add isl_obj_pw_qpolynomial_fold

[isl.git] / isl_affine_hull.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 *
 * Use of this software is governed by the GNU LGPLv2.1 license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 */

#include "isl_ctx.h"
#include "isl_seq.h"
#include "isl_set.h"
#include "isl_lp.h"
#include "isl_map.h"
#include "isl_map_private.h"
#include "isl_equalities.h"
#include "isl_sample.h"
#include "isl_tab.h"

struct isl_basic_map *isl_basic_map_implicit_equalities(
                                                struct isl_basic_map *bmap)
{
        struct isl_tab *tab;

        if (!bmap)
                return bmap;

        bmap = isl_basic_map_gauss(bmap, NULL);
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))
                return bmap;
        if (bmap->n_ineq <= 1)
                return bmap;

        tab = isl_tab_from_basic_map(bmap);
        tab = isl_tab_detect_implicit_equalities(tab);
        bmap = isl_basic_map_update_from_tab(bmap, tab);
        isl_tab_free(tab);
        bmap = isl_basic_map_gauss(bmap, NULL);
        ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
        return bmap;
}

struct isl_basic_set *isl_basic_set_implicit_equalities(
                                                struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
                isl_basic_map_implicit_equalities((struct isl_basic_map*)bset);
}

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

        if (!map)
                return map;

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

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

/* Make eq[row][col] of both bmaps equal so we can add the row
 * add the column to the common matrix.
 * Note that because of the echelon form, the columns of row row
 * after column col are zero.
 */
static void set_common_multiple(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2,
        unsigned row, unsigned col)
{
        isl_int m, c;

        if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col]))
                return;

        isl_int_init(c);
        isl_int_init(m);
        isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]);
        isl_int_divexact(c, m, bset1->eq[row][col]);
        isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1);
        isl_int_divexact(c, m, bset2->eq[row][col]);
        isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1);
        isl_int_clear(c);
        isl_int_clear(m);
}

/* Delete a given equality, moving all the following equalities one up.
 */
static void delete_row(struct isl_basic_set *bset, unsigned row)
{
        isl_int *t;
        int r;

        t = bset->eq[row];
        bset->n_eq--;
        for (r = row; r < bset->n_eq; ++r)
                bset->eq[r] = bset->eq[r+1];
        bset->eq[bset->n_eq] = t;
}

/* Make first row entries in column col of bset1 identical to
 * those of bset2, using the fact that entry bset1->eq[row][col]=a
 * is non-zero.  Initially, these elements of bset1 are all zero.
 * For each row i < row, we set
 *              A[i] = a * A[i] + B[i][col] * A[row]
 *              B[i] = a * B[i]
 * so that
 *              A[i][col] = B[i][col] = a * old(B[i][col])
 */
static void construct_column(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2,
        unsigned row, unsigned col)
{
        int r;
        isl_int a;
        isl_int b;
        unsigned total;

        isl_int_init(a);
        isl_int_init(b);
        total = 1 + isl_basic_set_n_dim(bset1);
        for (r = 0; r < row; ++r) {
                if (isl_int_is_zero(bset2->eq[r][col]))
                        continue;
                isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]);
                isl_int_divexact(a, bset1->eq[row][col], b);
                isl_int_divexact(b, bset2->eq[r][col], b);
                isl_seq_combine(bset1->eq[r], a, bset1->eq[r],
                                              b, bset1->eq[row], total);
                isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total);
        }
        isl_int_clear(a);
        isl_int_clear(b);
        delete_row(bset1, row);
}

