isl_map_simplify.c

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/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2012-2013 Ecole Normale Superieure
 * Copyright 2014-2015 INRIA Rocquencourt
 * Copyright 2016      Sven Verdoolaege
 * Copyright 2021,2023 Cerebras Systems
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
 * B.P. 105 - 78153 Le Chesnay, France
 * and Cerebras Systems, 1237 E Arques Ave, Sunnyvale, CA, USA
 */
 
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include "isl_equalities.h"
#include <isl/map.h>
#include <isl_seq.h>
#include "isl_tab.h"
#include <isl_space_private.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
 
#include <bset_to_bmap.c>
#include <bset_from_bmap.c>
#include <set_to_map.c>
#include <set_from_map.c>
 
/* Mark "bmap" as having one or more inequality constraints modified.
 * If "equivalent" is set, then this modification was done based
 * on an equality constraint already available in "bmap".
 *
 * Any modification may result in the constraints no longer being sorted and
 * may also undo the effect of reduce_coefficients.
 *
 * A modification that uses extra information may also result
 * in the modified constraint(s) becoming redundant or
 * turning into an implicit equality constraint.
 */
static __isl_give isl_basic_map *isl_basic_map_modify_inequality(
  __isl_take isl_basic_map *bmap, int equivalent)
{
  if (!bmap)
    return NULL;
  ISL_F_CLR(bmap, ISL_BASIC_MAP_SORTED);
  ISL_F_CLR(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS);
  if (equivalent)
    return bmap;
  ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
  ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_IMPLICIT);
  return bmap;
}
 
static void swap_equality(__isl_keep isl_basic_map *bmap, int a, int b)
{
  isl_int *t = bmap->eq[a];
  bmap->eq[a] = bmap->eq[b];
  bmap->eq[b] = t;
}
 
static void swap_inequality(__isl_keep isl_basic_map *bmap, int a, int b)
{
  if (a != b) {
    isl_int *t = bmap->ineq[a];
    bmap->ineq[a] = bmap->ineq[b];
    bmap->ineq[b] = t;
  }
}
 
/* Scale down the inequality constraint "ineq" of length "len"
 * by a factor of "f".
 * All the coefficients, except the constant term,
 * are assumed to be multiples of "f".
 */
static void scale_down_inequality(isl_int *ineq, isl_int f, unsigned len)
{
  isl_int_fdiv_q(ineq[0], ineq[0], f);
  isl_seq_scale_down(ineq + 1, ineq + 1, f, len);
}
 
/* Scale down the inequality constraint "ineq" of length "len"
 * by a factor of "f".
 * All the coefficients, except the constant term,
 * are assumed to be multiples of "f".
 *
 * If the factor is 0 or 1, then no scaling needs to be performed.
 *
 * If scaling is performed then take into account that the constraint
 * is modified (not simply based on an equality constraint).
 */
static __isl_give isl_basic_map *scale_down_bmap_inequality(
  __isl_take isl_basic_map *bmap, int ineq, isl_int f, unsigned len)
{
  if (!bmap)
    return NULL;
 
  if (isl_int_is_zero(f) || isl_int_is_one(f))
    return bmap;
 
  scale_down_inequality(bmap->ineq[ineq], f, len);
 
  bmap = isl_basic_map_modify_inequality(bmap, 0);
 
  return bmap;
}
 
__isl_give isl_basic_map *isl_basic_map_normalize_constraints(
  __isl_take isl_basic_map *bmap)
{
  int i;
  isl_int gcd;
  isl_size total = isl_basic_map_dim(bmap, isl_dim_all);
 
  if (total < 0)
    return isl_basic_map_free(bmap);
 
  isl_int_init(gcd);
  for (i = bmap->n_eq - 1; i >= 0; --i) {
    isl_seq_gcd(bmap->eq[i]+1, total, &gcd);
    if (isl_int_is_zero(gcd)) {
      if (!isl_int_is_zero(bmap->eq[i][0])) {
        bmap = isl_basic_map_set_to_empty(bmap);
        break;
      }
      if (isl_basic_map_drop_equality(bmap, i) < 0)
        goto error;
      continue;
    }
    if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
      isl_int_gcd(gcd, gcd, bmap->eq[i][0]);
    if (isl_int_is_one(gcd))
      continue;
    if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) {
      bmap = isl_basic_map_set_to_empty(bmap);
      break;
    }
    isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total);
  }
 
  for (i = bmap->n_ineq - 1; i >= 0; --i) {
    isl_seq_gcd(bmap->ineq[i]+1, total, &gcd);
    if (isl_int_is_zero(gcd)) {
      if (isl_int_is_neg(bmap->ineq[i][0])) {
        bmap = isl_basic_map_set_to_empty(bmap);
        break;
      }
      if (isl_basic_map_drop_inequality(bmap, i) < 0)
        goto error;
      continue;
    }
    if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL))
      isl_int_gcd(gcd, gcd, bmap->ineq[i][0]);
    bmap = scale_down_bmap_inequality(bmap, i, gcd, total);
    if (!bmap)
      goto error;
  }
  isl_int_clear(gcd);
 
  return bmap;
error:
  isl_int_clear(gcd);
  isl_basic_map_free(bmap);
  return NULL;
}
 
__isl_give isl_basic_set *isl_basic_set_normalize_constraints(
  __isl_take isl_basic_set *bset)
{
  isl_basic_map *bmap = bset_to_bmap(bset);
  return bset_from_bmap(isl_basic_map_normalize_constraints(bmap));
}
 
/* Reduce the coefficient of the variable at position "pos"
 * in integer division "div", such that it lies in the half-open
 * interval (1/2,1/2], extracting any excess value from this integer division.
 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
 * corresponds to the constant term.
 *
 * That is, the integer division is of the form
 *
 *  floor((... + (c * d + r) * x_pos + ...)/d)
 *
 * with -d < 2 * r <= d.
 * Replace it by
 *
 *  floor((... + r * x_pos + ...)/d) + c * x_pos
 *
 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
 * Otherwise, c = floor((c * d + r)/d) + 1.
 *
 * This is the same normalization that is performed by isl_aff_floor.
 */
static __isl_give isl_basic_map *reduce_coefficient_in_div(
  __isl_take isl_basic_map *bmap, int div, int pos)
{
  isl_int shift;
  int add_one;
 
  isl_int_init(shift);
  isl_int_fdiv_r(shift, bmap->div[div][1 + pos], bmap->div[div][0]);
  isl_int_mul_ui(shift, shift, 2);
  add_one = isl_int_gt(shift, bmap->div[div][0]);
  isl_int_fdiv_q(shift, bmap->div[div][1 + pos], bmap->div[div][0]);
  if (add_one)
    isl_int_add_ui(shift, shift, 1);
  isl_int_neg(shift, shift);
  bmap = isl_basic_map_shift_div(bmap, div, pos, shift);
  isl_int_clear(shift);
 
  return bmap;
}
 
/* Does the coefficient of the variable at position "pos"
 * in integer division "div" need to be reduced?
 * That is, does it lie outside the half-open interval (1/2,1/2]?
 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
 * 2 * c != d.
 */
static isl_bool needs_reduction(__isl_keep isl_basic_map *bmap, int div,
  int pos)
{
  isl_bool r;
 
  if (isl_int_is_zero(bmap->div[div][1 + pos]))
    return isl_bool_false;
 
  isl_int_mul_ui(bmap->div[div][1 + pos], bmap->div[div][1 + pos], 2);
  r = isl_int_abs_ge(bmap->div[div][1 + pos], bmap->div[div][0]) &&
      !isl_int_eq(bmap->div[div][1 + pos], bmap->div[div][0]);
  isl_int_divexact_ui(bmap->div[div][1 + pos],
          bmap->div[div][1 + pos], 2);
 
  return r;
}
 
/* Reduce the coefficients (including the constant term) of
 * integer division "div", if needed.
 * In particular, make sure all coefficients lie in
 * the half-open interval (1/2,1/2].
 */
static __isl_give isl_basic_map *reduce_div_coefficients_of_div(
  __isl_take isl_basic_map *bmap, int div)
{
  int i;
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_basic_map_free(bmap);
  for (i = 0; i < 1 + total; ++i) {
    isl_bool reduce;
 
    reduce = needs_reduction(bmap, div, i);
    if (reduce < 0)
      return isl_basic_map_free(bmap);
    if (!reduce)
      continue;
    bmap = reduce_coefficient_in_div(bmap, div, i);
    if (!bmap)
      break;
  }
 
  return bmap;
}
 
/* Reduce the coefficients (including the constant term) of
 * the known integer divisions, if needed
 * In particular, make sure all coefficients lie in
 * the half-open interval (1/2,1/2].
 */
static __isl_give isl_basic_map *reduce_div_coefficients(
  __isl_take isl_basic_map *bmap)
{
  int i;
 
  if (!bmap)
    return NULL;
  if (bmap->n_div == 0)
    return bmap;
 
  for (i = 0; i < bmap->n_div; ++i) {
    if (isl_int_is_zero(bmap->div[i][0]))
      continue;
    bmap = reduce_div_coefficients_of_div(bmap, i);
    if (!bmap)
      break;
  }
 
  return bmap;
}
 
/* Remove any common factor in numerator and denominator of the div expression,
 * not taking into account the constant term.
 * That is, if the div is of the form
 *
 *  floor((a + m f(x))/(m d))
 *
 * then replace it by
 *
 *  floor((floor(a/m) + f(x))/d)
 *
 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
 * and can therefore not influence the result of the floor.
 */
static __isl_give isl_basic_map *normalize_div_expression(
  __isl_take isl_basic_map *bmap, int div)
{
  isl_size total = isl_basic_map_dim(bmap, isl_dim_all);
  isl_ctx *ctx = bmap->ctx;
 
  if (total < 0)
    return isl_basic_map_free(bmap);
  if (isl_int_is_zero(bmap->div[div][0]))
    return bmap;
  isl_seq_gcd(bmap->div[div] + 2, total, &ctx->normalize_gcd);
  isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, bmap->div[div][0]);
  if (isl_int_is_one(ctx->normalize_gcd))
    return bmap;
  isl_int_fdiv_q(bmap->div[div][1], bmap->div[div][1],
      ctx->normalize_gcd);
  isl_int_divexact(bmap->div[div][0], bmap->div[div][0],
      ctx->normalize_gcd);
  isl_seq_scale_down(bmap->div[div] + 2, bmap->div[div] + 2,
      ctx->normalize_gcd, total);
 
  return bmap;
}
 
/* Remove any common factor in numerator and denominator of a div expression,
 * not taking into account the constant term.
 * That is, look for any div of the form
 *
 *  floor((a + m f(x))/(m d))
 *
 * and replace it by
 *
 *  floor((floor(a/m) + f(x))/d)
 *
 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
 * and can therefore not influence the result of the floor.
 */
static __isl_give isl_basic_map *normalize_div_expressions(
  __isl_take isl_basic_map *bmap)
{
  int i;
 
  if (!bmap)
    return NULL;
  if (bmap->n_div == 0)
    return bmap;
 
  for (i = 0; i < bmap->n_div; ++i)
    bmap = normalize_div_expression(bmap, i);
 
  return bmap;
}
 
/* Some progress has been made.
 * Set *progress if "progress" is not NULL.
 */
static void mark_progress(int *progress)
{
  if (progress)
    *progress = 1;
}
 
/* Eliminate the variable at position "pos" from the constraints of "bmap"
 * using the equality constraint "eq".
 * If "keep_divs" is set, then try and preserve
 * the integer division expressions.  In this case, these expressions
 * are assumed to have been ordered.
 * If "equivalent" is set, then the elimination is performed
 * using an equality constraint of "bmap", meaning that the meaning
 * of the constraints is preserved.
 */
static __isl_give isl_basic_map *eliminate_var_using_equality(
  __isl_take isl_basic_map *bmap,
  unsigned pos, isl_int *eq, int keep_divs, int equivalent, int *progress)
{
  isl_size total;
  isl_size v_div;
  int k;
  int last_div;
  isl_ctx *ctx;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (total < 0 || v_div < 0)
    return isl_basic_map_free(bmap);
  ctx = isl_basic_map_get_ctx(bmap);
  last_div = isl_seq_last_non_zero(eq + 1 + v_div, bmap->n_div);
  for (k = 0; k < bmap->n_eq; ++k) {
    if (bmap->eq[k] == eq)
      continue;
    if (isl_int_is_zero(bmap->eq[k][1+pos]))
      continue;
    mark_progress(progress);
    isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL);
    isl_seq_normalize(ctx, bmap->eq[k], 1 + total);
  }
 
  for (k = 0; k < bmap->n_ineq; ++k) {
    if (isl_int_is_zero(bmap->ineq[k][1+pos]))
      continue;
    mark_progress(progress);
    isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL);
    isl_seq_gcd(bmap->ineq[k], 1 + total, &ctx->normalize_gcd);
    bmap = scale_down_bmap_inequality(bmap, k, ctx->normalize_gcd,
            total);
    bmap = isl_basic_map_modify_inequality(bmap, equivalent);
    if (!bmap)
      return NULL;
  }
 
  for (k = 0; k < bmap->n_div; ++k) {
    if (isl_int_is_zero(bmap->div[k][0]))
      continue;
    if (isl_int_is_zero(bmap->div[k][1+1+pos]))
      continue;
    mark_progress(progress);
    /* We need to be careful about circular definitions,
     * so for now we just remove the definition of div k
     * if the equality contains any divs.
     * If keep_divs is set, then the divs have been ordered
     * and we can keep the definition as long as the result
     * is still ordered.
     */
    if (last_div == -1 || (keep_divs && last_div < k)) {
      isl_seq_elim(bmap->div[k]+1, eq,
          1+pos, 1+total, &bmap->div[k][0]);
      bmap = normalize_div_expression(bmap, k);
      if (!bmap)
        return NULL;
    } else
      isl_seq_clr(bmap->div[k], 1 + total);
  }
 
  return bmap;
}
 
/* Eliminate and remove the local variable at position "pos" of "bmap"
 * using the equality constraint "eq".
 * If "keep_divs" is set, then try and preserve
 * the integer division expressions.  In this case, these expressions
 * are assumed to have been ordered.
 * If "equivalent" is set, then the elimination is performed
 * using an equality constraint of "bmap", meaning that the meaning
 * of the constraints is preserved.
 */
static __isl_give isl_basic_map *eliminate_div(__isl_take isl_basic_map *bmap,
  isl_int *eq, unsigned div, int keep_divs, int equivalent)
{
  isl_size v_div;
  unsigned pos;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_basic_map_free(bmap);
  pos = v_div + div;
  bmap = eliminate_var_using_equality(bmap, pos, eq, keep_divs,
          equivalent, NULL);
 
  bmap = isl_basic_map_drop_div(bmap, div);
 
  return bmap;
}
 
/* Check if elimination of div "div" using equality "eq" would not
 * result in a div depending on a later div.
 */
static isl_bool ok_to_eliminate_div(__isl_keep isl_basic_map *bmap, isl_int *eq,
  unsigned div)
{
  int k;
  int last_div;
  isl_size v_div;
  unsigned pos;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_bool_error;
  pos = v_div + div;
 
  last_div = isl_seq_last_non_zero(eq + 1 + v_div, bmap->n_div);
  if (last_div < 0 || last_div <= div)
    return isl_bool_true;
 
  for (k = 0; k <= last_div; ++k) {
    if (isl_int_is_zero(bmap->div[k][0]))
      continue;
    if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos]))
      return isl_bool_false;
  }
 
  return isl_bool_true;
}
 
/* Eliminate divs based on equalities
 */
static __isl_give isl_basic_map *eliminate_divs_eq(
  __isl_take isl_basic_map *bmap, int *progress)
{
  int d;
  int i;
  int modified = 0;
  unsigned off;
 
  bmap = isl_basic_map_order_divs(bmap);
 
  if (!bmap)
    return NULL;
 
  off = isl_basic_map_offset(bmap, isl_dim_div);
 
  for (d = bmap->n_div - 1; d >= 0 ; --d) {
    for (i = 0; i < bmap->n_eq; ++i) {
      isl_bool ok;
 
      if (!isl_int_is_one(bmap->eq[i][off + d]) &&
          !isl_int_is_negone(bmap->eq[i][off + d]))
        continue;
      ok = ok_to_eliminate_div(bmap, bmap->eq[i], d);
      if (ok < 0)
        return isl_basic_map_free(bmap);
      if (!ok)
        continue;
      modified = 1;
      mark_progress(progress);
      bmap = eliminate_div(bmap, bmap->eq[i], d, 1, 1);
      if (isl_basic_map_drop_equality(bmap, i) < 0)
        return isl_basic_map_free(bmap);
      break;
    }
  }
  if (modified)
    return eliminate_divs_eq(bmap, progress);
  return bmap;
}
 
/* Eliminate divs based on inequalities
 */
static __isl_give isl_basic_map *eliminate_divs_ineq(
  __isl_take isl_basic_map *bmap, int *progress)
{
  int d;
  int i;
  unsigned off;
  struct isl_ctx *ctx;
 
  if (!bmap)
    return NULL;
 
  ctx = bmap->ctx;
  off = isl_basic_map_offset(bmap, isl_dim_div);
 
  for (d = bmap->n_div - 1; d >= 0 ; --d) {
    for (i = 0; i < bmap->n_eq; ++i)
      if (!isl_int_is_zero(bmap->eq[i][off + d]))
        break;
    if (i < bmap->n_eq)
      continue;
    for (i = 0; i < bmap->n_ineq; ++i)
      if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one))
        break;
    if (i < bmap->n_ineq)
      continue;
    mark_progress(progress);
    bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1);
    if (!bmap || ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
      break;
    bmap = isl_basic_map_drop_div(bmap, d);
    if (!bmap)
      break;
  }
  return bmap;
}
 
/* Does the equality constraint at position "eq" in "bmap" involve
 * any local variables in the range [first, first + n)
 * that are not marked as having an explicit representation?
 */
static isl_bool bmap_eq_involves_unknown_divs(__isl_keep isl_basic_map *bmap,
  int eq, unsigned first, unsigned n)
{
  unsigned o_div;
  int i;
 
  if (!bmap)
    return isl_bool_error;
 
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  for (i = 0; i < n; ++i) {
    isl_bool unknown;
 
    if (isl_int_is_zero(bmap->eq[eq][o_div + first + i]))
      continue;
    unknown = isl_basic_map_div_is_marked_unknown(bmap, first + i);
    if (unknown < 0)
      return isl_bool_error;
    if (unknown)
      return isl_bool_true;
  }
 
  return isl_bool_false;
}
 
/* The last local variable involved in the equality constraint
 * at position "eq" in "bmap" is the local variable at position "div".
 * It can therefore be used to extract an explicit representation
 * for that variable.
 * Do so unless the local variable already has an explicit representation or
 * the explicit representation would involve any other local variables
 * that in turn do not have an explicit representation.
 * An equality constraint involving local variables without an explicit
 * representation can be used in isl_basic_map_drop_redundant_divs
 * to separate out an independent local variable.  Introducing
 * an explicit representation here would block this transformation,
 * while the partial explicit representation in itself is not very useful.
 * Set *progress if anything is changed.
 *
 * The equality constraint is of the form
 *
 *  f(x) + n e >= 0
 *
 * with n a positive number.  The explicit representation derived from
 * this constraint is
 *
 *  floor((-f(x))/n)
 */
static __isl_give isl_basic_map *set_div_from_eq(__isl_take isl_basic_map *bmap,
  int div, int eq, int *progress)
{
  isl_size total;
  unsigned o_div;
  isl_bool involves;
 
  if (!bmap)
    return NULL;
 
  if (!isl_int_is_zero(bmap->div[div][0]))
    return bmap;
 
  involves = bmap_eq_involves_unknown_divs(bmap, eq, 0, div);
  if (involves < 0)
    return isl_basic_map_free(bmap);
  if (involves)
    return bmap;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_basic_map_free(bmap);
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  isl_seq_neg(bmap->div[div] + 1, bmap->eq[eq], 1 + total);
  isl_int_set_si(bmap->div[div][1 + o_div + div], 0);
  isl_int_set(bmap->div[div][0], bmap->eq[eq][o_div + div]);
  mark_progress(progress);
 
  return bmap;
}
 
/* Perform fangcheng (Gaussian elimination) on the equality
 * constraints of "bmap".
 * That is, put them into row-echelon form, starting from the last column
 * backward and use them to eliminate the corresponding coefficients
 * from all constraints.
 *
 * If "progress" is not NULL, then it gets set if the elimination
 * results in any changes.
 * The elimination process may result in some equality constraints
 * getting interchanged or removed.
 * If "swap" or "drop" are not NULL, then they get called when
 * two equality constraints get interchanged or
 * when a number of final equality constraints get removed.
 * As a special case, if the input turns out to be empty,
 * then drop gets called with the number of removed equality
 * constraints set to the total number of equality constraints.
 * If "swap" or "drop" are not NULL, then the local variables (if any)
 * are assumed to be in a valid order.
 */
__isl_give isl_basic_map *isl_basic_map_gauss5(__isl_take isl_basic_map *bmap,
  int *progress,
  isl_stat (*swap)(unsigned a, unsigned b, void *user),
  isl_stat (*drop)(unsigned n, void *user), void *user)
{
  int k;
  int done;
  int last_var;
  unsigned total_var;
  isl_size total;
  unsigned n_drop;
 
  if (!swap && !drop)
    bmap = isl_basic_map_order_divs(bmap);
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_basic_map_free(bmap);
 
  total_var = total - bmap->n_div;
 
  last_var = total - 1;
  for (done = 0; done < bmap->n_eq; ++done) {
    for (; last_var >= 0; --last_var) {
      for (k = done; k < bmap->n_eq; ++k)
        if (!isl_int_is_zero(bmap->eq[k][1+last_var]))
          break;
      if (k < bmap->n_eq)
        break;
    }
    if (last_var < 0)
      break;
    if (k != done) {
      swap_equality(bmap, k, done);
      if (swap && swap(k, done, user) < 0)
        return isl_basic_map_free(bmap);
    }
    if (isl_int_is_neg(bmap->eq[done][1+last_var]))
      isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total);
 
    bmap = eliminate_var_using_equality(bmap, last_var,
            bmap->eq[done], 1, 1, progress);
 
    if (last_var >= total_var)
      bmap = set_div_from_eq(bmap, last_var - total_var,
            done, progress);
    if (!bmap)
      return NULL;
  }
  if (done == bmap->n_eq)
    return bmap;
  for (k = done; k < bmap->n_eq; ++k) {
    if (isl_int_is_zero(bmap->eq[k][0]))
      continue;
    if (drop && drop(bmap->n_eq, user) < 0)
      return isl_basic_map_free(bmap);
    return isl_basic_map_set_to_empty(bmap);
  }
  n_drop = bmap->n_eq - done;
  bmap = isl_basic_map_free_equality(bmap, n_drop);
  if (drop && drop(n_drop, user) < 0)
    return isl_basic_map_free(bmap);
  return bmap;
}
 
__isl_give isl_basic_map *isl_basic_map_gauss(__isl_take isl_basic_map *bmap,
  int *progress)
{
  return isl_basic_map_gauss5(bmap, progress, NULL, NULL, NULL);
}
 
__isl_give isl_basic_set *isl_basic_set_gauss(
  __isl_take isl_basic_set *bset, int *progress)
{
  return bset_from_bmap(isl_basic_map_gauss(bset_to_bmap(bset),
              progress));
}
 
 
static unsigned int round_up(unsigned int v)
{
  int old_v = v;
 
  while (v) {
    old_v = v;
    v ^= v & -v;
  }
  return old_v << 1;
}
 
/* Hash table of inequalities in a basic map.
 * "index" is an array of addresses of inequalities in the basic map, some
 * of which are NULL.  The inequalities are hashed on the coefficients
 * except the constant term.
 * "size" is the number of elements in the array and is always a power of two
 * "bits" is the number of bits need to represent an index into the array.
 * "total" is the total dimension of the basic map.
 */
struct isl_constraint_index {
  unsigned int size;
  int bits;
  isl_int ***index;
  isl_size total;
};
 
/* Fill in the "ci" data structure for holding the inequalities of "bmap".
 */
static isl_stat create_constraint_index(struct isl_constraint_index *ci,
  __isl_keep isl_basic_map *bmap)
{
  isl_ctx *ctx;
 
  ci->index = NULL;
  if (!bmap)
    return isl_stat_error;
  ci->total = isl_basic_map_dim(bmap, isl_dim_all);
  if (ci->total < 0)
    return isl_stat_error;
  if (bmap->n_ineq == 0)
    return isl_stat_ok;
  ci->size = round_up(4 * (bmap->n_ineq + 1) / 3 - 1);
  ci->bits = ffs(ci->size) - 1;
  ctx = isl_basic_map_get_ctx(bmap);
  ci->index = isl_calloc_array(ctx, isl_int **, ci->size);
  if (!ci->index)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Free the memory allocated by create_constraint_index.
 */
static void constraint_index_free(struct isl_constraint_index *ci)
{
  free(ci->index);
}
 
/* Return the position in ci->index that contains the address of
 * an inequality that is equal to *ineq up to the constant term,
 * provided this address is not identical to "ineq".
 * If there is no such inequality, then return the position where
 * such an inequality should be inserted.
 */
static int hash_index_ineq(struct isl_constraint_index *ci, isl_int **ineq)
{
  int h;
  uint32_t hash = isl_seq_get_hash_bits((*ineq) + 1, ci->total, ci->bits);
  for (h = hash; ci->index[h]; h = (h+1) % ci->size)
    if (ineq != ci->index[h] &&
        isl_seq_eq((*ineq) + 1, ci->index[h][0]+1, ci->total))
      break;
  return h;
}
 
