isl_tab.c

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/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2013      Ecole Normale Superieure
 * Copyright 2014      INRIA Rocquencourt
 * Copyright 2016      Sven Verdoolaege
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France
 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
 * B.P. 105 - 78153 Le Chesnay, France
 */
 
#include <isl_ctx_private.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include "isl_map_private.h"
#include "isl_tab.h"
#include <isl_seq.h>
#include <isl_config.h>
 
#include <bset_to_bmap.c>
#include <bset_from_bmap.c>
 
/*
 * The implementation of tableaus in this file was inspired by Section 8
 * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem
 * prover for program checking".
 */
 
struct isl_tab *isl_tab_alloc(struct isl_ctx *ctx,
  unsigned n_row, unsigned n_var, unsigned M)
{
  int i;
  struct isl_tab *tab;
  unsigned off = 2 + M;
 
  tab = isl_calloc_type(ctx, struct isl_tab);
  if (!tab)
    return NULL;
  tab->mat = isl_mat_alloc(ctx, n_row, off + n_var);
  if (!tab->mat)
    goto error;
  tab->var = isl_alloc_array(ctx, struct isl_tab_var, n_var);
  if (n_var && !tab->var)
    goto error;
  tab->con = isl_alloc_array(ctx, struct isl_tab_var, n_row);
  if (n_row && !tab->con)
    goto error;
  tab->col_var = isl_alloc_array(ctx, int, n_var);
  if (n_var && !tab->col_var)
    goto error;
  tab->row_var = isl_alloc_array(ctx, int, n_row);
  if (n_row && !tab->row_var)
    goto error;
  for (i = 0; i < n_var; ++i) {
    tab->var[i].index = i;
    tab->var[i].is_row = 0;
    tab->var[i].is_nonneg = 0;
    tab->var[i].is_zero = 0;
    tab->var[i].is_redundant = 0;
    tab->var[i].frozen = 0;
    tab->var[i].negated = 0;
    tab->col_var[i] = i;
  }
  tab->n_row = 0;
  tab->n_con = 0;
  tab->n_eq = 0;
  tab->max_con = n_row;
  tab->n_col = n_var;
  tab->n_var = n_var;
  tab->max_var = n_var;
  tab->n_param = 0;
  tab->n_div = 0;
  tab->n_dead = 0;
  tab->n_redundant = 0;
  tab->strict_redundant = 0;
  tab->need_undo = 0;
  tab->rational = 0;
  tab->empty = 0;
  tab->in_undo = 0;
  tab->M = M;
  tab->cone = 0;
  tab->bottom.type = isl_tab_undo_bottom;
  tab->bottom.next = NULL;
  tab->top = &tab->bottom;
 
  tab->n_zero = 0;
  tab->n_unbounded = 0;
  tab->basis = NULL;
 
  return tab;
error:
  isl_tab_free(tab);
  return NULL;
}
 
isl_ctx *isl_tab_get_ctx(struct isl_tab *tab)
{
  return tab ? isl_mat_get_ctx(tab->mat) : NULL;
}
 
int isl_tab_extend_cons(struct isl_tab *tab, unsigned n_new)
{
  unsigned off;
 
  if (!tab)
    return -1;
 
  off = 2 + tab->M;
 
  if (tab->max_con < tab->n_con + n_new) {
    struct isl_tab_var *con;
 
    con = isl_realloc_array(tab->mat->ctx, tab->con,
            struct isl_tab_var, tab->max_con + n_new);
    if (!con)
      return -1;
    tab->con = con;
    tab->max_con += n_new;
  }
  if (tab->mat->n_row < tab->n_row + n_new) {
    int *row_var;
 
    tab->mat = isl_mat_extend(tab->mat,
          tab->n_row + n_new, off + tab->n_col);
    if (!tab->mat)
      return -1;
    row_var = isl_realloc_array(tab->mat->ctx, tab->row_var,
              int, tab->mat->n_row);
    if (!row_var)
      return -1;
    tab->row_var = row_var;
    if (tab->row_sign) {
      enum isl_tab_row_sign *s;
      s = isl_realloc_array(tab->mat->ctx, tab->row_sign,
          enum isl_tab_row_sign, tab->mat->n_row);
      if (!s)
        return -1;
      tab->row_sign = s;
    }
  }
  return 0;
}
 
/* Make room for at least n_new extra variables.
 * Return -1 if anything went wrong.
 */
int isl_tab_extend_vars(struct isl_tab *tab, unsigned n_new)
{
  struct isl_tab_var *var;
  unsigned off = 2 + tab->M;
 
  if (tab->max_var < tab->n_var + n_new) {
    var = isl_realloc_array(tab->mat->ctx, tab->var,
            struct isl_tab_var, tab->n_var + n_new);
    if (!var)
      return -1;
    tab->var = var;
    tab->max_var = tab->n_var + n_new;
  }
 
  if (tab->mat->n_col < off + tab->n_col + n_new) {
    int *p;
 
    tab->mat = isl_mat_extend(tab->mat,
            tab->mat->n_row, off + tab->n_col + n_new);
    if (!tab->mat)
      return -1;
    p = isl_realloc_array(tab->mat->ctx, tab->col_var,
              int, tab->n_col + n_new);
    if (!p)
      return -1;
    tab->col_var = p;
  }
 
  return 0;
}
 
static void free_undo_record(struct isl_tab_undo *undo)
{
  switch (undo->type) {
  case isl_tab_undo_saved_basis:
    free(undo->u.col_var);
    break;
  default:;
  }
  free(undo);
}
 
static void free_undo(struct isl_tab *tab)
{
  struct isl_tab_undo *undo, *next;
 
  for (undo = tab->top; undo && undo != &tab->bottom; undo = next) {
    next = undo->next;
    free_undo_record(undo);
  }
  tab->top = undo;
}
 
void isl_tab_free(struct isl_tab *tab)
{
  if (!tab)
    return;
  free_undo(tab);
  isl_mat_free(tab->mat);
  isl_vec_free(tab->dual);
  isl_basic_map_free(tab->bmap);
  free(tab->var);
  free(tab->con);
  free(tab->row_var);
  free(tab->col_var);
  free(tab->row_sign);
  isl_mat_free(tab->samples);
  free(tab->sample_index);
  isl_mat_free(tab->basis);
  free(tab);
}
 
struct isl_tab *isl_tab_dup(struct isl_tab *tab)
{
  int i;
  struct isl_tab *dup;
  unsigned off;
 
  if (!tab)
    return NULL;
 
  off = 2 + tab->M;
  dup = isl_calloc_type(tab->mat->ctx, struct isl_tab);
  if (!dup)
    return NULL;
  dup->mat = isl_mat_dup(tab->mat);
  if (!dup->mat)
    goto error;
  dup->var = isl_alloc_array(tab->mat->ctx, struct isl_tab_var, tab->max_var);
  if (tab->max_var && !dup->var)
    goto error;
  for (i = 0; i < tab->n_var; ++i)
    dup->var[i] = tab->var[i];
  dup->con = isl_alloc_array(tab->mat->ctx, struct isl_tab_var, tab->max_con);
  if (tab->max_con && !dup->con)
    goto error;
  for (i = 0; i < tab->n_con; ++i)
    dup->con[i] = tab->con[i];
  dup->col_var = isl_alloc_array(tab->mat->ctx, int, tab->mat->n_col - off);
  if ((tab->mat->n_col - off) && !dup->col_var)
    goto error;
  for (i = 0; i < tab->n_col; ++i)
    dup->col_var[i] = tab->col_var[i];
  dup->row_var = isl_alloc_array(tab->mat->ctx, int, tab->mat->n_row);
  if (tab->mat->n_row && !dup->row_var)
    goto error;
  for (i = 0; i < tab->n_row; ++i)
    dup->row_var[i] = tab->row_var[i];
  if (tab->row_sign) {
    dup->row_sign = isl_alloc_array(tab->mat->ctx, enum isl_tab_row_sign,
            tab->mat->n_row);
    if (tab->mat->n_row && !dup->row_sign)
      goto error;
    for (i = 0; i < tab->n_row; ++i)
      dup->row_sign[i] = tab->row_sign[i];
  }
  if (tab->samples) {
    dup->samples = isl_mat_dup(tab->samples);
    if (!dup->samples)
      goto error;
    dup->sample_index = isl_alloc_array(tab->mat->ctx, int,
              tab->samples->n_row);
    if (tab->samples->n_row && !dup->sample_index)
      goto error;
    dup->n_sample = tab->n_sample;
    dup->n_outside = tab->n_outside;
  }
  dup->n_row = tab->n_row;
  dup->n_con = tab->n_con;
  dup->n_eq = tab->n_eq;
  dup->max_con = tab->max_con;
  dup->n_col = tab->n_col;
  dup->n_var = tab->n_var;
  dup->max_var = tab->max_var;
  dup->n_param = tab->n_param;
  dup->n_div = tab->n_div;
  dup->n_dead = tab->n_dead;
  dup->n_redundant = tab->n_redundant;
  dup->rational = tab->rational;
  dup->empty = tab->empty;
  dup->strict_redundant = 0;
  dup->need_undo = 0;
  dup->in_undo = 0;
  dup->M = tab->M;
  dup->cone = tab->cone;
  dup->bottom.type = isl_tab_undo_bottom;
  dup->bottom.next = NULL;
  dup->top = &dup->bottom;
 
