return isl_basic_map_universe(space);

        bmap = isl_basic_map_alloc_space(isl_space_copy(space), 0, 1, 0);
        i = isl_basic_map_alloc_equality(bmap);
        if (i < 0)
                goto error;
        isl_seq_clr(bmap->eq[i], 1 + isl_basic_map_total_dim(bmap));
        pos1 += isl_basic_map_offset(bmap, type1);
        pos2 += isl_basic_map_offset(bmap, type2);
        isl_int_set_si(bmap->eq[i][pos1], -1);
        isl_int_set_si(bmap->eq[i][pos2], 1);
        bmap = isl_basic_map_finalize(bmap);
        isl_space_free(space);
        return bmap;
error:
        isl_space_free(space);
        isl_basic_map_free(bmap);
        return NULL;
}

/* Add a constraint imposing that the given two dimensions are equal.
 */
__isl_give isl_basic_map *isl_basic_map_equate(__isl_take isl_basic_map *bmap,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *eq;

        eq = equator(isl_basic_map_get_space(bmap), type1, pos1, type2, pos2);

        bmap = isl_basic_map_intersect(bmap, eq);

        return bmap;
}

/* Add a constraint imposing that the given two dimensions are equal.
 */
__isl_give isl_map *isl_map_equate(__isl_take isl_map *map,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *bmap;

        bmap = equator(isl_map_get_space(map), type1, pos1, type2, pos2);

        map = isl_map_intersect(map, isl_map_from_basic_map(bmap));

        return map;
}

/* Add a constraint imposing that the given two dimensions have opposite values.
 */
__isl_give isl_map *isl_map_oppose(__isl_take isl_map *map,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *bmap = NULL;
        int i;

        if (!map)
                return NULL;

        if (pos1 >= isl_map_dim(map, type1))
                isl_die(map->ctx, isl_error_invalid,
                        "index out of bounds", goto error);
        if (pos2 >= isl_map_dim(map, type2))
                isl_die(map->ctx, isl_error_invalid,
                        "index out of bounds", goto error);

        bmap = isl_basic_map_alloc_space(isl_map_get_space(map), 0, 1, 0);
        i = isl_basic_map_alloc_equality(bmap);
        if (i < 0)
                goto error;
        isl_seq_clr(bmap->eq[i], 1 + isl_basic_map_total_dim(bmap));
        pos1 += isl_basic_map_offset(bmap, type1);
        pos2 += isl_basic_map_offset(bmap, type2);
        isl_int_set_si(bmap->eq[i][pos1], 1);
        isl_int_set_si(bmap->eq[i][pos2], 1);
        bmap = isl_basic_map_finalize(bmap);

        map = isl_map_intersect(map, isl_map_from_basic_map(bmap));

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

/* Construct a constraint imposing that the value of the first dimension is
 * greater than or equal to that of the second.
 */
static __isl_give isl_constraint *constraint_order_ge(
        __isl_take isl_space *space, enum isl_dim_type type1, int pos1,
        enum isl_dim_type type2, int pos2)
{
        isl_constraint *c;

        if (!space)
                return NULL;

        c = isl_constraint_alloc_inequality(isl_local_space_from_space(space));

        if (pos1 >= isl_constraint_dim(c, type1))
                isl_die(isl_constraint_get_ctx(c), isl_error_invalid,
                        "index out of bounds", return isl_constraint_free(c));
        if (pos2 >= isl_constraint_dim(c, type2))
                isl_die(isl_constraint_get_ctx(c), isl_error_invalid,
                        "index out of bounds", return isl_constraint_free(c));

        if (type1 == type2 && pos1 == pos2)
                return c;

        c = isl_constraint_set_coefficient_si(c, type1, pos1, 1);
        c = isl_constraint_set_coefficient_si(c, type2, pos2, -1);

        return c;
}

/* Add a constraint imposing that the value of the first dimension is
 * greater than or equal to that of the second.
 */
__isl_give isl_basic_map *isl_basic_map_order_ge(__isl_take isl_basic_map *bmap,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_constraint *c;
        isl_space *space;

        if (type1 == type2 && pos1 == pos2)
                return bmap;
        space = isl_basic_map_get_space(bmap);
        c = constraint_order_ge(space, type1, pos1, type2, pos2);
        bmap = isl_basic_map_add_constraint(bmap, c);

        return bmap;
}

/* Add a constraint imposing that the value of the first dimension is
 * greater than or equal to that of the second.
 */
__isl_give isl_map *isl_map_order_ge(__isl_take isl_map *map,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_constraint *c;
        isl_space *space;

        if (type1 == type2 && pos1 == pos2)
                return map;
        space = isl_map_get_space(map);
        c = constraint_order_ge(space, type1, pos1, type2, pos2);
        map = isl_map_add_constraint(map, c);

        return map;
}

/* Add a constraint imposing that the value of the first dimension is
 * less than or equal to that of the second.
 */
__isl_give isl_map *isl_map_order_le(__isl_take isl_map *map,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        return isl_map_order_ge(map, type2, pos2, type1, pos1);
}