/* Make first row entries in column col of bset1 identical to
 * those of bset2, using only these entries of the two matrices.
 * Let t be the last row with different entries.
 * For each row i < t, we set
 *      A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t]
 *      B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t]
 * so that
 *      A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col])
 */
static int transform_column(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2,
        unsigned row, unsigned col)
{
        int i, t;
        isl_int a, b, g;
        unsigned total;

        for (t = row-1; t >= 0; --t)
                if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))
                        break;
        if (t < 0)
                return 0;

        total = 1 + isl_basic_set_n_dim(bset1);
        isl_int_init(a);
        isl_int_init(b);
        isl_int_init(g);
        isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]);
        for (i = 0; i < t; ++i) {
                isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]);
                isl_int_gcd(g, a, b);
                isl_int_divexact(a, a, g);
                isl_int_divexact(g, b, g);
                isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t],
                                total);
                isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t],
                                total);
        }
        isl_int_clear(a);
        isl_int_clear(b);
        isl_int_clear(g);
        delete_row(bset1, t);
        delete_row(bset2, t);
        return 1;
}

/* The implementation is based on Section 5.2 of Michael Karr,
 * "Affine Relationships Among Variables of a Program",
 * except that the echelon form we use starts from the last column
 * and that we are dealing with integer coefficients.
 */
static struct isl_basic_set *affine_hull(
        struct isl_basic_set *bset1, struct isl_basic_set *bset2)
{
        unsigned total;
        int col;
        int row;

        total = 1 + isl_basic_set_n_dim(bset1);

        row = 0;
        for (col = total-1; col >= 0; --col) {
                int is_zero1 = row >= bset1->n_eq ||
                        isl_int_is_zero(bset1->eq[row][col]);
                int is_zero2 = row >= bset2->n_eq ||
                        isl_int_is_zero(bset2->eq[row][col]);
                if (!is_zero1 && !is_zero2) {
                        set_common_multiple(bset1, bset2, row, col);
                        ++row;
                } else if (!is_zero1 && is_zero2) {
                        construct_column(bset1, bset2, row, col);
                } else if (is_zero1 && !is_zero2) {
                        construct_column(bset2, bset1, row, col);
                } else {
                        if (transform_column(bset1, bset2, row, col))
                                --row;
                }
        }
        isl_basic_set_free(bset2);
        isl_assert(bset1->ctx, row == bset1->n_eq, goto error);
        bset1 = isl_basic_set_normalize_constraints(bset1);
        return bset1;
error:
        isl_basic_set_free(bset1);
        return NULL;
}

/* Find an integer point in the set represented by "tab"
 * that lies outside of the equality "eq" e(x) = 0.
 * If "up" is true, look for a point satisfying e(x) - 1 >= 0.
 * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1).
 * The point, if found, is returned.
 * If no point can be found, a zero-length vector is returned.
 *
 * Before solving an ILP problem, we first check if simply
 * adding the normal of the constraint to one of the known
 * integer points in the basic set represented by "tab"
 * yields another point inside the basic set.
 *
 * The caller of this function ensures that the tableau is bounded or
 * that tab->basis and tab->n_unbounded have been set appropriately.
 */
static struct isl_vec *outside_point(struct isl_tab *tab, isl_int *eq, int up)
{
        struct isl_ctx *ctx;
        struct isl_vec *sample = NULL;
        struct isl_tab_undo *snap;
        unsigned dim;
        int k;

        if (!tab)
                return NULL;
        ctx = tab->mat->ctx;

        dim = tab->n_var;
        sample = isl_vec_alloc(ctx, 1 + dim);
        if (!sample)
                return NULL;
        isl_int_set_si(sample->el[0], 1);
        isl_seq_combine(sample->el + 1,
                ctx->one, tab->bmap->sample->el + 1,
                up ? ctx->one : ctx->negone, eq + 1, dim);
        if (isl_basic_map_contains(tab->bmap, sample))
                return sample;
        isl_vec_free(sample);
        sample = NULL;

        snap = isl_tab_snap(tab);

        if (!up)
                isl_seq_neg(eq, eq, 1 + dim);
        isl_int_sub_ui(eq[0], eq[0], 1);

        if (isl_tab_extend_cons(tab, 1) < 0)
                goto error;
        if (isl_tab_add_ineq(tab, eq) < 0)
                goto error;

        sample = isl_tab_sample(tab);

        isl_int_add_ui(eq[0], eq[0], 1);
        if (!up)
                isl_seq_neg(eq, eq, 1 + dim);