  dup->n_zero = tab->n_zero;
  dup->n_unbounded = tab->n_unbounded;
  dup->basis = isl_mat_dup(tab->basis);
 
  return dup;
error:
  isl_tab_free(dup);
  return NULL;
}
 
/* Construct the coefficient matrix of the product tableau
 * of two tableaus.
 * mat{1,2} is the coefficient matrix of tableau {1,2}
 * row{1,2} is the number of rows in tableau {1,2}
 * col{1,2} is the number of columns in tableau {1,2}
 * off is the offset to the coefficient column (skipping the
 *  denominator, the constant term and the big parameter if any)
 * r{1,2} is the number of redundant rows in tableau {1,2}
 * d{1,2} is the number of dead columns in tableau {1,2}
 *
 * The order of the rows and columns in the result is as explained
 * in isl_tab_product.
 */
static __isl_give isl_mat *tab_mat_product(__isl_keep isl_mat *mat1,
  __isl_keep isl_mat *mat2, unsigned row1, unsigned row2,
  unsigned col1, unsigned col2,
  unsigned off, unsigned r1, unsigned r2, unsigned d1, unsigned d2)
{
  int i;
  struct isl_mat *prod;
  unsigned n;
 
  prod = isl_mat_alloc(mat1->ctx, mat1->n_row + mat2->n_row,
          off + col1 + col2);
  if (!prod)
    return NULL;
 
  n = 0;
  for (i = 0; i < r1; ++i) {
    isl_seq_cpy(prod->row[n + i], mat1->row[i], off + d1);
    isl_seq_clr(prod->row[n + i] + off + d1, d2);
    isl_seq_cpy(prod->row[n + i] + off + d1 + d2,
        mat1->row[i] + off + d1, col1 - d1);
    isl_seq_clr(prod->row[n + i] + off + col1 + d1, col2 - d2);
  }
 
  n += r1;
  for (i = 0; i < r2; ++i) {
    isl_seq_cpy(prod->row[n + i], mat2->row[i], off);
    isl_seq_clr(prod->row[n + i] + off, d1);
    isl_seq_cpy(prod->row[n + i] + off + d1,
          mat2->row[i] + off, d2);
    isl_seq_clr(prod->row[n + i] + off + d1 + d2, col1 - d1);
    isl_seq_cpy(prod->row[n + i] + off + col1 + d1,
          mat2->row[i] + off + d2, col2 - d2);
  }
 
  n += r2;
  for (i = 0; i < row1 - r1; ++i) {
    isl_seq_cpy(prod->row[n + i], mat1->row[r1 + i], off + d1);
    isl_seq_clr(prod->row[n + i] + off + d1, d2);
    isl_seq_cpy(prod->row[n + i] + off + d1 + d2,
        mat1->row[r1 + i] + off + d1, col1 - d1);
    isl_seq_clr(prod->row[n + i] + off + col1 + d1, col2 - d2);
  }
 
  n += row1 - r1;
  for (i = 0; i < row2 - r2; ++i) {
    isl_seq_cpy(prod->row[n + i], mat2->row[r2 + i], off);
    isl_seq_clr(prod->row[n + i] + off, d1);
    isl_seq_cpy(prod->row[n + i] + off + d1,
          mat2->row[r2 + i] + off, d2);
    isl_seq_clr(prod->row[n + i] + off + d1 + d2, col1 - d1);
    isl_seq_cpy(prod->row[n + i] + off + col1 + d1,
          mat2->row[r2 + i] + off + d2, col2 - d2);
  }
 
  return prod;
}
 
/* Update the row or column index of a variable that corresponds
 * to a variable in the first input tableau.
 */
static void update_index1(struct isl_tab_var *var,
  unsigned r1, unsigned r2, unsigned d1, unsigned d2)
{
  if (var->index == -1)
    return;
  if (var->is_row && var->index >= r1)
    var->index += r2;
  if (!var->is_row && var->index >= d1)
    var->index += d2;
}
 
/* Update the row or column index of a variable that corresponds
 * to a variable in the second input tableau.
 */
static void update_index2(struct isl_tab_var *var,
  unsigned row1, unsigned col1,
  unsigned r1, unsigned r2, unsigned d1, unsigned d2)
{
  if (var->index == -1)
    return;
  if (var->is_row) {
    if (var->index < r2)
      var->index += r1;
    else
      var->index += row1;
  } else {
    if (var->index < d2)
      var->index += d1;
    else
      var->index += col1;
  }
}
 
/* Create a tableau that represents the Cartesian product of the sets
 * represented by tableaus tab1 and tab2.
 * The order of the rows in the product is
 *  - redundant rows of tab1
 *  - redundant rows of tab2
 *  - non-redundant rows of tab1
 *  - non-redundant rows of tab2
 * The order of the columns is
 *  - denominator
 *  - constant term
 *  - coefficient of big parameter, if any
 *  - dead columns of tab1
 *  - dead columns of tab2
 *  - live columns of tab1
 *  - live columns of tab2
 * The order of the variables and the constraints is a concatenation
 * of order in the two input tableaus.
 */
struct isl_tab *isl_tab_product(struct isl_tab *tab1, struct isl_tab *tab2)
{
  int i;
  struct isl_tab *prod;
  unsigned off;
  unsigned r1, r2, d1, d2;
 
  if (!tab1 || !tab2)
    return NULL;
 
  isl_assert(tab1->mat->ctx, tab1->M == tab2->M, return NULL);
  isl_assert(tab1->mat->ctx, tab1->rational == tab2->rational, return NULL);
  isl_assert(tab1->mat->ctx, tab1->cone == tab2->cone, return NULL);
  isl_assert(tab1->mat->ctx, !tab1->row_sign, return NULL);
  isl_assert(tab1->mat->ctx, !tab2->row_sign, return NULL);
  isl_assert(tab1->mat->ctx, tab1->n_param == 0, return NULL);
  isl_assert(tab1->mat->ctx, tab2->n_param == 0, return NULL);
  isl_assert(tab1->mat->ctx, tab1->n_div == 0, return NULL);
  isl_assert(tab1->mat->ctx, tab2->n_div == 0, return NULL);
 