/* Construct a basic map where the value of the first dimension is
 * greater than that of the second.
 */
static __isl_give isl_basic_map *greator(__isl_take isl_space *space,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *bmap = NULL;
        int i;

        if (!space)
                return NULL;

        if (pos1 >= isl_space_dim(space, type1))
                isl_die(isl_space_get_ctx(space), isl_error_invalid,
                        "index out of bounds", goto error);
        if (pos2 >= isl_space_dim(space, type2))
                isl_die(isl_space_get_ctx(space), isl_error_invalid,
                        "index out of bounds", goto error);

        if (type1 == type2 && pos1 == pos2)
                return isl_basic_map_empty(space);

        bmap = isl_basic_map_alloc_space(space, 0, 0, 1);
        i = isl_basic_map_alloc_inequality(bmap);
        if (i < 0)
                return isl_basic_map_free(bmap);
        isl_seq_clr(bmap->ineq[i], 1 + isl_basic_map_total_dim(bmap));
        pos1 += isl_basic_map_offset(bmap, type1);
        pos2 += isl_basic_map_offset(bmap, type2);
        isl_int_set_si(bmap->ineq[i][pos1], 1);
        isl_int_set_si(bmap->ineq[i][pos2], -1);
        isl_int_set_si(bmap->ineq[i][0], -1);
        bmap = isl_basic_map_finalize(bmap);

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

/* Add a constraint imposing that the value of the first dimension is
 * greater than that of the second.
 */
__isl_give isl_basic_map *isl_basic_map_order_gt(__isl_take isl_basic_map *bmap,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *gt;

        gt = greator(isl_basic_map_get_space(bmap), type1, pos1, type2, pos2);

        bmap = isl_basic_map_intersect(bmap, gt);

        return bmap;
}

/* Add a constraint imposing that the value of the first dimension is
 * greater than that of the second.
 */
__isl_give isl_map *isl_map_order_gt(__isl_take isl_map *map,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        isl_basic_map *bmap;

        bmap = greator(isl_map_get_space(map), type1, pos1, type2, pos2);

        map = isl_map_intersect(map, isl_map_from_basic_map(bmap));

        return map;
}

/* Add a constraint imposing that the value of the first dimension is
 * smaller than that of the second.
 */
__isl_give isl_map *isl_map_order_lt(__isl_take isl_map *map,
        enum isl_dim_type type1, int pos1, enum isl_dim_type type2, int pos2)
{
        return isl_map_order_gt(map, type2, pos2, type1, pos1);
}

__isl_give isl_aff *isl_basic_map_get_div(__isl_keep isl_basic_map *bmap,
        int pos)
{
        isl_aff *div;
        isl_local_space *ls;

        if (!bmap)
                return NULL;

        if (!isl_basic_map_divs_known(bmap))
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "some divs are unknown", return NULL);

        ls = isl_basic_map_get_local_space(bmap);
        div = isl_local_space_get_div(ls, pos);
        isl_local_space_free(ls);

        return div;
}

__isl_give isl_aff *isl_basic_set_get_div(__isl_keep isl_basic_set *bset,
        int pos)
{
        return isl_basic_map_get_div(bset, pos);
}

/* Plug in "subs" for dimension "type", "pos" of "bset".
 *
 * Let i be the dimension to replace and let "subs" be of the form
 *
 *      f/d
 *
 * Any integer division with a non-zero coefficient for i,
 *
 *      floor((a i + g)/m)
 *
 * is replaced by
 *
 *      floor((a f + d g)/(m d))
 *
 * Constraints of the form
 *
 *      a i + g
 *
 * are replaced by
 *
 *      a f + d g
 *
 * We currently require that "subs" is an integral expression.
 * Handling rational expressions may require us to add stride constraints
 * as we do in isl_basic_set_preimage_multi_aff.
 */
__isl_give isl_basic_set *isl_basic_set_substitute(
        __isl_take isl_basic_set *bset,
        enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
{
        int i;
        isl_int v;
        isl_ctx *ctx;

        if (bset && isl_basic_set_plain_is_empty(bset))
                return bset;