/* Return the position in ci->index that contains the address of
 * an inequality that is equal to the k'th inequality of "bmap"
 * up to the constant term, provided it does not point to the very
 * same inequality.
 * If there is no such inequality, then return the position where
 * such an inequality should be inserted.
 */
static int hash_index(struct isl_constraint_index *ci,
  __isl_keep isl_basic_map *bmap, int k)
{
  return hash_index_ineq(ci, &bmap->ineq[k]);
}
 
static int set_hash_index(struct isl_constraint_index *ci,
  __isl_keep isl_basic_set *bset, int k)
{
  return hash_index(ci, bset, k);
}
 
/* Fill in the "ci" data structure with the inequalities of "bset".
 */
static isl_stat setup_constraint_index(struct isl_constraint_index *ci,
  __isl_keep isl_basic_set *bset)
{
  int k, h;
 
  if (create_constraint_index(ci, bset) < 0)
    return isl_stat_error;
 
  for (k = 0; k < bset->n_ineq; ++k) {
    h = set_hash_index(ci, bset, k);
    ci->index[h] = &bset->ineq[k];
  }
 
  return isl_stat_ok;
}
 
/* Is the inequality ineq (obviously) redundant with respect
 * to the constraints in "ci"?
 *
 * Look for an inequality in "ci" with the same coefficients and then
 * check if the contant term of "ineq" is greater than or equal
 * to the constant term of that inequality.  If so, "ineq" is clearly
 * redundant.
 *
 * Note that hash_index_ineq ignores a stored constraint if it has
 * the same address as the passed inequality.  It is ok to pass
 * the address of a local variable here since it will never be
 * the same as the address of a constraint in "ci".
 */
static isl_bool constraint_index_is_redundant(struct isl_constraint_index *ci,
  isl_int *ineq)
{
  int h;
 
  h = hash_index_ineq(ci, &ineq);
  if (!ci->index[h])
    return isl_bool_false;
  return isl_int_ge(ineq[0], (*ci->index[h])[0]);
}
 
/* Look for pairs of inequality constraints in "bmap"
 * that are opposite to each other apart from the constant term.
 * "ci" contains a hash table of the inequality constraints of "bmap".
 * Return an array of size equal to the number of inequality constraints
 * with as entries either
 * - zero, if the constraint has no opposite, or
 * - 1 + the position of the opposite constraint.
 */
static int *detect_opposites(struct isl_constraint_index *ci,
  __isl_keep isl_basic_map *bmap)
{
  isl_size n_ineq;
  int k, l, h;
  isl_ctx *ctx;
  int *opposite;
 
  n_ineq = isl_basic_map_n_inequality(bmap);
  if (n_ineq < 0)
    return NULL;
  ctx = isl_basic_map_get_ctx(bmap);
  opposite = isl_calloc_array(ctx, int, n_ineq);
  if (!opposite)
    return NULL;
 
  for (k = 0; k < n_ineq - 1; ++k) {
    if (opposite[k])
      continue;
    isl_seq_neg(bmap->ineq[k] + 1, bmap->ineq[k] + 1, ci->total);
    h = hash_index(ci, bmap, k);
    isl_seq_neg(bmap->ineq[k] + 1, bmap->ineq[k] + 1, ci->total);
    if (!ci->index[h])
      continue;
    l = ci->index[h] - &bmap->ineq[0];
 
    opposite[k] = 1 + l;
    opposite[l] = 1 + k;
  }
 
  return opposite;
}
 
/* If we can eliminate more than one div, then we need to make
 * sure we do it from last div to first div, in order not to
 * change the position of the other divs that still need to
 * be removed.
 */
static __isl_give isl_basic_map *remove_duplicate_divs(
  __isl_take isl_basic_map *bmap, int *progress)
{
  unsigned int size;
  int *index;
  int *elim_for;
  int k, l, h;
  int bits;
  struct isl_blk eq;
  isl_size v_div;
  unsigned total;
  struct isl_ctx *ctx;
 
  bmap = isl_basic_map_order_divs(bmap);
  if (!bmap || bmap->n_div <= 1)
    return bmap;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_basic_map_free(bmap);
  total = v_div + bmap->n_div;
 
  ctx = bmap->ctx;
  for (k = bmap->n_div - 1; k >= 0; --k)
    if (!isl_int_is_zero(bmap->div[k][0]))
      break;
  if (k <= 0)
    return bmap;
 
  size = round_up(4 * bmap->n_div / 3 - 1);
  if (size == 0)
    return bmap;
  elim_for = isl_calloc_array(ctx, int, bmap->n_div);
  bits = ffs(size) - 1;
  index = isl_calloc_array(ctx, int, size);
  if (!elim_for || !index)
    goto out;
  eq = isl_blk_alloc(ctx, 1+total);
  if (isl_blk_is_error(eq))
    goto out;
 
  isl_seq_clr(eq.data, 1+total);
  index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1;
  for (--k; k >= 0; --k) {
    uint32_t hash;
 
    if (isl_int_is_zero(bmap->div[k][0]))
      continue;
 
    hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits);
    for (h = hash; index[h]; h = (h+1) % size)
      if (isl_seq_eq(bmap->div[k],
               bmap->div[index[h]-1], 2+total))
        break;
    if (index[h]) {
      mark_progress(progress);
      l = index[h] - 1;
      elim_for[l] = k + 1;
    }
    index[h] = k+1;
  }
  for (l = bmap->n_div - 1; l >= 0; --l) {
    if (!elim_for[l])
      continue;
    k = elim_for[l] - 1;
    isl_int_set_si(eq.data[1 + v_div + k], -1);
    isl_int_set_si(eq.data[1 + v_div + l], 1);
    bmap = eliminate_div(bmap, eq.data, l, 1, 0);
    if (!bmap)
      break;
    isl_int_set_si(eq.data[1 + v_div + k], 0);
    isl_int_set_si(eq.data[1 + v_div + l], 0);
  }
 
  isl_blk_free(ctx, eq);
out:
  free(index);
  free(elim_for);
  return bmap;
}
 
/* Is the local variable at position "div" of "bmap"
 * an integral integer division?
 */
static isl_bool is_known_integral_div(__isl_keep isl_basic_map *bmap, int div)
{
  isl_bool unknown;
 
  unknown = isl_basic_map_div_is_marked_unknown(bmap, div);
  if (unknown < 0 || unknown)
    return isl_bool_not(unknown);
  return isl_basic_map_div_is_integral(bmap, div);
}
 
/* Eliminate local variable "div" from "bmap", given
 * that it represents an integer division with denominator 1.
 *
 * Construct an equality constraint that equates the local variable
 * to the argument of the integer division and use that to eliminate
 * the local variable.
 */
static __isl_give isl_basic_map *eliminate_integral_div(
  __isl_take isl_basic_map *bmap, int div)
{
  isl_size total, v_div;
  isl_vec *v;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (v_div < 0 || total < 0)
    return isl_basic_map_free(bmap);
  v = isl_vec_alloc(isl_basic_map_get_ctx(bmap), 1 + total);
  if (!v)
    return isl_basic_map_free(bmap);
  isl_seq_cpy(v->el, bmap->div[div] + 1, 1 + total);
  isl_int_set_si(v->el[1 + v_div + div], -1);
  bmap = eliminate_div(bmap, v->el, div, 1, 0);
  isl_vec_free(v);
 
  return bmap;
}
 
/* Eliminate all integer divisions with denominator 1.
 */
static __isl_give isl_basic_map *eliminate_integral_divs(
  __isl_take isl_basic_map *bmap, int *progress)
{
  int i;
  isl_size n_div;
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    return isl_basic_map_free(bmap);
 
  for (i = 0; i < n_div; ++i) {
    isl_bool eliminate;
 
    eliminate = is_known_integral_div(bmap, i);
    if (eliminate < 0)
      return isl_basic_map_free(bmap);
    if (!eliminate)
      continue;
 
    bmap = eliminate_integral_div(bmap, i);
    mark_progress(progress);
    i--;
    n_div--;
  }
 
  return bmap;
}
 
static int n_pure_div_eq(__isl_keep isl_basic_map *bmap)
{
  int i, j;
  isl_size v_div;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return -1;
  for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) {
    while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + v_div + j]))
      --j;
    if (j < 0)
      break;
    if (isl_seq_any_non_zero(bmap->eq[i] + 1 + v_div, j))
      return 0;
  }
  return i;
}
 
/* Normalize divs that appear in equalities.
 *
 * In particular, we assume that bmap contains some equalities
 * of the form
 *
 *  a x = m * e_i
 *
 * and we want to replace the set of e_i by a minimal set and
 * such that the new e_i have a canonical representation in terms
 * of the vector x.
 * If any of the equalities involves more than one divs, then
 * we currently simply bail out.
 *
 * Let us first additionally assume that all equalities involve
 * a div.  The equalities then express modulo constraints on the
 * remaining variables and we can use "parameter compression"
 * to find a minimal set of constraints.  The result is a transformation
 *
 *  x = T(x') = x_0 + G x'
 *
 * with G a lower-triangular matrix with all elements below the diagonal
 * non-negative and smaller than the diagonal element on the same row.
 * We first normalize x_0 by making the same property hold in the affine
 * T matrix.
 * The rows i of G with a 1 on the diagonal do not impose any modulo
 * constraint and simply express x_i = x'_i.
 * For each of the remaining rows i, we introduce a div and a corresponding
 * equality.  In particular
 *
 *  g_ii e_j = x_i - g_i(x')
 *
 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
 * corresponding div (if g_kk != 1).
 *
 * If there are any equalities not involving any div, then we
 * first apply a variable compression on the variables x:
 *
 *  x = C x'' x'' = C_2 x
 *
 * and perform the above parameter compression on A C instead of on A.
 * The resulting compression is then of the form
 *
 *  x'' = T(x') = x_0 + G x'
 *
 * and in constructing the new divs and the corresponding equalities,
 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
 * by the corresponding row from C_2.
 */
static __isl_give isl_basic_map *normalize_divs(__isl_take isl_basic_map *bmap,
  int *progress)
{
  int i, j, k;
  isl_size v_div;
  int div_eq;
  struct isl_mat *B;
  struct isl_vec *d;
  struct isl_mat *T = NULL;
  struct isl_mat *C = NULL;
  struct isl_mat *C2 = NULL;
  isl_int v;
  int *pos = NULL;
  int dropped, needed;
 
  if (!bmap)
    return NULL;
 
  if (bmap->n_div == 0)
    return bmap;
 
  if (bmap->n_eq == 0)
    return bmap;
 
  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS))
    return bmap;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  div_eq = n_pure_div_eq(bmap);
  if (v_div < 0 || div_eq < 0)
    return isl_basic_map_free(bmap);
  if (div_eq == 0)
    return bmap;
 
  if (div_eq < bmap->n_eq) {
    B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, div_eq,
          bmap->n_eq - div_eq, 0, 1 + v_div);
    C = isl_mat_variable_compression(B, &C2);
    if (!C || !C2)
      goto error;
    if (C->n_col == 0) {
      bmap = isl_basic_map_set_to_empty(bmap);
      isl_mat_free(C);
      isl_mat_free(C2);
      goto done;
    }
  }
 
  d = isl_vec_alloc(bmap->ctx, div_eq);
  if (!d)
    goto error;
  for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) {
    while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + v_div + j]))
      --j;
    isl_int_set(d->block.data[i], bmap->eq[i][1 + v_div + j]);
  }
  B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + v_div);
 
  if (C) {
    B = isl_mat_product(B, C);
    C = NULL;
  }
 
  T = isl_mat_parameter_compression(B, d);
  if (!T)
    goto error;
  if (T->n_col == 0) {
    bmap = isl_basic_map_set_to_empty(bmap);
    isl_mat_free(C2);
    isl_mat_free(T);
    goto done;
  }
  isl_int_init(v);
  for (i = 0; i < T->n_row - 1; ++i) {
    isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]);
    if (isl_int_is_zero(v))
      continue;
    isl_mat_col_submul(T, 0, v, 1 + i);
  }
  isl_int_clear(v);
  pos = isl_alloc_array(bmap->ctx, int, T->n_row);
  if (!pos)
    goto error;
  /* We have to be careful because dropping equalities may reorder them */
  dropped = 0;
  for (j = bmap->n_div - 1; j >= 0; --j) {
    for (i = 0; i < bmap->n_eq; ++i)
      if (!isl_int_is_zero(bmap->eq[i][1 + v_div + j]))
        break;
    if (i < bmap->n_eq) {
      bmap = isl_basic_map_drop_div(bmap, j);
      if (isl_basic_map_drop_equality(bmap, i) < 0)
        goto error;
      ++dropped;
    }
  }
  pos[0] = 0;
  needed = 0;
  for (i = 1; i < T->n_row; ++i) {
    if (isl_int_is_one(T->row[i][i]))
      pos[i] = i;
    else
      needed++;
  }
  if (needed > dropped) {
    bmap = isl_basic_map_extend(bmap, needed, needed, 0);
    if (!bmap)
      goto error;
  }
  for (i = 1; i < T->n_row; ++i) {
    if (isl_int_is_one(T->row[i][i]))
      continue;
    k = isl_basic_map_alloc_div(bmap);
    pos[i] = 1 + v_div + k;
    isl_seq_clr(bmap->div[k] + 1, 1 + v_div + bmap->n_div);
    isl_int_set(bmap->div[k][0], T->row[i][i]);
    if (C2)
      isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + v_div);
    else
      isl_int_set_si(bmap->div[k][1 + i], 1);
    for (j = 0; j < i; ++j) {
      if (isl_int_is_zero(T->row[i][j]))
        continue;
      if (pos[j] < T->n_row && C2)
        isl_seq_submul(bmap->div[k] + 1, T->row[i][j],
            C2->row[pos[j]], 1 + v_div);
      else
        isl_int_neg(bmap->div[k][1 + pos[j]],
                T->row[i][j]);
    }
    j = isl_basic_map_alloc_equality(bmap);
    isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+v_div+bmap->n_div);
    isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]);
  }
  free(pos);
  isl_mat_free(C2);
  isl_mat_free(T);
 
  mark_progress(progress);
done:
  ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS);
 
  return bmap;
error:
  free(pos);
  isl_mat_free(C);
  isl_mat_free(C2);
  isl_mat_free(T);
  isl_basic_map_free(bmap);
  return NULL;
}
 
static __isl_give isl_basic_map *set_div_from_lower_bound(
  __isl_take isl_basic_map *bmap, int div, isl_int *ineq)
{
  unsigned total = isl_basic_map_offset(bmap, isl_dim_div);
 
  isl_seq_neg(bmap->div[div] + 1, ineq, total + bmap->n_div);
  isl_int_set(bmap->div[div][0], ineq[total + div]);
  isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]);
  isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
  isl_int_set_si(bmap->div[div][1 + total + div], 0);
 
  return bmap;
}
 
/* Check whether it is ok to define a div based on an inequality.
 * To avoid the introduction of circular definitions of divs, we
 * do not allow such a definition if the resulting expression would refer to
 * any other undefined divs or if any known div is defined in
 * terms of the unknown div.
 */
static isl_bool ok_to_set_div_from_bound(__isl_keep isl_basic_map *bmap,
  int div, isl_int *ineq)
{
  int j;
  isl_size v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  isl_size n_div = isl_basic_map_dim(bmap, isl_dim_div);
 
  if (v_div < 0 || n_div < 0)
    return isl_bool_error;
 
  /* Not defined in terms of unknown divs */
  for (j = 0; j < n_div; ++j) {
    isl_bool unknown;
 
    if (div == j)
      continue;
    if (isl_int_is_zero(ineq[1 + v_div + j]))
      continue;
    unknown = isl_basic_map_div_is_marked_unknown(bmap, j);
    if (unknown < 0 || unknown)
      return isl_bool_not(unknown);
  }
 
  /* No other div defined in terms of this one => avoid loops */
  for (j = 0; j < n_div; ++j) {
    if (div == j)
      continue;
    if (isl_int_is_zero(bmap->div[j][0]))
      continue;
    if (!isl_int_is_zero(bmap->div[j][1 + 1 + v_div + div]))
      return isl_bool_false;
  }
 
  return isl_bool_true;
}
 
/* Would an expression for div "div" based on inequality "ineq"
 * be a better expression than the current one in "bmap"?
 *
 * If we do not have any expression yet, then any expression would be better.
 * Otherwise we check if the last variable involved in the inequality
 * (disregarding the div that it would define) is in an earlier position
 * than the last variable involved in the current div expression.
 */
static isl_bool better_div_constraint(__isl_keep isl_basic_map *bmap,
  int div, isl_int *ineq)
{
  unsigned total = isl_basic_map_offset(bmap, isl_dim_div);
  isl_size n_div = isl_basic_map_dim(bmap, isl_dim_div);
  int last_div;
  int last_ineq;
  isl_bool unknown;
 
  if (n_div < 0)
    return isl_bool_error;
  unknown = isl_basic_map_div_is_marked_unknown(bmap, div);
  if (unknown < 0 || unknown)
    return unknown;
 
  if (isl_seq_any_non_zero(ineq + total + div + 1, n_div - (div + 1)))
    return isl_bool_false;
 
  last_ineq = isl_seq_last_non_zero(ineq, total + div);
  last_div = isl_seq_last_non_zero(bmap->div[div] + 1, total + n_div);
 
  return last_ineq < last_div;
}
 
/* Given a lower bound "ineq" on local variable "div" of "bmap"
 * that could in theory be used to define an integer division expression,
 * do so if it is better than the current expression (if any) and
 * if there is no risk of introducing circular definitions.
 * Set *progress if anything is changed.
 */
static __isl_give isl_basic_map *set_div_from_lower_bound_if_better(
  __isl_take isl_basic_map *bmap, int div, isl_int *ineq, int *progress)
{
  isl_bool set_div;
 
  set_div = better_div_constraint(bmap, div, ineq);
  if (set_div >= 0 && set_div)
    set_div = ok_to_set_div_from_bound(bmap, div, ineq);
  if (set_div < 0)
    return isl_basic_map_free(bmap);
  if (!set_div)
    return bmap;
  bmap = set_div_from_lower_bound(bmap, div, ineq);
  mark_progress(progress);
  return bmap;
}
 
/* Is the sequence of "len" coefficients "ineq" equal to "res"
 * plus some non-trivial coefficients that are all a multiple of some number
 * greater than "sum"?
 * If so, this factor is stored in "gcd".
 *
 * The current implementation requires the coefficients
 * in "res" to appear directly in "ineq", so that "gcd"
 * is the gcd of the remaining coefficients.
 * The same assumption is used in has_nested_unit_div.
 */
static int is_residue(isl_int *res, isl_int *ineq, isl_int sum, unsigned len,
  isl_int *gcd)
{
  int j;
 
  isl_int_set_si(*gcd, 0);
  for (j = 0; j < len; ++j) {
    if (!isl_int_is_zero(res[1 + j])) {
      if (isl_int_eq(res[1 + j], ineq[1 + j]))
        continue;
      return 0;
    }
    if (!isl_int_is_zero(ineq[1 + j])) {
      isl_int_gcd(*gcd, *gcd, ineq[1 + j]);
      if (isl_int_le(*gcd, sum))
        return 0;
    }
  }
 
  return !isl_int_is_zero(*gcd);
}
 
/* Is
 *
 *  (cst - cst2) mod n + sum
 *
 * greater than or equal to n?
 */
static int residue_exceeded(isl_int cst, isl_int cst2, isl_int n, isl_int sum)
{
  int exceeded;
  isl_int t;
 
  isl_int_init(t);
  isl_int_sub(t, cst, cst2);
  isl_int_fdiv_r(t, t, n);
  isl_int_add(t, t, sum);
  exceeded = isl_int_ge(t, n);
  isl_int_clear(t);
 
  return exceeded;
}
 
/* Given two constraints "k" and "l" that are opposite to each other,
 * except for the constant term, with "sum" the sum of these constant terms,
 * check if they can be used to simplify any integer division expression.
 *
 * In particular, let "k" and "l" be of the form
 *
 *   f(x) >= 0
 *  -f(x) + c >= 0
 *
 * That is,
 *
 *  0 <= f(x) <= c
 *
 * Note that the same constraint holds for "k" and "l" interchanged, i.e.,
 *
 *  0 <= -f(x) + c <= c
 *
 * That is, the reasoning below holds for both "f(x)" and "-f(x) + c".
 *
 * If there is an integer division definition of the form
 *
 *  floor((f(x) + n h(x) + c')/(n * m))
 *
 * with
 *
 *  c' % n < n - c
 *
 * then it is equal to
 *
 *  floor((h(x) + floor(c'/n))/m)
 *
 * because
 *
 *  floor((f(x) + n h(x) + c')/(n * m))
 *  = floor((f(x) + c' % n + n (h(x) + floor(c'/n)))/(n * m))
 *
 * and
 *
 *  0 <= f(x) + c' % n < n
 *
 * Note that h(x) may be equal to zero, in which case the denominator
 * of the integer division can be used as n (and m = 1).
 */
static __isl_give isl_basic_map *check_for_residues_in_divs(
  __isl_take isl_basic_map *bmap, int k, int l, isl_int sum,
  int *progress)
{
  int i;
  int p;
  isl_ctx *ctx;
  isl_size n_div, total;
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (n_div < 0 || total < 0)
    return isl_basic_map_free(bmap);
 
  ctx = isl_basic_map_get_ctx(bmap);
  p = isl_seq_last_non_zero(bmap->ineq[k] + 1, total);
  for (i = 0; i < n_div; ++i) {
    int c;
    isl_bool unknown;
 
    unknown = isl_basic_map_div_is_marked_unknown(bmap, i);
    if (unknown < 0)
      return isl_basic_map_free(bmap);
    if (unknown)
      continue;
    if (isl_int_le(bmap->div[i][0], sum))
      continue;
    if (isl_int_eq(bmap->div[i][2 + p], bmap->ineq[k][1 + p]))
      c = k;
    else if (isl_int_eq(bmap->div[i][2 + p], bmap->ineq[l][1 + p]))
      c = l;
    else
      continue;
 
    if (isl_seq_eq(bmap->div[i] + 2, bmap->ineq[c] + 1, total))
      isl_int_set(ctx->normalize_gcd, bmap->div[i][0]);
    else if (!is_residue(bmap->ineq[c], bmap->div[i] + 1, sum,
          total, &ctx->normalize_gcd))
      continue;
 
    if (residue_exceeded(bmap->div[i][1], bmap->ineq[c][0],
            ctx->normalize_gcd, sum))
      continue;
 
    if (!isl_int_is_divisible_by(bmap->div[i][0],
              ctx->normalize_gcd))
      continue;
 
    isl_seq_sub(bmap->div[i] + 1, bmap->ineq[c], 1 + total);
    mark_progress(progress);
  }
 
  return bmap;
}
 
/* Is inequality "ineq" of "bmap" a constraint defining an integer division?
 * "v_div" is the position of the first local variable.
 * "n_div" is the number of local variables.
 *
 * A constraint defining an integer division must involve some local variable
 * and could only possibly define the last local variable involved since
 * it can only be defined in terms of earlier variables.
 */
static isl_bool is_div_constraint(__isl_keep isl_basic_map *bmap, int ineq,
  unsigned v_div, unsigned n_div)
{
  int last;
 
  last = isl_seq_last_non_zero(bmap->ineq[ineq] + 1 + v_div, n_div);
  if (last < 0)
    return isl_bool_false;
  return isl_basic_map_is_div_constraint(bmap, bmap->ineq[ineq], last);
}
 