  off = 2 + tab1->M;
  r1 = tab1->n_redundant;
  r2 = tab2->n_redundant;
  d1 = tab1->n_dead;
  d2 = tab2->n_dead;
  prod = isl_calloc_type(tab1->mat->ctx, struct isl_tab);
  if (!prod)
    return NULL;
  prod->mat = tab_mat_product(tab1->mat, tab2->mat,
        tab1->n_row, tab2->n_row,
        tab1->n_col, tab2->n_col, off, r1, r2, d1, d2);
  if (!prod->mat)
    goto error;
  prod->var = isl_alloc_array(tab1->mat->ctx, struct isl_tab_var,
          tab1->max_var + tab2->max_var);
  if ((tab1->max_var + tab2->max_var) && !prod->var)
    goto error;
  for (i = 0; i < tab1->n_var; ++i) {
    prod->var[i] = tab1->var[i];
    update_index1(&prod->var[i], r1, r2, d1, d2);
  }
  for (i = 0; i < tab2->n_var; ++i) {
    prod->var[tab1->n_var + i] = tab2->var[i];
    update_index2(&prod->var[tab1->n_var + i],
        tab1->n_row, tab1->n_col,
        r1, r2, d1, d2);
  }
  prod->con = isl_alloc_array(tab1->mat->ctx, struct isl_tab_var,
          tab1->max_con +  tab2->max_con);
  if ((tab1->max_con + tab2->max_con) && !prod->con)
    goto error;
  for (i = 0; i < tab1->n_con; ++i) {
    prod->con[i] = tab1->con[i];
    update_index1(&prod->con[i], r1, r2, d1, d2);
  }
  for (i = 0; i < tab2->n_con; ++i) {
    prod->con[tab1->n_con + i] = tab2->con[i];
    update_index2(&prod->con[tab1->n_con + i],
        tab1->n_row, tab1->n_col,
        r1, r2, d1, d2);
  }
  prod->col_var = isl_alloc_array(tab1->mat->ctx, int,
          tab1->n_col + tab2->n_col);
  if ((tab1->n_col + tab2->n_col) && !prod->col_var)
    goto error;
  for (i = 0; i < tab1->n_col; ++i) {
    int pos = i < d1 ? i : i + d2;
    prod->col_var[pos] = tab1->col_var[i];
  }
  for (i = 0; i < tab2->n_col; ++i) {
    int pos = i < d2 ? d1 + i : tab1->n_col + i;
    int t = tab2->col_var[i];
    if (t >= 0)
      t += tab1->n_var;
    else
      t -= tab1->n_con;
    prod->col_var[pos] = t;
  }
  prod->row_var = isl_alloc_array(tab1->mat->ctx, int,
          tab1->mat->n_row + tab2->mat->n_row);
  if ((tab1->mat->n_row + tab2->mat->n_row) && !prod->row_var)
    goto error;
  for (i = 0; i < tab1->n_row; ++i) {
    int pos = i < r1 ? i : i + r2;
    prod->row_var[pos] = tab1->row_var[i];
  }
  for (i = 0; i < tab2->n_row; ++i) {
    int pos = i < r2 ? r1 + i : tab1->n_row + i;
    int t = tab2->row_var[i];
    if (t >= 0)
      t += tab1->n_var;
    else
      t -= tab1->n_con;
    prod->row_var[pos] = t;
  }
  prod->samples = NULL;
  prod->sample_index = NULL;
  prod->n_row = tab1->n_row + tab2->n_row;
  prod->n_con = tab1->n_con + tab2->n_con;
  prod->n_eq = 0;
  prod->max_con = tab1->max_con + tab2->max_con;
  prod->n_col = tab1->n_col + tab2->n_col;
  prod->n_var = tab1->n_var + tab2->n_var;
  prod->max_var = tab1->max_var + tab2->max_var;
  prod->n_param = 0;
  prod->n_div = 0;
  prod->n_dead = tab1->n_dead + tab2->n_dead;
  prod->n_redundant = tab1->n_redundant + tab2->n_redundant;
  prod->rational = tab1->rational;
  prod->empty = tab1->empty || tab2->empty;
  prod->strict_redundant = tab1->strict_redundant || tab2->strict_redundant;
  prod->need_undo = 0;
  prod->in_undo = 0;
  prod->M = tab1->M;
  prod->cone = tab1->cone;
  prod->bottom.type = isl_tab_undo_bottom;
  prod->bottom.next = NULL;
  prod->top = &prod->bottom;
 
  prod->n_zero = 0;
  prod->n_unbounded = 0;
  prod->basis = NULL;
 
  return prod;
error:
  isl_tab_free(prod);
  return NULL;
}
 
static struct isl_tab_var *var_from_index(struct isl_tab *tab, int i)
{
  if (i >= 0)
    return &tab->var[i];
  else
    return &tab->con[~i];
}
 
struct isl_tab_var *isl_tab_var_from_row(struct isl_tab *tab, int i)
{
  return var_from_index(tab, tab->row_var[i]);
}
 
static struct isl_tab_var *var_from_col(struct isl_tab *tab, int i)
{
  return var_from_index(tab, tab->col_var[i]);
}
 
/* Check if there are any upper bounds on column variable "var",
 * i.e., non-negative rows where var appears with a negative coefficient.
 * Return 1 if there are no such bounds.
 */
static int max_is_manifestly_unbounded(struct isl_tab *tab,
  struct isl_tab_var *var)
{
  int i;
  unsigned off = 2 + tab->M;
 
  if (var->is_row)
    return 0;
  for (i = tab->n_redundant; i < tab->n_row; ++i) {
    if (!isl_int_is_neg(tab->mat->row[i][off + var->index]))
      continue;
    if (isl_tab_var_from_row(tab, i)->is_nonneg)
      return 0;
  }
  return 1;
}
 
/* Check if there are any lower bounds on column variable "var",
 * i.e., non-negative rows where var appears with a positive coefficient.
 * Return 1 if there are no such bounds.
 */
static int min_is_manifestly_unbounded(struct isl_tab *tab,
  struct isl_tab_var *var)
{
  int i;
  unsigned off = 2 + tab->M;
 
  if (var->is_row)
    return 0;
  for (i = tab->n_redundant; i < tab->n_row; ++i) {
    if (!isl_int_is_pos(tab->mat->row[i][off + var->index]))
      continue;
    if (isl_tab_var_from_row(tab, i)->is_nonneg)
      return 0;
  }
  return 1;
}
 
static int row_cmp(struct isl_tab *tab, int r1, int r2, int c, isl_int *t)
{
  unsigned off = 2 + tab->M;
 
  if (tab->M) {
    int s;
    isl_int_mul(*t, tab->mat->row[r1][2], tab->mat->row[r2][off+c]);
    isl_int_submul(*t, tab->mat->row[r2][2], tab->mat->row[r1][off+c]);
    s = isl_int_sgn(*t);
    if (s)
      return s;
  }
  isl_int_mul(*t, tab->mat->row[r1][1], tab->mat->row[r2][off + c]);
  isl_int_submul(*t, tab->mat->row[r2][1], tab->mat->row[r1][off + c]);
  return isl_int_sgn(*t);
}
 
/* Given the index of a column "c", return the index of a row
 * that can be used to pivot the column in, with either an increase
 * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable.
 * If "var" is not NULL, then the row returned will be different from
 * the one associated with "var".
 *
 * Each row in the tableau is of the form
 *
 *  x_r = a_r0 + \sum_i a_ri x_i
 *
 * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn"
 * impose any limit on the increase or decrease in the value of x_c
 * and this bound is equal to a_r0 / |a_rc|.  We are therefore looking
 * for the row with the smallest (most stringent) such bound.
 * Note that the common denominator of each row drops out of the fraction.
 * To check if row j has a smaller bound than row r, i.e.,
 * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|,
 * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0,
 * where -sign(a_jc) is equal to "sgn".
 */
static int pivot_row(struct isl_tab *tab,
  struct isl_tab_var *var, int sgn, int c)
{
  int j, r, tsgn;
  isl_int t;
  unsigned off = 2 + tab->M;
 
  isl_int_init(t);
  r = -1;
  for (j = tab->n_redundant; j < tab->n_row; ++j) {
    if (var && j == var->index)
      continue;
    if (!isl_tab_var_from_row(tab, j)->is_nonneg)
      continue;
    if (sgn * isl_int_sgn(tab->mat->row[j][off + c]) >= 0)
      continue;
    if (r < 0) {
      r = j;
      continue;
    }
    tsgn = sgn * row_cmp(tab, r, j, c, &t);
    if (tsgn < 0 || (tsgn == 0 &&
              tab->row_var[j] < tab->row_var[r]))
      r = j;
  }
  isl_int_clear(t);
  return r;
}
 
/* Find a pivot (row and col) that will increase (sgn > 0) or decrease
 * (sgn < 0) the value of row variable var.
 * If not NULL, then skip_var is a row variable that should be ignored
 * while looking for a pivot row.  It is usually equal to var.
 *
 * As the given row in the tableau is of the form
 *
 *  x_r = a_r0 + \sum_i a_ri x_i
 *
 * we need to find a column such that the sign of a_ri is equal to "sgn"
 * (such that an increase in x_i will have the desired effect) or a
 * column with a variable that may attain negative values.
 * If a_ri is positive, then we need to move x_i in the same direction
 * to obtain the desired effect.  Otherwise, x_i has to move in the
 * opposite direction.
 */
static void find_pivot(struct isl_tab *tab,
  struct isl_tab_var *var, struct isl_tab_var *skip_var,
  int sgn, int *row, int *col)
{
  int j, r, c;
  isl_int *tr;
 
  *row = *col = -1;
 
  isl_assert(tab->mat->ctx, var->is_row, return);
  tr = tab->mat->row[var->index] + 2 + tab->M;
 
  c = -1;
  for (j = tab->n_dead; j < tab->n_col; ++j) {
    if (isl_int_is_zero(tr[j]))
      continue;
    if (isl_int_sgn(tr[j]) != sgn &&
        var_from_col(tab, j)->is_nonneg)
      continue;
    if (c < 0 || tab->col_var[j] < tab->col_var[c])
      c = j;
  }
  if (c < 0)
    return;
 
  sgn *= isl_int_sgn(tr[c]);
  r = pivot_row(tab, skip_var, sgn, c);
  *row = r < 0 ? var->index : r;
  *col = c;
}
 