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

        ctx = isl_basic_set_get_ctx(bset);
        if (!isl_space_is_equal(bset->dim, subs->ls->dim))
                isl_die(ctx, isl_error_invalid,
                        "spaces don't match", goto error);
        if (isl_local_space_dim(subs->ls, isl_dim_div) != 0)
                isl_die(ctx, isl_error_unsupported,
                        "cannot handle divs yet", goto error);
        if (!isl_int_is_one(subs->v->el[0]))
                isl_die(ctx, isl_error_invalid,
                        "can only substitute integer expressions", goto error);

        pos += isl_basic_set_offset(bset, type);

        isl_int_init(v);

        for (i = 0; i < bset->n_eq; ++i) {
                if (isl_int_is_zero(bset->eq[i][pos]))
                        continue;
                isl_int_set(v, bset->eq[i][pos]);
                isl_int_set_si(bset->eq[i][pos], 0);
                isl_seq_combine(bset->eq[i], subs->v->el[0], bset->eq[i],
                                v, subs->v->el + 1, subs->v->size - 1);
        }

        for (i = 0; i < bset->n_ineq; ++i) {
                if (isl_int_is_zero(bset->ineq[i][pos]))
                        continue;
                isl_int_set(v, bset->ineq[i][pos]);
                isl_int_set_si(bset->ineq[i][pos], 0);
                isl_seq_combine(bset->ineq[i], subs->v->el[0], bset->ineq[i],
                                v, subs->v->el + 1, subs->v->size - 1);
        }

        for (i = 0; i < bset->n_div; ++i) {
                if (isl_int_is_zero(bset->div[i][1 + pos]))
                        continue;
                isl_int_set(v, bset->div[i][1 + pos]);
                isl_int_set_si(bset->div[i][1 + pos], 0);
                isl_seq_combine(bset->div[i] + 1,
                                subs->v->el[0], bset->div[i] + 1,
                                v, subs->v->el + 1, subs->v->size - 1);
                isl_int_mul(bset->div[i][0], bset->div[i][0], subs->v->el[0]);
        }

        isl_int_clear(v);

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

/* Plug in "subs" for dimension "type", "pos" of "set".
 */
__isl_give isl_set *isl_set_substitute(__isl_take isl_set *set,
        enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
{
        int i;

        if (set && isl_set_plain_is_empty(set))
                return set;

        set = isl_set_cow(set);
        if (!set || !subs)
                goto error;

        for (i = set->n - 1; i >= 0; --i) {
                set->p[i] = isl_basic_set_substitute(set->p[i], type, pos, subs);
                if (remove_if_empty(set, i) < 0)
                        goto error;
        }

        return set;
error:
        isl_set_free(set);
        return NULL;
}

/* Check if the range of "ma" is compatible with the domain or range
 * (depending on "type") of "bmap".
 * Return -1 if anything is wrong.
 */
static int check_basic_map_compatible_range_multi_aff(
        __isl_keep isl_basic_map *bmap, enum isl_dim_type type,
        __isl_keep isl_multi_aff *ma)
{
        int m;
        isl_space *ma_space;

        ma_space = isl_multi_aff_get_space(ma);

        m = isl_space_match(bmap->dim, isl_dim_param, ma_space, isl_dim_param);
        if (m < 0)
                goto error;
        if (!m)
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "parameters don't match", goto error);
        m = isl_space_tuple_is_equal(bmap->dim, type, ma_space, isl_dim_out);
        if (m < 0)
                goto error;
        if (!m)
                isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
                        "spaces don't match", goto error);

        isl_space_free(ma_space);
        return m;
error:
        isl_space_free(ma_space);
        return -1;
}

/* Copy the divs from "ma" to "bmap", adding zeros for the "n_before"
 * coefficients before the transformed range of dimensions,
 * the "n_after" coefficients after the transformed range of dimensions
 * and the coefficients of the other divs in "bmap".
 */
static int set_ma_divs(__isl_keep isl_basic_map *bmap,
        __isl_keep isl_multi_aff *ma, int n_before, int n_after, int n_div)
{
        int i;
        int n_param;
        int n_set;
        isl_local_space *ls;

        if (n_div == 0)
                return 0;

        ls = isl_aff_get_domain_local_space(ma->p[0]);
        if (!ls)
                return -1;

        n_param = isl_local_space_dim(ls, isl_dim_param);
        n_set = isl_local_space_dim(ls, isl_dim_set);
        for (i = 0; i < n_div; ++i) {
                int o_bmap = 0, o_ls = 0;

                isl_seq_cpy(bmap->div[i], ls->div->row[i], 1 + 1 + n_param);
                o_bmap += 1 + 1 + n_param;
                o_ls += 1 + 1 + n_param;
                isl_seq_clr(bmap->div[i] + o_bmap, n_before);
                o_bmap += n_before;
                isl_seq_cpy(bmap->div[i] + o_bmap,
                            ls->div->row[i] + o_ls, n_set);
                o_bmap += n_set;
                o_ls += n_set;
                isl_seq_clr(bmap->div[i] + o_bmap, n_after);
                o_bmap += n_after;
                isl_seq_cpy(bmap->div[i] + o_bmap,
                            ls->div->row[i] + o_ls, n_div);
                o_bmap += n_div;
                o_ls += n_div;
                isl_seq_clr(bmap->div[i] + o_bmap, bmap->n_div - n_div);
                if (isl_basic_set_add_div_constraints(bmap, i) < 0)
                        goto error;
        }

        isl_local_space_free(ls);
        return 0;
error:
        isl_local_space_free(ls);
        return -1;
}

