/*

 * Copyright 2008-2009 Katholieke Universiteit Leuven

 * Copyright 2010      INRIA Saclay

 * Copyright 2012      Ecole Normale Superieure

 *

 * 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

 */

#include <isl_ctx_private.h>

#include <isl_map_private.h>

#include <isl_seq.h>

#include <isl/set.h>

#include <isl/lp.h>

#include <isl/map.h>

#include "isl_equalities.h"

#include "isl_sample.h"

#include "isl_tab.h"

#include <isl_mat_private.h>

#include <isl_vec_private.h>

#include <bset_to_bmap.c>

#include <bset_from_bmap.c>

#include <set_to_map.c>

#include <set_from_map.c>

__isl_give isl_basic_map *isl_basic_map_implicit_equalities(

  __isl_take 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 bmap785
;

  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT))

    
return bmap5.13k
;

  if (bmap->n_ineq <= 1)

    return bmap;

  tab = isl_tab_from_basic_map(bmap, 0);

  if (isl_tab_detect_implicit_equalities(tab) < 0)

    goto error;

  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;

error:

  isl_tab_free(tab);

  isl_basic_map_free(bmap);

  return NULL;

}

struct isl_basic_set *isl_basic_set_implicit_equalities(

            struct isl_basic_set *bset)

{

  return bset_from_bmap(

    isl_basic_map_implicit_equalities(bset_to_bmap(bset)));

}

/* 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]))

    
return1.45M
;

  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; 
++r1.86M
)

    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; 
++r1.89M
) {

    if (isl_int_is_zero(bset2->eq[r][col]))

      
continue1.85M
;

    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; 
--t678k
)

    if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col]))

      
break557k
;

  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; 
++i850k
) {

    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;

  if (!bset1 || !bset2)

    goto error;

  total = 1 + isl_basic_set_n_dim(bset1);

  row = 0;

  for (col = total-1; col >= 0; 
--col3.97M
) {

    int is_zero1 = row >= bset1->n_eq ||

      
isl_int_is_zero1.93M
(bset1->eq[row][col]);

    int is_zero2 = row >= bset2->n_eq ||

      
isl_int_is_zero3.03M
(bset2->eq[row][col]);

    if (!is_zero1 && 
!is_zero21.60M
) {

      set_common_multiple(bset1, bset2, row, col);

      ++row;

    } else if (!is_zero1 && 
is_zero21.55k
) {

      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_assert(bset1->ctx, row == bset1->n_eq, goto error);

  isl_basic_set_free(bset2);

  bset1 = isl_basic_set_normalize_constraints(bset1);

  return bset1;

error:

  isl_basic_set_free(bset1);

  isl_basic_set_free(bset2);

  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;

  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->one27.6k
 : 
ctx->negone8.00k
, 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 (sample && isl_tab_rollback(tab, snap) < 0)

    goto error;

  return sample;

error:

  isl_vec_free(sample);

  return NULL;

}

__isl_give isl_basic_set *isl_basic_set_recession_cone(

  __isl_take 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; 361k
i < bset->n_eq; 
++i522
)

    
isl_int_set_si522
(bset->eq[i][0], 0);

  for (i = 0; i < bset->n_ineq; 
++i2.50M
)

    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;

}

/* Move "sample" to a point that is one up (or down) from the original

 * point in dimension "pos".

 */

static void adjacent_point(__isl_keep isl_vec *sample, int pos, int up)

{

  if (up)

    
isl_int_add_ui1.30M
(sample->el[1 + pos], sample->el[1 + pos], 1);

  else

    
isl_int_sub_ui1.30M
(sample->el[1 + pos], sample->el[1 + pos], 1);

}

/* Check if any points that are adjacent to "sample" also belong to "bset".

 * If so, add them to "hull" and return the updated hull.

 *

 * Before checking whether and adjacent point belongs to "bset", we first

 * check whether it already belongs to "hull" as this test is typically

 * much cheaper.