/* Does the inequality constraint "ineq" of "bmap" involve nested integer
 * divisions with unit coefficient after removing the coefficients
 * of inequality constraint "base"?
 * "v_div" is the position of the first local variable.
 * "n_div" is the number of local variables.
 * "n" is the factor with which the constraint will be scaled down.
 *
 * Use the same simplifying assumption as "is_residue" that
 * the coefficients in "base" appear directly in "ineq".
 * Look for an integer division involving nested integer divisions
 * that does not appear in "base" and does appear in "ineq"
 * with a coefficient equal to "n" (up to a change of sign).
 */
static isl_bool has_nested_unit_div(__isl_keep isl_basic_map *bmap,
  int base, int ineq, unsigned v_div, unsigned n_div, isl_int n)
{
  int j;
 
  for (j = 0; j < n_div; ++j) {
    isl_bool nested;
 
    if (!isl_int_is_zero(bmap->ineq[base][1 + v_div + j]))
      continue;
    if (!isl_int_abs_eq(bmap->ineq[ineq][1 + v_div + j], n))
      continue;
    nested = isl_basic_map_div_expr_involves_vars(bmap, j,
                v_div, n_div);
    if (nested < 0 || nested)
      return nested;
  }
 
  return isl_bool_false;
}
 
/* Given two constraints "k" and "l" that are opposite to each other,
 * except for the constant term, with "sum" the sum of these constant terms,
 * check if they can be used to simplify other constraints.
 * Only do this for integer basic maps.
 *
 * In particular, let "k" and "l" be of the form
 *
 *   f(x) >= 0
 *  -f(x) + c >= 0
 *
 * That is,
 *
 *  0 <= f(x) <= c
 *
 * Note that the same constraint holds for "k" and "l" interchanged, i.e.,
 *
 *  0 <= -f(x) + c <= c
 *
 * That is, the reasoning below holds for both "f(x)" and "-f(x) + c".
 *
 * If there is some other constraint
 *
 *  g(x) >= 0
 *
 * such that
 *
 *  g(x) - f(x) = n h(x) + c'
 *
 * with
 *
 *  0 <= c' < n - c
 *
 * in particular, for the constant term,
 *
 *  (g(x) - f(x)) mod n + c < n
 *
 * then this other constraint is equivalent to
 *
 *  h(x) >= 0
 *
 * (given "k" and "l") since
 *
 *  0 <= f(x) + c' < n
 *
 * Note that the constraint does not necessarily need to be scaled down here
 * since it would otherwise also be scaled down
 * by isl_basic_map_normalize_constraints, but since the scaling factor
 * is already known here, it might as well be done immediately.
 * Also note that the current implementation only checks for constraints
 * where the coefficients of f(x) appear directly in g(x), while it would
 * be sufficient for the differences with the corresponding coefficients
 * in g(x) to be multiples of n.
 *
 * Similarly, if there is an integer division definition of the form
 *
 *  floor((f(x) + n h(x) + c')/(n * m))
 *
 * with
 *
 *  c' % n < n - c
 *
 * then it is equal to
 *
 *  floor((h(x) + floor(c'/n))/m)
 *
 *
 * Do not apply any simplification to constraint(s) defining integer divisions.
 * Such constraints would first be simplified to be of the form
 *
 *  e + c'' >= 0
 *
 * and then the integer division definition would be plugged into
 * this constraint, resulting in the original constraint and
 * causing an infinite loop.
 * Simplifying the integer division definitions as well mitigates
 * some (possibly all) of this effect, but it is too fragile to rely on
 * for avoiding infinite loops.
 *
 * If any nested integer divisions are involved, then a similar effect
 * may be obtained even on constraints that do not (obviously)
 * define an integer division through multiple steps of such substitutions.
 * Any constraint that would result in an integer division with nested
 * integer divisions and a unit coefficient is therefore also left untouched.
 */
static __isl_give isl_basic_map *check_for_residues(
  __isl_take isl_basic_map *bmap, int k, int l, isl_int sum,
  int *progress)
{
  int i;
  int p;
  isl_ctx *ctx;
  isl_bool rat;
  isl_size n_ineq, total, v_div, n_div;
 
  rat = isl_basic_map_is_rational(bmap);
  n_ineq = isl_basic_map_n_inequality(bmap);
  total = isl_basic_map_dim(bmap, isl_dim_all);
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (rat < 0 || n_ineq < 0 || total < 0 || v_div < 0 || n_div < 0)
    return isl_basic_map_free(bmap);
  if (rat)
    return bmap;
 
  bmap = check_for_residues_in_divs(bmap, k, l, sum, progress);
  if (!bmap)
    return bmap;
 
  ctx = isl_basic_map_get_ctx(bmap);
  p = isl_seq_last_non_zero(bmap->ineq[k] + 1, total);
  for (i = 0; i < n_ineq; ++i) {
    isl_bool skip;
    int c;
 
    if (i == k || i == l)
      continue;
    if (isl_int_is_zero(bmap->ineq[i][1 + p]))
      continue;
    skip = is_div_constraint(bmap, i, v_div, n_div);
    if (skip < 0)
      return isl_basic_map_free(bmap);
    if (skip)
      continue;
    if (isl_int_eq(bmap->ineq[i][1 + p], bmap->ineq[k][1 + p]))
      c = k;
    else if (isl_int_eq(bmap->ineq[i][1 + p], bmap->ineq[l][1 + p]))
      c = l;
    else
      continue;
 
    if (!is_residue(bmap->ineq[c], bmap->ineq[i], sum, total,
        &ctx->normalize_gcd))
      continue;
 
    skip = has_nested_unit_div(bmap, c, i, v_div, n_div,
          ctx->normalize_gcd);
    if (skip < 0)
      return isl_basic_map_free(bmap);
    if (skip)
      continue;
    if (residue_exceeded(bmap->ineq[i][0], bmap->ineq[c][0],
            ctx->normalize_gcd, sum))
      continue;
 
    isl_seq_sub(bmap->ineq[i], bmap->ineq[c], 1 + total);
    bmap = scale_down_bmap_inequality(bmap, i, ctx->normalize_gcd,
            total);
    if (!bmap)
      return NULL;
    mark_progress(progress);
  }
 
  return bmap;
}
 
/* Given constraints of the form
 *
 *  -n f + g + m n e + c1 >= 0  (l)
 *  g + c3 >= 0     (i)
 *  -g + c4 >= 0      (j)
 *
 * with e the local variable at position "div",
 * does the following condition hold?
 *
 *  c3 + c4 + (c1 - c3) % n < m n
 *
 * "v_div" is the position of the first local variable.
 */
static int is_residue_div_pair(__isl_take isl_basic_map *bmap,
  int l, int div, unsigned v_div, int i, int j, isl_int n)
{
  isl_int tmp;
  int ok;
 
  isl_int_init(tmp);
  isl_int_sub(tmp, bmap->ineq[l][0], bmap->ineq[i][0]);
  isl_int_fdiv_r(tmp, tmp, n);
  isl_int_add(tmp, tmp, bmap->ineq[i][0]);
  isl_int_add(tmp, tmp, bmap->ineq[j][0]);
  ok = isl_int_lt(tmp, bmap->ineq[l][1 + v_div + div]);
  isl_int_clear(tmp);
 
  return ok;
}
 
/* Given a lower bound constraint at position "l" of "bmap"
 * on local variable "div" such that the sum of the constant terms
 * of this constraint and the corresponding upper bound is equal to "sum",
 * as well as a separate pair of opposite constraints "i" and "j"
 * such that the sum of their constant terms is smaller than
 * the coefficient of "div" in "l",
 * check if this pair can be used to derive a reduced expression
 * for the local variable.
 * "total" is the total number of variables.
 * "v_div" is the position of the first local variable.
 * Set *progress if anything is changed.
 *
 * In particular, let l be the constraint
 *
 *  -n f + g + m n e + c1 >= 0
 *
 * The opposite constraint is
 *
 *  n f - g - m n e + c2 >= 0
 *
 * with c1 + c2 equal to "sum".
 *
 * The constraints i and j are of the form
 *
 *  h + c3 >= 0
 *  -h + c4 >= 0
 *
 * with c3 + c4 < m n.
 * First check that h is equal to g for some n greater than "sum".
 * The constraints i and j are then of the form
 *
 *  g + c3 >= 0
 *  -g + c4 >= 0
 *
 * From constraint "l" and its opposite,
 *
 *  g + m n e - c2 <= n f <= g + m n e + c1
 *
 * combined with the fact that c1 + c2 < n, the following expression
 * can be obtained:
 *
 *  f = floor((g + m n e + c1)/n) = floor((g + c1)/n) + m e
 *
 * Replacing c1 with
 *
 *  c1 = c3 + (c1 - c3) = c3 + n floor((c1 - c3)/n) + (c1 - c3) % n
 *
 * results in
 *
 *  f = floor((g + c3 + (c1 - c3) % n)/n) + floor((c1 - c3)/n) + m e
 *
 * From the constraints i and j, together with c3 + c4 < m n,
 *
 *  0 <= g + c3 < m n
 *
 * Since (c1 - c3) % n is non-negative,
 *
 *  0 <= g + c3 + (c1 - c3) % n
 *
 * holds as well.  However,
 *
 *  g + c3 + (c1 - c3) % n < m n
 *
 * may not necessarily hold.  It is however the case if
 *
 *  c3 + c4 + (c1 - c3) % n < m n
 *
 * holds, which is checked in is_residue_div_pair.
 *
 * Given
 *
 *  0 <= g + c3 + (c1 - c3) % n < m n
 *
 * also
 *
 *  0 <= floor((g + c3 + (c1 - c3) % n) / n) < m
 *
 * holds.
 *
 * So,
 *
 *  f = floor((g + c3 + (c1 - c3) % n)/n) + floor((c1 - c3)/n) + m e
 *
 * and
 *
 *  0 <= floor((g + c3 + (c1 - c3) % n) / n) < m
 *
 * This means
 *
 *  f - floor((c1 - c3)/n)
 *
 * is equal to a multiple of m plus a value between 0 and m - 1.
 * That is,
 *
 *  e = floor((f - floor((c1 - c3)/n))/m)
 *
 * Construct the inequality constraint
 *
 *  -n f + m n e + c1 - c3 >= 0
 *
 * scale it down to
 *
 *  -f + m e + floor((c1 - c3)/n) >= 0
 *
 * and add m - 1, to obtain
 *
 *  -f + m e + floor((c1 - c3)/n) + m - 1 >= 0
 *
 * This constraint can then be used by set_div_from_lower_bound_if_better
 * to obtain
 *
 *  e = floor((f - floor((c1 - c3)/n))/m)
 *
 * Adding m - 1 is needed to be able to reuse set_div_from_lower_bound,
 * which subtracts this amount from the constraint (by adding it
 * to the negated constraint).
 */
static __isl_give isl_basic_map *set_residue_div(__isl_take isl_basic_map *bmap,
  int l, isl_int sum, int div,
  unsigned v_div, unsigned total, int i, int j, int *progress)
{
  isl_ctx *ctx;
  isl_vec *v;
 
  if (!bmap)
    return NULL;
 
  ctx = isl_basic_map_get_ctx(bmap);
  if (!is_residue(bmap->ineq[i], bmap->ineq[l], sum, total,
      &ctx->normalize_gcd))
    return bmap;
 
  if (!is_residue_div_pair(bmap, l, div, v_div, i, j, ctx->normalize_gcd))
    return bmap;
 
  v = isl_vec_alloc(ctx, 1 + total);
  if (!v)
    return isl_basic_map_free(bmap);
  isl_seq_cpy(v->el, bmap->ineq[l], 1 + total);
  isl_seq_sub(v->el, bmap->ineq[i], 1 + total);
  scale_down_inequality(v->el, ctx->normalize_gcd, total);
  isl_int_add(v->el[0], v->el[0], v->el[1 + v_div + div]);
  isl_int_sub_ui(v->el[0], v->el[0], 1);
  bmap = set_div_from_lower_bound_if_better(bmap, div, v->el, progress);
  isl_vec_free(v);
 
  return bmap;
}
 
/* Given a lower bound constraint at position "l" of "bmap"
 * on local variable "div" such that the sum of the constant terms
 * of this constraint and the corresponding upper bound is equal to "sum",
 * check if some other pair of constraints can be found in "opposite"
 * that can be used to derive a reduced expression for the local variable.
 * "opposite" describes all pairs of opposite constraints.
 * It is an array of size equal to the number of inequality constraints
 * with as entries either
 * - zero, if the constraint has no opposite, or
 * - 1 + the position of the opposite constraint.
 * Set *progress if anything is changed.
 *
 * In particular, let l be the constraint
 *
 *  -n f + g + m n e + c1 >= 0
 *
 * The opposite constraint is
 *
 *  n f - g - m n e + c2 >= 0
 *
 * with c1 + c2 equal to "sum".
 *
 * Look for a pair of constraints
 *
 *  g + c3 >= 0
 *  -g + c4 >= 0
 *
 * with c3 + c4 < m n that can be used to obtain a reduced expression for e.
 * First look for any pair of constraints with constant terms
 * that satisfy c3 + c4 < m n and then check whether either of them
 * can be used in this way.
 */
static __isl_give isl_basic_map *check_for_residue_div(
  __isl_take isl_basic_map *bmap, int *opposite, int l, isl_int sum,
  int div, int *progress)
{
  int i;
  isl_size n, v_div, total;
  isl_ctx *ctx;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  total = isl_basic_map_dim(bmap, isl_dim_all);
  n = isl_basic_map_n_inequality(bmap);
  if (v_div < 0 || total < 0 || n < 0)
    return isl_basic_map_free(bmap);
 
  ctx = isl_basic_map_get_ctx(bmap);
  for (i = 0; i < n; ++i) {
    int j;
 
    if (i == l)
      continue;
    if (!opposite[i])
      continue;
    j = opposite[i] - 1;
    if (j < i)
      continue;
    if (j == l)
      continue;
    if (!isl_int_is_zero(bmap->ineq[i][1 + v_div + div]))
      continue;
    isl_int_add(ctx->normalize_gcd,
        bmap->ineq[i][0], bmap->ineq[j][0]);
    if (isl_int_ge(ctx->normalize_gcd,
        bmap->ineq[l][1 + v_div + div]))
      continue;
    bmap = set_residue_div(bmap, l, sum, div, v_div, total, i, j,
          progress);
    bmap = set_residue_div(bmap, l, sum, div, v_div, total, j, i,
          progress);
  }
 
  return bmap;
}
 
/* Given two constraints "k" and "l" that are opposite to each other,
 * except for the constant term, check if we can use them
 * to obtain an expression for one of the hitherto unknown divs or
 * a "better" expression for a div for which we already have an expression.
 * "sum" is the sum of the constant terms of the constraints.
 * If this sum is strictly smaller than the coefficient of one
 * of the divs, then this pair can be used to define the div.
 * Use the lower bound of this pair if it is better than
 * any previously known expression and
 * if there is no risk of introducing circular definitions.
 * If, moreover, this coefficient is a non-trivial multiple
 * of a value greater than the sum and if some other pair of constraints
 * can be found that can be used to eliminate coefficients
 * that are not a multiple of this value, then a more simplified
 * expression can be obtained.
 * "opposite" describes all pairs of opposite constraints.
 * It is an array of size equal to the number of inequality constraints
 * with as entries either
 * - zero, if the constraint has no opposite, or
 * - 1 + the position of the opposite constraint.
 */
static __isl_give isl_basic_map *check_for_div_constraints(
  __isl_take isl_basic_map *bmap, int *opposite,
  int k, int l, isl_int sum, int *progress)
{
  int i;
  unsigned total = isl_basic_map_offset(bmap, isl_dim_div);
  isl_size n_div = isl_basic_map_dim(bmap, isl_dim_div);
 
  if (n_div < 0)
    return isl_basic_map_free(bmap);
 
  for (i = 0; i < n_div; ++i) {
    int div_set = 0;
    int c;
 
    if (isl_int_is_zero(bmap->ineq[k][total + i]))
      continue;
    if (isl_int_is_pos(bmap->ineq[k][total + i]))
      c = k;
    else
      c = l;
    if (isl_int_ge(sum, bmap->ineq[c][total + i]))
      continue;
    bmap = check_for_residue_div(bmap, opposite, c, sum, i,
            &div_set);
    bmap = set_div_from_lower_bound_if_better(bmap, i,
            bmap->ineq[c], &div_set);
    if (!bmap)
      return NULL;
    if (!div_set)
      continue;
    mark_progress(progress);
    break;
  }
  return bmap;
}
 
/* Exploit the pairs of inequality constraints of "bmap"
 * with opposite coefficients.  They are described by "opposite",
 * an array of size equal to the number of inequality constraints
 * with as entries either
 * - zero, if the constraint has no opposite, or
 * - 1 + the position of the opposite constraint.
 * Detect (better) integer division expressions if "detect_divs" is set.
 * Set *progress if any progress is made.
 *
 * For each pair of constraints with opposite coefficients,
 * consider the sum of the constant terms.
 * If the sum is smaller than zero, then the constraints conflict.
 * If the sum is equal to zero, then the constraints form
 * an equality constraint.
 * If the sum is greater than zero, then check whether this pair
 * can be used to simplify any other constraints and/or,
 * if "detect_divs" is set, whether a (better) integer division definition
 * can be read off from the pair.
 *
 * Only check each pair once (with k < l).
 */
static __isl_give isl_basic_map *exploit_opposite_constraints(
  __isl_take isl_basic_map *bmap, int *opposite,
  int *progress, int detect_divs)
{
  int k, l;
  isl_int sum;
 
  if (!opposite)
    return bmap;
  isl_int_init(sum);
  for (k = 0; bmap && k < bmap->n_ineq-1; ++k) {
    if (!opposite[k])
      continue;
    l = opposite[k] - 1;
    if (l < k)
      continue;
    isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]);
    if (isl_int_is_pos(sum)) {
      int residue = 0;
 
      bmap = check_for_residues(bmap, k, l, sum, &residue);
      if (detect_divs)
        bmap = check_for_div_constraints(bmap, opposite,
              k, l, sum, progress);
      if (!residue)
        continue;
      mark_progress(progress);
      break;
    }
    if (isl_int_is_zero(sum)) {
      /* We need to break out of the loop after these
       * changes since the contents of the hash
       * will no longer be valid.
       * Plus, we probably we want to regauss first.
       */
      mark_progress(progress);
      isl_basic_map_drop_inequality(bmap, l);
      isl_basic_map_inequality_to_equality(bmap, k);
    } else
      bmap = isl_basic_map_set_to_empty(bmap);
    break;
  }
  isl_int_clear(sum);
  free(opposite);
 
  return bmap;
}
 
/* Look for pairs of constraints that have equal or opposite coefficients.
 * Detect (better) integer division expressions if "detect_divs" is set.
 *
 * For each pair of constraints with equal coefficients, only keep
 * the one which imposes the most stringent constraint, i.e.,
 * the one with the smallest constant term.
 * Detect opposite constraints and handle them
 * in exploit_opposite_constraints.
 */
__isl_give isl_basic_map *isl_basic_map_remove_duplicate_constraints(
  __isl_take isl_basic_map *bmap, int *progress, int detect_divs)
{
  struct isl_constraint_index ci;
  int k, l, h;
  isl_size total = isl_basic_map_dim(bmap, isl_dim_all);
  int *opposite;
 
  if (total < 0 || bmap->n_ineq <= 1)
    return bmap;
 
  if (create_constraint_index(&ci, bmap) < 0)
    return bmap;
 
  h = isl_seq_get_hash_bits(bmap->ineq[0] + 1, total, ci.bits);
  ci.index[h] = &bmap->ineq[0];
  for (k = 1; k < bmap->n_ineq; ++k) {
    h = hash_index(&ci, bmap, k);
    if (!ci.index[h]) {
      ci.index[h] = &bmap->ineq[k];
      continue;
    }
    l = ci.index[h] - &bmap->ineq[0];
    if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0]))
      swap_inequality(bmap, k, l);
    isl_basic_map_drop_inequality(bmap, k);
    --k;
  }
  opposite = detect_opposites(&ci, bmap);
  constraint_index_free(&ci);
  return exploit_opposite_constraints(bmap, opposite, progress,
                detect_divs);
}
 
/* Detect all pairs of inequalities that form an equality.
 *
 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
 * Call it repeatedly while it is making progress.
 */
__isl_give isl_basic_map *isl_basic_map_detect_inequality_pairs(
  __isl_take isl_basic_map *bmap, int *progress)
{
  int duplicate;
 
  do {
    duplicate = 0;
    bmap = isl_basic_map_remove_duplicate_constraints(bmap,
                &duplicate, 0);
    if (duplicate)
      mark_progress(progress);
  } while (duplicate);
 
  return bmap;
}
 
/* Given a known integer division "div" that is not integral
 * (with denominator 1), eliminate it from the constraints in "bmap"
 * where it appears with a (positive or negative) unit coefficient.
 * If "progress" is not NULL, then it gets set if the elimination
 * results in any changes.
 *
 * That is, replace
 *
 *  floor(e/m) + f >= 0
 *
 * by
 *
 *  e + m f >= 0
 *
 * and
 *
 *  -floor(e/m) + f >= 0
 *
 * by
 *
 *  -e + m f + m - 1 >= 0
 *
 * The first conversion is valid because floor(e/m) >= -f is equivalent
 * to e/m >= -f because -f is an integral expression.
 * The second conversion follows from the fact that
 *
 *  -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
 *
 *
 * Note that one of the div constraints may have been eliminated
 * due to being redundant with respect to the constraint that is
 * being modified by this function.  The modified constraint may
 * no longer imply this div constraint, so we add it back to make
 * sure we do not lose any information.
 */
static __isl_give isl_basic_map *eliminate_unit_div(
  __isl_take isl_basic_map *bmap, int div, int *progress)
{
  int j;
  isl_size v_div, dim;
  isl_ctx *ctx;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  dim = isl_basic_map_dim(bmap, isl_dim_all);
  if (v_div < 0 || dim < 0)
    return isl_basic_map_free(bmap);
 
  ctx = isl_basic_map_get_ctx(bmap);
 
  for (j = 0; j < bmap->n_ineq; ++j) {
    int s;
 
    if (!isl_int_is_one(bmap->ineq[j][1 + v_div + div]) &&
        !isl_int_is_negone(bmap->ineq[j][1 + v_div + div]))
      continue;
 
    mark_progress(progress);
 
    s = isl_int_sgn(bmap->ineq[j][1 + v_div + div]);
    isl_int_set_si(bmap->ineq[j][1 + v_div + div], 0);
    if (s < 0)
      isl_seq_combine(bmap->ineq[j],
        ctx->negone, bmap->div[div] + 1,
        bmap->div[div][0], bmap->ineq[j], 1 + dim);
    else
      isl_seq_combine(bmap->ineq[j],
        ctx->one, bmap->div[div] + 1,
        bmap->div[div][0], bmap->ineq[j], 1 + dim);
    if (s < 0) {
      isl_int_add(bmap->ineq[j][0],
        bmap->ineq[j][0], bmap->div[div][0]);
      isl_int_sub_ui(bmap->ineq[j][0],
        bmap->ineq[j][0], 1);
    }
 
    bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
    bmap = isl_basic_map_add_div_constraint(bmap, div, s);
    if (!bmap)
      return NULL;
  }
 
  return bmap;
}
 
/* Eliminate selected known divs from constraints where they appear with
 * a (positive or negative) unit coefficient.
 * In particular, only handle those for which "select" returns isl_bool_true.
 * If "progress" is not NULL, then it gets set if the elimination
 * results in any changes.
 *
 * We skip integral divs, i.e., those with denominator 1, as we would
 * risk eliminating the div from the div constraints.
 * They are eliminated in eliminate_integral_divs instead.
 */
static __isl_give isl_basic_map *eliminate_selected_unit_divs(
  __isl_take isl_basic_map *bmap,
  isl_bool (*select)(__isl_keep isl_basic_map *bmap, int div),
  int *progress)
{
  int i;
  isl_size n_div;
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    return isl_basic_map_free(bmap);
 
  for (i = 0; i < n_div; ++i) {
    isl_bool skip;
    isl_bool selected;
 
    skip = isl_basic_map_div_is_marked_unknown(bmap, i);
    if (skip >= 0 && !skip)
      skip = isl_basic_map_div_is_integral(bmap, i);
    if (skip < 0)
      return isl_basic_map_free(bmap);
    if (skip)
      continue;
    selected = select(bmap, i);
    if (selected < 0)
      return isl_basic_map_free(bmap);
    if (!selected)
      continue;
    bmap = eliminate_unit_div(bmap, i, progress);
    if (!bmap)
      return NULL;
  }
 
  return bmap;
}
 
/* eliminate_selected_unit_divs callback that selects every
 * integer division.
 */
static isl_bool is_any_div(__isl_keep isl_basic_map *bmap, int div)
{
  return isl_bool_true;
}
 
/* Eliminate known divs from constraints where they appear with
 * a (positive or negative) unit coefficient.
 * If "progress" is not NULL, then it gets set if the elimination
 * results in any changes.
 */
static __isl_give isl_basic_map *eliminate_unit_divs(
  __isl_take isl_basic_map *bmap, int *progress)
{
  return eliminate_selected_unit_divs(bmap, &is_any_div, progress);
}
 
/* eliminate_selected_unit_divs callback that selects
 * integer divisions that only appear with
 * a (positive or negative) unit coefficient
 * (outside their div constraints).
 */
static isl_bool is_pure_unit_div(__isl_keep isl_basic_map *bmap, int div)
{
  int i;
  isl_size v_div, n_ineq;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  n_ineq = isl_basic_map_n_inequality(bmap);
  if (v_div < 0 || n_ineq < 0)
    return isl_bool_error;
 
  for (i = 0; i < n_ineq; ++i) {
    isl_bool skip;
 
    if (isl_int_is_zero(bmap->ineq[i][1 + v_div + div]))
      continue;
    skip = isl_basic_map_is_div_constraint(bmap,
              bmap->ineq[i], div);
    if (skip < 0)
      return isl_bool_error;
    if (skip)
      continue;
    if (!isl_int_is_one(bmap->ineq[i][1 + v_div + div]) &&
        !isl_int_is_negone(bmap->ineq[i][1 + v_div + div]))
      return isl_bool_false;
  }
 
  return isl_bool_true;
}
 
/* Eliminate known divs from constraints where they appear with
 * a (positive or negative) unit coefficient,
 * but only if they do not appear in any other constraints
 * (other than the div constraints).
 */
__isl_give isl_basic_map *isl_basic_map_eliminate_pure_unit_divs(
  __isl_take isl_basic_map *bmap)
{
  return eliminate_selected_unit_divs(bmap, &is_pure_unit_div, NULL);
}
 
__isl_give isl_basic_map *isl_basic_map_simplify(__isl_take isl_basic_map *bmap)
{
  int progress = 1;
  if (!bmap)
    return NULL;
  while (progress) {
    isl_bool empty;
 
    progress = 0;
    empty = isl_basic_map_plain_is_empty(bmap);
    if (empty < 0)
      return isl_basic_map_free(bmap);
    if (empty)
      break;
    bmap = isl_basic_map_normalize_constraints(bmap);
    bmap = reduce_div_coefficients(bmap);
    bmap = normalize_div_expressions(bmap);
    bmap = remove_duplicate_divs(bmap, &progress);
    bmap = eliminate_unit_divs(bmap, &progress);
    bmap = eliminate_divs_eq(bmap, &progress);
    bmap = eliminate_divs_ineq(bmap, &progress);
    bmap = eliminate_integral_divs(bmap, &progress);
    bmap = isl_basic_map_gauss(bmap, &progress);
    /* requires equalities in normal form */
    bmap = normalize_divs(bmap, &progress);
    bmap = isl_basic_map_remove_duplicate_constraints(bmap,
                &progress, 1);
  }
  return bmap;
}
 