        if (isl_tab_rollback(tab, snap) < 0)
                goto error;

        return sample;
error:
        isl_vec_free(sample);
        return NULL;
}

struct isl_basic_set *isl_basic_set_recession_cone(struct isl_basic_set *bset)
{
        int i;

        bset = isl_basic_set_cow(bset);
        if (!bset)
                return NULL;
        isl_assert(bset->ctx, bset->n_div == 0, goto error);

        for (i = 0; i < bset->n_eq; ++i)
                isl_int_set_si(bset->eq[i][0], 0);

        for (i = 0; i < bset->n_ineq; ++i)
                isl_int_set_si(bset->ineq[i][0], 0);

        ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT);
        return isl_basic_set_implicit_equalities(bset);
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Extend an initial (under-)approximation of the affine hull of basic
 * set represented by the tableau "tab"
 * by looking for points that do not satisfy one of the equalities
 * in the current approximation and adding them to that approximation
 * until no such points can be found any more.
 *
 * The caller of this function ensures that "tab" is bounded or
 * that tab->basis and tab->n_unbounded have been set appropriately.
 */
static struct isl_basic_set *extend_affine_hull(struct isl_tab *tab,
        struct isl_basic_set *hull)
{
        int i, j, k;
        unsigned dim;

        if (!tab || !hull)
                goto error;

        dim = tab->n_var;

        if (isl_tab_extend_cons(tab, 2 * dim + 1) < 0)
                goto error;

        for (i = 0; i < dim; ++i) {
                struct isl_vec *sample;
                struct isl_basic_set *point;
                for (j = 0; j < hull->n_eq; ++j) {
                        sample = outside_point(tab, hull->eq[j], 1);
                        if (!sample)
                                goto error;
                        if (sample->size > 0)
                                break;
                        isl_vec_free(sample);
                        sample = outside_point(tab, hull->eq[j], 0);
                        if (!sample)
                                goto error;
                        if (sample->size > 0)
                                break;
                        isl_vec_free(sample);

                        tab = isl_tab_add_eq(tab, hull->eq[j]);
                        if (!tab)
                                goto error;
                }
                if (j == hull->n_eq)
                        break;
                if (tab->samples)
                        tab = isl_tab_add_sample(tab, isl_vec_copy(sample));
                if (!tab)
                        goto error;
                point = isl_basic_set_from_vec(sample);
                hull = affine_hull(hull, point);
        }

        return hull;
error:
        isl_basic_set_free(hull);
        return NULL;
}

/* Drop all constraints in bset that involve any of the dimensions
 * first to first+n-1.
 */
static struct isl_basic_set *drop_constraints_involving
        (struct isl_basic_set *bset, unsigned first, unsigned n)
{
        int i;

        if (!bset)
                return NULL;

        bset = isl_basic_set_cow(bset);

        for (i = bset->n_eq - 1; i >= 0; --i) {
                if (isl_seq_first_non_zero(bset->eq[i] + 1 + first, n) == -1)
                        continue;
                isl_basic_set_drop_equality(bset, i);
        }

        for (i = bset->n_ineq - 1; i >= 0; --i) {
                if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
                        continue;
                isl_basic_set_drop_inequality(bset, i);
        }

        return bset;
}

/* Look for all equalities satisfied by the integer points in bset,
 * which is assumed to be bounded.
 *
 * The equalities are obtained by successively looking for
 * a point that is affinely independent of the points found so far.
 * In particular, for each equality satisfied by the points so far,
 * we check if there is any point on a hyperplane parallel to the
 * corresponding hyperplane shifted by at least one (in either direction).
 */
static struct isl_basic_set *uset_affine_hull_bounded(struct isl_basic_set *bset)
{
        struct isl_vec *sample = NULL;
        struct isl_basic_set *hull;
        struct isl_tab *tab = NULL;
        unsigned dim;

        if (isl_basic_set_fast_is_empty(bset))
                return bset;

        dim = isl_basic_set_n_dim(bset);