/* How many stride constraints does "ma" enforce?
 * That is, how many of the affine expressions have a denominator
 * different from one?
 */
static int multi_aff_strides(__isl_keep isl_multi_aff *ma)
{
        int i;
        int strides = 0;

        for (i = 0; i < ma->n; ++i)
                if (!isl_int_is_one(ma->p[i]->v->el[0]))
                        strides++;

        return strides;
}

/* For each affine expression in ma of the form
 *
 *      x_i = (f_i y + h_i)/m_i
 *
 * with m_i different from one, add a constraint to "bmap"
 * of the form
 *
 *      f_i y + h_i = m_i alpha_i
 *
 * with alpha_i an additional existentially quantified variable.
 *
 * The input variables of "ma" correspond to a subset of the variables
 * of "bmap".  There are "n_before" variables in "bmap" before this
 * subset and "n_after" variables after this subset.
 * The integer divisions of the affine expressions in "ma" are assumed
 * to have been aligned.  There are "n_div_ma" of them and
 * they appear first in "bmap", straight after the "n_after" variables.
 */
static __isl_give isl_basic_map *add_ma_strides(
        __isl_take isl_basic_map *bmap, __isl_keep isl_multi_aff *ma,
        int n_before, int n_after, int n_div_ma)
{
        int i, k;
        int div;
        int total;
        int n_param;
        int n_in;

        total = isl_basic_map_total_dim(bmap);
        n_param = isl_multi_aff_dim(ma, isl_dim_param);
        n_in = isl_multi_aff_dim(ma, isl_dim_in);
        for (i = 0; i < ma->n; ++i) {
                int o_bmap = 0, o_ma = 1;

                if (isl_int_is_one(ma->p[i]->v->el[0]))
                        continue;
                div = isl_basic_map_alloc_div(bmap);
                k = isl_basic_map_alloc_equality(bmap);
                if (div < 0 || k < 0)
                        goto error;
                isl_int_set_si(bmap->div[div][0], 0);
                isl_seq_cpy(bmap->eq[k] + o_bmap,
                            ma->p[i]->v->el + o_ma, 1 + n_param);
                o_bmap += 1 + n_param;
                o_ma += 1 + n_param;
                isl_seq_clr(bmap->eq[k] + o_bmap, n_before);
                o_bmap += n_before;
                isl_seq_cpy(bmap->eq[k] + o_bmap,
                            ma->p[i]->v->el + o_ma, n_in);
                o_bmap += n_in;
                o_ma += n_in;
                isl_seq_clr(bmap->eq[k] + o_bmap, n_after);
                o_bmap += n_after;
                isl_seq_cpy(bmap->eq[k] + o_bmap,
                            ma->p[i]->v->el + o_ma, n_div_ma);
                o_bmap += n_div_ma;
                o_ma += n_div_ma;
                isl_seq_clr(bmap->eq[k] + o_bmap, 1 + total - o_bmap);
                isl_int_neg(bmap->eq[k][1 + total], ma->p[i]->v->el[0]);
                total++;
        }

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

/* Replace the domain or range space (depending on "type) of "space" by "set".
 */
static __isl_give isl_space *isl_space_set(__isl_take isl_space *space,
        enum isl_dim_type type, __isl_take isl_space *set)
{
        if (type == isl_dim_in) {
                space = isl_space_range(space);
                space = isl_space_map_from_domain_and_range(set, space);
        } else {
                space = isl_space_domain(space);
                space = isl_space_map_from_domain_and_range(space, set);
        }

        return space;
}

/* Compute the preimage of the domain or range (depending on "type")
 * of "bmap" under the function represented by "ma".
 * In other words, plug in "ma" in the domain or range of "bmap".
 * The result is a basic map that lives in the same space as "bmap"
 * except that the domain or range has been replaced by
 * the domain space of "ma".
 *
 * If bmap is represented by
 *
 *      A(p) + S u + B x + T v + C(divs) >= 0,
 *
 * where u and x are input and output dimensions if type == isl_dim_out
 * while x and v are input and output dimensions if type == isl_dim_in,
 * and ma is represented by
 *
 *      x = D(p) + F(y) + G(divs')
 *
 * then the result is
 *
 *      A(p) + B D(p) + S u + B F(y) + T v + B G(divs') + C(divs) >= 0
 *
 * The divs in the input set are similarly adjusted.
 * In particular
 *
 *      floor((a_i(p) + s u + b_i x + t v + c_i(divs))/n_i)
 *
 * becomes
 *
 *      floor((a_i(p) + b_i D(p) + s u + b_i F(y) + t v +
 *              B_i G(divs') + c_i(divs))/n_i)
 *
 * If bmap is not a rational map and if F(y) involves any denominators
 *
 *      x_i = (f_i y + h_i)/m_i
 *
 * then additional constraints are added to ensure that we only
 * map back integer points.  That is we enforce
 *
 *      f_i y + h_i = m_i alpha_i
 *
 * with alpha_i an additional existentially quantified variable.
 *
 * We first copy over the divs from "ma".
 * Then we add the modified constraints and divs from "bmap".
 * Finally, we add the stride constraints, if needed.
 */
__isl_give isl_basic_map *isl_basic_map_preimage_multi_aff(
        __isl_take isl_basic_map *bmap, enum isl_dim_type type,
        __isl_take isl_multi_aff *ma)
{
        int i, k;
        isl_space *space;
        isl_basic_map *res = NULL;
        int n_before, n_after, n_div_bmap, n_div_ma;
        isl_int f, c1, c2, g;
        int rational, strides;