__isl_give isl_basic_set *isl_basic_set_simplify(
  __isl_take isl_basic_set *bset)
{
  return bset_from_bmap(isl_basic_map_simplify(bset_to_bmap(bset)));
}
 
 
isl_bool isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap,
  isl_int *constraint, unsigned div)
{
  unsigned pos;
 
  if (!bmap)
    return isl_bool_error;
 
  pos = isl_basic_map_offset(bmap, isl_dim_div) + div;
 
  if (isl_int_eq(constraint[pos], bmap->div[div][0])) {
    int neg;
    isl_int_sub(bmap->div[div][1],
        bmap->div[div][1], bmap->div[div][0]);
    isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1);
    neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos);
    isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1);
    isl_int_add(bmap->div[div][1],
        bmap->div[div][1], bmap->div[div][0]);
    if (!neg)
      return isl_bool_false;
    if (isl_seq_any_non_zero(constraint+pos+1,
              bmap->n_div-div-1))
      return isl_bool_false;
  } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) {
    if (!isl_seq_eq(constraint, bmap->div[div]+1, pos))
      return isl_bool_false;
    if (isl_seq_any_non_zero(constraint+pos+1,
              bmap->n_div-div-1))
      return isl_bool_false;
  } else
    return isl_bool_false;
 
  return isl_bool_true;
}
 
/* If the only constraints a div d=floor(f/m)
 * appears in are its two defining constraints
 *
 *  f - m d >=0
 *  -(f - (m - 1)) + m d >= 0
 *
 * then it can safely be removed.
 */
static isl_bool div_is_redundant(__isl_keep isl_basic_map *bmap, int div)
{
  int i;
  isl_bool involves;
  isl_size v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  unsigned pos = 1 + v_div + div;
 
  if (v_div < 0)
    return isl_bool_error;
 
  for (i = 0; i < bmap->n_eq; ++i)
    if (!isl_int_is_zero(bmap->eq[i][pos]))
      return isl_bool_false;
 
  for (i = 0; i < bmap->n_ineq; ++i) {
    isl_bool red;
 
    if (isl_int_is_zero(bmap->ineq[i][pos]))
      continue;
    red = isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div);
    if (red < 0 || !red)
      return red;
  }
 
  involves = isl_basic_map_any_div_involves_vars(bmap, v_div + div, 1);
  if (involves < 0 || involves)
    return isl_bool_not(involves);
 
  return isl_bool_true;
}
 
/*
 * Remove divs that don't occur in any of the constraints or other divs.
 * These can arise when dropping constraints from a basic map or
 * when the divs of a basic map have been temporarily aligned
 * with the divs of another basic map.
 */
static __isl_give isl_basic_map *remove_redundant_divs(
  __isl_take isl_basic_map *bmap)
{
  int i;
  isl_size v_div;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_basic_map_free(bmap);
 
  for (i = bmap->n_div-1; i >= 0; --i) {
    isl_bool redundant;
 
    redundant = div_is_redundant(bmap, i);
    if (redundant < 0)
      return isl_basic_map_free(bmap);
    if (!redundant)
      continue;
    bmap = isl_basic_map_drop_constraints_involving(bmap,
                v_div + i, 1);
    bmap = isl_basic_map_drop_div(bmap, i);
  }
  return bmap;
}
 
/* Mark "bmap" as final, without checking for obviously redundant
 * integer divisions.  This function should be used when "bmap"
 * is known not to involve any such integer divisions.
 */
__isl_give isl_basic_map *isl_basic_map_mark_final(
  __isl_take isl_basic_map *bmap)
{
  if (!bmap)
    return NULL;
  ISL_F_SET(bmap, ISL_BASIC_SET_FINAL);
  return bmap;
}
 
/* Mark "bmap" as final, after removing obviously redundant integer divisions.
 */
__isl_give isl_basic_map *isl_basic_map_finalize(__isl_take isl_basic_map *bmap)
{
  bmap = remove_redundant_divs(bmap);
  bmap = isl_basic_map_mark_final(bmap);
  return bmap;
}
 
__isl_give isl_basic_set *isl_basic_set_finalize(
  __isl_take isl_basic_set *bset)
{
  return bset_from_bmap(isl_basic_map_finalize(bset_to_bmap(bset)));
}
 
/* Remove definition of any div that is defined in terms of the given variable.
 * The div itself is not removed.  Functions such as
 * eliminate_divs_ineq depend on the other divs remaining in place.
 */
static __isl_give isl_basic_map *remove_dependent_vars(
  __isl_take isl_basic_map *bmap, int pos)
{
  int i;
 
  if (!bmap)
    return NULL;
 
  for (i = 0; i < bmap->n_div; ++i) {
    if (isl_int_is_zero(bmap->div[i][0]))
      continue;
    if (isl_int_is_zero(bmap->div[i][1+1+pos]))
      continue;
    bmap = isl_basic_map_mark_div_unknown(bmap, i);
    if (!bmap)
      return NULL;
  }
  return bmap;
}
 
/* Eliminate the specified variables from the constraints using
 * Fourier-Motzkin.  The variables themselves are not removed.
 */
__isl_give isl_basic_map *isl_basic_map_eliminate_vars(
  __isl_take isl_basic_map *bmap, unsigned pos, unsigned n)
{
  int d;
  int i, j, k;
  isl_size total;
  int need_gauss = 0;
 
  if (n == 0)
    return bmap;
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_basic_map_free(bmap);
 
  bmap = isl_basic_map_cow(bmap);
  for (d = pos + n - 1; d >= 0 && d >= pos; --d)
    bmap = remove_dependent_vars(bmap, d);
  if (!bmap)
    return NULL;
 
  for (d = pos + n - 1;
       d >= 0 && d >= total - bmap->n_div && d >= pos; --d)
    isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total);
  for (d = pos + n - 1; d >= 0 && d >= pos; --d) {
    int n_lower, n_upper;
    if (!bmap)
      return NULL;
    for (i = 0; i < bmap->n_eq; ++i) {
      if (isl_int_is_zero(bmap->eq[i][1+d]))
        continue;
      bmap = eliminate_var_using_equality(bmap, d,
                bmap->eq[i], 0, 1, NULL);
      if (isl_basic_map_drop_equality(bmap, i) < 0)
        return isl_basic_map_free(bmap);
      need_gauss = 1;
      break;
    }
    if (i < bmap->n_eq)
      continue;
    n_lower = 0;
    n_upper = 0;
    for (i = 0; i < bmap->n_ineq; ++i) {
      if (isl_int_is_pos(bmap->ineq[i][1+d]))
        n_lower++;
      else if (isl_int_is_neg(bmap->ineq[i][1+d]))
        n_upper++;
    }
    bmap = isl_basic_map_extend_constraints(bmap,
        0, n_lower * n_upper);
    if (!bmap)
      goto error;
    for (i = bmap->n_ineq - 1; i >= 0; --i) {
      int last;
      if (isl_int_is_zero(bmap->ineq[i][1+d]))
        continue;
      last = -1;
      for (j = 0; j < i; ++j) {
        if (isl_int_is_zero(bmap->ineq[j][1+d]))
          continue;
        last = j;
        if (isl_int_sgn(bmap->ineq[i][1+d]) ==
            isl_int_sgn(bmap->ineq[j][1+d]))
          continue;
        k = isl_basic_map_alloc_inequality(bmap);
        if (k < 0)
          goto error;
        isl_seq_cpy(bmap->ineq[k], bmap->ineq[i],
            1+total);
        isl_seq_elim(bmap->ineq[k], bmap->ineq[j],
            1+d, 1+total, NULL);
      }
      isl_basic_map_drop_inequality(bmap, i);
      i = last + 1;
    }
    if (n_lower > 0 && n_upper > 0) {
      bmap = isl_basic_map_normalize_constraints(bmap);
      bmap = isl_basic_map_remove_duplicate_constraints(bmap,
                    NULL, 0);
      bmap = isl_basic_map_gauss(bmap, NULL);
      bmap = isl_basic_map_remove_redundancies(bmap);
      need_gauss = 0;
      if (!bmap)
        goto error;
      if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
        break;
    }
  }
  if (need_gauss)
    bmap = isl_basic_map_gauss(bmap, NULL);
  return bmap;
error:
  isl_basic_map_free(bmap);
  return NULL;
}
 
__isl_give isl_basic_set *isl_basic_set_eliminate_vars(
  __isl_take isl_basic_set *bset, unsigned pos, unsigned n)
{
  return bset_from_bmap(isl_basic_map_eliminate_vars(bset_to_bmap(bset),
                pos, n));
}
 
/* Eliminate the specified n dimensions starting at first from the
 * constraints, without removing the dimensions from the space.
 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
 * Otherwise, they are projected out and the original space is restored.
 */
__isl_give isl_basic_map *isl_basic_map_eliminate(
  __isl_take isl_basic_map *bmap,
  enum isl_dim_type type, unsigned first, unsigned n)
{
  isl_space *space;
 
  if (!bmap)
    return NULL;
  if (n == 0)
    return bmap;
 
  if (isl_basic_map_check_range(bmap, type, first, n) < 0)
    return isl_basic_map_free(bmap);
 
  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) {
    first += isl_basic_map_offset(bmap, type) - 1;
    bmap = isl_basic_map_eliminate_vars(bmap, first, n);
    return isl_basic_map_finalize(bmap);
  }
 
  space = isl_basic_map_get_space(bmap);
  bmap = isl_basic_map_project_out(bmap, type, first, n);
  bmap = isl_basic_map_insert_dims(bmap, type, first, n);
  bmap = isl_basic_map_reset_space(bmap, space);
  return bmap;
}
 
__isl_give isl_basic_set *isl_basic_set_eliminate(
  __isl_take isl_basic_set *bset,
  enum isl_dim_type type, unsigned first, unsigned n)
{
  return isl_basic_map_eliminate(bset, type, first, n);
}
 
/* Remove all constraints from "bmap" that reference any unknown local
 * variables (directly or indirectly).
 *
 * Dropping all constraints on a local variable will make it redundant,
 * so it will get removed implicitly by
 * isl_basic_map_drop_constraints_involving_dims.  Some other local
 * variables may also end up becoming redundant if they only appear
 * in constraints together with the unknown local variable.
 * Therefore, start over after calling
 * isl_basic_map_drop_constraints_involving_dims.
 */
__isl_give isl_basic_map *isl_basic_map_drop_constraints_involving_unknown_divs(
  __isl_take isl_basic_map *bmap)
{
  isl_bool known;
  isl_size n_div;
  int i, o_div;
 
  known = isl_basic_map_divs_known(bmap);
  if (known < 0)
    return isl_basic_map_free(bmap);
  if (known)
    return bmap;
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    return isl_basic_map_free(bmap);
  o_div = isl_basic_map_offset(bmap, isl_dim_div) - 1;
 
  for (i = 0; i < n_div; ++i) {
    known = isl_basic_map_div_is_known(bmap, i);
    if (known < 0)
      return isl_basic_map_free(bmap);
    if (known)
      continue;
    bmap = remove_dependent_vars(bmap, o_div + i);
    bmap = isl_basic_map_drop_constraints_involving_dims(bmap,
                  isl_dim_div, i, 1);
    n_div = isl_basic_map_dim(bmap, isl_dim_div);
    if (n_div < 0)
      return isl_basic_map_free(bmap);
    i = -1;
  }
 
  return bmap;
}
 
/* Remove all constraints from "bset" that reference any unknown local
 * variables (directly or indirectly).
 */
__isl_give isl_basic_set *isl_basic_set_drop_constraints_involving_unknown_divs(
  __isl_take isl_basic_set *bset)
{
  isl_basic_map *bmap;
 
  bmap = bset_to_bmap(bset);
  bmap = isl_basic_map_drop_constraints_involving_unknown_divs(bmap);
  return bset_from_bmap(bmap);
}
 
/* Remove all constraints from "map" that reference any unknown local
 * variables (directly or indirectly).
 *
 * Since constraints may get dropped from the basic maps,
 * they may no longer be disjoint from each other.
 */
__isl_give isl_map *isl_map_drop_constraints_involving_unknown_divs(
  __isl_take isl_map *map)
{
  int i;
  isl_bool known;
 
  known = isl_map_divs_known(map);
  if (known < 0)
    return isl_map_free(map);
  if (known)
    return map;
 
  map = isl_map_cow(map);
  if (!map)
    return NULL;
 
  for (i = 0; i < map->n; ++i) {
    map->p[i] =
        isl_basic_map_drop_constraints_involving_unknown_divs(
                    map->p[i]);
    if (!map->p[i])
      return isl_map_free(map);
  }
 
  if (map->n > 1)
    ISL_F_CLR(map, ISL_MAP_DISJOINT);
 
  return map;
}
 
/* Don't assume equalities are in order, because align_divs
 * may have changed the order of the divs.
 */
static void compute_elimination_index(__isl_keep isl_basic_map *bmap, int *elim,
  unsigned len)
{
  int d, i;
 
  for (d = 0; d < len; ++d)
    elim[d] = -1;
  for (i = 0; i < bmap->n_eq; ++i) {
    for (d = len - 1; d >= 0; --d) {
      if (isl_int_is_zero(bmap->eq[i][1+d]))
        continue;
      elim[d] = i;
      break;
    }
  }
}
 
static void set_compute_elimination_index(__isl_keep isl_basic_set *bset,
  int *elim, unsigned len)
{
  compute_elimination_index(bset_to_bmap(bset), elim, len);
}
 
static int reduced_using_equalities(isl_int *dst, isl_int *src,
  __isl_keep isl_basic_map *bmap, int *elim, unsigned total)
{
  int d;
  int copied = 0;
 
  for (d = total - 1; d >= 0; --d) {
    if (isl_int_is_zero(src[1+d]))
      continue;
    if (elim[d] == -1)
      continue;
    if (!copied) {
      isl_seq_cpy(dst, src, 1 + total);
      copied = 1;
    }
    isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL);
  }
  return copied;
}
 
static int set_reduced_using_equalities(isl_int *dst, isl_int *src,
  __isl_keep isl_basic_set *bset, int *elim, unsigned total)
{
  return reduced_using_equalities(dst, src,
          bset_to_bmap(bset), elim, total);
}
 
static __isl_give isl_basic_set *isl_basic_set_reduce_using_equalities(
  __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context)
{
  int i;
  int *elim;
  isl_size dim;
 
  if (!bset || !context)
    goto error;
 
  if (context->n_eq == 0) {
    isl_basic_set_free(context);
    return bset;
  }
 
  bset = isl_basic_set_cow(bset);
  dim = isl_basic_set_dim(bset, isl_dim_set);
  if (dim < 0)
    goto error;
 
  elim = isl_alloc_array(bset->ctx, int, dim);
  if (!elim)
    goto error;
  set_compute_elimination_index(context, elim, dim);
  for (i = 0; i < bset->n_eq; ++i)
    set_reduced_using_equalities(bset->eq[i], bset->eq[i],
              context, elim, dim);
  for (i = 0; i < bset->n_ineq; ++i)
    set_reduced_using_equalities(bset->ineq[i], bset->ineq[i],
              context, elim, dim);
  isl_basic_set_free(context);
  free(elim);
  bset = isl_basic_set_simplify(bset);
  bset = isl_basic_set_finalize(bset);
  return bset;
error:
  isl_basic_set_free(bset);
  isl_basic_set_free(context);
  return NULL;
}
 
/* For each inequality in "ineq" that is a shifted (more relaxed)
 * copy of an inequality in "context", mark the corresponding entry
 * in "row" with -1.
 * If an inequality only has a non-negative constant term, then
 * mark it as well.
 */
static isl_stat mark_shifted_constraints(__isl_keep isl_mat *ineq,
  __isl_keep isl_basic_set *context, int *row)
{
  struct isl_constraint_index ci;
  isl_size n_ineq, cols;
  unsigned total;
  int k;
 
  if (!ineq || !context)
    return isl_stat_error;
  if (context->n_ineq == 0)
    return isl_stat_ok;
  if (setup_constraint_index(&ci, context) < 0)
    return isl_stat_error;
 
  n_ineq = isl_mat_rows(ineq);
  cols = isl_mat_cols(ineq);
  if (n_ineq < 0 || cols < 0)
    return isl_stat_error;
  total = cols - 1;
  for (k = 0; k < n_ineq; ++k) {
    int l;
    isl_bool redundant;
 
    l = isl_seq_first_non_zero(ineq->row[k] + 1, total);
    if (l < 0 && isl_int_is_nonneg(ineq->row[k][0])) {
      row[k] = -1;
      continue;
    }
    redundant = constraint_index_is_redundant(&ci, ineq->row[k]);
    if (redundant < 0)
      goto error;
    if (!redundant)
      continue;
    row[k] = -1;
  }
  constraint_index_free(&ci);
  return isl_stat_ok;
error:
  constraint_index_free(&ci);
  return isl_stat_error;
}
 
static __isl_give isl_basic_set *remove_shifted_constraints(
  __isl_take isl_basic_set *bset, __isl_keep isl_basic_set *context)
{
  struct isl_constraint_index ci;
  int k;
 
  if (!bset || !context)
    return bset;
 
  if (context->n_ineq == 0)
    return bset;
  if (setup_constraint_index(&ci, context) < 0)
    return bset;
 
  for (k = 0; k < bset->n_ineq; ++k) {
    isl_bool redundant;
 
    redundant = constraint_index_is_redundant(&ci, bset->ineq[k]);
    if (redundant < 0)
      goto error;
    if (!redundant)
      continue;
    bset = isl_basic_set_cow(bset);
    if (!bset)
      goto error;
    isl_basic_set_drop_inequality(bset, k);
    --k;
  }
  constraint_index_free(&ci);
  return bset;
error:
  constraint_index_free(&ci);
  return bset;
}
 
/* Remove constraints from "bmap" that are identical to constraints
 * in "context" or that are more relaxed (greater constant term).
 *
 * We perform the test for shifted copies on the pure constraints
 * in remove_shifted_constraints.
 */
static __isl_give isl_basic_map *isl_basic_map_remove_shifted_constraints(
  __isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context)
{
  isl_basic_set *bset, *bset_context;
 
  if (!bmap || !context)
    goto error;
 
  if (bmap->n_ineq == 0 || context->n_ineq == 0) {
    isl_basic_map_free(context);
    return bmap;
  }
 
  context = isl_basic_map_drop_constraints_involving_unknown_divs(
                context);
  context = isl_basic_map_remove_unknown_divs(context);
 
  context = isl_basic_map_order_divs(context);
  bmap = isl_basic_map_align_divs(bmap, context);
  context = isl_basic_map_align_divs(context, bmap);
 
  bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
  bset_context = isl_basic_map_underlying_set(context);
  bset = remove_shifted_constraints(bset, bset_context);
  isl_basic_set_free(bset_context);
 
  bmap = isl_basic_map_overlying_set(bset, bmap);
 
  return bmap;
error:
  isl_basic_map_free(bmap);
  isl_basic_map_free(context);
  return NULL;
}
 
/* Does the (linear part of a) constraint "c" involve any of the "len"
 * "relevant" dimensions?
 */
static int is_related(isl_int *c, int len, int *relevant)
{
  int i;
 
  for (i = 0; i < len; ++i) {
    if (!relevant[i])
      continue;
    if (!isl_int_is_zero(c[i]))
      return 1;
  }
 
  return 0;
}
 
/* Drop constraints from "bmap" that do not involve any of
 * the dimensions marked "relevant".
 */
static __isl_give isl_basic_map *drop_unrelated_constraints(
  __isl_take isl_basic_map *bmap, int *relevant)
{
  int i;
  isl_size dim;
 
  dim = isl_basic_map_dim(bmap, isl_dim_all);
  if (dim < 0)
    return isl_basic_map_free(bmap);
  for (i = 0; i < dim; ++i)
    if (!relevant[i])
      break;
  if (i >= dim)
    return bmap;
 
  for (i = bmap->n_eq - 1; i >= 0; --i)
    if (!is_related(bmap->eq[i] + 1, dim, relevant)) {
      bmap = isl_basic_map_cow(bmap);
      if (isl_basic_map_drop_equality(bmap, i) < 0)
        return isl_basic_map_free(bmap);
    }
 
  for (i = bmap->n_ineq - 1; i >= 0; --i)
    if (!is_related(bmap->ineq[i] + 1, dim, relevant)) {
      bmap = isl_basic_map_cow(bmap);
      if (isl_basic_map_drop_inequality(bmap, i) < 0)
        return isl_basic_map_free(bmap);
    }
 
  return bmap;
}
 
/* Update the groups in "group" based on the (linear part of a) constraint "c".
 *
 * In particular, for any variable involved in the constraint,
 * find the actual group id from before and replace the group
 * of the corresponding variable by the minimal group of all
 * the variables involved in the constraint considered so far
 * (if this minimum is smaller) or replace the minimum by this group
 * (if the minimum is larger).
 *
 * At the end, all the variables in "c" will (indirectly) point
 * to the minimal of the groups that they referred to originally.
 */
static void update_groups(int dim, int *group, isl_int *c)
{
  int j;
  int min = dim;
 
  for (j = 0; j < dim; ++j) {
    if (isl_int_is_zero(c[j]))
      continue;
    while (group[j] >= 0 && group[group[j]] != group[j])
      group[j] = group[group[j]];
    if (group[j] == min)
      continue;
    if (group[j] < min) {
      if (min >= 0 && min < dim)
        group[min] = group[j];
      min = group[j];
    } else
      group[group[j]] = min;
  }
}
 
/* Allocate an array of groups of variables, one for each variable
 * in "context", initialized to zero.
 */
static int *alloc_groups(__isl_keep isl_basic_set *context)
{
  isl_ctx *ctx;
  isl_size dim;
 
  dim = isl_basic_set_dim(context, isl_dim_set);
  if (dim < 0)
    return NULL;
  ctx = isl_basic_set_get_ctx(context);
  return isl_calloc_array(ctx, int, dim);
}
 
/* Drop constraints from "bmap" that only involve variables that are
 * not related to any of the variables marked with a "-1" in "group".
 *
 * We construct groups of variables that collect variables that
 * (indirectly) appear in some common constraint of "bmap".
 * Each group is identified by the first variable in the group,
 * except for the special group of variables that was already identified
 * in the input as -1 (or are related to those variables).
 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
 * otherwise the group of i is the group of group[i].
 *
 * We first initialize groups for the remaining variables.
 * Then we iterate over the constraints of "bmap" and update the
 * group of the variables in the constraint by the smallest group.
 * Finally, we resolve indirect references to groups by running over
 * the variables.
 *
 * After computing the groups, we drop constraints that do not involve
 * any variables in the -1 group.
 */
__isl_give isl_basic_map *isl_basic_map_drop_unrelated_constraints(
  __isl_take isl_basic_map *bmap, __isl_take int *group)
{
  isl_size dim;
  int i;
  int last;
 
  dim = isl_basic_map_dim(bmap, isl_dim_all);
  if (dim < 0)
    goto error;
 
  last = -1;
  for (i = 0; i < dim; ++i)
    if (group[i] >= 0)
      last = group[i] = i;
  if (last < 0) {
    free(group);
    return bmap;
  }
 
  for (i = 0; i < bmap->n_eq; ++i)
    update_groups(dim, group, bmap->eq[i] + 1);
  for (i = 0; i < bmap->n_ineq; ++i)
    update_groups(dim, group, bmap->ineq[i] + 1);
 
  for (i = 0; i < dim; ++i)
    if (group[i] >= 0)
      group[i] = group[group[i]];
 
  for (i = 0; i < dim; ++i)
    group[i] = group[i] == -1;
 
  bmap = drop_unrelated_constraints(bmap, group);
 
  free(group);
  return bmap;
error:
  free(group);
  isl_basic_map_free(bmap);
  return NULL;
}
 
/* Drop constraints from "context" that are irrelevant for computing
 * the gist of "bset".
 *
 * In particular, drop constraints in variables that are not related
 * to any of the variables involved in the constraints of "bset"
 * in the sense that there is no sequence of constraints that connects them.
 *
 * We first mark all variables that appear in "bset" as belonging
 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
 */
static __isl_give isl_basic_set *drop_irrelevant_constraints(
  __isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset)
{
  int *group;
  isl_size dim;
  int i, j;
 
  dim = isl_basic_set_dim(bset, isl_dim_set);
  if (!context || dim < 0)
    return isl_basic_set_free(context);
 
  group = alloc_groups(context);
 
  if (!group)
    return isl_basic_set_free(context);
 
  for (i = 0; i < dim; ++i) {
    for (j = 0; j < bset->n_eq; ++j)
      if (!isl_int_is_zero(bset->eq[j][1 + i]))
        break;
    if (j < bset->n_eq) {
      group[i] = -1;
      continue;
    }
    for (j = 0; j < bset->n_ineq; ++j)
      if (!isl_int_is_zero(bset->ineq[j][1 + i]))
        break;
    if (j < bset->n_ineq)
      group[i] = -1;
  }
 
  return isl_basic_map_drop_unrelated_constraints(context, group);
}
 
/* Drop constraints from "context" that are irrelevant for computing
 * the gist of the inequalities "ineq".
 * Inequalities in "ineq" for which the corresponding element of row
 * is set to -1 have already been marked for removal and should be ignored.
 *
 * In particular, drop constraints in variables that are not related
 * to any of the variables involved in "ineq"
 * in the sense that there is no sequence of constraints that connects them.
 *
 * We first mark all variables that appear in "bset" as belonging
 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
 */
static __isl_give isl_basic_set *drop_irrelevant_constraints_marked(
  __isl_take isl_basic_set *context, __isl_keep isl_mat *ineq, int *row)
{
  int *group;
  isl_size dim;
  int i, j;
  isl_size n;
 
  dim = isl_basic_set_dim(context, isl_dim_set);
  n = isl_mat_rows(ineq);
  if (dim < 0 || n < 0)
    return isl_basic_set_free(context);
 
  group = alloc_groups(context);
 
  if (!group)
    return isl_basic_set_free(context);
 
  for (i = 0; i < dim; ++i) {
    for (j = 0; j < n; ++j) {
      if (row[j] < 0)
        continue;
      if (!isl_int_is_zero(ineq->row[j][1 + i]))
        break;
    }
    if (j < n)
      group[i] = -1;
  }
 
  return isl_basic_map_drop_unrelated_constraints(context, group);
}
 
/* Do all "n" entries of "row" contain a negative value?
 */
static int all_neg(int *row, int n)
{
  int i;
 
  for (i = 0; i < n; ++i)
    if (row[i] >= 0)
      return 0;
 
  return 1;
}
 
/* Update the inequalities in "bset" based on the information in "row"
 * and "tab".
 *
 * In particular, the array "row" contains either -1, meaning that
 * the corresponding inequality of "bset" is redundant, or the index
 * of an inequality in "tab".
 *
 * If the row entry is -1, then drop the inequality.
 * Otherwise, if the constraint is marked redundant in the tableau,
 * then drop the inequality.  Similarly, if it is marked as an equality
 * in the tableau, then turn the inequality into an equality and
 * perform Gaussian elimination.
 */
static __isl_give isl_basic_set *update_ineq(__isl_take isl_basic_set *bset,
  __isl_keep int *row, struct isl_tab *tab)
{
  int i;
  unsigned n_ineq;
  unsigned n_eq;
  int found_equality = 0;
 
  if (!bset)
    return NULL;
  if (tab && tab->empty)
    return isl_basic_set_set_to_empty(bset);
 
  n_ineq = bset->n_ineq;
  for (i = n_ineq - 1; i >= 0; --i) {
    if (row[i] < 0) {
      if (isl_basic_set_drop_inequality(bset, i) < 0)
        return isl_basic_set_free(bset);
      continue;
    }
    if (!tab)
      continue;
    n_eq = tab->n_eq;
    if (isl_tab_is_equality(tab, n_eq + row[i])) {
      isl_basic_map_inequality_to_equality(bset, i);
      found_equality = 1;
    } else if (isl_tab_is_redundant(tab, n_eq + row[i])) {
      if (isl_basic_set_drop_inequality(bset, i) < 0)
        return isl_basic_set_free(bset);
    }
  }
 
  if (found_equality)
    bset = isl_basic_set_gauss(bset, NULL);
  bset = isl_basic_set_finalize(bset);
  return bset;
}
 