/* Return 1 if row "row" represents an obviously redundant inequality.
 * This means
 *  - it represents an inequality or a variable
 *  - that is the sum of a non-negative sample value and a positive
 *    combination of zero or more non-negative constraints.
 */
int isl_tab_row_is_redundant(struct isl_tab *tab, int row)
{
  int i;
  unsigned off = 2 + tab->M;
 
  if (tab->row_var[row] < 0 && !isl_tab_var_from_row(tab, row)->is_nonneg)
    return 0;
 
  if (isl_int_is_neg(tab->mat->row[row][1]))
    return 0;
  if (tab->strict_redundant && isl_int_is_zero(tab->mat->row[row][1]))
    return 0;
  if (tab->M && isl_int_is_neg(tab->mat->row[row][2]))
    return 0;
 
  for (i = tab->n_dead; i < tab->n_col; ++i) {
    if (isl_int_is_zero(tab->mat->row[row][off + i]))
      continue;
    if (tab->col_var[i] >= 0)
      return 0;
    if (isl_int_is_neg(tab->mat->row[row][off + i]))
      return 0;
    if (!var_from_col(tab, i)->is_nonneg)
      return 0;
  }
  return 1;
}
 
static void swap_rows(struct isl_tab *tab, int row1, int row2)
{
  int t;
  enum isl_tab_row_sign s;
 
  t = tab->row_var[row1];
  tab->row_var[row1] = tab->row_var[row2];
  tab->row_var[row2] = t;
  isl_tab_var_from_row(tab, row1)->index = row1;
  isl_tab_var_from_row(tab, row2)->index = row2;
  tab->mat = isl_mat_swap_rows(tab->mat, row1, row2);
 
  if (!tab->row_sign)
    return;
  s = tab->row_sign[row1];
  tab->row_sign[row1] = tab->row_sign[row2];
  tab->row_sign[row2] = s;
}
 
static isl_stat push_union(struct isl_tab *tab,
  enum isl_tab_undo_type type, union isl_tab_undo_val u) WARN_UNUSED;
 
/* Push record "u" onto the undo stack of "tab", provided "tab"
 * keeps track of undo information.
 *
 * If the record cannot be pushed, then mark the undo stack as invalid
 * such that a later rollback attempt will not try to undo earlier
 * records without having been able to undo the current record.
 */
static isl_stat push_union(struct isl_tab *tab,
  enum isl_tab_undo_type type, union isl_tab_undo_val u)
{
  struct isl_tab_undo *undo;
 
  if (!tab)
    return isl_stat_error;
  if (!tab->need_undo)
    return isl_stat_ok;
 
  undo = isl_alloc_type(tab->mat->ctx, struct isl_tab_undo);
  if (!undo)
    goto error;
  undo->type = type;
  undo->u = u;
  undo->next = tab->top;
  tab->top = undo;
 
  return isl_stat_ok;
error:
  free_undo(tab);
  tab->top = NULL;
  return isl_stat_error;
}
 
isl_stat isl_tab_push_var(struct isl_tab *tab,
  enum isl_tab_undo_type type, struct isl_tab_var *var)
{
  union isl_tab_undo_val u;
  if (var->is_row)
    u.var_index = tab->row_var[var->index];
  else
    u.var_index = tab->col_var[var->index];
  return push_union(tab, type, u);
}
 
isl_stat isl_tab_push(struct isl_tab *tab, enum isl_tab_undo_type type)
{
  union isl_tab_undo_val u = { 0 };
  return push_union(tab, type, u);
}
 
/* Push a record on the undo stack describing the current basic
 * variables, so that the this state can be restored during rollback.
 */
isl_stat isl_tab_push_basis(struct isl_tab *tab)
{
  int i;
  union isl_tab_undo_val u;
 
  u.col_var = isl_alloc_array(tab->mat->ctx, int, tab->n_col);
  if (tab->n_col && !u.col_var)
    return isl_stat_error;
  for (i = 0; i < tab->n_col; ++i)
    u.col_var[i] = tab->col_var[i];
  return push_union(tab, isl_tab_undo_saved_basis, u);
}
 
isl_stat isl_tab_push_callback(struct isl_tab *tab,
  struct isl_tab_callback *callback)
{
  union isl_tab_undo_val u;
  u.callback = callback;
  return push_union(tab, isl_tab_undo_callback, u);
}
 
/* Push a record onto the undo stack indicating that inequality "ineq"
 * has been turned into an equality constraint (in the first position).
 */
static isl_stat isl_tab_push_ineq_to_eq(struct isl_tab *tab, int ineq)
{
  union isl_tab_undo_val u = { .n = ineq };
 
  return push_union(tab, isl_tab_undo_ineq_to_eq, u);
}
 
struct isl_tab *isl_tab_init_samples(struct isl_tab *tab)
{
  if (!tab)
    return NULL;
 
  tab->n_sample = 0;
  tab->n_outside = 0;
  tab->samples = isl_mat_alloc(tab->mat->ctx, 1, 1 + tab->n_var);
  if (!tab->samples)
    goto error;
  tab->sample_index = isl_alloc_array(tab->mat->ctx, int, 1);
  if (!tab->sample_index)
    goto error;
  return tab;
error:
  isl_tab_free(tab);
  return NULL;
}
 
int isl_tab_add_sample(struct isl_tab *tab, __isl_take isl_vec *sample)
{
  if (!tab || !sample)
    goto error;
 
  if (tab->n_sample + 1 > tab->samples->n_row) {
    int *t = isl_realloc_array(tab->mat->ctx,
          tab->sample_index, int, tab->n_sample + 1);
    if (!t)
      goto error;
    tab->sample_index = t;
  }
 
  tab->samples = isl_mat_extend(tab->samples,
        tab->n_sample + 1, tab->samples->n_col);
  if (!tab->samples)
    goto error;
 
  isl_seq_cpy(tab->samples->row[tab->n_sample], sample->el, sample->size);
  isl_vec_free(sample);
  tab->sample_index[tab->n_sample] = tab->n_sample;
  tab->n_sample++;
 
  return 0;
error:
  isl_vec_free(sample);
  return -1;
}
 
struct isl_tab *isl_tab_drop_sample(struct isl_tab *tab, int s)
{
  if (s != tab->n_outside) {
    int t = tab->sample_index[tab->n_outside];
    tab->sample_index[tab->n_outside] = tab->sample_index[s];
    tab->sample_index[s] = t;
    isl_mat_swap_rows(tab->samples, tab->n_outside, s);
  }
  tab->n_outside++;
  if (isl_tab_push(tab, isl_tab_undo_drop_sample) < 0) {
    isl_tab_free(tab);
    return NULL;
  }
 
  return tab;
}
 
/* Record the current number of samples so that we can remove newer
 * samples during a rollback.
 */
isl_stat isl_tab_save_samples(struct isl_tab *tab)
{
  union isl_tab_undo_val u;
 
  if (!tab)
    return isl_stat_error;
 
  u.n = tab->n_sample;
  return push_union(tab, isl_tab_undo_saved_samples, u);
}
 
/* Mark row with index "row" as being redundant.
 * If we may need to undo the operation or if the row represents
 * a variable of the original problem, the row is kept,
 * but no longer considered when looking for a pivot row.
 * Otherwise, the row is simply removed.
 *
 * The row may be interchanged with some other row.  If it
 * is interchanged with a later row, return 1.  Otherwise return 0.
 * If the rows are checked in order in the calling function,
 * then a return value of 1 means that the row with the given
 * row number may now contain a different row that hasn't been checked yet.
 */
int isl_tab_mark_redundant(struct isl_tab *tab, int row)
{
  struct isl_tab_var *var = isl_tab_var_from_row(tab, row);
  var->is_redundant = 1;
  isl_assert(tab->mat->ctx, row >= tab->n_redundant, return -1);
  if (tab->preserve || tab->need_undo || tab->row_var[row] >= 0) {
    if (tab->row_var[row] >= 0 && !var->is_nonneg) {
      var->is_nonneg = 1;
      if (isl_tab_push_var(tab, isl_tab_undo_nonneg, var) < 0)
        return -1;
    }
    if (row != tab->n_redundant)
      swap_rows(tab, row, tab->n_redundant);
    tab->n_redundant++;
    return isl_tab_push_var(tab, isl_tab_undo_redundant, var);
  } else {
    if (row != tab->n_row - 1)
      swap_rows(tab, row, tab->n_row - 1);
    isl_tab_var_from_row(tab, tab->n_row - 1)->index = -1;
    tab->n_row--;
    return 1;
  }
}
 