        isl_int_init(f);
        isl_int_init(c1);
        isl_int_init(c2);
        isl_int_init(g);

        ma = isl_multi_aff_align_divs(ma);
        if (!bmap || !ma)
                goto error;
        if (check_basic_map_compatible_range_multi_aff(bmap, type, ma) < 0)
                goto error;

        if (type == isl_dim_in) {
                n_before = 0;
                n_after = isl_basic_map_dim(bmap, isl_dim_out);
        } else {
                n_before = isl_basic_map_dim(bmap, isl_dim_in);
                n_after = 0;
        }
        n_div_bmap = isl_basic_map_dim(bmap, isl_dim_div);
        n_div_ma = ma->n ? isl_aff_dim(ma->p[0], isl_dim_div) : 0;

        space = isl_multi_aff_get_domain_space(ma);
        space = isl_space_set(isl_basic_map_get_space(bmap), type, space);
        rational = isl_basic_map_is_rational(bmap);
        strides = rational ? 0 : multi_aff_strides(ma);
        res = isl_basic_map_alloc_space(space, n_div_ma + n_div_bmap + strides,
                            bmap->n_eq + strides, bmap->n_ineq + 2 * n_div_ma);
        if (rational)
                res = isl_basic_map_set_rational(res);

        for (i = 0; i < n_div_ma + n_div_bmap; ++i)
                if (isl_basic_map_alloc_div(res) < 0)
                        goto error;

        if (set_ma_divs(res, ma, n_before, n_after, n_div_ma) < 0)
                goto error;

        for (i = 0; i < bmap->n_eq; ++i) {
                k = isl_basic_map_alloc_equality(res);
                if (k < 0)
                        goto error;
                isl_seq_preimage(res->eq[k], bmap->eq[i], ma, n_before,
                                n_after, n_div_ma, n_div_bmap, f, c1, c2, g, 0);
        }

        for (i = 0; i < bmap->n_ineq; ++i) {
                k = isl_basic_map_alloc_inequality(res);
                if (k < 0)
                        goto error;
                isl_seq_preimage(res->ineq[k], bmap->ineq[i], ma, n_before,
                                n_after, n_div_ma, n_div_bmap, f, c1, c2, g, 0);
        }

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0])) {
                        isl_int_set_si(res->div[n_div_ma + i][0], 0);
                        continue;
                }
                isl_seq_preimage(res->div[n_div_ma + i], bmap->div[i], ma,
                                    n_before, n_after, n_div_ma, n_div_bmap,
                                    f, c1, c2, g, 1);
        }

        if (strides)
                res = add_ma_strides(res, ma, n_before, n_after, n_div_ma);

        isl_int_clear(f);
        isl_int_clear(c1);
        isl_int_clear(c2);
        isl_int_clear(g);
        isl_basic_map_free(bmap);
        isl_multi_aff_free(ma);
        res = isl_basic_set_simplify(res);
        return isl_basic_map_finalize(res);
error:
        isl_int_clear(f);
        isl_int_clear(c1);
        isl_int_clear(c2);
        isl_int_clear(g);
        isl_basic_map_free(bmap);
        isl_multi_aff_free(ma);
        isl_basic_map_free(res);
        return NULL;
}

/* Compute the preimage of "bset" under the function represented by "ma".
 * In other words, plug in "ma" in "bset".  The result is a basic set
 * that lives in the domain space of "ma".
 */
__isl_give isl_basic_set *isl_basic_set_preimage_multi_aff(
        __isl_take isl_basic_set *bset, __isl_take isl_multi_aff *ma)
{
        return isl_basic_map_preimage_multi_aff(bset, isl_dim_set, ma);
}

/* Compute the preimage of the domain of "bmap" under the function
 * represented by "ma".
 * In other words, plug in "ma" in the domain of "bmap".
 * The result is a basic map that lives in the same space as "bmap"
 * except that the domain has been replaced by the domain space of "ma".
 */
__isl_give isl_basic_map *isl_basic_map_preimage_domain_multi_aff(
        __isl_take isl_basic_map *bmap, __isl_take isl_multi_aff *ma)
{
        return isl_basic_map_preimage_multi_aff(bmap, isl_dim_in, ma);
}