        if (bset->sample && bset->sample->size == 1 + dim) {
                int contains = isl_basic_set_contains(bset, bset->sample);
                if (contains < 0)
                        goto error;
                if (contains) {
                        if (dim == 0)
                                return bset;
                        sample = isl_vec_copy(bset->sample);
                } else {
                        isl_vec_free(bset->sample);
                        bset->sample = NULL;
                }
        }

        tab = isl_tab_from_basic_set(bset);
        if (!tab)
                goto error;
        if (tab->empty) {
                isl_tab_free(tab);
                isl_vec_free(sample);
                return isl_basic_set_set_to_empty(bset);
        }
        if (isl_tab_track_bset(tab, isl_basic_set_copy(bset)) < 0)
                goto error;

        if (!sample) {
                struct isl_tab_undo *snap;
                snap = isl_tab_snap(tab);
                sample = isl_tab_sample(tab);
                if (isl_tab_rollback(tab, snap) < 0)
                        goto error;
                isl_vec_free(tab->bmap->sample);
                tab->bmap->sample = isl_vec_copy(sample);
        }

        if (!sample)
                goto error;
        if (sample->size == 0) {
                isl_tab_free(tab);
                isl_vec_free(sample);
                return isl_basic_set_set_to_empty(bset);
        }

        hull = isl_basic_set_from_vec(sample);

        isl_basic_set_free(bset);
        hull = extend_affine_hull(tab, hull);
        isl_tab_free(tab);

        return hull;
error:
        isl_vec_free(sample);
        isl_tab_free(tab);
        isl_basic_set_free(bset);
        return NULL;
}

/* Given an unbounded tableau and an integer point satisfying the tableau,
 * construct an intial affine hull containing the recession cone
 * shifted to the given point.
 *
 * The unbounded directions are taken from the last rows of the basis,
 * which is assumed to have been initialized appropriately.
 */
static __isl_give isl_basic_set *initial_hull(struct isl_tab *tab,
        __isl_take isl_vec *vec)
{
        int i;
        int k;
        struct isl_basic_set *bset = NULL;
        struct isl_ctx *ctx;
        unsigned dim;

        if (!vec || !tab)
                return NULL;
        ctx = vec->ctx;
        isl_assert(ctx, vec->size != 0, goto error);

        bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
        if (!bset)
                goto error;
        dim = isl_basic_set_n_dim(bset) - tab->n_unbounded;
        for (i = 0; i < dim; ++i) {
                k = isl_basic_set_alloc_equality(bset);
                if (k < 0)
                        goto error;
                isl_seq_cpy(bset->eq[k] + 1, tab->basis->row[1 + i] + 1,
                            vec->size - 1);
                isl_seq_inner_product(bset->eq[k] + 1, vec->el +1,
                                      vec->size - 1, &bset->eq[k][0]);
                isl_int_neg(bset->eq[k][0], bset->eq[k][0]);
        }
        bset->sample = vec;
        bset = isl_basic_set_gauss(bset, NULL);

        return bset;
error:
        isl_basic_set_free(bset);
        isl_vec_free(vec);
        return NULL;
}

/* Given a tableau of a set and a tableau of the corresponding
 * recession cone, detect and add all equalities to the tableau.
 * If the tableau is bounded, then we can simply keep the
 * tableau in its state after the return from extend_affine_hull.
 * However, if the tableau is unbounded, then
 * isl_tab_set_initial_basis_with_cone will add some additional
 * constraints to the tableau that have to be removed again.
 * In this case, we therefore rollback to the state before
 * any constraints were added and then add the eqaulities back in.
 */
struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab,
        struct isl_tab *tab_cone)
{
        int j;
        struct isl_vec *sample;
        struct isl_basic_set *hull;
        struct isl_tab_undo *snap;

        if (!tab || !tab_cone)
                goto error;

        snap = isl_tab_snap(tab);

        isl_mat_free(tab->basis);
        tab->basis = NULL;

        isl_assert(tab->mat->ctx, tab->bmap, goto error);
        isl_assert(tab->mat->ctx, tab->samples, goto error);
        isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
        isl_assert(tab->mat->ctx, tab->n_sample > tab->n_outside, goto error);