/* Mark "tab" as a rational tableau.
 * If it wasn't marked as a rational tableau already and if we may
 * need to undo changes, then arrange for the marking to be undone
 * during the undo.
 */
int isl_tab_mark_rational(struct isl_tab *tab)
{
  if (!tab)
    return -1;
  if (!tab->rational && tab->need_undo)
    if (isl_tab_push(tab, isl_tab_undo_rational) < 0)
      return -1;
  tab->rational = 1;
  return 0;
}
 
isl_stat isl_tab_mark_empty(struct isl_tab *tab)
{
  if (!tab)
    return isl_stat_error;
  if (!tab->empty && tab->need_undo)
    if (isl_tab_push(tab, isl_tab_undo_empty) < 0)
      return isl_stat_error;
  tab->empty = 1;
  return isl_stat_ok;
}
 
int isl_tab_freeze_constraint(struct isl_tab *tab, int con)
{
  struct isl_tab_var *var;
 
  if (!tab)
    return -1;
 
  var = &tab->con[con];
  if (var->frozen)
    return 0;
  if (var->index < 0)
    return 0;
  var->frozen = 1;
 
  if (tab->need_undo)
    return isl_tab_push_var(tab, isl_tab_undo_freeze, var);
 
  return 0;
}
 
/* Update the rows signs after a pivot of "row" and "col", with "row_sgn"
 * the original sign of the pivot element.
 * We only keep track of row signs during PILP solving and in this case
 * we only pivot a row with negative sign (meaning the value is always
 * non-positive) using a positive pivot element.
 *
 * For each row j, the new value of the parametric constant is equal to
 *
 *  a_j0 - a_jc a_r0/a_rc
 *
 * where a_j0 is the original parametric constant, a_rc is the pivot element,
 * a_r0 is the parametric constant of the pivot row and a_jc is the
 * pivot column entry of the row j.
 * Since a_r0 is non-positive and a_rc is positive, the sign of row j
 * remains the same if a_jc has the same sign as the row j or if
 * a_jc is zero.  In all other cases, we reset the sign to "unknown".
 */
static void update_row_sign(struct isl_tab *tab, int row, int col, int row_sgn)
{
  int i;
  struct isl_mat *mat = tab->mat;
  unsigned off = 2 + tab->M;
 
  if (!tab->row_sign)
    return;
 
  if (tab->row_sign[row] == 0)
    return;
  isl_assert(mat->ctx, row_sgn > 0, return);
  isl_assert(mat->ctx, tab->row_sign[row] == isl_tab_row_neg, return);
  tab->row_sign[row] = isl_tab_row_pos;
  for (i = 0; i < tab->n_row; ++i) {
    int s;
    if (i == row)
      continue;
    s = isl_int_sgn(mat->row[i][off + col]);
    if (!s)
      continue;
    if (!tab->row_sign[i])
      continue;
    if (s < 0 && tab->row_sign[i] == isl_tab_row_neg)
      continue;
    if (s > 0 && tab->row_sign[i] == isl_tab_row_pos)
      continue;
    tab->row_sign[i] = isl_tab_row_unknown;
  }
}
 
/* Given a row number "row" and a column number "col", pivot the tableau
 * such that the associated variables are interchanged.
 * The given row in the tableau expresses
 *
 *  x_r = a_r0 + \sum_i a_ri x_i
 *
 * or
 *
 *  x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
 *
 * Substituting this equality into the other rows
 *
 *  x_j = a_j0 + \sum_i a_ji x_i
 *
 * with a_jc \ne 0, we obtain
 *
 *  x_j = a_jc/a_rc x_r + a_j0 - a_jc a_r0/a_rc + sum a_ji - a_jc a_ri/a_rc 
 *
 * The tableau
 *
 *  n_rc/d_r    n_ri/d_r
 *  n_jc/d_j    n_ji/d_j
 *
 * where i is any other column and j is any other row,
 * is therefore transformed into
 *
 * s(n_rc)d_r/|n_rc|    -s(n_rc)n_ri/|n_rc|
 * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
 *
 * The transformation is performed along the following steps
 *
 *  d_r/n_rc    n_ri/n_rc
 *  n_jc/d_j    n_ji/d_j
 *
 *  s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
 *  n_jc/d_j    n_ji/d_j
 *
 *  s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
 *  n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j)
 *
 *  s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
 *  n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j)
 *
 *  s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
 *  n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
 *
 * s(n_rc)d_r/|n_rc|    -s(n_rc)n_ri/|n_rc|
 * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
 *
 */
int isl_tab_pivot(struct isl_tab *tab, int row, int col)
{
  int i, j;
  int sgn;
  int t;
  isl_ctx *ctx;
  struct isl_mat *mat = tab->mat;
  struct isl_tab_var *var;
  unsigned off = 2 + tab->M;
 
  ctx = isl_tab_get_ctx(tab);
  if (isl_ctx_next_operation(ctx) < 0)
    return -1;
 
  isl_int_swap(mat->row[row][0], mat->row[row][off + col]);
  sgn = isl_int_sgn(mat->row[row][0]);
  if (sgn < 0) {
    isl_int_neg(mat->row[row][0], mat->row[row][0]);
    isl_int_neg(mat->row[row][off + col], mat->row[row][off + col]);
  } else
    for (j = 0; j < off - 1 + tab->n_col; ++j) {
      if (j == off - 1 + col)
        continue;
      isl_int_neg(mat->row[row][1 + j], mat->row[row][1 + j]);
    }
  if (!isl_int_is_one(mat->row[row][0]))
    isl_seq_normalize(mat->ctx, mat->row[row], off + tab->n_col);
  for (i = 0; i < tab->n_row; ++i) {
    if (i == row)
      continue;
    if (isl_int_is_zero(mat->row[i][off + col]))
      continue;
    isl_int_mul(mat->row[i][0], mat->row[i][0], mat->row[row][0]);
    for (j = 0; j < off - 1 + tab->n_col; ++j) {
      if (j == off - 1 + col)
        continue;
      isl_int_mul(mat->row[i][1 + j],
            mat->row[i][1 + j], mat->row[row][0]);
      isl_int_addmul(mat->row[i][1 + j],
            mat->row[i][off + col], mat->row[row][1 + j]);
    }
    isl_int_mul(mat->row[i][off + col],
          mat->row[i][off + col], mat->row[row][off + col]);
    if (!isl_int_is_one(mat->row[i][0]))
      isl_seq_normalize(mat->ctx, mat->row[i], off + tab->n_col);
  }
  t = tab->row_var[row];
  tab->row_var[row] = tab->col_var[col];
  tab->col_var[col] = t;
  var = isl_tab_var_from_row(tab, row);
  var->is_row = 1;
  var->index = row;
  var = var_from_col(tab, col);
  var->is_row = 0;
  var->index = col;
  update_row_sign(tab, row, col, sgn);
  if (tab->in_undo)
    return 0;
  for (i = tab->n_redundant; i < tab->n_row; ++i) {
    if (isl_int_is_zero(mat->row[i][off + col]))
      continue;
    if (!isl_tab_var_from_row(tab, i)->frozen &&
        isl_tab_row_is_redundant(tab, i)) {
      int redo = isl_tab_mark_redundant(tab, i);
      if (redo < 0)
        return -1;
      if (redo)
        --i;
    }
  }
  return 0;
}
 