/* Update the inequalities in "bset" based on the information in "row"
 * and "tab" and free all arguments (other than "bset").
 */
static __isl_give isl_basic_set *update_ineq_free(
  __isl_take isl_basic_set *bset, __isl_take isl_mat *ineq,
  __isl_take isl_basic_set *context, __isl_take int *row,
  struct isl_tab *tab)
{
  isl_mat_free(ineq);
  isl_basic_set_free(context);
 
  bset = update_ineq(bset, row, tab);
 
  free(row);
  isl_tab_free(tab);
  return bset;
}
 
/* Remove all information from bset that is redundant in the context
 * of context.
 * "ineq" contains the (possibly transformed) inequalities of "bset",
 * in the same order.
 * The (explicit) equalities of "bset" are assumed to have been taken
 * into account by the transformation such that only the inequalities
 * are relevant.
 * "context" is assumed not to be empty.
 *
 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
 * A value of -1 means that the inequality is obviously redundant and may
 * not even appear in  "tab".
 *
 * We first mark the inequalities of "bset"
 * that are obviously redundant with respect to some inequality in "context".
 * Then we remove those constraints from "context" that have become
 * irrelevant for computing the gist of "bset".
 * Note that this removal of constraints cannot be replaced by
 * a factorization because factors in "bset" may still be connected
 * to each other through constraints in "context".
 *
 * If there are any inequalities left, we construct a tableau for
 * the context and then add the inequalities of "bset".
 * Before adding these inequalities, we freeze all constraints such that
 * they won't be considered redundant in terms of the constraints of "bset".
 * Then we detect all redundant constraints (among the
 * constraints that weren't frozen), first by checking for redundancy in the
 * the tableau and then by checking if replacing a constraint by its negation
 * would lead to an empty set.  This last step is fairly expensive
 * and could be optimized by more reuse of the tableau.
 * Finally, we update bset according to the results.
 */
static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset,
  __isl_take isl_mat *ineq, __isl_take isl_basic_set *context)
{
  int i, r;
  int *row = NULL;
  isl_ctx *ctx;
  isl_basic_set *combined = NULL;
  struct isl_tab *tab = NULL;
  unsigned n_eq, context_ineq;
 
  if (!bset || !ineq || !context)
    goto error;
 
  if (bset->n_ineq == 0 || isl_basic_set_plain_is_universe(context)) {
    isl_basic_set_free(context);
    isl_mat_free(ineq);
    return bset;
  }
 
  ctx = isl_basic_set_get_ctx(context);
  row = isl_calloc_array(ctx, int, bset->n_ineq);
  if (!row)
    goto error;
 
  if (mark_shifted_constraints(ineq, context, row) < 0)
    goto error;
  if (all_neg(row, bset->n_ineq))
    return update_ineq_free(bset, ineq, context, row, NULL);
 
  context = drop_irrelevant_constraints_marked(context, ineq, row);
  if (!context)
    goto error;
  if (isl_basic_set_plain_is_universe(context))
    return update_ineq_free(bset, ineq, context, row, NULL);
 
  n_eq = context->n_eq;
  context_ineq = context->n_ineq;
  combined = isl_basic_set_cow(isl_basic_set_copy(context));
  combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq);
  tab = isl_tab_from_basic_set(combined, 0);
  for (i = 0; i < context_ineq; ++i)
    if (isl_tab_freeze_constraint(tab, n_eq + i) < 0)
      goto error;
  if (isl_tab_extend_cons(tab, bset->n_ineq) < 0)
    goto error;
  r = context_ineq;
  for (i = 0; i < bset->n_ineq; ++i) {
    if (row[i] < 0)
      continue;
    combined = isl_basic_set_add_ineq(combined, ineq->row[i]);
    if (isl_tab_add_ineq(tab, ineq->row[i]) < 0)
      goto error;
    row[i] = r++;
  }
  if (isl_tab_detect_implicit_equalities(tab) < 0)
    goto error;
  if (isl_tab_detect_redundant(tab) < 0)
    goto error;
  for (i = bset->n_ineq - 1; i >= 0; --i) {
    isl_basic_set *test;
    int is_empty;
 
    if (row[i] < 0)
      continue;
    r = row[i];
    if (tab->con[n_eq + r].is_redundant)
      continue;
    test = isl_basic_set_dup(combined);
    test = isl_inequality_negate(test, r);
    test = isl_basic_set_update_from_tab(test, tab);
    is_empty = isl_basic_set_is_empty(test);
    isl_basic_set_free(test);
    if (is_empty < 0)
      goto error;
    if (is_empty)
      tab->con[n_eq + r].is_redundant = 1;
  }
  bset = update_ineq_free(bset, ineq, context, row, tab);
  if (bset) {
    ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
    ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
  }
 
  isl_basic_set_free(combined);
  return bset;
error:
  free(row);
  isl_mat_free(ineq);
  isl_tab_free(tab);
  isl_basic_set_free(combined);
  isl_basic_set_free(context);
  isl_basic_set_free(bset);
  return NULL;
}
 
/* Extract the inequalities of "bset" as an isl_mat.
 */
static __isl_give isl_mat *extract_ineq(__isl_keep isl_basic_set *bset)
{
  isl_size total;
  isl_ctx *ctx;
  isl_mat *ineq;
 
  total = isl_basic_set_dim(bset, isl_dim_all);
  if (total < 0)
    return NULL;
 
  ctx = isl_basic_set_get_ctx(bset);
  ineq = isl_mat_sub_alloc6(ctx, bset->ineq, 0, bset->n_ineq,
            0, 1 + total);
 
  return ineq;
}
 
/* Remove all information from "bset" that is redundant in the context
 * of "context", for the case where both "bset" and "context" are
 * full-dimensional.
 */
static __isl_give isl_basic_set *uset_gist_uncompressed(
  __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context)
{
  isl_mat *ineq;
 
  ineq = extract_ineq(bset);
  return uset_gist_full(bset, ineq, context);
}
 
/* Replace "bset" by an empty basic set in the same space.
 */
static __isl_give isl_basic_set *replace_by_empty(
  __isl_take isl_basic_set *bset)
{
  isl_space *space;
 
  space = isl_basic_set_get_space(bset);
  isl_basic_set_free(bset);
  return isl_basic_set_empty(space);
}
 
/* Remove all information from "bset" that is redundant in the context
 * of "context", for the case where the combined equalities of
 * "bset" and "context" allow for a compression that can be obtained
 * by preapplication of "T".
 * If the compression of "context" is empty, meaning that "bset" and
 * "context" do not intersect, then return the empty set.
 *
 * "bset" itself is not transformed by "T".  Instead, the inequalities
 * are extracted from "bset" and those are transformed by "T".
 * uset_gist_full then determines which of the transformed inequalities
 * are redundant with respect to the transformed "context" and removes
 * the corresponding inequalities from "bset".
 *
 * After preapplying "T" to the inequalities, any common factor is
 * removed from the coefficients.  If this results in a tightening
 * of the constant term, then the same tightening is applied to
 * the corresponding untransformed inequality in "bset".
 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
 *
 *  g f'(x) + r >= 0
 *
 * with 0 <= r < g, then it is equivalent to
 *
 *  f'(x) >= 0
 *
 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
 * subspace compressed by T since the latter would be transformed to
 *
 *  g f'(x) >= 0
 */
static __isl_give isl_basic_set *uset_gist_compressed(
  __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context,
  __isl_take isl_mat *T)
{
  isl_ctx *ctx;
  isl_mat *ineq;
  int i;
  isl_size n_row, n_col;
  isl_int rem;
 
  ineq = extract_ineq(bset);
  ineq = isl_mat_product(ineq, isl_mat_copy(T));
  context = isl_basic_set_preimage(context, T);
 
  if (!ineq || !context)
    goto error;
  if (isl_basic_set_plain_is_empty(context)) {
    isl_mat_free(ineq);
    isl_basic_set_free(context);
    return replace_by_empty(bset);
  }
 
  ctx = isl_mat_get_ctx(ineq);
  n_row = isl_mat_rows(ineq);
  n_col = isl_mat_cols(ineq);
  if (n_row < 0 || n_col < 0)
    goto error;
  isl_int_init(rem);
  for (i = 0; i < n_row; ++i) {
    isl_seq_gcd(ineq->row[i] + 1, n_col - 1, &ctx->normalize_gcd);
    if (isl_int_is_zero(ctx->normalize_gcd))
      continue;
    if (isl_int_is_one(ctx->normalize_gcd))
      continue;
    isl_seq_scale_down(ineq->row[i] + 1, ineq->row[i] + 1,
            ctx->normalize_gcd, n_col - 1);
    isl_int_fdiv_r(rem, ineq->row[i][0], ctx->normalize_gcd);
    isl_int_fdiv_q(ineq->row[i][0],
        ineq->row[i][0], ctx->normalize_gcd);
    if (isl_int_is_zero(rem))
      continue;
    bset = isl_basic_set_cow(bset);
    if (!bset)
      break;
    isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], rem);
  }
  isl_int_clear(rem);
 
  return uset_gist_full(bset, ineq, context);
error:
  isl_mat_free(ineq);
  isl_basic_set_free(context);
  isl_basic_set_free(bset);
  return NULL;
}
 
/* Project "bset" onto the variables that are involved in "template".
 */
static __isl_give isl_basic_set *project_onto_involved(
  __isl_take isl_basic_set *bset, __isl_keep isl_basic_set *template)
{
  int i;
  isl_size n;
 
  n = isl_basic_set_dim(template, isl_dim_set);
  if (n < 0 || !template)
    return isl_basic_set_free(bset);
 
  for (i = 0; i < n; ++i) {
    isl_bool involved;
 
    involved = isl_basic_set_involves_dims(template,
              isl_dim_set, i, 1);
    if (involved < 0)
      return isl_basic_set_free(bset);
    if (involved)
      continue;
    bset = isl_basic_set_eliminate_vars(bset, i, 1);
  }
 
  return bset;
}
 
/* Remove all information from bset that is redundant in the context
 * of context.  In particular, equalities that are linear combinations
 * of those in context are removed.  Then the inequalities that are
 * redundant in the context of the equalities and inequalities of
 * context are removed.
 *
 * First of all, we drop those constraints from "context"
 * that are irrelevant for computing the gist of "bset".
 * Alternatively, we could factorize the intersection of "context" and "bset".
 *
 * We first compute the intersection of the integer affine hulls
 * of "bset" and "context",
 * compute the gist inside this intersection and then reduce
 * the constraints with respect to the equalities of the context
 * that only involve variables already involved in the input.
 * If the intersection of the affine hulls turns out to be empty,
 * then return the empty set.
 *
 * If two constraints are mutually redundant, then uset_gist_full
 * will remove the second of those constraints.  We therefore first
 * sort the constraints so that constraints not involving existentially
 * quantified variables are given precedence over those that do.
 * We have to perform this sorting before the variable compression,
 * because that may effect the order of the variables.
 */
static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset,
  __isl_take isl_basic_set *context)
{
  isl_mat *eq;
  isl_mat *T;
  isl_basic_set *aff;
  isl_basic_set *aff_context;
  isl_size total;
 
  total = isl_basic_set_dim(bset, isl_dim_all);
  if (total < 0 || !context)
    goto error;
 
  context = drop_irrelevant_constraints(context, bset);
 
  bset = isl_basic_set_detect_equalities(bset);
  aff = isl_basic_set_copy(bset);
  aff = isl_basic_set_plain_affine_hull(aff);
  context = isl_basic_set_detect_equalities(context);
  aff_context = isl_basic_set_copy(context);
  aff_context = isl_basic_set_plain_affine_hull(aff_context);
  aff = isl_basic_set_intersect(aff, aff_context);
  if (!aff)
    goto error;
  if (isl_basic_set_plain_is_empty(aff)) {
    isl_basic_set_free(bset);
    isl_basic_set_free(context);
    return aff;
  }
  bset = isl_basic_set_sort_constraints(bset);
  if (aff->n_eq == 0) {
    isl_basic_set_free(aff);
    return uset_gist_uncompressed(bset, context);
  }
  eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total);
  eq = isl_mat_cow(eq);
  T = isl_mat_variable_compression(eq, NULL);
  isl_basic_set_free(aff);
  if (T && T->n_col == 0) {
    isl_mat_free(T);
    isl_basic_set_free(context);
    return replace_by_empty(bset);
  }
 
  aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context));
  aff_context = project_onto_involved(aff_context, bset);
 
  bset = uset_gist_compressed(bset, context, T);
  bset = isl_basic_set_reduce_using_equalities(bset, aff_context);
 
  if (bset) {
    ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT);
    ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT);
  }
 
  return bset;
error:
  isl_basic_set_free(bset);
  isl_basic_set_free(context);
  return NULL;
}
 
/* Return the number of equality constraints in "bmap" that involve
 * local variables.  This function assumes that Gaussian elimination
 * has been applied to the equality constraints.
 */
static int n_div_eq(__isl_keep isl_basic_map *bmap)
{
  int i;
  isl_size total, n_div;
 
  if (!bmap)
    return -1;
 
  if (bmap->n_eq == 0)
    return 0;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (total < 0 || n_div < 0)
    return -1;
  total -= n_div;
 
  for (i = 0; i < bmap->n_eq; ++i)
    if (!isl_seq_any_non_zero(bmap->eq[i] + 1 + total, n_div))
      return i;
 
  return bmap->n_eq;
}
 
/* Construct a basic map in "space" defined by the equality constraints in "eq".
 * The constraints are assumed not to involve any local variables.
 */
static __isl_give isl_basic_map *basic_map_from_equalities(
  __isl_take isl_space *space, __isl_take isl_mat *eq)
{
  int i, k;
  isl_size total;
  isl_basic_map *bmap = NULL;
 
  total = isl_space_dim(space, isl_dim_all);
  if (total < 0 || !eq)
    goto error;
 
  if (1 + total != eq->n_col)
    isl_die(isl_space_get_ctx(space), isl_error_internal,
      "unexpected number of columns", goto error);
 
  bmap = isl_basic_map_alloc_space(isl_space_copy(space),
              0, eq->n_row, 0);
  for (i = 0; i < eq->n_row; ++i) {
    k = isl_basic_map_alloc_equality(bmap);
    if (k < 0)
      goto error;
    isl_seq_cpy(bmap->eq[k], eq->row[i], eq->n_col);
  }
 
  isl_space_free(space);
  isl_mat_free(eq);
  return bmap;
error:
  isl_space_free(space);
  isl_mat_free(eq);
  isl_basic_map_free(bmap);
  return NULL;
}
 
/* Construct and return a variable compression based on the equality
 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
 * "n1" is the number of (initial) equality constraints in "bmap1"
 * that do involve local variables.
 * "n2" is the number of (initial) equality constraints in "bmap2"
 * that do involve local variables.
 * "total" is the total number of other variables.
 * This function assumes that Gaussian elimination
 * has been applied to the equality constraints in both "bmap1" and "bmap2"
 * such that the equality constraints not involving local variables
 * are those that start at "n1" or "n2".
 *
 * If either of "bmap1" and "bmap2" does not have such equality constraints,
 * then simply compute the compression based on the equality constraints
 * in the other basic map.
 * Otherwise, combine the equality constraints from both into a new
 * basic map such that Gaussian elimination can be applied to this combination
 * and then construct a variable compression from the resulting
 * equality constraints.
 */
static __isl_give isl_mat *combined_variable_compression(
  __isl_keep isl_basic_map *bmap1, int n1,
  __isl_keep isl_basic_map *bmap2, int n2, int total)
{
  isl_ctx *ctx;
  isl_mat *E1, *E2, *V;
  isl_basic_map *bmap;
 
  ctx = isl_basic_map_get_ctx(bmap1);
  if (bmap1->n_eq == n1) {
    E2 = isl_mat_sub_alloc6(ctx, bmap2->eq,
          n2, bmap2->n_eq - n2, 0, 1 + total);
    return isl_mat_variable_compression(E2, NULL);
  }
  if (bmap2->n_eq == n2) {
    E1 = isl_mat_sub_alloc6(ctx, bmap1->eq,
          n1, bmap1->n_eq - n1, 0, 1 + total);
    return isl_mat_variable_compression(E1, NULL);
  }
  E1 = isl_mat_sub_alloc6(ctx, bmap1->eq,
        n1, bmap1->n_eq - n1, 0, 1 + total);
  E2 = isl_mat_sub_alloc6(ctx, bmap2->eq,
        n2, bmap2->n_eq - n2, 0, 1 + total);
  E1 = isl_mat_concat(E1, E2);
  bmap = basic_map_from_equalities(isl_basic_map_get_space(bmap1), E1);
  bmap = isl_basic_map_gauss(bmap, NULL);
  if (!bmap)
    return NULL;
  E1 = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap->n_eq, 0, 1 + total);
  V = isl_mat_variable_compression(E1, NULL);
  isl_basic_map_free(bmap);
 
  return V;
}
 
/* Extract the stride constraints from "bmap", compressed
 * with respect to both the stride constraints in "context" and
 * the remaining equality constraints in both "bmap" and "context".
 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
 * "context_n_eq" is the number of (initial) stride constraints in "context".
 *
 * Let x be all variables in "bmap" (and "context") other than the local
 * variables.  First compute a variable compression
 *
 *  x = V x'
 *
 * based on the non-stride equality constraints in "bmap" and "context".
 * Consider the stride constraints of "context",
 *
 *  A(x) + B(y) = 0
 *
 * with y the local variables and plug in the variable compression,
 * resulting in
 *
 *  A(V x') + B(y) = 0
 *
 * Use these constraints to compute a parameter compression on x'
 *
 *  x' = T x''
 *
 * Now consider the stride constraints of "bmap"
 *
 *  C(x) + D(y) = 0
 *
 * and plug in x = V*T x''.
 * That is, return A = [C*V*T D].
 */
static __isl_give isl_mat *extract_compressed_stride_constraints(
  __isl_keep isl_basic_map *bmap, int bmap_n_eq,
  __isl_keep isl_basic_map *context, int context_n_eq)
{
  isl_size total, n_div;
  isl_ctx *ctx;
  isl_mat *A, *B, *T, *V;
 
  total = isl_basic_map_dim(context, isl_dim_all);
  n_div = isl_basic_map_dim(context, isl_dim_div);
  if (total < 0 || n_div < 0)
    return NULL;
  total -= n_div;
 
  ctx = isl_basic_map_get_ctx(bmap);
 
  V = combined_variable_compression(bmap, bmap_n_eq,
            context, context_n_eq, total);
 
  A = isl_mat_sub_alloc6(ctx, context->eq, 0, context_n_eq, 0, 1 + total);
  B = isl_mat_sub_alloc6(ctx, context->eq,
        0, context_n_eq, 1 + total, n_div);
  A = isl_mat_product(A, isl_mat_copy(V));
  T = isl_mat_parameter_compression_ext(A, B);
  T = isl_mat_product(V, T);
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    T = isl_mat_free(T);
  else
    T = isl_mat_diagonal(T, isl_mat_identity(ctx, n_div));
 
  A = isl_mat_sub_alloc6(ctx, bmap->eq,
        0, bmap_n_eq, 0, 1 + total + n_div);
  A = isl_mat_product(A, T);
 
  return A;
}
 
/* Remove the prime factors from *g that have an exponent that
 * is strictly smaller than the exponent in "c".
 * All exponents in *g are known to be smaller than or equal
 * to those in "c".
 *
 * That is, if *g is equal to
 *
 *  p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
 *
 * and "c" is equal to
 *
 *  p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
 *
 * then update *g to
 *
 *  p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
 *    p_n^{e_n * (e_n = f_n)}
 *
 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
 * neither does the gcd of *g and c / *g.
 * If e_i < f_i, then the gcd of *g and c / *g has a positive
 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
 * Dividing *g by this gcd therefore strictly reduces the exponent
 * of the prime factors that need to be removed, while leaving the
 * other prime factors untouched.
 * Repeating this process until gcd(*g, c / *g) = 1 therefore
 * removes all undesired factors, without removing any others.
 */
static void remove_incomplete_powers(isl_int *g, isl_int c)
{
  isl_int t;
 
  isl_int_init(t);
  for (;;) {
    isl_int_divexact(t, c, *g);
    isl_int_gcd(t, t, *g);
    if (isl_int_is_one(t))
      break;
    isl_int_divexact(*g, *g, t);
  }
  isl_int_clear(t);
}
 
/* Reduce the "n" stride constraints in "bmap" based on a copy "A"
 * of the same stride constraints in a compressed space that exploits
 * all equalities in the context and the other equalities in "bmap".
 *
 * If the stride constraints of "bmap" are of the form
 *
 *  C(x) + D(y) = 0
 *
 * then A is of the form
 *
 *  B(x') + D(y) = 0
 *
 * If any of these constraints involves only a single local variable y,
 * then the constraint appears as
 *
 *  f(x) + m y_i = 0
 *
 * in "bmap" and as
 *
 *  h(x') + m y_i = 0
 *
 * in "A".
 *
 * Let g be the gcd of m and the coefficients of h.
 * Then, in particular, g is a divisor of the coefficients of h and
 *
 *  f(x) = h(x')
 *
 * is known to be a multiple of g.
 * If some prime factor in m appears with the same exponent in g,
 * then it can be removed from m because f(x) is already known
 * to be a multiple of g and therefore in particular of this power
 * of the prime factors.
 * Prime factors that appear with a smaller exponent in g cannot
 * be removed from m.
 * Let g' be the divisor of g containing all prime factors that
 * appear with the same exponent in m and g, then
 *
 *  f(x) + m y_i = 0
 *
 * can be replaced by
 *
 *  f(x) + m/g' y_i' = 0
 *
 * Note that (if g' != 1) this changes the explicit representation
 * of y_i to that of y_i', so the integer division at position i
 * is marked unknown and later recomputed by a call to
 * isl_basic_map_gauss.
 */
static __isl_give isl_basic_map *reduce_stride_constraints(
  __isl_take isl_basic_map *bmap, int n, __isl_keep isl_mat *A)
{
  int i;
  isl_size total, n_div;
  int any = 0;
  isl_int gcd;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (total < 0 || n_div < 0 || !A)
    return isl_basic_map_free(bmap);
  total -= n_div;
 
  isl_int_init(gcd);
  for (i = 0; i < n; ++i) {
    int div;
 
    div = isl_seq_first_non_zero(bmap->eq[i] + 1 + total, n_div);
    if (div < 0)
      isl_die(isl_basic_map_get_ctx(bmap), isl_error_internal,
        "equality constraints modified unexpectedly",
        goto error);
    if (isl_seq_any_non_zero(bmap->eq[i] + 1 + total + div + 1,
            n_div - div - 1))
      continue;
    if (isl_mat_row_gcd(A, i, &gcd) < 0)
      goto error;
    if (isl_int_is_one(gcd))
      continue;
    remove_incomplete_powers(&gcd, bmap->eq[i][1 + total + div]);
    if (isl_int_is_one(gcd))
      continue;
    isl_int_divexact(bmap->eq[i][1 + total + div],
        bmap->eq[i][1 + total + div], gcd);
    bmap = isl_basic_map_mark_div_unknown(bmap, div);
    if (!bmap)
      goto error;
    any = 1;
  }
  isl_int_clear(gcd);
 
  if (any)
    bmap = isl_basic_map_gauss(bmap, NULL);
 
  return bmap;
error:
  isl_int_clear(gcd);
  isl_basic_map_free(bmap);
  return NULL;
}
 