        if (isl_tab_set_initial_basis_with_cone(tab, tab_cone) < 0)
                goto error;

        sample = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
        if (!sample)
                goto error;

        isl_seq_cpy(sample->el, tab->samples->row[tab->n_outside], sample->size);

        isl_vec_free(tab->bmap->sample);
        tab->bmap->sample = isl_vec_copy(sample);

        if (tab->n_unbounded == 0)
                hull = isl_basic_set_from_vec(isl_vec_copy(sample));
        else
                hull = initial_hull(tab, isl_vec_copy(sample));

        for (j = tab->n_outside + 1; j < tab->n_sample; ++j) {
                isl_seq_cpy(sample->el, tab->samples->row[j], sample->size);
                hull = affine_hull(hull,
                                isl_basic_set_from_vec(isl_vec_copy(sample)));
        }

        isl_vec_free(sample);

        hull = extend_affine_hull(tab, hull);
        if (!hull)
                goto error;

        if (tab->n_unbounded == 0) {
                isl_basic_set_free(hull);
                return tab;
        }

        if (isl_tab_rollback(tab, snap) < 0)
                goto error;

        if (hull->n_eq > tab->n_zero) {
                for (j = 0; j < hull->n_eq; ++j) {
                        isl_seq_normalize(tab->mat->ctx, hull->eq[j], 1 + tab->n_var);
                        tab = isl_tab_add_eq(tab, hull->eq[j]);
                }
        }

        isl_basic_set_free(hull);

        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

/* Compute the affine hull of "bset", where "cone" is the recession cone
 * of "bset".
 *
 * We first compute a unimodular transformation that puts the unbounded
 * directions in the last dimensions.  In particular, we take a transformation
 * that maps all equalities to equalities (in HNF) on the first dimensions.
 * Let x be the original dimensions and y the transformed, with y_1 bounded
 * and y_2 unbounded.
 *
 *             [ y_1 ]                  [ y_1 ]   [ Q_1 ]
 *      x = U  [ y_2 ]                  [ y_2 ] = [ Q_2 ] x
 *
 * Let's call the input basic set S.  We compute S' = preimage(S, U)
 * and drop the final dimensions including any constraints involving them.
 * This results in set S''.
 * Then we compute the affine hull A'' of S''.
 * Let F y_1 >= g be the constraint system of A''.  In the transformed
 * space the y_2 are unbounded, so we can add them back without any constraints,
 * resulting in
 *
 *                      [ y_1 ]
 *              [ F 0 ] [ y_2 ] >= g
 * or
 *                      [ Q_1 ]
 *              [ F 0 ] [ Q_2 ] x >= g
 * or
 *              F Q_1 x >= g
 *
 * The affine hull in the original space is then obtained as
 * A = preimage(A'', Q_1).
 */
static struct isl_basic_set *affine_hull_with_cone(struct isl_basic_set *bset,
        struct isl_basic_set *cone)
{
        unsigned total;
        unsigned cone_dim;
        struct isl_basic_set *hull;
        struct isl_mat *M, *U, *Q;

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

        total = isl_basic_set_total_dim(cone);
        cone_dim = total - cone->n_eq;

        M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
        M = isl_mat_left_hermite(M, 0, &U, &Q);
        if (!M)
                goto error;
        isl_mat_free(M);

        U = isl_mat_lin_to_aff(U);
        bset = isl_basic_set_preimage(bset, isl_mat_copy(U));

        bset = drop_constraints_involving(bset, total - cone_dim, cone_dim);
        bset = isl_basic_set_drop_dims(bset, total - cone_dim, cone_dim);