/* If "var" represents a column variable, then pivot is up (sgn > 0)
 * or down (sgn < 0) to a row.  The variable is assumed not to be
 * unbounded in the specified direction.
 * If sgn = 0, then the variable is unbounded in both directions,
 * and we pivot with any row we can find.
 */
static int to_row(struct isl_tab *tab, struct isl_tab_var *var, int sign) WARN_UNUSED;
static int to_row(struct isl_tab *tab, struct isl_tab_var *var, int sign)
{
  int r;
  unsigned off = 2 + tab->M;
 
  if (var->is_row)
    return 0;
 
  if (sign == 0) {
    for (r = tab->n_redundant; r < tab->n_row; ++r)
      if (!isl_int_is_zero(tab->mat->row[r][off+var->index]))
        break;
    isl_assert(tab->mat->ctx, r < tab->n_row, return -1);
  } else {
    r = pivot_row(tab, NULL, sign, var->index);
    isl_assert(tab->mat->ctx, r >= 0, return -1);
  }
 
  return isl_tab_pivot(tab, r, var->index);
}
 
/* Check whether all variables that are marked as non-negative
 * also have a non-negative sample value.  This function is not
 * called from the current code but is useful during debugging.
 */
static void check_table(struct isl_tab *tab) __attribute__ ((unused));
static void check_table(struct isl_tab *tab)
{
  int i;
 
  if (tab->empty)
    return;
  for (i = tab->n_redundant; i < tab->n_row; ++i) {
    struct isl_tab_var *var;
    var = isl_tab_var_from_row(tab, i);
    if (!var->is_nonneg)
      continue;
    if (tab->M) {
      isl_assert(tab->mat->ctx,
        !isl_int_is_neg(tab->mat->row[i][2]), abort());
      if (isl_int_is_pos(tab->mat->row[i][2]))
        continue;
    }
    isl_assert(tab->mat->ctx, !isl_int_is_neg(tab->mat->row[i][1]),
        abort());
  }
}
 
/* Return the sign of the maximal value of "var".
 * If the sign is not negative, then on return from this function,
 * the sample value will also be non-negative.
 *
 * If "var" is manifestly unbounded wrt positive values, we are done.
 * Otherwise, we pivot the variable up to a row if needed.
 * Then we continue pivoting up until either
 *  - no more up pivots can be performed
 *  - the sample value is positive
 *  - the variable is pivoted into a manifestly unbounded column
 */
static int sign_of_max(struct isl_tab *tab, struct isl_tab_var *var)
{
  int row, col;
 
  if (max_is_manifestly_unbounded(tab, var))
    return 1;
  if (to_row(tab, var, 1) < 0)
    return -2;
  while (!isl_int_is_pos(tab->mat->row[var->index][1])) {
    find_pivot(tab, var, var, 1, &row, &col);
    if (row == -1)
      return isl_int_sgn(tab->mat->row[var->index][1]);
    if (isl_tab_pivot(tab, row, col) < 0)
      return -2;
    if (!var->is_row) /* manifestly unbounded */
      return 1;
  }
  return 1;
}
 
int isl_tab_sign_of_max(struct isl_tab *tab, int con)
{
  struct isl_tab_var *var;
 
  if (!tab)
    return -2;
 
  var = &tab->con[con];
  isl_assert(tab->mat->ctx, !var->is_redundant, return -2);
  isl_assert(tab->mat->ctx, !var->is_zero, return -2);
 
  return sign_of_max(tab, var);
}
 
static int row_is_neg(struct isl_tab *tab, int row)
{
  if (!tab->M)
    return isl_int_is_neg(tab->mat->row[row][1]);
  if (isl_int_is_pos(tab->mat->row[row][2]))
    return 0;
  if (isl_int_is_neg(tab->mat->row[row][2]))
    return 1;
  return isl_int_is_neg(tab->mat->row[row][1]);
}
 
static int row_sgn(struct isl_tab *tab, int row)
{
  if (!tab->M)
    return isl_int_sgn(tab->mat->row[row][1]);
  if (!isl_int_is_zero(tab->mat->row[row][2]))
    return isl_int_sgn(tab->mat->row[row][2]);
  else
    return isl_int_sgn(tab->mat->row[row][1]);
}
 
/* Perform pivots until the row variable "var" has a non-negative
 * sample value or until no more upward pivots can be performed.
 * Return the sign of the sample value after the pivots have been
 * performed.
 */
static int restore_row(struct isl_tab *tab, struct isl_tab_var *var)
{
  int row, col;
 
  while (row_is_neg(tab, var->index)) {
    find_pivot(tab, var, var, 1, &row, &col);
    if (row == -1)
      break;
    if (isl_tab_pivot(tab, row, col) < 0)
      return -2;
    if (!var->is_row) /* manifestly unbounded */
      return 1;
  }
  return row_sgn(tab, var->index);
}
 
/* Perform pivots until we are sure that the row variable "var"
 * can attain non-negative values.  After return from this
 * function, "var" is still a row variable, but its sample
 * value may not be non-negative, even if the function returns 1.
 */
static int at_least_zero(struct isl_tab *tab, struct isl_tab_var *var)
{
  int row, col;
 
  while (isl_int_is_neg(tab->mat->row[var->index][1])) {
    find_pivot(tab, var, var, 1, &row, &col);
    if (row == -1)
      break;
    if (row == var->index) /* manifestly unbounded */
      return 1;
    if (isl_tab_pivot(tab, row, col) < 0)
      return -1;
  }
  return !isl_int_is_neg(tab->mat->row[var->index][1]);
}
 
/* Return a negative value if "var" can attain negative values.
 * Return a non-negative value otherwise.
 *
 * If "var" is manifestly unbounded wrt negative values, we are done.
 * Otherwise, if var is in a column, we can pivot it down to a row.
 * Then we continue pivoting down until either
 *  - the pivot would result in a manifestly unbounded column
 *    => we don't perform the pivot, but simply return -1
 *  - no more down pivots can be performed
 *  - the sample value is negative
 * If the sample value becomes negative and the variable is supposed
 * to be nonnegative, then we undo the last pivot.
 * However, if the last pivot has made the pivoting variable
 * obviously redundant, then it may have moved to another row.
 * In that case we look for upward pivots until we reach a non-negative
 * value again.
 */
static int sign_of_min(struct isl_tab *tab, struct isl_tab_var *var)
{
  int row, col;
  struct isl_tab_var *pivot_var = NULL;
 
  if (min_is_manifestly_unbounded(tab, var))
    return -1;
  if (!var->is_row) {
    col = var->index;
    row = pivot_row(tab, NULL, -1, col);
    pivot_var = var_from_col(tab, col);
    if (isl_tab_pivot(tab, row, col) < 0)
      return -2;
    if (var->is_redundant)
      return 0;
    if (isl_int_is_neg(tab->mat->row[var->index][1])) {
      if (var->is_nonneg) {
        if (!pivot_var->is_redundant &&
            pivot_var->index == row) {
          if (isl_tab_pivot(tab, row, col) < 0)
            return -2;
        } else
          if (restore_row(tab, var) < -1)
            return -2;
      }
      return -1;
    }
  }
  if (var->is_redundant)
    return 0;
  while (!isl_int_is_neg(tab->mat->row[var->index][1])) {
    find_pivot(tab, var, var, -1, &row, &col);
    if (row == var->index)
      return -1;
    if (row == -1)
      return isl_int_sgn(tab->mat->row[var->index][1]);
    pivot_var = var_from_col(tab, col);
    if (isl_tab_pivot(tab, row, col) < 0)
      return -2;
    if (var->is_redundant)
      return 0;
  }
  if (pivot_var && var->is_nonneg) {
    /* pivot back to non-negative value */
    if (!pivot_var->is_redundant && pivot_var->index == row) {
      if (isl_tab_pivot(tab, row, col) < 0)
        return -2;
    } else
      if (restore_row(tab, var) < -1)
        return -2;
  }
  return -1;
}
 
static int row_at_most_neg_one(struct isl_tab *tab, int row)
{
  if (tab->M) {
    if (isl_int_is_pos(tab->mat->row[row][2]))
      return 0;
    if (isl_int_is_neg(tab->mat->row[row][2]))
      return 1;
  }
  return isl_int_is_neg(tab->mat->row[row][1]) &&
         isl_int_abs_ge(tab->mat->row[row][1],
            tab->mat->row[row][0]);
}
 
/* Return 1 if "var" can attain values <= -1.
 * Return 0 otherwise.
 *
 * If the variable "var" is supposed to be non-negative (is_nonneg is set),
 * then the sample value of "var" is assumed to be non-negative when the
 * the function is called.  If 1 is returned then the constraint
 * is not redundant and the sample value is made non-negative again before
 * the function returns.
 */
int isl_tab_min_at_most_neg_one(struct isl_tab *tab, struct isl_tab_var *var)
{
  int row, col;
  struct isl_tab_var *pivot_var;
 
  if (min_is_manifestly_unbounded(tab, var))
    return 1;
  if (!var->is_row) {
    col = var->index;
    row = pivot_row(tab, NULL, -1, col);
    pivot_var = var_from_col(tab, col);
    if (isl_tab_pivot(tab, row, col) < 0)
      return -1;
    if (var->is_redundant)
      return 0;
    if (row_at_most_neg_one(tab, var->index)) {
      if (var->is_nonneg) {
        if (!pivot_var->is_redundant &&
            pivot_var->index == row) {
          if (isl_tab_pivot(tab, row, col) < 0)
            return -1;
        } else
          if (restore_row(tab, var) < -1)
            return -1;
      }
      return 1;
    }
  }
  if (var->is_redundant)
    return 0;
  do {
    find_pivot(tab, var, var, -1, &row, &col);
    if (row == var->index) {
      if (var->is_nonneg && restore_row(tab, var) < -1)
        return -1;
      return 1;
    }
    if (row == -1)
      return 0;
    pivot_var = var_from_col(tab, col);
    if (isl_tab_pivot(tab, row, col) < 0)
      return -1;
    if (var->is_redundant)
      return 0;
  } while (!row_at_most_neg_one(tab, var->index));
  if (var->is_nonneg) {
    /* pivot back to non-negative value */
    if (!pivot_var->is_redundant && pivot_var->index == row)
      if (isl_tab_pivot(tab, row, col) < 0)
        return -1;
    if (restore_row(tab, var) < -1)
      return -1;
  }
  return 1;
}
 