 */

static __isl_give isl_basic_set *add_adjacent_points(

  __isl_take isl_basic_set *hull, __isl_take isl_vec *sample,

  __isl_keep isl_basic_set *bset)

{

  int i, up;

  int dim;

  if (!sample)

    goto error;

  dim = isl_basic_set_dim(hull, isl_dim_set);

  for (i = 0; i < dim; 
++i686k
) {

    for (up = 0; up <= 1; 
++up751k
) {

      int contains;

      isl_basic_set *point;

      adjacent_point(sample, i, up);

      contains = isl_basic_set_contains(hull, sample);

      if (contains < 0)

        goto error;

      if (contains) {

        adjacent_point(sample, i, !up);

        continue;

      }

      contains = isl_basic_set_contains(bset, sample);

      if (contains < 0)

        goto error;

      if (contains) {

        point = isl_basic_set_from_vec(

              isl_vec_copy(sample));

        hull = affine_hull(hull, point);

      }

      adjacent_point(sample, i, !up);

      if (contains)

        break;

    }

  }

  isl_vec_free(sample);

  return hull;

error:

  isl_vec_free(sample);

  isl_basic_set_free(hull);

  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.

 *

 * "bset" may be either NULL or the basic set represented by "tab".

 * If "bset" is not NULL, we check for any point we find if any

 * of its adjacent points also belong to "bset".

 */

static __isl_give isl_basic_set *extend_affine_hull(struct isl_tab *tab,

  __isl_take isl_basic_set *hull, __isl_keep isl_basic_set *bset)

{

  int i, j;

  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; 165k
i < dim; 
++i21.8k
) {

    struct isl_vec *sample;

    struct isl_basic_set *point;

    for (j = 0; j < hull->n_eq; 
++j5.82k
) {

      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);

      if (isl_tab_add_eq(tab, hull->eq[j]) < 0)

        goto error;

    }

    if (j == hull->n_eq)

      break;

    if (tab->samples &&

        
isl_tab_add_sample(tab, isl_vec_copy(sample)) < 01.31k
)

      hull = isl_basic_set_free(hull);

    if (bset)

      hull = add_adjacent_points(hull, isl_vec_copy(sample),

                bset);

    point = isl_basic_set_from_vec(sample);

    hull = affine_hull(hull, point);

    if (!hull)

      return NULL;

  }

  return hull;

error:

  isl_basic_set_free(hull);

  return NULL;

}

/* Construct an initial underapproximation of the hull of "bset"

 * from "sample" and any of its adjacent points that also belong to "bset".

 */

static __isl_give isl_basic_set *initialize_hull(__isl_keep isl_basic_set *bset,

  __isl_take isl_vec *sample)

{

  isl_basic_set *hull;

  hull = isl_basic_set_from_vec(isl_vec_copy(sample));

  hull = add_adjacent_points(hull, sample, bset);

  return hull;

}

/* 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_plain_is_empty(bset))

    return bset;

  dim = isl_basic_set_n_dim(bset);

  if (bset->sample && 
bset->sample->size == 1 + dim93.4k
) {

    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, 1);

  if (!tab)

    goto error;

  if (tab->empty) {

    isl_tab_free(tab);

    isl_vec_free(sample);

    return isl_basic_set_set_to_empty(bset);

  }

  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 = initialize_hull(bset, sample);

  hull = extend_affine_hull(tab, hull, bset);

  isl_basic_set_free(bset);

  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 initial 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; 
++i1.81k
) {

    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 equalities 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 = NULL;

  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; 
++j562
) {

    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, NULL);

  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; 
++j188
) {

      isl_seq_normalize(tab->mat->ctx, hull->eq[j], 1 + tab->n_var);

      if (isl_tab_add_eq(tab, hull->eq[j]) < 0)

        goto error;

    }

  }

  isl_basic_set_free(hull);

  return tab;

error:

  isl_basic_set_free(hull);

  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_alloc6(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 = isl_basic_set_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 + total83.9k
)

    bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample);

  hull = uset_affine_hull_bounded(bset);

  if (!hull) {

    isl_mat_free(Q);