        Q = isl_mat_lin_to_aff(Q);
        Q = isl_mat_drop_rows(Q, 1 + total - cone_dim, cone_dim);

        if (bset && bset->sample && bset->sample->size == 1 + total)
                bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample);

        hull = uset_affine_hull_bounded(bset);

        if (!hull)
                isl_mat_free(U);
        else {
                struct isl_vec *sample = isl_vec_copy(hull->sample);
                U = isl_mat_drop_cols(U, 1 + total - cone_dim, cone_dim);
                if (sample && sample->size > 0)
                        sample = isl_mat_vec_product(U, sample);
                else
                        isl_mat_free(U);
                hull = isl_basic_set_preimage(hull, Q);
                isl_vec_free(hull->sample);
                hull->sample = sample;
        }

        isl_basic_set_free(cone);

        return hull;
error:
        isl_basic_set_free(bset);
        isl_basic_set_free(cone);
        return NULL;
}

/* Look for all equalities satisfied by the integer points in bset,
 * which is assumed not to have any explicit equalities.
 *
 * The equalities are obtained by successively looking for
 * a point that is affinely independent of the points found so far.
 * In particular, for each equality satisfied by the points so far,
 * we check if there is any point on a hyperplane parallel to the
 * corresponding hyperplane shifted by at least one (in either direction).
 *
 * Before looking for any outside points, we first compute the recession
 * cone.  The directions of this recession cone will always be part
 * of the affine hull, so there is no need for looking for any points
 * in these directions.
 * In particular, if the recession cone is full-dimensional, then
 * the affine hull is simply the whole universe.
 */
static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset)
{
        struct isl_basic_set *cone;

        if (isl_basic_set_fast_is_empty(bset))
                return bset;

        cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
        if (!cone)
                goto error;
        if (cone->n_eq == 0) {
                struct isl_basic_set *hull;
                isl_basic_set_free(cone);
                hull = isl_basic_set_universe_like(bset);
                isl_basic_set_free(bset);
                return hull;
        }

        if (cone->n_eq < isl_basic_set_total_dim(cone))
                return affine_hull_with_cone(bset, cone);

        isl_basic_set_free(cone);
        return uset_affine_hull_bounded(bset);
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Look for all equalities satisfied by the integer points in bmap
 * that are independent of the equalities already explicitly available
 * in bmap.
 *
 * We first remove all equalities already explicitly available,
 * then look for additional equalities in the reduced space
 * and then transform the result to the original space.
 * The original equalities are _not_ added to this set.  This is
 * the responsibility of the calling function.
 * The resulting basic set has all meaning about the dimensions removed.
 * In particular, dimensions that correspond to existential variables
 * in bmap and that are found to be fixed are not removed.
 */
static struct isl_basic_set *equalities_in_underlying_set(
                                                struct isl_basic_map *bmap)
{
        struct isl_mat *T1 = NULL;
        struct isl_mat *T2 = NULL;
        struct isl_basic_set *bset = NULL;
        struct isl_basic_set *hull = NULL;

        bset = isl_basic_map_underlying_set(bmap);
        if (!bset)
                return NULL;
        if (bset->n_eq)
                bset = isl_basic_set_remove_equalities(bset, &T1, &T2);
        if (!bset)
                goto error;

        hull = uset_affine_hull(bset);
        if (!T2)
                return hull;

        if (!hull)
                isl_mat_free(T1);
        else {
                struct isl_vec *sample = isl_vec_copy(hull->sample);
                if (sample && sample->size > 0)
                        sample = isl_mat_vec_product(T1, sample);
                else
                        isl_mat_free(T1);
                hull = isl_basic_set_preimage(hull, T2);
                isl_vec_free(hull->sample);
                hull->sample = sample;
        }

        return hull;
error:
        isl_mat_free(T2);
        isl_basic_set_free(bset);
        isl_basic_set_free(hull);
        return NULL;
}