/* Return 1 if "var" can attain values >= 1.
 * Return 0 otherwise.
 */
static int at_least_one(struct isl_tab *tab, struct isl_tab_var *var)
{
  int row, col;
  isl_int *r;
 
  if (max_is_manifestly_unbounded(tab, var))
    return 1;
  if (to_row(tab, var, 1) < 0)
    return -1;
  r = tab->mat->row[var->index];
  while (isl_int_lt(r[1], r[0])) {
    find_pivot(tab, var, var, 1, &row, &col);
    if (row == -1)
      return isl_int_ge(r[1], r[0]);
    if (row == var->index) /* manifestly unbounded */
      return 1;
    if (isl_tab_pivot(tab, row, col) < 0)
      return -1;
  }
  return 1;
}
 
static void swap_cols(struct isl_tab *tab, int col1, int col2)
{
  int t;
  unsigned off = 2 + tab->M;
  t = tab->col_var[col1];
  tab->col_var[col1] = tab->col_var[col2];
  tab->col_var[col2] = t;
  var_from_col(tab, col1)->index = col1;
  var_from_col(tab, col2)->index = col2;
  tab->mat = isl_mat_swap_cols(tab->mat, off + col1, off + col2);
}
 
/* Mark column with index "col" as representing a zero variable.
 * If we may need to undo the operation the column is kept,
 * but no longer considered.
 * Otherwise, the column is simply removed.
 *
 * The column may be interchanged with some other column.  If it
 * is interchanged with a later column, return 1.  Otherwise return 0.
 * If the columns are checked in order in the calling function,
 * then a return value of 1 means that the column with the given
 * column number may now contain a different column that
 * hasn't been checked yet.
 */
int isl_tab_kill_col(struct isl_tab *tab, int col)
{
  var_from_col(tab, col)->is_zero = 1;
  if (tab->need_undo) {
    if (isl_tab_push_var(tab, isl_tab_undo_zero,
              var_from_col(tab, col)) < 0)
      return -1;
    if (col != tab->n_dead)
      swap_cols(tab, col, tab->n_dead);
    tab->n_dead++;
    return 0;
  } else {
    if (col != tab->n_col - 1)
      swap_cols(tab, col, tab->n_col - 1);
    var_from_col(tab, tab->n_col - 1)->index = -1;
    tab->n_col--;
    return 1;
  }
}
 
static int row_is_manifestly_non_integral(struct isl_tab *tab, int row)
{
  unsigned off = 2 + tab->M;
 
  if (tab->M && !isl_int_eq(tab->mat->row[row][2],
          tab->mat->row[row][0]))
    return 0;
  if (isl_seq_any_non_zero(tab->mat->row[row] + off + tab->n_dead,
            tab->n_col - tab->n_dead))
    return 0;
 
  return !isl_int_is_divisible_by(tab->mat->row[row][1],
          tab->mat->row[row][0]);
}
 
/* For integer tableaus, check if any of the coordinates are stuck
 * at a non-integral value.
 */
static int tab_is_manifestly_empty(struct isl_tab *tab)
{
  int i;
 
  if (tab->empty)
    return 1;
  if (tab->rational)
    return 0;
 
  for (i = 0; i < tab->n_var; ++i) {
    if (!tab->var[i].is_row)
      continue;
    if (row_is_manifestly_non_integral(tab, tab->var[i].index))
      return 1;
  }
 
  return 0;
}
 
/* Row variable "var" is non-negative and cannot attain any values
 * larger than zero.  This means that the coefficients of the unrestricted
 * column variables are zero and that the coefficients of the non-negative
 * column variables are zero or negative.
 * Each of the non-negative variables with a negative coefficient can
 * then also be written as the negative sum of non-negative variables
 * and must therefore also be zero.
 *
 * If "temp_var" is set, then "var" is a temporary variable that
 * will be removed after this function returns and for which
 * no information is recorded on the undo stack.
 * Do not add any undo records involving this variable in this case
 * since the variable will have been removed before any future undo
 * operations.  Also avoid marking the variable as redundant,
 * since that either adds an undo record or needlessly removes the row
 * (the caller will take care of removing the row).
 */
static isl_stat close_row(struct isl_tab *tab, struct isl_tab_var *var,
  int temp_var) WARN_UNUSED;
static isl_stat close_row(struct isl_tab *tab, struct isl_tab_var *var,
  int temp_var)
{
  int j;
  struct isl_mat *mat = tab->mat;
  unsigned off = 2 + tab->M;
 
  if (!var->is_nonneg)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "expecting non-negative variable",
      return isl_stat_error);
  var->is_zero = 1;
  if (!temp_var && tab->need_undo)
    if (isl_tab_push_var(tab, isl_tab_undo_zero, var) < 0)
      return isl_stat_error;
  for (j = tab->n_dead; j < tab->n_col; ++j) {
    int recheck;
    if (isl_int_is_zero(mat->row[var->index][off + j]))
      continue;
    if (isl_int_is_pos(mat->row[var->index][off + j]))
      isl_die(isl_tab_get_ctx(tab), isl_error_internal,
        "row cannot have positive coefficients",
        return isl_stat_error);
    recheck = isl_tab_kill_col(tab, j);
    if (recheck < 0)
      return isl_stat_error;
    if (recheck)
      --j;
  }
  if (!temp_var && isl_tab_mark_redundant(tab, var->index) < 0)
    return isl_stat_error;
  if (tab_is_manifestly_empty(tab) && isl_tab_mark_empty(tab) < 0)
    return isl_stat_error;
  return isl_stat_ok;
}
 
/* Add a constraint to the tableau and allocate a row for it.
 * Return the index into the constraint array "con".
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
int isl_tab_allocate_con(struct isl_tab *tab)
{
  int r;
 
  isl_assert(tab->mat->ctx, tab->n_row < tab->mat->n_row, return -1);
  isl_assert(tab->mat->ctx, tab->n_con < tab->max_con, return -1);
 
  r = tab->n_con;
  tab->con[r].index = tab->n_row;
  tab->con[r].is_row = 1;
  tab->con[r].is_nonneg = 0;
  tab->con[r].is_zero = 0;
  tab->con[r].is_redundant = 0;
  tab->con[r].frozen = 0;
  tab->con[r].negated = 0;
  tab->row_var[tab->n_row] = ~r;
 
  tab->n_row++;
  tab->n_con++;
  if (isl_tab_push_var(tab, isl_tab_undo_allocate, &tab->con[r]) < 0)
    return -1;
 
  return r;
}
 
/* Move the entries in tab->var up one position, starting at "first",
 * creating room for an extra entry at position "first".
 * Since some of the entries of tab->row_var and tab->col_var contain
 * indices into this array, they have to be updated accordingly.
 */
static int var_insert_entry(struct isl_tab *tab, int first)
{
  int i;
 
  if (tab->n_var >= tab->max_var)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "not enough room for new variable", return -1);
  if (first > tab->n_var)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "invalid initial position", return -1);
 
  for (i = tab->n_var - 1; i >= first; --i) {
    tab->var[i + 1] = tab->var[i];
    if (tab->var[i + 1].is_row)
      tab->row_var[tab->var[i + 1].index]++;
    else
      tab->col_var[tab->var[i + 1].index]++;
  }
 
  tab->n_var++;
 
  return 0;
}
 
/* Drop the entry at position "first" in tab->var, moving all
 * subsequent entries down.
 * Since some of the entries of tab->row_var and tab->col_var contain
 * indices into this array, they have to be updated accordingly.
 */
static int var_drop_entry(struct isl_tab *tab, int first)
{
  int i;
 