/* Compute the preimage of the range of "bmap" under the function
 * represented by "ma".
 * In other words, plug in "ma" in the range of "bmap".
 * The result is a basic map that lives in the same space as "bmap"
 * except that the range has been replaced by the domain space of "ma".
 */
__isl_give isl_basic_map *isl_basic_map_preimage_range_multi_aff(
        __isl_take isl_basic_map *bmap, __isl_take isl_multi_aff *ma)
{
        return isl_basic_map_preimage_multi_aff(bmap, isl_dim_out, ma);
}

/* Check if the range of "ma" is compatible with the domain or range
 * (depending on "type") of "map".
 * Return -1 if anything is wrong.
 */
static int check_map_compatible_range_multi_aff(
        __isl_keep isl_map *map, enum isl_dim_type type,
        __isl_keep isl_multi_aff *ma)
{
        int m;
        isl_space *ma_space;

        ma_space = isl_multi_aff_get_space(ma);
        m = isl_space_tuple_is_equal(map->dim, type, ma_space, isl_dim_out);
        isl_space_free(ma_space);
        if (m >= 0 && !m)
                isl_die(isl_map_get_ctx(map), isl_error_invalid,
                        "spaces don't match", return -1);
        return m;
}

/* Compute the preimage of the domain or range (depending on "type")
 * of "map" under the function represented by "ma".
 * In other words, plug in "ma" in the domain or range of "map".
 * The result is a map that lives in the same space as "map"
 * except that the domain or range has been replaced by
 * the domain space of "ma".
 *
 * The parameters are assumed to have been aligned.
 */
static __isl_give isl_map *map_preimage_multi_aff(__isl_take isl_map *map,
        enum isl_dim_type type, __isl_take isl_multi_aff *ma)
{
        int i;
        isl_space *space;

        map = isl_map_cow(map);
        ma = isl_multi_aff_align_divs(ma);
        if (!map || !ma)
                goto error;
        if (check_map_compatible_range_multi_aff(map, type, ma) < 0)
                goto error;

        for (i = 0; i < map->n; ++i) {
                map->p[i] = isl_basic_map_preimage_multi_aff(map->p[i], type,
                                                        isl_multi_aff_copy(ma));
                if (!map->p[i])
                        goto error;
        }

        space = isl_multi_aff_get_domain_space(ma);
        space = isl_space_set(isl_map_get_space(map), type, space);

        isl_space_free(map->dim);
        map->dim = space;
        if (!map->dim)
                goto error;

        isl_multi_aff_free(ma);
        if (map->n > 1)
                ISL_F_CLR(map, ISL_MAP_DISJOINT);
        ISL_F_CLR(map, ISL_SET_NORMALIZED);
        return map;
error:
        isl_multi_aff_free(ma);
        isl_map_free(map);
        return NULL;
}

/* Compute the preimage of the domain or range (depending on "type")
 * of "map" under the function represented by "ma".
 * In other words, plug in "ma" in the domain or range of "map".
 * The result is a map that lives in the same space as "map"
 * except that the domain or range has been replaced by
 * the domain space of "ma".
 */
__isl_give isl_map *isl_map_preimage_multi_aff(__isl_take isl_map *map,
        enum isl_dim_type type, __isl_take isl_multi_aff *ma)
{
        if (!map || !ma)
                goto error;

        if (isl_space_match(map->dim, isl_dim_param, ma->space, isl_dim_param))
                return map_preimage_multi_aff(map, type, ma);

        if (!isl_space_has_named_params(map->dim) ||
            !isl_space_has_named_params(ma->space))
                isl_die(map->ctx, isl_error_invalid,
                        "unaligned unnamed parameters", goto error);
        map = isl_map_align_params(map, isl_multi_aff_get_space(ma));
        ma = isl_multi_aff_align_params(ma, isl_map_get_space(map));

        return map_preimage_multi_aff(map, type, ma);
error:
        isl_multi_aff_free(ma);
        return isl_map_free(map);
}

/* Compute the preimage of "set" under the function represented by "ma".
 * In other words, plug in "ma" in "set".  The result is a set
 * that lives in the domain space of "ma".
 */
__isl_give isl_set *isl_set_preimage_multi_aff(__isl_take isl_set *set,
        __isl_take isl_multi_aff *ma)
{
        return isl_map_preimage_multi_aff(set, isl_dim_set, ma);
}

/* Compute the preimage of the domain of "map" under the function
 * represented by "ma".
 * In other words, plug in "ma" in the domain of "map".
 * The result is a map that lives in the same space as "map"
 * except that the domain has been replaced by the domain space of "ma".
 */
__isl_give isl_map *isl_map_preimage_domain_multi_aff(__isl_take isl_map *map,
        __isl_take isl_multi_aff *ma)
{
        return isl_map_preimage_multi_aff(map, isl_dim_in, ma);
}