    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 > 0152k
)

      sample = isl_mat_vec_product(U, sample);

    else

      isl_mat_free(U);

    hull = isl_basic_set_preimage(hull, Q);

    if (hull) {

      isl_vec_free(hull->sample);

      hull->sample = sample;

    } else

      isl_vec_free(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_plain_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) {

    isl_space *space;

    space = isl_basic_set_get_space(bset);

    isl_basic_set_free(cone);

    isl_basic_set_free(bset);

    return isl_basic_set_universe(space);

  }

  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);

    isl_mat_free(T2);

  } else {

    struct isl_vec *sample = isl_vec_copy(hull->sample);

    if (sample && 
sample->size > 0117k
)

      sample = isl_mat_vec_product(T1, sample);

    else

      isl_mat_free(T1);

    hull = isl_basic_set_preimage(hull, T2);

    if (hull) {

      isl_vec_free(hull->sample);

      hull->sample = sample;

    } else

      isl_vec_free(sample);

  }

  return hull;

error:

  isl_mat_free(T1);

  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.

 */

__isl_give isl_basic_map *isl_basic_map_detect_equalities(

  __isl_take 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 bmap0
;

  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES))

    
return bmap291k
;

  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))

    
return isl_basic_map_implicit_equalities(bmap)18.1k
;

  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_space(bmap, isl_space_copy(bmap->dim), 0,

          hull->n_eq, 0);

  for (i = 0; i < hull->n_eq; 
++i4.54k
) {

    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 bset_from_bmap(

    isl_basic_map_detect_equalities(bset_to_bmap(bset)));

}

__isl_give isl_map *isl_map_detect_equalities(__isl_take isl_map *map)

{

  return isl_map_inline_foreach_basic_map(map,

              &isl_basic_map_detect_equalities);

}

__isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set)

{

  return set_from_map(isl_map_detect_equalities(set_to_map(set)));

}

/* Return the superset of "bmap" described by the equalities

 * satisfied by "bmap" that are already known.

 */

__isl_give isl_basic_map *isl_basic_map_plain_affine_hull(

  __isl_take isl_basic_map *bmap)

{

  bmap = isl_basic_map_cow(bmap);

  if (bmap)

    isl_basic_map_free_inequality(bmap, bmap->n_ineq);

  bmap = isl_basic_map_finalize(bmap);

  return bmap;

}

/* Return the superset of "bset" described by the equalities

 * satisfied by "bset" that are already known.

 */

__isl_give isl_basic_set *isl_basic_set_plain_affine_hull(

  __isl_take isl_basic_set *bset)

{

  return isl_basic_map_plain_affine_hull(bset);

}

/* 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.

 */

__isl_give isl_basic_map *isl_basic_map_affine_hull(

  __isl_take isl_basic_map *bmap)

{

  bmap = isl_basic_map_detect_equalities(bmap);

  bmap = isl_basic_map_plain_affine_hull(bmap);

  return bmap;

}

struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset)

{

  return bset_from_bmap(isl_basic_map_affine_hull(bset_to_bmap(bset)));

}

/* Given a rational affine matrix "M", add stride constraints to "bmap"

 * that ensure that

 *

 *    M(x)

 *

 * is an integer vector.  The variables x include all the variables

 * of "bmap" except the unknown divs.

 *

 * If d is the common denominator of M, then we need to impose that

 *

 *    d M(x) = 0  mod d

 *

 * or

 *

 *    exists alpha : d M(x) = d alpha

 *

 * This function is similar to add_strides in isl_morph.c

 */

static __isl_give isl_basic_map *add_strides(__isl_take isl_basic_map *bmap,

  __isl_keep isl_mat *M, int n_known)

{

  int i, div, k;

  isl_int gcd;

  if (isl_int_is_one(M->row[0][0]))

    return bmap;

  bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),

          M->n_row - 1, M->n_row - 1, 0);

  isl_int_init(gcd);

  for (i = 1; i < M->n_row; ++i) {

    isl_seq_gcd(M->row[i], M->n_col, &gcd);