/* Simplify the stride constraints in "bmap" based on
 * the remaining equality constraints in "bmap" and all equality
 * constraints in "context".
 * Only do this if both "bmap" and "context" have stride constraints.
 *
 * First extract a copy of the stride constraints in "bmap" in a compressed
 * space exploiting all the other equality constraints and then
 * use this compressed copy to simplify the original stride constraints.
 */
static __isl_give isl_basic_map *gist_strides(__isl_take isl_basic_map *bmap,
  __isl_keep isl_basic_map *context)
{
  int bmap_n_eq, context_n_eq;
  isl_mat *A;
 
  if (!bmap || !context)
    return isl_basic_map_free(bmap);
 
  bmap_n_eq = n_div_eq(bmap);
  context_n_eq = n_div_eq(context);
 
  if (bmap_n_eq < 0 || context_n_eq < 0)
    return isl_basic_map_free(bmap);
  if (bmap_n_eq == 0 || context_n_eq == 0)
    return bmap;
 
  A = extract_compressed_stride_constraints(bmap, bmap_n_eq,
                context, context_n_eq);
  bmap = reduce_stride_constraints(bmap, bmap_n_eq, A);
 
  isl_mat_free(A);
 
  return bmap;
}
 
/* Return a basic map that has the same intersection with "context" as "bmap"
 * and that is as "simple" as possible.
 *
 * The core computation is performed on the pure constraints.
 * When we add back the meaning of the integer divisions, we need
 * to (re)introduce the div constraints.  If we happen to have
 * discovered that some of these integer divisions are equal to
 * some affine combination of other variables, then these div
 * constraints may end up getting simplified in terms of the equalities,
 * resulting in extra inequalities on the other variables that
 * may have been removed already or that may not even have been
 * part of the input.  We try and remove those constraints of
 * this form that are most obviously redundant with respect to
 * the context.  We also remove those div constraints that are
 * redundant with respect to the other constraints in the result.
 *
 * The stride constraints among the equality constraints in "bmap" are
 * also simplified with respecting to the other equality constraints
 * in "bmap" and with respect to all equality constraints in "context".
 */
__isl_give isl_basic_map *isl_basic_map_gist(__isl_take isl_basic_map *bmap,
  __isl_take isl_basic_map *context)
{
  isl_basic_set *bset, *eq;
  isl_basic_map *eq_bmap;
  isl_size total, n_div, n_div_bmap;
  unsigned extra, n_eq, n_ineq;
 
  if (!bmap || !context)
    goto error;
 
  if (isl_basic_map_plain_is_universe(bmap)) {
    isl_basic_map_free(context);
    return bmap;
  }
  if (isl_basic_map_plain_is_empty(context)) {
    isl_space *space = isl_basic_map_get_space(bmap);
    isl_basic_map_free(bmap);
    isl_basic_map_free(context);
    return isl_basic_map_universe(space);
  }
  if (isl_basic_map_plain_is_empty(bmap)) {
    isl_basic_map_free(context);
    return bmap;
  }
 
  bmap = isl_basic_map_remove_redundancies(bmap);
  context = isl_basic_map_remove_redundancies(context);
  bmap = isl_basic_map_order_divs(bmap);
  context = isl_basic_map_align_divs(context, bmap);
 
  n_div = isl_basic_map_dim(context, isl_dim_div);
  total = isl_basic_map_dim(bmap, isl_dim_all);
  n_div_bmap = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0 || total < 0 || n_div_bmap < 0)
    goto error;
  extra = n_div - n_div_bmap;
 
  bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
  bset = isl_basic_set_add_dims(bset, isl_dim_set, extra);
  bset = uset_gist(bset,
        isl_basic_map_underlying_set(isl_basic_map_copy(context)));
  bset = isl_basic_set_project_out(bset, isl_dim_set, total, extra);
 
  if (!bset || bset->n_eq == 0 || n_div == 0 ||
      isl_basic_set_plain_is_empty(bset)) {
    isl_basic_map_free(context);
    return isl_basic_map_overlying_set(bset, bmap);
  }
 
  n_eq = bset->n_eq;
  n_ineq = bset->n_ineq;
  eq = isl_basic_set_copy(bset);
  eq = isl_basic_set_cow(eq);
  eq = isl_basic_set_free_inequality(eq, n_ineq);
  bset = isl_basic_set_free_equality(bset, n_eq);
 
  eq_bmap = isl_basic_map_overlying_set(eq, isl_basic_map_copy(bmap));
  eq_bmap = gist_strides(eq_bmap, context);
  eq_bmap = isl_basic_map_remove_shifted_constraints(eq_bmap, context);
  bmap = isl_basic_map_overlying_set(bset, bmap);
  bmap = isl_basic_map_intersect(bmap, eq_bmap);
  bmap = isl_basic_map_remove_redundancies(bmap);
 
  return bmap;
error:
  isl_basic_map_free(bmap);
  isl_basic_map_free(context);
  return NULL;
}
 
/*
 * Assumes context has no implicit divs.
 */
__isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map,
  __isl_take isl_basic_map *context)
{
  int i;
 
  if (!map || !context)
    goto error;
 
  if (isl_basic_map_plain_is_empty(context)) {
    isl_space *space = isl_map_get_space(map);
    isl_map_free(map);
    isl_basic_map_free(context);
    return isl_map_universe(space);
  }
 
  context = isl_basic_map_remove_redundancies(context);
  map = isl_map_cow(map);
  if (isl_map_basic_map_check_equal_space(map, context) < 0)
    goto error;
  map = isl_map_compute_divs(map);
  if (!map)
    goto error;
  for (i = map->n - 1; i >= 0; --i) {
    map->p[i] = isl_basic_map_gist(map->p[i],
            isl_basic_map_copy(context));
    if (!map->p[i])
      goto error;
    if (isl_basic_map_plain_is_empty(map->p[i])) {
      isl_basic_map_free(map->p[i]);
      if (i != map->n - 1)
        map->p[i] = map->p[map->n - 1];
      map->n--;
    }
  }
  isl_basic_map_free(context);
  ISL_F_CLR(map, ISL_MAP_NORMALIZED);
  return map;
error:
  isl_map_free(map);
  isl_basic_map_free(context);
  return NULL;
}
 
/* Drop all inequalities from "bmap" that also appear in "context".
 * "context" is assumed to have only known local variables and
 * the initial local variables of "bmap" are assumed to be the same
 * as those of "context".
 * The constraints of both "bmap" and "context" are assumed
 * to have been sorted using isl_basic_map_sort_constraints.
 *
 * Run through the inequality constraints of "bmap" and "context"
 * in sorted order.
 * If a constraint of "bmap" involves variables not in "context",
 * then it cannot appear in "context".
 * If a matching constraint is found, it is removed from "bmap".
 */
static __isl_give isl_basic_map *drop_inequalities(
  __isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context)
{
  int i1, i2;
  isl_size total, bmap_total;
  unsigned extra;
 
  total = isl_basic_map_dim(context, isl_dim_all);
  bmap_total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0 || bmap_total < 0)
    return isl_basic_map_free(bmap);
 
  extra = bmap_total - total;
 
  i1 = bmap->n_ineq - 1;
  i2 = context->n_ineq - 1;
  while (bmap && i1 >= 0 && i2 >= 0) {
    int cmp;
 
    if (isl_seq_any_non_zero(bmap->ineq[i1] + 1 + total, extra)) {
      --i1;
      continue;
    }
    cmp = isl_basic_map_constraint_cmp(context, bmap->ineq[i1],
              context->ineq[i2]);
    if (cmp < 0) {
      --i2;
      continue;
    }
    if (cmp > 0) {
      --i1;
      continue;
    }
    if (isl_int_eq(bmap->ineq[i1][0], context->ineq[i2][0])) {
      bmap = isl_basic_map_cow(bmap);
      if (isl_basic_map_drop_inequality(bmap, i1) < 0)
        bmap = isl_basic_map_free(bmap);
    }
    --i1;
    --i2;
  }
 
  return bmap;
}
 
/* Drop all equalities from "bmap" that also appear in "context".
 * "context" is assumed to have only known local variables and
 * the initial local variables of "bmap" are assumed to be the same
 * as those of "context".
 *
 * Run through the equality constraints of "bmap" and "context"
 * in sorted order.
 * If a constraint of "bmap" involves variables not in "context",
 * then it cannot appear in "context".
 * If a matching constraint is found, it is removed from "bmap".
 */
static __isl_give isl_basic_map *drop_equalities(
  __isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context)
{
  int i1, i2;
  isl_size total, bmap_total;
  unsigned extra;
 
  total = isl_basic_map_dim(context, isl_dim_all);
  bmap_total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0 || bmap_total < 0)
    return isl_basic_map_free(bmap);
 
  extra = bmap_total - total;
 
  i1 = bmap->n_eq - 1;
  i2 = context->n_eq - 1;
 
  while (bmap && i1 >= 0 && i2 >= 0) {
    int last1, last2;
 
    if (isl_seq_any_non_zero(bmap->eq[i1] + 1 + total, extra))
      break;
    last1 = isl_seq_last_non_zero(bmap->eq[i1] + 1, total);
    last2 = isl_seq_last_non_zero(context->eq[i2] + 1, total);
    if (last1 > last2) {
      --i2;
      continue;
    }
    if (last1 < last2) {
      --i1;
      continue;
    }
    if (isl_seq_eq(bmap->eq[i1], context->eq[i2], 1 + total)) {
      bmap = isl_basic_map_cow(bmap);
      if (isl_basic_map_drop_equality(bmap, i1) < 0)
        bmap = isl_basic_map_free(bmap);
    }
    --i1;
    --i2;
  }
 
  return bmap;
}
 
/* Remove the constraints in "context" from "bmap".
 * "context" is assumed to have explicit representations
 * for all local variables.
 *
 * First align the divs of "bmap" to those of "context" and
 * sort the constraints.  Then drop all constraints from "bmap"
 * that appear in "context".
 */
__isl_give isl_basic_map *isl_basic_map_plain_gist(
  __isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context)
{
  isl_bool done, known;
 
  done = isl_basic_map_plain_is_universe(context);
  if (done == isl_bool_false)
    done = isl_basic_map_plain_is_universe(bmap);
  if (done == isl_bool_false)
    done = isl_basic_map_plain_is_empty(context);
  if (done == isl_bool_false)
    done = isl_basic_map_plain_is_empty(bmap);
  if (done < 0)
    goto error;
  if (done) {
    isl_basic_map_free(context);
    return bmap;
  }
  known = isl_basic_map_divs_known(context);
  if (known < 0)
    goto error;
  if (!known)
    isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
      "context has unknown divs", goto error);
 
  context = isl_basic_map_order_divs(context);
  bmap = isl_basic_map_align_divs(bmap, context);
  bmap = isl_basic_map_gauss(bmap, NULL);
  bmap = isl_basic_map_sort_constraints(bmap);
  context = isl_basic_map_sort_constraints(context);
 
  bmap = drop_inequalities(bmap, context);
  bmap = drop_equalities(bmap, context);
 
  isl_basic_map_free(context);
  bmap = isl_basic_map_finalize(bmap);
  return bmap;
error:
  isl_basic_map_free(bmap);
  isl_basic_map_free(context);
  return NULL;
}
 
/* Replace "map" by the disjunct at position "pos" and free "context".
 */
static __isl_give isl_map *replace_by_disjunct(__isl_take isl_map *map,
  int pos, __isl_take isl_basic_map *context)
{
  isl_basic_map *bmap;
 
  bmap = isl_basic_map_copy(map->p[pos]);
  isl_map_free(map);
  isl_basic_map_free(context);
  return isl_map_from_basic_map(bmap);
}
 
/* Remove the constraints in "context" from "map".
 * If any of the disjuncts in the result turns out to be the universe,
 * then return this universe.
 * "context" is assumed to have explicit representations
 * for all local variables.
 */
__isl_give isl_map *isl_map_plain_gist_basic_map(__isl_take isl_map *map,
  __isl_take isl_basic_map *context)
{
  int i;
  isl_bool univ, known;
 
  univ = isl_basic_map_plain_is_universe(context);
  if (univ < 0)
    goto error;
  if (univ) {
    isl_basic_map_free(context);
    return map;
  }
  known = isl_basic_map_divs_known(context);
  if (known < 0)
    goto error;
  if (!known)
    isl_die(isl_map_get_ctx(map), isl_error_invalid,
      "context has unknown divs", goto error);
 
  map = isl_map_cow(map);
  if (!map)
    goto error;
  for (i = 0; i < map->n; ++i) {
    map->p[i] = isl_basic_map_plain_gist(map->p[i],
            isl_basic_map_copy(context));
    univ = isl_basic_map_plain_is_universe(map->p[i]);
    if (univ < 0)
      goto error;
    if (univ && map->n > 1)
      return replace_by_disjunct(map, i, context);
  }
 
  isl_basic_map_free(context);
  ISL_F_CLR(map, ISL_MAP_NORMALIZED);
  if (map->n > 1)
    ISL_F_CLR(map, ISL_MAP_DISJOINT);
  return map;
error:
  isl_map_free(map);
  isl_basic_map_free(context);
  return NULL;
}
 
/* Remove the constraints in "context" from "set".
 * If any of the disjuncts in the result turns out to be the universe,
 * then return this universe.
 * "context" is assumed to have explicit representations
 * for all local variables.
 */
__isl_give isl_set *isl_set_plain_gist_basic_set(__isl_take isl_set *set,
  __isl_take isl_basic_set *context)
{
  return set_from_map(isl_map_plain_gist_basic_map(set_to_map(set),
              bset_to_bmap(context)));
}
 
/* Remove the constraints in "context" from "map".
 * If any of the disjuncts in the result turns out to be the universe,
 * then return this universe.
 * "context" is assumed to consist of a single disjunct and
 * to have explicit representations for all local variables.
 */
__isl_give isl_map *isl_map_plain_gist(__isl_take isl_map *map,
  __isl_take isl_map *context)
{
  isl_basic_map *hull;
 
  hull = isl_map_unshifted_simple_hull(context);
  return isl_map_plain_gist_basic_map(map, hull);
}
 
/* Replace "map" by a universe map in the same space and free "drop".
 */
static __isl_give isl_map *replace_by_universe(__isl_take isl_map *map,
  __isl_take isl_map *drop)
{
  isl_map *res;
 
  res = isl_map_universe(isl_map_get_space(map));
  isl_map_free(map);
  isl_map_free(drop);
  return res;
}
 
/* Return a map that has the same intersection with "context" as "map"
 * and that is as "simple" as possible.
 *
 * If "map" is already the universe, then we cannot make it any simpler.
 * Similarly, if "context" is the universe, then we cannot exploit it
 * to simplify "map"
 * If "map" and "context" are identical to each other, then we can
 * return the corresponding universe.
 *
 * If either "map" or "context" consists of multiple disjuncts,
 * then check if "context" happens to be a subset of "map",
 * in which case all constraints can be removed.
 * In case of multiple disjuncts, the standard procedure
 * may not be able to detect that all constraints can be removed.
 *
 * If none of these cases apply, we have to work a bit harder.
 * During this computation, we make use of a single disjunct context,
 * so if the original context consists of more than one disjunct
 * then we need to approximate the context by a single disjunct set.
 * Simply taking the simple hull may drop constraints that are
 * only implicitly available in each disjunct.  We therefore also
 * look for constraints among those defining "map" that are valid
 * for the context.  These can then be used to simplify away
 * the corresponding constraints in "map".
 */
__isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
  __isl_take isl_map *context)
{
  int equal;
  int is_universe;
  isl_size n_disjunct_map, n_disjunct_context;
  isl_bool subset;
  isl_basic_map *hull;
 
  is_universe = isl_map_plain_is_universe(map);
  if (is_universe >= 0 && !is_universe)
    is_universe = isl_map_plain_is_universe(context);
  if (is_universe < 0)
    goto error;
  if (is_universe) {
    isl_map_free(context);
    return map;
  }
 
  isl_map_align_params_bin(&map, &context);
  equal = isl_map_plain_is_equal(map, context);
  if (equal < 0)
    goto error;
  if (equal)
    return replace_by_universe(map, context);
 
  n_disjunct_map = isl_map_n_basic_map(map);
  n_disjunct_context = isl_map_n_basic_map(context);
  if (n_disjunct_map < 0 || n_disjunct_context < 0)
    goto error;
  if (n_disjunct_map != 1 || n_disjunct_context != 1) {
    subset = isl_map_is_subset(context, map);
    if (subset < 0)
      goto error;
    if (subset)
      return replace_by_universe(map, context);
  }
 
  context = isl_map_compute_divs(context);
  if (!context)
    goto error;
  if (n_disjunct_context == 1) {
    hull = isl_map_simple_hull(context);
  } else {
    isl_ctx *ctx;
    isl_map_list *list;
 
    ctx = isl_map_get_ctx(map);
    list = isl_map_list_alloc(ctx, 2);
    list = isl_map_list_add(list, isl_map_copy(context));
    list = isl_map_list_add(list, isl_map_copy(map));
    hull = isl_map_unshifted_simple_hull_from_map_list(context,
                    list);
  }
  return isl_map_gist_basic_map(map, hull);
error:
  isl_map_free(map);
  isl_map_free(context);
  return NULL;
}
 
__isl_give isl_basic_set *isl_basic_set_gist(__isl_take isl_basic_set *bset,
  __isl_take isl_basic_set *context)
{
  return bset_from_bmap(isl_basic_map_gist(bset_to_bmap(bset),
            bset_to_bmap(context)));
}
 
__isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set,
  __isl_take isl_basic_set *context)
{
  return set_from_map(isl_map_gist_basic_map(set_to_map(set),
          bset_to_bmap(context)));
}
 
__isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set,
  __isl_take isl_basic_set *context)
{
  isl_space *space = isl_set_get_space(set);
  isl_basic_set *dom_context = isl_basic_set_universe(space);
  dom_context = isl_basic_set_intersect_params(dom_context, context);
  return isl_set_gist_basic_set(set, dom_context);
}
 
__isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
  __isl_take isl_set *context)
{
  return set_from_map(isl_map_gist(set_to_map(set), set_to_map(context)));
}
 
/* Compute the gist of "bmap" with respect to the constraints "context"
 * on the domain.
 */
__isl_give isl_basic_map *isl_basic_map_gist_domain(
  __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *context)
{
  isl_space *space = isl_basic_map_get_space(bmap);
  isl_basic_map *bmap_context = isl_basic_map_universe(space);
 
  bmap_context = isl_basic_map_intersect_domain(bmap_context, context);
  return isl_basic_map_gist(bmap, bmap_context);
}
 
__isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map,
  __isl_take isl_set *context)
{
  isl_map *map_context = isl_map_universe(isl_map_get_space(map));
  map_context = isl_map_intersect_domain(map_context, context);
  return isl_map_gist(map, map_context);
}
 
__isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map,
  __isl_take isl_set *context)
{
  isl_map *map_context = isl_map_universe(isl_map_get_space(map));
  map_context = isl_map_intersect_range(map_context, context);
  return isl_map_gist(map, map_context);
}
 
__isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map,
  __isl_take isl_set *context)
{
  isl_map *map_context = isl_map_universe(isl_map_get_space(map));
  map_context = isl_map_intersect_params(map_context, context);
  return isl_map_gist(map, map_context);
}
 
__isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set,
  __isl_take isl_set *context)
{
  return isl_map_gist_params(set, context);
}
 
/* Quick check to see if two basic maps are disjoint.
 * In particular, we reduce the equalities and inequalities of
 * one basic map in the context of the equalities of the other
 * basic map and check if we get a contradiction.
 */
isl_bool isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1,
  __isl_keep isl_basic_map *bmap2)
{
  struct isl_vec *v = NULL;
  int *elim = NULL;
  isl_size total;
  int i;
 
  if (isl_basic_map_check_equal_space(bmap1, bmap2) < 0)
    return isl_bool_error;
  if (bmap1->n_div || bmap2->n_div)
    return isl_bool_false;
  if (!bmap1->n_eq && !bmap2->n_eq)
    return isl_bool_false;
 
  total = isl_space_dim(bmap1->dim, isl_dim_all);
  if (total < 0)
    return isl_bool_error;
  if (total == 0)
    return isl_bool_false;
  v = isl_vec_alloc(bmap1->ctx, 1 + total);
  if (!v)
    goto error;
  elim = isl_alloc_array(bmap1->ctx, int, total);
  if (!elim)
    goto error;
  compute_elimination_index(bmap1, elim, total);
  for (i = 0; i < bmap2->n_eq; ++i) {
    int reduced;
    reduced = reduced_using_equalities(v->block.data, bmap2->eq[i],
              bmap1, elim, total);
    if (reduced && !isl_int_is_zero(v->block.data[0]) &&
        !isl_seq_any_non_zero(v->block.data + 1, total))
      goto disjoint;
  }
  for (i = 0; i < bmap2->n_ineq; ++i) {
    int reduced;
    reduced = reduced_using_equalities(v->block.data,
          bmap2->ineq[i], bmap1, elim, total);
    if (reduced && isl_int_is_neg(v->block.data[0]) &&
        !isl_seq_any_non_zero(v->block.data + 1, total))
      goto disjoint;
  }
  compute_elimination_index(bmap2, elim, total);
  for (i = 0; i < bmap1->n_ineq; ++i) {
    int reduced;
    reduced = reduced_using_equalities(v->block.data,
          bmap1->ineq[i], bmap2, elim, total);
    if (reduced && isl_int_is_neg(v->block.data[0]) &&
        !isl_seq_any_non_zero(v->block.data + 1, total))
      goto disjoint;
  }
  isl_vec_free(v);
  free(elim);
  return isl_bool_false;
disjoint:
  isl_vec_free(v);
  free(elim);
  return isl_bool_true;
error:
  isl_vec_free(v);
  free(elim);
  return isl_bool_error;
}
 
int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1,
  __isl_keep isl_basic_set *bset2)
{
  return isl_basic_map_plain_is_disjoint(bset_to_bmap(bset1),
                bset_to_bmap(bset2));
}
 
/* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
 */
static isl_bool all_pairs(__isl_keep isl_map *map1, __isl_keep isl_map *map2,
  isl_bool (*test)(__isl_keep isl_basic_map *bmap1,
    __isl_keep isl_basic_map *bmap2))
{
  int i, j;
 
  if (!map1 || !map2)
    return isl_bool_error;
 
  for (i = 0; i < map1->n; ++i) {
    for (j = 0; j < map2->n; ++j) {
      isl_bool d = test(map1->p[i], map2->p[j]);
      if (d != isl_bool_true)
        return d;
    }
  }
 
  return isl_bool_true;
}
 
/* Are "map1" and "map2" obviously disjoint, based on information
 * that can be derived without looking at the individual basic maps?
 *
 * In particular, if one of them is empty or if they live in different spaces
 * (ignoring parameters), then they are clearly disjoint.
 */
static isl_bool isl_map_plain_is_disjoint_global(__isl_keep isl_map *map1,
  __isl_keep isl_map *map2)
{
  isl_bool disjoint;
  isl_bool match;
 
  if (!map1 || !map2)
    return isl_bool_error;
 
  disjoint = isl_map_plain_is_empty(map1);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  disjoint = isl_map_plain_is_empty(map2);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  match = isl_map_tuple_is_equal(map1, isl_dim_in, map2, isl_dim_in);
  if (match < 0 || !match)
    return match < 0 ? isl_bool_error : isl_bool_true;
 
  match = isl_map_tuple_is_equal(map1, isl_dim_out, map2, isl_dim_out);
  if (match < 0 || !match)
    return match < 0 ? isl_bool_error : isl_bool_true;
 
  return isl_bool_false;
}
 
/* Are "map1" and "map2" obviously disjoint?
 *
 * If one of them is empty or if they live in different spaces (ignoring
 * parameters), then they are clearly disjoint.
 * This is checked by isl_map_plain_is_disjoint_global.
 *
 * If they have different parameters, then we skip any further tests.
 *
 * If they are obviously equal, but not obviously empty, then we will
 * not be able to detect if they are disjoint.
 *
 * Otherwise we check if each basic map in "map1" is obviously disjoint
 * from each basic map in "map2".
 */
isl_bool isl_map_plain_is_disjoint(__isl_keep isl_map *map1,
  __isl_keep isl_map *map2)
{
  isl_bool disjoint;
  isl_bool intersect;
  isl_bool match;
 
  disjoint = isl_map_plain_is_disjoint_global(map1, map2);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  match = isl_map_has_equal_params(map1, map2);
  if (match < 0 || !match)
    return match < 0 ? isl_bool_error : isl_bool_false;
 
  intersect = isl_map_plain_is_equal(map1, map2);
  if (intersect < 0 || intersect)
    return intersect < 0 ? isl_bool_error : isl_bool_false;
 
  return all_pairs(map1, map2, &isl_basic_map_plain_is_disjoint);
}
 
/* Are "map1" and "map2" disjoint?
 * The parameters are assumed to have been aligned.
 *
 * In particular, check whether all pairs of basic maps are disjoint.
 */
static isl_bool isl_map_is_disjoint_aligned(__isl_keep isl_map *map1,
  __isl_keep isl_map *map2)
{
  return all_pairs(map1, map2, &isl_basic_map_is_disjoint);
}
 
/* Are "map1" and "map2" disjoint?
 *
 * They are disjoint if they are "obviously disjoint" or if one of them
 * is empty.  Otherwise, they are not disjoint if one of them is universal.
 * If the two inputs are (obviously) equal and not empty, then they are
 * not disjoint.
 * If none of these cases apply, then check if all pairs of basic maps
 * are disjoint after aligning the parameters.
 */
isl_bool isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{
  isl_bool disjoint;
  isl_bool intersect;
 
  disjoint = isl_map_plain_is_disjoint_global(map1, map2);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  disjoint = isl_map_is_empty(map1);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  disjoint = isl_map_is_empty(map2);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  intersect = isl_map_plain_is_universe(map1);
  if (intersect < 0 || intersect)
    return isl_bool_not(intersect);
 
  intersect = isl_map_plain_is_universe(map2);
  if (intersect < 0 || intersect)
    return isl_bool_not(intersect);
 
  intersect = isl_map_plain_is_equal(map1, map2);
  if (intersect < 0 || intersect)
    return isl_bool_not(intersect);
 
  return isl_map_align_params_map_map_and_test(map1, map2,
            &isl_map_is_disjoint_aligned);
}
 