/* Compute the preimage of the range of "map" under the function
 * represented by "ma".
 * In other words, plug in "ma" in the range of "map".
 * The result is a map that lives in the same space as "map"
 * except that the range has been replaced by the domain space of "ma".
 */
__isl_give isl_map *isl_map_preimage_range_multi_aff(__isl_take isl_map *map,
        __isl_take isl_multi_aff *ma)
{
        return isl_map_preimage_multi_aff(map, isl_dim_out, ma);
}

/* Compute the preimage of "map" under the function represented by "pma".
 * In other words, plug in "pma" in the domain or range of "map".
 * The result is a map that lives in the same space as "map",
 * except that the space of type "type" has been replaced by
 * the domain space of "pma".
 *
 * The parameters of "map" and "pma" are assumed to have been aligned.
 */
static __isl_give isl_map *isl_map_preimage_pw_multi_aff_aligned(
        __isl_take isl_map *map, enum isl_dim_type type,
        __isl_take isl_pw_multi_aff *pma)
{
        int i;
        isl_map *res;

        if (!pma)
                goto error;

        if (pma->n == 0) {
                isl_pw_multi_aff_free(pma);
                res = isl_map_empty(isl_map_get_space(map));
                isl_map_free(map);
                return res;
        }

        res = isl_map_preimage_multi_aff(isl_map_copy(map), type,
                                        isl_multi_aff_copy(pma->p[0].maff));
        if (type == isl_dim_in)
                res = isl_map_intersect_domain(res,
                                                isl_map_copy(pma->p[0].set));
        else
                res = isl_map_intersect_range(res,
                                                isl_map_copy(pma->p[0].set));

        for (i = 1; i < pma->n; ++i) {
                isl_map *res_i;

                res_i = isl_map_preimage_multi_aff(isl_map_copy(map), type,
                                        isl_multi_aff_copy(pma->p[i].maff));
                if (type == isl_dim_in)
                        res_i = isl_map_intersect_domain(res_i,
                                                isl_map_copy(pma->p[i].set));
                else
                        res_i = isl_map_intersect_range(res_i,
                                                isl_map_copy(pma->p[i].set));
                res = isl_map_union(res, res_i);
        }

        isl_pw_multi_aff_free(pma);
        isl_map_free(map);
        return res;
error:
        isl_pw_multi_aff_free(pma);
        isl_map_free(map);
        return NULL;
}

/* Compute the preimage of "map" under the function represented by "pma".
 * In other words, plug in "pma" in the domain or range of "map".
 * The result is a map that lives in the same space as "map",
 * except that the space of type "type" has been replaced by
 * the domain space of "pma".
 */
__isl_give isl_map *isl_map_preimage_pw_multi_aff(__isl_take isl_map *map,
        enum isl_dim_type type, __isl_take isl_pw_multi_aff *pma)
{
        if (!map || !pma)
                goto error;

        if (isl_space_match(map->dim, isl_dim_param, pma->dim, isl_dim_param))
                return isl_map_preimage_pw_multi_aff_aligned(map, type, pma);

        if (!isl_space_has_named_params(map->dim) ||
            !isl_space_has_named_params(pma->dim))
                isl_die(map->ctx, isl_error_invalid,
                        "unaligned unnamed parameters", goto error);
        map = isl_map_align_params(map, isl_pw_multi_aff_get_space(pma));
        pma = isl_pw_multi_aff_align_params(pma, isl_map_get_space(map));

        return isl_map_preimage_pw_multi_aff_aligned(map, type, pma);
error:
        isl_pw_multi_aff_free(pma);
        return isl_map_free(map);
}

/* Compute the preimage of "set" under the function represented by "pma".
 * In other words, plug in "pma" in "set".  The result is a set
 * that lives in the domain space of "pma".
 */
__isl_give isl_set *isl_set_preimage_pw_multi_aff(__isl_take isl_set *set,
        __isl_take isl_pw_multi_aff *pma)
{
        return isl_map_preimage_pw_multi_aff(set, isl_dim_set, pma);
}

/* Compute the preimage of the domain of "map" under the function
 * represented by "pma".
 * In other words, plug in "pma" in the domain of "map".
 * The result is a map that lives in the same space as "map",
 * except that domain space has been replaced by the domain space of "pma".
 */
__isl_give isl_map *isl_map_preimage_domain_pw_multi_aff(
        __isl_take isl_map *map, __isl_take isl_pw_multi_aff *pma)
{
        return isl_map_preimage_pw_multi_aff(map, isl_dim_in, pma);
}

/* Compute the preimage of the range of "map" under the function
 * represented by "pma".
 * In other words, plug in "pma" in the range of "map".
 * The result is a map that lives in the same space as "map",
 * except that range space has been replaced by the domain space of "pma".
 */
__isl_give isl_map *isl_map_preimage_range_pw_multi_aff(
        __isl_take isl_map *map, __isl_take isl_pw_multi_aff *pma)
{
        return isl_map_preimage_pw_multi_aff(map, isl_dim_out, pma);
}