/* Detect and make explicit all equalities satisfied by the (integer)
 * points in bmap.
 */
struct isl_basic_map *isl_basic_map_detect_equalities(
                                                struct isl_basic_map *bmap)
{
        int i, j;
        struct isl_basic_set *hull = NULL;

        if (!bmap)
                return NULL;
        if (bmap->n_ineq == 0)
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))
                return bmap;
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
                return isl_basic_map_implicit_equalities(bmap);

        hull = equalities_in_underlying_set(isl_basic_map_copy(bmap));
        if (!hull)
                goto error;
        if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) {
                isl_basic_set_free(hull);
                return isl_basic_map_set_to_empty(bmap);
        }
        bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim), 0,
                                        hull->n_eq, 0);
        for (i = 0; i < hull->n_eq; ++i) {
                j = isl_basic_map_alloc_equality(bmap);
                if (j < 0)
                        goto error;
                isl_seq_cpy(bmap->eq[j], hull->eq[i],
                                1 + isl_basic_set_total_dim(hull));
        }
        isl_vec_free(bmap->sample);
        bmap->sample = isl_vec_copy(hull->sample);
        isl_basic_set_free(hull);
        ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES);
        bmap = isl_basic_map_simplify(bmap);
        return isl_basic_map_finalize(bmap);
error:
        isl_basic_set_free(hull);
        isl_basic_map_free(bmap);
        return NULL;
}

__isl_give isl_basic_set *isl_basic_set_detect_equalities(
                                                __isl_take isl_basic_set *bset)
{
        return (isl_basic_set *)
                isl_basic_map_detect_equalities((isl_basic_map *)bset);
}

struct isl_map *isl_map_detect_equalities(struct isl_map *map)
{
        struct isl_basic_map *bmap;
        int i;

        if (!map)
                return NULL;

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

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

__isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set)
{
        return (isl_set *)isl_map_detect_equalities((isl_map *)set);
}

/* After computing the rational affine hull (by detecting the implicit
 * equalities), we compute the additional equalities satisfied by
 * the integer points (if any) and add the original equalities back in.
 */
struct isl_basic_map *isl_basic_map_affine_hull(struct isl_basic_map *bmap)
{
        bmap = isl_basic_map_detect_equalities(bmap);
        bmap = isl_basic_map_cow(bmap);
        isl_basic_map_free_inequality(bmap, bmap->n_ineq);
        return bmap;
}

struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)
{
        return (struct isl_basic_set *)
                isl_basic_map_affine_hull((struct isl_basic_map *)bset);
}

struct isl_basic_map *isl_map_affine_hull(struct isl_map *map)
{
        int i;
        struct isl_basic_map *model = NULL;
        struct isl_basic_map *hull = NULL;
        struct isl_set *set;

        if (!map)
                return NULL;

        if (map->n == 0) {
                hull = isl_basic_map_empty_like_map(map);
                isl_map_free(map);
                return hull;
        }

        map = isl_map_detect_equalities(map);
        map = isl_map_align_divs(map);
        if (!map)
                return NULL;
        model = isl_basic_map_copy(map->p[0]);
        set = isl_map_underlying_set(map);
        set = isl_set_cow(set);
        if (!set)
                goto error;

        for (i = 0; i < set->n; ++i) {
                set->p[i] = isl_basic_set_cow(set->p[i]);
                set->p[i] = isl_basic_set_affine_hull(set->p[i]);
                set->p[i] = isl_basic_set_gauss(set->p[i], NULL);
                if (!set->p[i])
                        goto error;
        }
        set = isl_set_remove_empty_parts(set);
        if (set->n == 0) {
                hull = isl_basic_map_empty_like(model);
                isl_basic_map_free(model);
        } else {
                struct isl_basic_set *bset;
                while (set->n > 1) {
                        set->p[0] = affine_hull(set->p[0], set->p[--set->n]);
                        if (!set->p[0])
                                goto error;
                }
                bset = isl_basic_set_copy(set->p[0]);
                hull = isl_basic_map_overlying_set(bset, model);
        }
        isl_set_free(set);
        hull = isl_basic_map_simplify(hull);
        return isl_basic_map_finalize(hull);
error:
        isl_basic_map_free(model);
        isl_set_free(set);
        return NULL;
}

struct isl_basic_set *isl_set_affine_hull(struct isl_set *set)
{
        return (struct isl_basic_set *)
                isl_map_affine_hull((struct isl_map *)set);
}