/* Are "bmap1" and "bmap2" disjoint?
 *
 * They are disjoint if they are "obviously disjoint" or if one of them
 * is empty.  Otherwise, they are not disjoint if one of them is universal.
 * If none of these cases apply, we compute the intersection and see if
 * the result is empty.
 */
isl_bool isl_basic_map_is_disjoint(__isl_keep isl_basic_map *bmap1,
  __isl_keep isl_basic_map *bmap2)
{
  isl_bool disjoint;
  isl_bool intersect;
  isl_basic_map *test;
 
  disjoint = isl_basic_map_plain_is_disjoint(bmap1, bmap2);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  disjoint = isl_basic_map_is_empty(bmap1);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  disjoint = isl_basic_map_is_empty(bmap2);
  if (disjoint < 0 || disjoint)
    return disjoint;
 
  intersect = isl_basic_map_plain_is_universe(bmap1);
  if (intersect < 0 || intersect)
    return isl_bool_not(intersect);
 
  intersect = isl_basic_map_plain_is_universe(bmap2);
  if (intersect < 0 || intersect)
    return isl_bool_not(intersect);
 
  test = isl_basic_map_intersect(isl_basic_map_copy(bmap1),
    isl_basic_map_copy(bmap2));
  disjoint = isl_basic_map_is_empty(test);
  isl_basic_map_free(test);
 
  return disjoint;
}
 
/* Are "bset1" and "bset2" disjoint?
 */
isl_bool isl_basic_set_is_disjoint(__isl_keep isl_basic_set *bset1,
  __isl_keep isl_basic_set *bset2)
{
  return isl_basic_map_is_disjoint(bset1, bset2);
}
 
isl_bool isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
  __isl_keep isl_set *set2)
{
  return isl_map_plain_is_disjoint(set_to_map(set1), set_to_map(set2));
}
 
/* Are "set1" and "set2" disjoint?
 */
isl_bool isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
{
  return isl_map_is_disjoint(set1, set2);
}
 
/* Is "v" equal to 0, 1 or -1?
 */
static int is_zero_or_one(isl_int v)
{
  return isl_int_is_zero(v) || isl_int_is_one(v) || isl_int_is_negone(v);
}
 
/* Are the "n" coefficients starting at "first" of inequality constraints
 * "i" and "j" of "bmap" opposite to each other?
 */
static int is_opposite_part(__isl_keep isl_basic_map *bmap, int i, int j,
  int first, int n)
{
  return isl_seq_is_neg(bmap->ineq[i] + first, bmap->ineq[j] + first, n);
}
 
/* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
 * apart from the constant term?
 */
static isl_bool is_opposite(__isl_keep isl_basic_map *bmap, int i, int j)
{
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_bool_error;
  return is_opposite_part(bmap, i, j, 1, total);
}
 
/* Check if we can combine a given div with lower bound l and upper
 * bound u with some other div and if so return that other div.
 * Otherwise, return a position beyond the integer divisions.
 * Return isl_size_error on error.
 *
 * We first check that
 *  - the bounds are opposites of each other (except for the constant
 *    term)
 *  - the bounds do not reference any other div
 *  - no div is defined in terms of this div
 *
 * Let m be the size of the range allowed on the div by the bounds.
 * That is, the bounds are of the form
 *
 *  e <= a <= e + m - 1
 *
 * with e some expression in the other variables.
 * We look for another div b such that no third div is defined in terms
 * of this second div b and such that in any constraint that contains
 * a (except for the given lower and upper bound), also contains b
 * with a coefficient that is m times that of b.
 * That is, all constraints (except for the lower and upper bound)
 * are of the form
 *
 *  e + f (a + m b) >= 0
 *
 * Furthermore, in the constraints that only contain b, the coefficient
 * of b should be equal to 1 or -1.
 * If so, we return b so that "a + m b" can be replaced by
 * a single div "c = a + m b".
 */
static isl_size div_find_coalesce(__isl_keep isl_basic_map *bmap, int *pairs,
  unsigned div, unsigned l, unsigned u)
{
  int i, j;
  isl_size n_div;
  isl_size v_div;
  isl_size coalesce;
  isl_bool involves, opp;
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div <= 1)
    return n_div;
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_size_error;
  if (isl_seq_any_non_zero(bmap->ineq[l] + 1 + v_div, div))
    return n_div;
  if (isl_seq_any_non_zero(bmap->ineq[l] + 1 + v_div + div + 1,
           n_div - div - 1))
    return n_div;
  opp = is_opposite(bmap, l, u);
  if (opp < 0 || !opp)
    return opp < 0 ? isl_size_error : n_div;
 
  involves = isl_basic_map_any_div_involves_vars(bmap, v_div + div, 1);
  if (involves < 0 || involves)
    return involves < 0 ? isl_size_error : n_div;
 
  isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
  if (isl_int_is_neg(bmap->ineq[l][0])) {
    isl_int_sub(bmap->ineq[l][0],
          bmap->ineq[l][0], bmap->ineq[u][0]);
    bmap = isl_basic_map_copy(bmap);
    bmap = isl_basic_map_set_to_empty(bmap);
    isl_basic_map_free(bmap);
    return n_div;
  }
  isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
  coalesce = n_div;
  for (i = 0; i < n_div; ++i) {
    if (i == div)
      continue;
    if (!pairs[i])
      continue;
    involves = isl_basic_map_any_div_involves_vars(bmap,
                  v_div + i, 1);
    if (involves < 0)
      goto error;
    if (involves)
      continue;
    for (j = 0; j < bmap->n_ineq; ++j) {
      int valid;
      if (j == l || j == u)
        continue;
      if (isl_int_is_zero(bmap->ineq[j][1 + v_div + div])) {
        if (is_zero_or_one(bmap->ineq[j][1 + v_div + i]))
          continue;
        break;
      }
      if (isl_int_is_zero(bmap->ineq[j][1 + v_div + i]))
        break;
      isl_int_mul(bmap->ineq[j][1 + v_div + div],
            bmap->ineq[j][1 + v_div + div],
            bmap->ineq[l][0]);
      valid = isl_int_eq(bmap->ineq[j][1 + v_div + div],
             bmap->ineq[j][1 + v_div + i]);
      isl_int_divexact(bmap->ineq[j][1 + v_div + div],
           bmap->ineq[j][1 + v_div + div],
           bmap->ineq[l][0]);
      if (!valid)
        break;
    }
    if (j < bmap->n_ineq)
      continue;
    coalesce = i;
    break;
  }
  if (0)
error:    coalesce = isl_size_error;
  isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1);
  isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]);
  return coalesce;
}
 
/* Internal data structure used during the construction and/or evaluation of
 * an inequality that ensures that a pair of bounds always allows
 * for an integer value.
 *
 * "tab" is the tableau in which the inequality is evaluated.  It may
 * be NULL until it is actually needed.
 * "v" contains the inequality coefficients.
 * "g", "fl" and "fu" are temporary scalars used during the construction and
 * evaluation.
 */
struct test_ineq_data {
  struct isl_tab *tab;
  isl_vec *v;
  isl_int g;
  isl_int fl;
  isl_int fu;
};
 
/* Free all the memory allocated by the fields of "data".
 */
static void test_ineq_data_clear(struct test_ineq_data *data)
{
  isl_tab_free(data->tab);
  isl_vec_free(data->v);
  isl_int_clear(data->g);
  isl_int_clear(data->fl);
  isl_int_clear(data->fu);
}
 
/* Is the inequality stored in data->v satisfied by "bmap"?
 * That is, does it only attain non-negative values?
 * data->tab is a tableau corresponding to "bmap".
 */
static isl_bool test_ineq_is_satisfied(__isl_keep isl_basic_map *bmap,
  struct test_ineq_data *data)
{
  isl_ctx *ctx;
  enum isl_lp_result res;
 
  ctx = isl_basic_map_get_ctx(bmap);
  if (!data->tab)
    data->tab = isl_tab_from_basic_map(bmap, 0);
  res = isl_tab_min(data->tab, data->v->el, ctx->one, &data->g, NULL, 0);
  if (res == isl_lp_error)
    return isl_bool_error;
  return res == isl_lp_ok && isl_int_is_nonneg(data->g);
}
 
/* Given a lower and an upper bound on div i, do they always allow
 * for an integer value of the given div?
 * Determine this property by constructing an inequality
 * such that the property is guaranteed when the inequality is nonnegative.
 * The lower bound is inequality l, while the upper bound is inequality u.
 * The constructed inequality is stored in data->v.
 *
 * Let the upper bound be
 *
 *  -n_u a + e_u >= 0
 *
 * and the lower bound
 *
 *  n_l a + e_l >= 0
 *
 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
 * We have
 *
 *  - f_u e_l <= f_u f_l g a <= f_l e_u
 *
 * Since all variables are integer valued, this is equivalent to
 *
 *  - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
 *
 * If this interval is at least f_u f_l g, then it contains at least
 * one integer value for a.
 * That is, the test constraint is
 *
 *  f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
 *
 * or
 *
 *  f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
 *
 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
 * then the constraint can be scaled down by a factor g',
 * with the constant term replaced by
 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
 * Note that the result of applying Fourier-Motzkin to this pair
 * of constraints is
 *
 *  f_l e_u + f_u e_l >= 0
 *
 * If the constant term of the scaled down version of this constraint,
 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
 * term of the scaled down test constraint, then the test constraint
 * is known to hold and no explicit evaluation is required.
 * This is essentially the Omega test.
 *
 * If the test constraint consists of only a constant term, then
 * it is sufficient to look at the sign of this constant term.
 */
static isl_bool int_between_bounds(__isl_keep isl_basic_map *bmap, int i,
  int l, int u, struct test_ineq_data *data)
{
  unsigned offset;
  isl_size n_div;
 
  offset = isl_basic_map_offset(bmap, isl_dim_div);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    return isl_bool_error;
 
  isl_int_gcd(data->g,
        bmap->ineq[l][offset + i], bmap->ineq[u][offset + i]);
  isl_int_divexact(data->fl, bmap->ineq[l][offset + i], data->g);
  isl_int_divexact(data->fu, bmap->ineq[u][offset + i], data->g);
  isl_int_neg(data->fu, data->fu);
  isl_seq_combine(data->v->el, data->fl, bmap->ineq[u],
      data->fu, bmap->ineq[l], offset + n_div);
  isl_int_mul(data->g, data->g, data->fl);
  isl_int_mul(data->g, data->g, data->fu);
  isl_int_sub(data->g, data->g, data->fl);
  isl_int_sub(data->g, data->g, data->fu);
  isl_int_add_ui(data->g, data->g, 1);
  isl_int_sub(data->fl, data->v->el[0], data->g);
 
  isl_seq_gcd(data->v->el + 1, offset - 1 + n_div, &data->g);
  if (isl_int_is_zero(data->g))
    return isl_int_is_nonneg(data->fl);
  if (isl_int_is_one(data->g)) {
    isl_int_set(data->v->el[0], data->fl);
    return test_ineq_is_satisfied(bmap, data);
  }
  isl_int_fdiv_q(data->fl, data->fl, data->g);
  isl_int_fdiv_q(data->v->el[0], data->v->el[0], data->g);
  if (isl_int_eq(data->fl, data->v->el[0]))
    return isl_bool_true;
  isl_int_set(data->v->el[0], data->fl);
  isl_seq_scale_down(data->v->el + 1, data->v->el + 1, data->g,
          offset - 1 + n_div);
 
  return test_ineq_is_satisfied(bmap, data);
}
 
/* Remove more kinds of divs that are not strictly needed.
 * In particular, if all pairs of lower and upper bounds on a div
 * are such that they allow at least one integer value of the div,
 * then we can eliminate the div using Fourier-Motzkin without
 * introducing any spurious solutions.
 *
 * If at least one of the two constraints has a unit coefficient for the div,
 * then the presence of such a value is guaranteed so there is no need to check.
 * In particular, the value attained by the bound with unit coefficient
 * can serve as this intermediate value.
 */
static __isl_give isl_basic_map *drop_more_redundant_divs(
  __isl_take isl_basic_map *bmap, __isl_take int *pairs, int n)
{
  isl_ctx *ctx;
  struct test_ineq_data data = { NULL, NULL };
  unsigned off;
  isl_size n_div;
  int remove = -1;
 
  isl_int_init(data.g);
  isl_int_init(data.fl);
  isl_int_init(data.fu);
 
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    goto error;
 
  ctx = isl_basic_map_get_ctx(bmap);
  off = isl_basic_map_offset(bmap, isl_dim_div);
  data.v = isl_vec_alloc(ctx, off + n_div);
  if (!data.v)
    goto error;
 
  while (n > 0) {
    int i, l, u;
    int best = -1;
    isl_bool has_int;
 
    for (i = 0; i < n_div; ++i) {
      if (!pairs[i])
        continue;
      if (best >= 0 && pairs[best] <= pairs[i])
        continue;
      best = i;
    }
 
    i = best;
    for (l = 0; l < bmap->n_ineq; ++l) {
      if (!isl_int_is_pos(bmap->ineq[l][off + i]))
        continue;
      if (isl_int_is_one(bmap->ineq[l][off + i]))
        continue;
      for (u = 0; u < bmap->n_ineq; ++u) {
        if (!isl_int_is_neg(bmap->ineq[u][off + i]))
          continue;
        if (isl_int_is_negone(bmap->ineq[u][off + i]))
          continue;
        has_int = int_between_bounds(bmap, i, l, u,
                &data);
        if (has_int < 0)
          goto error;
        if (data.tab && data.tab->empty)
          break;
        if (!has_int)
          break;
      }
      if (u < bmap->n_ineq)
        break;
    }
    if (data.tab && data.tab->empty) {
      bmap = isl_basic_map_set_to_empty(bmap);
      break;
    }
    if (l == bmap->n_ineq) {
      remove = i;
      break;
    }
    pairs[i] = 0;
    --n;
  }
 
  test_ineq_data_clear(&data);
 
  free(pairs);
 
  if (remove < 0)
    return bmap;
 
  bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1);
  return isl_basic_map_drop_redundant_divs(bmap);
error:
  free(pairs);
  isl_basic_map_free(bmap);
  test_ineq_data_clear(&data);
  return NULL;
}
 
/* Given a pair of divs div1 and div2 such that, except for the lower bound l
 * and the upper bound u, div1 always occurs together with div2 in the form
 * (div1 + m div2), where m is the constant range on the variable div1
 * allowed by l and u, replace the pair div1 and div2 by a single
 * div that is equal to div1 + m div2.
 *
 * The new div will appear in the location that contains div2.
 * We need to modify all constraints that contain
 * div2 = (div - div1) / m
 * The coefficient of div2 is known to be equal to 1 or -1.
 * (If a constraint does not contain div2, it will also not contain div1.)
 * If the constraint also contains div1, then we know they appear
 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
 * i.e., the coefficient of div is f.
 *
 * Otherwise, we first need to introduce div1 into the constraint.
 * Let l be
 *
 *  div1 + f >=0
 *
 * and u
 *
 *  -div1 + f' >= 0
 *
 * A lower bound on div2
 *
 *  div2 + t >= 0
 *
 * can be replaced by
 *
 *  m div2 + div1 + m t + f >= 0
 *
 * An upper bound
 *
 *  -div2 + t >= 0
 *
 * can be replaced by
 *
 *  -(m div2 + div1) + m t + f' >= 0
 *
 * These constraint are those that we would obtain from eliminating
 * div1 using Fourier-Motzkin.
 *
 * After all constraints have been modified, we drop the lower and upper
 * bound and then drop div1.
 * Since the new div is only placed in the same location that used
 * to store div2, but otherwise has a different meaning, any possible
 * explicit representation of the original div2 is removed.
 */
static __isl_give isl_basic_map *coalesce_divs(__isl_take isl_basic_map *bmap,
  unsigned div1, unsigned div2, unsigned l, unsigned u)
{
  isl_ctx *ctx;
  isl_int m;
  isl_size v_div;
  unsigned total;
  int i;
 
  ctx = isl_basic_map_get_ctx(bmap);
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (v_div < 0)
    return isl_basic_map_free(bmap);
  total = 1 + v_div + bmap->n_div;
 
  isl_int_init(m);
  isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]);
  isl_int_add_ui(m, m, 1);
 
  for (i = 0; i < bmap->n_ineq; ++i) {
    if (i == l || i == u)
      continue;
    if (isl_int_is_zero(bmap->ineq[i][1 + v_div + div2]))
      continue;
    if (isl_int_is_zero(bmap->ineq[i][1 + v_div + div1])) {
      if (isl_int_is_pos(bmap->ineq[i][1 + v_div + div2]))
        isl_seq_combine(bmap->ineq[i], m, bmap->ineq[i],
            ctx->one, bmap->ineq[l], total);
      else
        isl_seq_combine(bmap->ineq[i], m, bmap->ineq[i],
            ctx->one, bmap->ineq[u], total);
    }
    isl_int_set(bmap->ineq[i][1 + v_div + div2],
          bmap->ineq[i][1 + v_div + div1]);
    isl_int_set_si(bmap->ineq[i][1 + v_div + div1], 0);
  }
 
  isl_int_clear(m);
  if (l > u) {
    isl_basic_map_drop_inequality(bmap, l);
    isl_basic_map_drop_inequality(bmap, u);
  } else {
    isl_basic_map_drop_inequality(bmap, u);
    isl_basic_map_drop_inequality(bmap, l);
  }
  bmap = isl_basic_map_mark_div_unknown(bmap, div2);
  bmap = isl_basic_map_drop_div(bmap, div1);
  return bmap;
}
 
/* First check if we can coalesce any pair of divs and
 * then continue with dropping more redundant divs.
 *
 * We loop over all pairs of lower and upper bounds on a div
 * with coefficient 1 and -1, respectively, check if there
 * is any other div "c" with which we can coalesce the div
 * and if so, perform the coalescing.
 */
static __isl_give isl_basic_map *coalesce_or_drop_more_redundant_divs(
  __isl_take isl_basic_map *bmap, int *pairs, int n)
{
  int i, l, u;
  isl_size v_div;
  isl_size n_div;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (v_div < 0 || n_div < 0)
    return isl_basic_map_free(bmap);
 
  for (i = 0; i < n_div; ++i) {
    if (!pairs[i])
      continue;
    for (l = 0; l < bmap->n_ineq; ++l) {
      if (!isl_int_is_one(bmap->ineq[l][1 + v_div + i]))
        continue;
      for (u = 0; u < bmap->n_ineq; ++u) {
        int c;
 
        if (!isl_int_is_negone(bmap->ineq[u][1+v_div+i]))
          continue;
        c = div_find_coalesce(bmap, pairs, i, l, u);
        if (c < 0)
          goto error;
        if (c >= n_div)
          continue;
        free(pairs);
        bmap = coalesce_divs(bmap, i, c, l, u);
        return isl_basic_map_drop_redundant_divs(bmap);
      }
    }
  }
 
  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
    free(pairs);
    return bmap;
  }
 
  return drop_more_redundant_divs(bmap, pairs, n);
error:
  free(pairs);
  isl_basic_map_free(bmap);
  return NULL;
}
 
/* Are the "n" coefficients starting at "first" of inequality constraints
 * "i" and "j" of "bmap" equal to each other?
 */
static int is_parallel_part(__isl_keep isl_basic_map *bmap, int i, int j,
  int first, int n)
{
  return isl_seq_eq(bmap->ineq[i] + first, bmap->ineq[j] + first, n);
}
 
/* Are inequality constraints "i" and "j" of "bmap" equal to each other,
 * apart from the constant term and the coefficient at position "pos"?
 */
static isl_bool is_parallel_except(__isl_keep isl_basic_map *bmap, int i, int j,
  int pos)
{
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_bool_error;
  return is_parallel_part(bmap, i, j, 1, pos - 1) &&
    is_parallel_part(bmap, i, j, pos + 1, total - pos);
}
 
/* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
 * apart from the constant term and the coefficient at position "pos"?
 */
static isl_bool is_opposite_except(__isl_keep isl_basic_map *bmap, int i, int j,
  int pos)
{
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_bool_error;
  return is_opposite_part(bmap, i, j, 1, pos - 1) &&
    is_opposite_part(bmap, i, j, pos + 1, total - pos);
}
 
/* Restart isl_basic_map_drop_redundant_divs after "bmap" has
 * been modified, simplying it if "simplify" is set.
 * Free the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 */
static __isl_give isl_basic_map *drop_redundant_divs_again(
  __isl_take isl_basic_map *bmap, __isl_take int *pairs, int simplify)
{
  if (simplify)
    bmap = isl_basic_map_simplify(bmap);
  free(pairs);
  return isl_basic_map_drop_redundant_divs(bmap);
}
 
/* Is "div" the single unknown existentially quantified variable
 * in inequality constraint "ineq" of "bmap"?
 * "div" is known to have a non-zero coefficient in "ineq".
 */
static isl_bool single_unknown(__isl_keep isl_basic_map *bmap, int ineq,
  int div)
{
  int i;
  isl_size n_div;
  unsigned o_div;
  isl_bool known;
 
  known = isl_basic_map_div_is_known(bmap, div);
  if (known < 0 || known)
    return isl_bool_not(known);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    return isl_bool_error;
  if (n_div == 1)
    return isl_bool_true;
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  for (i = 0; i < n_div; ++i) {
    isl_bool known;
 
    if (i == div)
      continue;
    if (isl_int_is_zero(bmap->ineq[ineq][o_div + i]))
      continue;
    known = isl_basic_map_div_is_known(bmap, i);
    if (known < 0 || !known)
      return known;
  }
 
  return isl_bool_true;
}
 
/* Does integer division "div" have coefficient 1 in inequality constraint
 * "ineq" of "map"?
 */
static isl_bool has_coef_one(__isl_keep isl_basic_map *bmap, int div, int ineq)
{
  unsigned o_div;
 
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  if (isl_int_is_one(bmap->ineq[ineq][o_div + div]))
    return isl_bool_true;
 
  return isl_bool_false;
}
 
/* Turn inequality constraint "ineq" of "bmap" into an equality and
 * then try and drop redundant divs again,
 * freeing the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 */
static __isl_give isl_basic_map *set_eq_and_try_again(
  __isl_take isl_basic_map *bmap, int ineq, __isl_take int *pairs)
{
  bmap = isl_basic_map_cow(bmap);
  isl_basic_map_inequality_to_equality(bmap, ineq);
  return drop_redundant_divs_again(bmap, pairs, 1);
}
 
/* Drop the integer division at position "div", along with the two
 * inequality constraints "ineq1" and "ineq2" in which it appears
 * from "bmap" and then try and drop redundant divs again,
 * freeing the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 */
static __isl_give isl_basic_map *drop_div_and_try_again(
  __isl_take isl_basic_map *bmap, int div, int ineq1, int ineq2,
  __isl_take int *pairs)
{
  if (ineq1 > ineq2) {
    isl_basic_map_drop_inequality(bmap, ineq1);
    isl_basic_map_drop_inequality(bmap, ineq2);
  } else {
    isl_basic_map_drop_inequality(bmap, ineq2);
    isl_basic_map_drop_inequality(bmap, ineq1);
  }
  bmap = isl_basic_map_drop_div(bmap, div);
  return drop_redundant_divs_again(bmap, pairs, 0);
}
 
/* Given two inequality constraints
 *
 *  f(x) + n d + c >= 0,    (ineq)
 *
 * with d the variable at position "pos", and
 *
 *  f(x) + c0 >= 0,     (lower)
 *
 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
 * determined by the first constraint.
 * That is, store
 *
 *  ceil((c0 - c)/n)
 *
 * in *l.
 */
static void lower_bound_from_parallel(__isl_keep isl_basic_map *bmap,
  int ineq, int lower, int pos, isl_int *l)
{
  isl_int_neg(*l, bmap->ineq[ineq][0]);
  isl_int_add(*l, *l, bmap->ineq[lower][0]);
  isl_int_cdiv_q(*l, *l, bmap->ineq[ineq][pos]);
}
 
/* Given two inequality constraints
 *
 *  f(x) + n d + c >= 0,    (ineq)
 *
 * with d the variable at position "pos", and
 *
 *  -f(x) - c0 >= 0,    (upper)
 *
 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
 * determined by the first constraint.
 * That is, store
 *
 *  ceil((-c1 - c)/n)
 *
 * in *u.
 */
static void lower_bound_from_opposite(__isl_keep isl_basic_map *bmap,
  int ineq, int upper, int pos, isl_int *u)
{
  isl_int_neg(*u, bmap->ineq[ineq][0]);
  isl_int_sub(*u, *u, bmap->ineq[upper][0]);
  isl_int_cdiv_q(*u, *u, bmap->ineq[ineq][pos]);
}
 
/* Given a lower bound constraint "ineq" on "div" in "bmap",
 * does the corresponding lower bound have a fixed value in "bmap"?
 *
 * In particular, "ineq" is of the form
 *
 *  f(x) + n d + c >= 0
 *
 * with n > 0, c the constant term and
 * d the existentially quantified variable "div".
 * That is, the lower bound is
 *
 *  ceil((-f(x) - c)/n)
 *
 * Look for a pair of constraints
 *
 *  f(x) + c0 >= 0
 *  -f(x) + c1 >= 0
 *
 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
 * That is, check that
 *
 *  ceil((-c1 - c)/n) = ceil((c0 - c)/n)
 *
 * If so, return the index of inequality f(x) + c0 >= 0.
 * Otherwise, return bmap->n_ineq.
 * Return -1 on error.
 */
static int lower_bound_is_cst(__isl_keep isl_basic_map *bmap, int div, int ineq)
{
  int i;
  int lower = -1, upper = -1;
  unsigned o_div;
  isl_int l, u;
  int equal;
 