/* Compute the preimage of "map" under the function represented by "mpa".
 * In other words, plug in "mpa" in the domain or range of "map".
 * The result is a map that lives in the same space as "map",
 * except that the space of type "type" has been replaced by
 * the domain space of "mpa".
 *
 * If the map does not involve any constraints that refer to the
 * dimensions of the substituted space, then the only possible
 * effect of "mpa" on the map is to map the space to a different space.
 * We create a separate isl_multi_aff to effectuate this change
 * in order to avoid spurious splitting of the map along the pieces
 * of "mpa".
 */
__isl_give isl_map *isl_map_preimage_multi_pw_aff(__isl_take isl_map *map,
        enum isl_dim_type type, __isl_take isl_multi_pw_aff *mpa)
{
        int n;
        isl_pw_multi_aff *pma;

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

        n = isl_map_dim(map, type);
        if (!isl_map_involves_dims(map, type, 0, n)) {
                isl_space *space;
                isl_multi_aff *ma;

                space = isl_multi_pw_aff_get_space(mpa);
                isl_multi_pw_aff_free(mpa);
                ma = isl_multi_aff_zero(space);
                return isl_map_preimage_multi_aff(map, type, ma);
        }

        pma = isl_pw_multi_aff_from_multi_pw_aff(mpa);
        return isl_map_preimage_pw_multi_aff(map, type, pma);
error:
        isl_map_free(map);
        isl_multi_pw_aff_free(mpa);
        return NULL;
}

/* Compute the preimage of "map" under the function represented by "mpa".
 * In other words, plug in "mpa" in the domain "map".
 * The result is a map that lives in the same space as "map",
 * except that domain space has been replaced by the domain space of "mpa".
 */
__isl_give isl_map *isl_map_preimage_domain_multi_pw_aff(
        __isl_take isl_map *map, __isl_take isl_multi_pw_aff *mpa)
{
        return isl_map_preimage_multi_pw_aff(map, isl_dim_in, mpa);
}

/* Compute the preimage of "set" by the function represented by "mpa".
 * In other words, plug in "mpa" in "set".
 */
__isl_give isl_set *isl_set_preimage_multi_pw_aff(__isl_take isl_set *set,
        __isl_take isl_multi_pw_aff *mpa)
{
        return isl_map_preimage_multi_pw_aff(set, isl_dim_set, mpa);
}

/* Is the point "inner" internal to inequality constraint "ineq"
 * of "bset"?
 * The point is considered to be internal to the inequality constraint,
 * if it strictly lies on the positive side of the inequality constraint,
 * or if it lies on the constraint and the constraint is lexico-positive.
 */
static isl_bool is_internal(__isl_keep isl_vec *inner,
        __isl_keep isl_basic_set *bset, int ineq)
{
        isl_ctx *ctx;
        int pos;
        unsigned total;

        if (!inner || !bset)
                return isl_bool_error;

        ctx = isl_basic_set_get_ctx(bset);
        isl_seq_inner_product(inner->el, bset->ineq[ineq], inner->size,
                                &ctx->normalize_gcd);
        if (!isl_int_is_zero(ctx->normalize_gcd))
                return isl_int_is_nonneg(ctx->normalize_gcd);

        total = isl_basic_set_dim(bset, isl_dim_all);
        pos = isl_seq_first_non_zero(bset->ineq[ineq] + 1, total);
        return isl_int_is_pos(bset->ineq[ineq][1 + pos]);
}

/* Tighten the inequality constraints of "bset" that are outward with respect
 * to the point "vec".
 * That is, tighten the constraints that are not satisfied by "vec".
 *
 * "vec" is a point internal to some superset S of "bset" that is used
 * to make the subsets of S disjoint, by tightening one half of the constraints
 * that separate two subsets.  In particular, the constraints of S
 * are all satisfied by "vec" and should not be tightened.
 * Of the internal constraints, those that have "vec" on the outside
 * are tightened.  The shared facet is included in the adjacent subset
 * with the opposite constraint.
 * For constraints that saturate "vec", this criterion cannot be used
 * to determine which of the two sides should be tightened.
 * Instead, the sign of the first non-zero coefficient is used
 * to make this choice.  Note that this second criterion is never used
 * on the constraints of S since "vec" is interior to "S".
 */
__isl_give isl_basic_set *isl_basic_set_tighten_outward(
        __isl_take isl_basic_set *bset, __isl_keep isl_vec *vec)
{
        int j;

        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;
        for (j = 0; j < bset->n_ineq; ++j) {
                isl_bool internal;

                internal = is_internal(vec, bset, j);
                if (internal < 0)
                        return isl_basic_set_free(bset);
                if (internal)
                        continue;
                isl_int_sub_ui(bset->ineq[j][0], bset->ineq[j][0], 1);
        }

        return bset;
}