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  for (i = 0; i < bmap->n_ineq && (lower < 0 || upper < 0); ++i) {
    isl_bool par, opp;
 
    if (i == ineq)
      continue;
    if (!isl_int_is_zero(bmap->ineq[i][o_div + div]))
      continue;
    par = isl_bool_false;
    if (lower < 0)
      par = is_parallel_except(bmap, ineq, i, o_div + div);
    if (par < 0)
      return -1;
    if (par) {
      lower = i;
      continue;
    }
    opp = isl_bool_false;
    if (upper < 0)
      opp = is_opposite_except(bmap, ineq, i, o_div + div);
    if (opp < 0)
      return -1;
    if (opp)
      upper = i;
  }
 
  if (lower < 0 || upper < 0)
    return bmap->n_ineq;
 
  isl_int_init(l);
  isl_int_init(u);
 
  lower_bound_from_parallel(bmap, ineq, lower, o_div + div, &l);
  lower_bound_from_opposite(bmap, ineq, upper, o_div + div, &u);
 
  equal = isl_int_eq(l, u);
 
  isl_int_clear(l);
  isl_int_clear(u);
 
  return equal ? lower : bmap->n_ineq;
}
 
/* Given a lower bound constraint "ineq" on the existentially quantified
 * variable "div", such that the corresponding lower bound has
 * a fixed value in "bmap", assign this fixed value to the variable and
 * then try and drop redundant divs again,
 * freeing the temporary data structure "pairs" that was associated
 * to the old version of "bmap".
 * "lower" determines the constant value for the lower bound.
 *
 * In particular, "ineq" is of the form
 *
 *  f(x) + n d + c >= 0,
 *
 * while "lower" is of the form
 *
 *  f(x) + c0 >= 0
 *
 * The lower bound is ceil((-f(x) - c)/n) and its constant value
 * is ceil((c0 - c)/n).
 */
static __isl_give isl_basic_map *fix_cst_lower(__isl_take isl_basic_map *bmap,
  int div, int ineq, int lower, int *pairs)
{
  isl_int c;
  unsigned o_div;
 
  isl_int_init(c);
 
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  lower_bound_from_parallel(bmap, ineq, lower, o_div + div, &c);
  bmap = isl_basic_map_fix(bmap, isl_dim_div, div, c);
  free(pairs);
 
  isl_int_clear(c);
 
  return isl_basic_map_drop_redundant_divs(bmap);
}
 
/* Do any of the integer divisions of "bmap" involve integer division "div"?
 *
 * The integer division "div" could only ever appear in any later
 * integer division (with an explicit representation).
 */
static isl_bool any_div_involves_div(__isl_keep isl_basic_map *bmap, int div)
{
  int i;
  isl_size v_div, n_div;
 
  v_div = isl_basic_map_var_offset(bmap, isl_dim_div);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (v_div < 0 || n_div < 0)
    return isl_bool_error;
 
  for (i = div + 1; i < n_div; ++i) {
    isl_bool involves;
 
    involves = isl_basic_map_div_expr_involves_vars(bmap, i,
                v_div + div, 1);
    if (involves < 0 || involves)
      return involves;
  }
 
  return isl_bool_false;
}
 
/* Remove divs that are not strictly needed based on the inequality
 * constraints.
 * In particular, if a div only occurs positively (or negatively)
 * in constraints, then it can simply be dropped.
 * Also, if a div occurs in only two constraints and if moreover
 * those two constraints are opposite to each other, except for the constant
 * term and if the sum of the constant terms is such that for any value
 * of the other values, there is always at least one integer value of the
 * div, i.e., if one plus this sum is greater than or equal to
 * the (absolute value) of the coefficient of the div in the constraints,
 * then we can also simply drop the div.
 *
 * If an existentially quantified variable does not have an explicit
 * representation, appears in only a single lower bound that does not
 * involve any other such existentially quantified variables and appears
 * in this lower bound with coefficient 1,
 * then fix the variable to the value of the lower bound.  That is,
 * turn the inequality into an equality.
 * If for any value of the other variables, there is any value
 * for the existentially quantified variable satisfying the constraints,
 * then this lower bound also satisfies the constraints.
 * It is therefore safe to pick this lower bound.
 *
 * The same reasoning holds even if the coefficient is not one.
 * However, fixing the variable to the value of the lower bound may
 * in general introduce an extra integer division, in which case
 * it may be better to pick another value.
 * If this integer division has a known constant value, then plugging
 * in this constant value removes the existentially quantified variable
 * completely.  In particular, if the lower bound is of the form
 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
 * then the existentially quantified variable can be assigned this
 * shared value.
 *
 * We skip divs that appear in equalities or in the definition of other divs.
 * Divs that appear in the definition of other divs usually occur in at least
 * 4 constraints, but the constraints may have been simplified.
 *
 * If any divs are left after these simple checks then we move on
 * to more complicated cases in drop_more_redundant_divs.
 */
static __isl_give isl_basic_map *isl_basic_map_drop_redundant_divs_ineq(
  __isl_take isl_basic_map *bmap)
{
  int i, j;
  isl_size off;
  int *pairs = NULL;
  int n = 0;
  isl_size n_ineq;
 
  if (!bmap)
    goto error;
  if (bmap->n_div == 0)
    return bmap;
 
  off = isl_basic_map_var_offset(bmap, isl_dim_div);
  if (off < 0)
    return isl_basic_map_free(bmap);
  pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div);
  if (!pairs)
    goto error;
 
  n_ineq = isl_basic_map_n_inequality(bmap);
  if (n_ineq < 0)
    goto error;
  for (i = 0; i < bmap->n_div; ++i) {
    int pos, neg;
    int last_pos, last_neg;
    int redundant;
    int defined;
    isl_bool involves, opp, set_div;
 
    defined = !isl_int_is_zero(bmap->div[i][0]);
    involves = any_div_involves_div(bmap, i);
    if (involves < 0)
      goto error;
    if (involves)
      continue;
    for (j = 0; j < bmap->n_eq; ++j)
      if (!isl_int_is_zero(bmap->eq[j][1 + off + i]))
        break;
    if (j < bmap->n_eq)
      continue;
    ++n;
    pos = neg = 0;
    for (j = 0; j < bmap->n_ineq; ++j) {
      if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) {
        last_pos = j;
        ++pos;
      }
      if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) {
        last_neg = j;
        ++neg;
      }
    }
    pairs[i] = pos * neg;
    if (pairs[i] == 0) {
      for (j = bmap->n_ineq - 1; j >= 0; --j)
        if (!isl_int_is_zero(bmap->ineq[j][1+off+i]))
          isl_basic_map_drop_inequality(bmap, j);
      bmap = isl_basic_map_drop_div(bmap, i);
      return drop_redundant_divs_again(bmap, pairs, 0);
    }
    if (pairs[i] != 1)
      opp = isl_bool_false;
    else
      opp = is_opposite(bmap, last_pos, last_neg);
    if (opp < 0)
      goto error;
    if (!opp) {
      int lower;
      isl_bool single, one;
 
      if (pos != 1)
        continue;
      single = single_unknown(bmap, last_pos, i);
      if (single < 0)
        goto error;
      if (!single)
        continue;
      one = has_coef_one(bmap, i, last_pos);
      if (one < 0)
        goto error;
      if (one)
        return set_eq_and_try_again(bmap, last_pos,
                  pairs);
      lower = lower_bound_is_cst(bmap, i, last_pos);
      if (lower < 0)
        goto error;
      if (lower < n_ineq)
        return fix_cst_lower(bmap, i, last_pos, lower,
            pairs);
      continue;
    }
 
    isl_int_add(bmap->ineq[last_pos][0],
          bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
    isl_int_add_ui(bmap->ineq[last_pos][0],
             bmap->ineq[last_pos][0], 1);
    redundant = isl_int_ge(bmap->ineq[last_pos][0],
        bmap->ineq[last_pos][1+off+i]);
    isl_int_sub_ui(bmap->ineq[last_pos][0],
             bmap->ineq[last_pos][0], 1);
    isl_int_sub(bmap->ineq[last_pos][0],
          bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]);
    if (redundant)
      return drop_div_and_try_again(bmap, i,
                last_pos, last_neg, pairs);
    if (defined)
      set_div = isl_bool_false;
    else
      set_div = ok_to_set_div_from_bound(bmap, i,
              bmap->ineq[last_pos]);
    if (set_div < 0)
      return isl_basic_map_free(bmap);
    if (set_div) {
      bmap = set_div_from_lower_bound(bmap, i,
              bmap->ineq[last_pos]);
      return drop_redundant_divs_again(bmap, pairs, 1);
    }
    pairs[i] = 0;
    --n;
  }
 
  if (n > 0)
    return coalesce_or_drop_more_redundant_divs(bmap, pairs, n);
 
  free(pairs);
  return bmap;
error:
  free(pairs);
  isl_basic_map_free(bmap);
  return NULL;
}
 
/* Consider the coefficients at "c" as a row vector and replace
 * them with their product with "T".  "T" is assumed to be a square matrix.
 */
static isl_stat preimage(isl_int *c, __isl_keep isl_mat *T)
{
  isl_size n;
  isl_ctx *ctx;
  isl_vec *v;
 
  n = isl_mat_rows(T);
  if (n < 0)
    return isl_stat_error;
  if (!isl_seq_any_non_zero(c, n))
    return isl_stat_ok;
  ctx = isl_mat_get_ctx(T);
  v = isl_vec_alloc(ctx, n);
  if (!v)
    return isl_stat_error;
  isl_seq_swp_or_cpy(v->el, c, n);
  v = isl_vec_mat_product(v, isl_mat_copy(T));
  if (!v)
    return isl_stat_error;
  isl_seq_swp_or_cpy(c, v->el, n);
  isl_vec_free(v);
 
  return isl_stat_ok;
}
 
/* Plug in T for the variables in "bmap" starting at "pos".
 * T is a linear unimodular matrix, i.e., without constant term.
 */
static __isl_give isl_basic_map *isl_basic_map_preimage_vars(
  __isl_take isl_basic_map *bmap, unsigned pos, __isl_take isl_mat *T)
{
  int i;
  isl_size n_row, n_col;
 
  bmap = isl_basic_map_cow(bmap);
  n_row = isl_mat_rows(T);
  n_col = isl_mat_cols(T);
  if (!bmap || n_row < 0 || n_col < 0)
    goto error;
 
  if (n_col != n_row)
    isl_die(isl_mat_get_ctx(T), isl_error_invalid,
      "expecting square matrix", goto error);
 
  if (isl_basic_map_check_range(bmap, isl_dim_all, pos, n_col) < 0)
    goto error;
 
  for (i = 0; i < bmap->n_eq; ++i)
    if (preimage(bmap->eq[i] + 1 + pos, T) < 0)
      goto error;
  for (i = 0; i < bmap->n_ineq; ++i)
    if (preimage(bmap->ineq[i] + 1 + pos, T) < 0)
      goto error;
  for (i = 0; i < bmap->n_div; ++i) {
    if (isl_basic_map_div_is_marked_unknown(bmap, i))
      continue;
    if (preimage(bmap->div[i] + 1 + 1 + pos, T) < 0)
      goto error;
  }
 
  isl_mat_free(T);
  return bmap;
error:
  isl_basic_map_free(bmap);
  isl_mat_free(T);
  return NULL;
}
 
/* Remove divs that are not strictly needed.
 *
 * First look for an equality constraint involving two or more
 * existentially quantified variables without an explicit
 * representation.  Replace the combination that appears
 * in the equality constraint by a single existentially quantified
 * variable such that the equality can be used to derive
 * an explicit representation for the variable.
 * If there are no more such equality constraints, then continue
 * with isl_basic_map_drop_redundant_divs_ineq.
 *
 * In particular, if the equality constraint is of the form
 *
 *  f(x) + \sum_i c_i a_i = 0
 *
 * with a_i existentially quantified variable without explicit
 * representation, then apply a transformation on the existentially
 * quantified variables to turn the constraint into
 *
 *  f(x) + g a_1' = 0
 *
 * with g the gcd of the c_i.
 * In order to easily identify which existentially quantified variables
 * have a complete explicit representation, i.e., without being defined
 * in terms of other existentially quantified variables without
 * an explicit representation, the existentially quantified variables
 * are first sorted.
 *
 * The variable transformation is computed by extending the row
 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
 *
 *  [a_1']   [c_1/g ... c_n/g]   [ a_1 ]
 *  [a_2']                       [ a_2 ]
 *   ...   =         U             ....
 *  [a_n']                   [ a_n ]
 *
 * with [c_1/g ... c_n/g] representing the first row of U.
 * The inverse of U is then plugged into the original constraints.
 * The call to isl_basic_map_simplify makes sure the explicit
 * representation for a_1' is extracted from the equality constraint.
 */
__isl_give isl_basic_map *isl_basic_map_drop_redundant_divs(
  __isl_take isl_basic_map *bmap)
{
  int first;
  int i;
  unsigned o_div;
  isl_size n_div;
  int l;
  isl_ctx *ctx;
  isl_mat *T;
 
  if (!bmap)
    return NULL;
  if (isl_basic_map_divs_known(bmap))
    return isl_basic_map_drop_redundant_divs_ineq(bmap);
  if (bmap->n_eq == 0)
    return isl_basic_map_drop_redundant_divs_ineq(bmap);
  bmap = isl_basic_map_sort_divs(bmap);
  if (!bmap)
    return NULL;
 
  first = isl_basic_map_first_unknown_div(bmap);
  if (first < 0)
    return isl_basic_map_free(bmap);
 
  o_div = isl_basic_map_offset(bmap, isl_dim_div);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  if (n_div < 0)
    return isl_basic_map_free(bmap);
 
  for (i = 0; i < bmap->n_eq; ++i) {
    l = isl_seq_first_non_zero(bmap->eq[i] + o_div + first,
              n_div - (first));
    if (l < 0)
      continue;
    l += first;
    if (!isl_seq_any_non_zero(bmap->eq[i] + o_div + l + 1,
              n_div - (l + 1)))
      continue;
    break;
  }
  if (i >= bmap->n_eq)
    return isl_basic_map_drop_redundant_divs_ineq(bmap);
 
  ctx = isl_basic_map_get_ctx(bmap);
  T = isl_mat_alloc(ctx, n_div - l, n_div - l);
  if (!T)
    return isl_basic_map_free(bmap);
  isl_seq_cpy(T->row[0], bmap->eq[i] + o_div + l, n_div - l);
  T = isl_mat_normalize_row(T, 0);
  T = isl_mat_unimodular_complete(T, 1);
  T = isl_mat_right_inverse(T);
 
  for (i = l; i < n_div; ++i)
    bmap = isl_basic_map_mark_div_unknown(bmap, i);
  bmap = isl_basic_map_preimage_vars(bmap, o_div - 1 + l, T);
  bmap = isl_basic_map_simplify(bmap);
 
  return isl_basic_map_drop_redundant_divs(bmap);
}
 
/* Does "bmap" satisfy any equality that involves more than 2 variables
 * and/or has coefficients different from -1 and 1?
 */
static isl_bool has_multiple_var_equality(__isl_keep isl_basic_map *bmap)
{
  int i;
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_bool_error;
 
  for (i = 0; i < bmap->n_eq; ++i) {
    int j, k;
 
    j = isl_seq_first_non_zero(bmap->eq[i] + 1, total);
    if (j < 0)
      continue;
    if (!isl_int_is_one(bmap->eq[i][1 + j]) &&
        !isl_int_is_negone(bmap->eq[i][1 + j]))
      return isl_bool_true;
 
    j += 1;
    k = isl_seq_first_non_zero(bmap->eq[i] + 1 + j, total - j);
    if (k < 0)
      continue;
    j += k;
    if (!isl_int_is_one(bmap->eq[i][1 + j]) &&
        !isl_int_is_negone(bmap->eq[i][1 + j]))
      return isl_bool_true;
 
    j += 1;
    if (isl_seq_any_non_zero(bmap->eq[i] + 1 + j, total - j))
      return isl_bool_true;
  }
 
  return isl_bool_false;
}
 
/* Remove any common factor g from the constraint coefficients in "v".
 * The constant term is stored in the first position and is replaced
 * by floor(c/g).  If any common factor is removed and if this results
 * in a tightening of the constraint, then set *tightened.
 */
static __isl_give isl_vec *normalize_constraint(__isl_take isl_vec *v,
  int *tightened)
{
  isl_ctx *ctx;
 
  if (!v)
    return NULL;
  ctx = isl_vec_get_ctx(v);
  isl_seq_gcd(v->el + 1, v->size - 1, &ctx->normalize_gcd);
  if (isl_int_is_zero(ctx->normalize_gcd))
    return v;
  if (isl_int_is_one(ctx->normalize_gcd))
    return v;
  v = isl_vec_cow(v);
  if (!v)
    return NULL;
  if (tightened && !isl_int_is_divisible_by(v->el[0], ctx->normalize_gcd))
    *tightened = 1;
  isl_int_fdiv_q(v->el[0], v->el[0], ctx->normalize_gcd);
  isl_seq_scale_down(v->el + 1, v->el + 1, ctx->normalize_gcd,
        v->size - 1);
  return v;
}
 
/* Internal representation used by isl_basic_map_reduce_coefficients.
 *
 * "total" is the total dimensionality of the original basic map.
 * "v" is a temporary vector of size 1 + total that can be used
 * to store constraint coefficients.
 * "T" is the variable compression.
 * "T2" is the inverse transformation.
 * "tightened" is set if any constant term got tightened
 * while reducing the coefficients.
 */
struct isl_reduce_coefficients_data {
  isl_size total;
  isl_vec *v;
  isl_mat *T;
  isl_mat *T2;
  int tightened;
};
 
/* Free all memory allocated in "data".
 */
static void isl_reduce_coefficients_data_clear(
  struct isl_reduce_coefficients_data *data)
{
  data->T = isl_mat_free(data->T);
  data->T2 = isl_mat_free(data->T2);
  data->v = isl_vec_free(data->v);
}
 
/* Initialize "data" for "bmap", freeing all allocated memory
 * if anything goes wrong.
 *
 * In particular, construct a variable compression
 * from the equality constraints of "bmap" and
 * allocate a temporary vector.
 */
static isl_stat isl_reduce_coefficients_data_init(
  __isl_keep isl_basic_map *bmap,
  struct isl_reduce_coefficients_data *data)
{
  isl_ctx *ctx;
  isl_mat *eq;
 
  data->v = NULL;
  data->T = NULL;
  data->T2 = NULL;
  data->tightened = 0;
 
  data->total = isl_basic_map_dim(bmap, isl_dim_all);
  if (data->total < 0)
    return isl_stat_error;
  ctx = isl_basic_map_get_ctx(bmap);
  data->v = isl_vec_alloc(ctx, 1 + data->total);
  if (!data->v)
    return isl_stat_error;
 
  eq = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap->n_eq,
        0, 1 + data->total);
  data->T = isl_mat_variable_compression(eq, &data->T2);
  if (!data->T || !data->T2)
    goto error;
 
  return isl_stat_ok;
error:
  isl_reduce_coefficients_data_clear(data);
  return isl_stat_error;
}
 
/* Reduce the coefficients of "bmap" by applying the variable compression
 * in "data".
 * In particular, apply the variable compression to each constraint,
 * factor out any common factor in the non-constant coefficients and
 * then apply the inverse of the compression.
 *
 * Only apply the reduction on a single copy of the basic map
 * since the reduction may leave the result in an inconsistent state.
 * In particular, the constraints may not be gaussed.
 */
static __isl_give isl_basic_map *reduce_coefficients(
  __isl_take isl_basic_map *bmap,
  struct isl_reduce_coefficients_data *data)
{
  int i;
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return isl_basic_map_free(bmap);
  if (total != data->total)
    isl_die(isl_basic_map_get_ctx(bmap), isl_error_internal,
      "total dimensionality changed unexpectedly",
      return isl_basic_map_free(bmap));
 
  bmap = isl_basic_map_cow(bmap);
  if (!bmap)
    return NULL;
 
  for (i = 0; i < bmap->n_ineq; ++i) {
    isl_seq_cpy(data->v->el, bmap->ineq[i], 1 + data->total);
    data->v = isl_vec_mat_product(data->v, isl_mat_copy(data->T));
    data->v = normalize_constraint(data->v, &data->tightened);
    data->v = isl_vec_mat_product(data->v, isl_mat_copy(data->T2));
    if (!data->v)
      return isl_basic_map_free(bmap);
    isl_seq_cpy(bmap->ineq[i], data->v->el, 1 + data->total);
  }
 
  ISL_F_SET(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS);
 
  return bmap;
}
 
/* If "bmap" is an integer set that satisfies any equality involving
 * more than 2 variables and/or has coefficients different from -1 and 1,
 * then use variable compression to reduce the coefficients by removing
 * any (hidden) common factor.
 * In particular, apply the variable compression to each constraint,
 * factor out any common factor in the non-constant coefficients and
 * then apply the inverse of the compression.
 * At the end, we mark the basic map as having reduced constants.
 * If this flag is still set on the next invocation of this function,
 * then we skip the computation.
 *
 * Removing a common factor may result in a tightening of some of
 * the constraints.  If this happens, then we may end up with two
 * opposite inequalities that can be replaced by an equality.
 * We therefore call isl_basic_map_detect_inequality_pairs,
 * which checks for such pairs of inequalities as well as eliminate_divs_eq
 * and isl_basic_map_gauss if such a pair was found.
 * This call to isl_basic_map_gauss may undo much of the effect
 * of the reduction on which isl_map_coalesce depends.
 * In particular, constraints in terms of (compressed) local variables
 * get reformulated in terms of the set variables again.
 * The reduction is therefore applied again afterwards.
 * This has to be done before the call to eliminate_divs_eq, however,
 * since that may remove some local variables, while
 * the data used during the reduction is formulated in terms
 * of the original variables.
 *
 * Tightening may also result in some other constraints becoming
 * (rationally) redundant with respect to the tightened constraint
 * (in combination with other constraints).  The basic map may
 * therefore no longer be assumed to have no redundant constraints.
 *
 * Note that this function may leave the result in an inconsistent state.
 * In particular, the constraints may not be gaussed.
 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
 * for some of the test cases to pass successfully.
 * Any potential modification of the representation is therefore only
 * performed on a single copy of the basic map.
 */
__isl_give isl_basic_map *isl_basic_map_reduce_coefficients(
  __isl_take isl_basic_map *bmap)
{
  struct isl_reduce_coefficients_data data;
  isl_bool multi;
 
  if (!bmap)
    return NULL;
  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS))
    return bmap;
  if (isl_basic_map_is_rational(bmap))
    return bmap;
  if (bmap->n_eq == 0)
    return bmap;
  multi = has_multiple_var_equality(bmap);
  if (multi < 0)
    return isl_basic_map_free(bmap);
  if (!multi)
    return bmap;
 
  if (isl_reduce_coefficients_data_init(bmap, &data) < 0)
    return isl_basic_map_free(bmap);
 
  if (data.T->n_col == 0) {
    isl_reduce_coefficients_data_clear(&data);
    return isl_basic_map_set_to_empty(bmap);
  }
 
  bmap = reduce_coefficients(bmap, &data);
  if (!bmap)
    goto error;
 
  if (data.tightened) {
    int progress = 0;
 
    ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT);
    bmap = isl_basic_map_detect_inequality_pairs(bmap, &progress);
    if (progress) {
      isl_bool empty;
 
      bmap = isl_basic_map_gauss(bmap, NULL);
      empty = isl_basic_map_plain_is_empty(bmap);
      if (empty < 0)
        goto error;
      if (!empty)
        bmap = reduce_coefficients(bmap, &data);
      bmap = eliminate_divs_eq(bmap, &progress);
    }
  }
 
  isl_reduce_coefficients_data_clear(&data);
 
  return bmap;
error:
  isl_reduce_coefficients_data_clear(&data);
  return isl_basic_map_free(bmap);
}
 
/* Shift the integer division at position "div" of "bmap"
 * by "shift" times the variable at position "pos".
 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
 * corresponds to the constant term.
 *
 * That is, if the integer division has the form
 *
 *  floor(f(x)/d)
 *
 * then replace it by
 *
 *  floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
 */
__isl_give isl_basic_map *isl_basic_map_shift_div(
  __isl_take isl_basic_map *bmap, int div, int pos, isl_int shift)
{
  int i;
  isl_size total, n_div;
 
  if (isl_int_is_zero(shift))
    return bmap;
  total = isl_basic_map_dim(bmap, isl_dim_all);
  n_div = isl_basic_map_dim(bmap, isl_dim_div);
  total -= n_div;
  if (total < 0 || n_div < 0)
    return isl_basic_map_free(bmap);
 
  isl_int_addmul(bmap->div[div][1 + pos], shift, bmap->div[div][0]);
 
  for (i = 0; i < bmap->n_eq; ++i) {
    if (isl_int_is_zero(bmap->eq[i][1 + total + div]))
      continue;
    isl_int_submul(bmap->eq[i][pos],
        shift, bmap->eq[i][1 + total + div]);
  }
  for (i = 0; i < bmap->n_ineq; ++i) {
    if (isl_int_is_zero(bmap->ineq[i][1 + total + div]))
      continue;
    isl_int_submul(bmap->ineq[i][pos],
        shift, bmap->ineq[i][1 + total + div]);
  }
  for (i = 0; i < bmap->n_div; ++i) {
    if (isl_int_is_zero(bmap->div[i][0]))
      continue;
    if (isl_int_is_zero(bmap->div[i][1 + 1 + total + div]))
      continue;
    isl_int_submul(bmap->div[i][1 + pos],
        shift, bmap->div[i][1 + 1 + total + div]);
  }
 
  return bmap;
}