  if (first >= tab->n_var)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "invalid initial position", return -1);
 
  tab->n_var--;
 
  for (i = first; i < tab->n_var; ++i) {
    tab->var[i] = tab->var[i + 1];
    if (tab->var[i + 1].is_row)
      tab->row_var[tab->var[i].index]--;
    else
      tab->col_var[tab->var[i].index]--;
  }
 
  return 0;
}
 
/* Add a variable to the tableau at position "r" and allocate a column for it.
 * Return the index into the variable array "var", i.e., "r",
 * or -1 on error.
 */
int isl_tab_insert_var(struct isl_tab *tab, int r)
{
  int i;
  unsigned off = 2 + tab->M;
 
  isl_assert(tab->mat->ctx, tab->n_col < tab->mat->n_col, return -1);
 
  if (var_insert_entry(tab, r) < 0)
    return -1;
 
  tab->var[r].index = tab->n_col;
  tab->var[r].is_row = 0;
  tab->var[r].is_nonneg = 0;
  tab->var[r].is_zero = 0;
  tab->var[r].is_redundant = 0;
  tab->var[r].frozen = 0;
  tab->var[r].negated = 0;
  tab->col_var[tab->n_col] = r;
 
  for (i = 0; i < tab->n_row; ++i)
    isl_int_set_si(tab->mat->row[i][off + tab->n_col], 0);
 
  tab->n_col++;
  if (isl_tab_push_var(tab, isl_tab_undo_allocate, &tab->var[r]) < 0)
    return -1;
 
  return r;
}
 
/* Add a row to the tableau.  The row is given as an affine combination
 * of the original variables and needs to be expressed in terms of the
 * column variables.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 *
 * We add each term in turn.
 * If r = n/d_r is the current sum and we need to add k x, then
 *  if x is a column variable, we increase the numerator of
 *    this column by k d_r
 *  if x = f/d_x is a row variable, then the new representation of r is
 *
 *     n    k f   d_x/g n + d_r/g k f   m/d_r n + m/d_g k f
 *    --- + --- = ------------------- = -------------------
 *    d_r   d_r        d_r d_x/g                m
 *
 *  with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
 *
 * If tab->M is set, then, internally, each variable x is represented
 * as x' - M.  We then also need no subtract k d_r from the coefficient of M.
 */
int isl_tab_add_row(struct isl_tab *tab, isl_int *line)
{
  int i;
  int r;
  isl_int *row;
  isl_int a, b;
  unsigned off = 2 + tab->M;
 
  r = isl_tab_allocate_con(tab);
  if (r < 0)
    return -1;
 
  isl_int_init(a);
  isl_int_init(b);
  row = tab->mat->row[tab->con[r].index];
  isl_int_set_si(row[0], 1);
  isl_int_set(row[1], line[0]);
  isl_seq_clr(row + 2, tab->M + tab->n_col);
  for (i = 0; i < tab->n_var; ++i) {
    if (tab->var[i].is_zero)
      continue;
    if (tab->var[i].is_row) {
      isl_int_lcm(a,
        row[0], tab->mat->row[tab->var[i].index][0]);
      isl_int_swap(a, row[0]);
      isl_int_divexact(a, row[0], a);
      isl_int_divexact(b,
        row[0], tab->mat->row[tab->var[i].index][0]);
      isl_int_mul(b, b, line[1 + i]);
      isl_seq_combine(row + 1, a, row + 1,
          b, tab->mat->row[tab->var[i].index] + 1,
          1 + tab->M + tab->n_col);
    } else
      isl_int_addmul(row[off + tab->var[i].index],
              line[1 + i], row[0]);
    if (tab->M && i >= tab->n_param && i < tab->n_var - tab->n_div)
      isl_int_submul(row[2], line[1 + i], row[0]);
  }
  isl_seq_normalize(tab->mat->ctx, row, off + tab->n_col);
  isl_int_clear(a);
  isl_int_clear(b);
 
  if (tab->row_sign)
    tab->row_sign[tab->con[r].index] = isl_tab_row_unknown;
 
  return r;
}
 
static isl_stat drop_row(struct isl_tab *tab, int row)
{
  isl_assert(tab->mat->ctx, ~tab->row_var[row] == tab->n_con - 1,
    return isl_stat_error);
  if (row != tab->n_row - 1)
    swap_rows(tab, row, tab->n_row - 1);
  tab->n_row--;
  tab->n_con--;
  return isl_stat_ok;
}
 
/* Drop the variable in column "col" along with the column.
 * The column is removed first because it may need to be moved
 * into the last position and this process requires
 * the contents of the col_var array in a state
 * before the removal of the variable.
 */
static isl_stat drop_col(struct isl_tab *tab, int col)
{
  int var;
 
  var = tab->col_var[col];
  if (col != tab->n_col - 1)
    swap_cols(tab, col, tab->n_col - 1);
  tab->n_col--;
  if (var_drop_entry(tab, var) < 0)
    return isl_stat_error;
  return isl_stat_ok;
}
 
/* Add inequality "ineq" and check if it conflicts with the
 * previously added constraints or if it is obviously redundant.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
isl_stat isl_tab_add_ineq(struct isl_tab *tab, isl_int *ineq)
{
  int r;
  int sgn;
  isl_int cst;
 
  if (!tab)
    return isl_stat_error;
  if (tab->bmap) {
    struct isl_basic_map *bmap = tab->bmap;
 
    isl_assert(tab->mat->ctx, tab->n_eq == bmap->n_eq,
      return isl_stat_error);
    isl_assert(tab->mat->ctx,
          tab->n_con == bmap->n_eq + bmap->n_ineq,
          return isl_stat_error);
    tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq);
    if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
      return isl_stat_error;
    if (!tab->bmap)
      return isl_stat_error;
  }
  if (tab->cone) {
    isl_int_init(cst);
    isl_int_set_si(cst, 0);
    isl_int_swap(ineq[0], cst);
  }
  r = isl_tab_add_row(tab, ineq);
  if (tab->cone) {
    isl_int_swap(ineq[0], cst);
    isl_int_clear(cst);
  }
  if (r < 0)
    return isl_stat_error;
  tab->con[r].is_nonneg = 1;
  if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
    return isl_stat_error;
  if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
    if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
      return isl_stat_error;
    return isl_stat_ok;
  }
 
  sgn = restore_row(tab, &tab->con[r]);
  if (sgn < -1)
    return isl_stat_error;
  if (sgn < 0)
    return isl_tab_mark_empty(tab);
  if (tab->con[r].is_row && isl_tab_row_is_redundant(tab, tab->con[r].index))
    if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
      return isl_stat_error;
  return isl_stat_ok;
}
 
/* Pivot a non-negative variable down until it reaches the value zero
 * and then pivot the variable into a column position.
 */
static int to_col(struct isl_tab *tab, struct isl_tab_var *var) WARN_UNUSED;
static int to_col(struct isl_tab *tab, struct isl_tab_var *var)
{
  int i;
  int row, col;
  unsigned off = 2 + tab->M;
 
  if (!var->is_row)
    return 0;
 
  while (isl_int_is_pos(tab->mat->row[var->index][1])) {
    find_pivot(tab, var, NULL, -1, &row, &col);
    isl_assert(tab->mat->ctx, row != -1, return -1);
    if (isl_tab_pivot(tab, row, col) < 0)
      return -1;
    if (!var->is_row)
      return 0;
  }
 
  for (i = tab->n_dead; i < tab->n_col; ++i)
    if (!isl_int_is_zero(tab->mat->row[var->index][off + i]))
      break;
 
  isl_assert(tab->mat->ctx, i < tab->n_col, return -1);
  if (isl_tab_pivot(tab, var->index, i) < 0)
    return -1;
 
  return 0;
}
 
/* We assume Gaussian elimination has been performed on the equalities.
 * The equalities can therefore never conflict.
 * Adding the equalities is currently only really useful for a later call
 * to isl_tab_ineq_type.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
static struct isl_tab *add_eq(struct isl_tab *tab, isl_int *eq)
{
  int i;
  int r;
 
  if (!tab)
    return NULL;
  r = isl_tab_add_row(tab, eq);
  if (r < 0)
    goto error;
 
  r = tab->con[r].index;
  i = isl_seq_first_non_zero(tab->mat->row[r] + 2 + tab->M + tab->n_dead,
          tab->n_col - tab->n_dead);
  isl_assert(tab->mat->ctx, i >= 0, goto error);
  i += tab->n_dead;
  if (isl_tab_pivot(tab, r, i) < 0)
    goto error;
  if (isl_tab_kill_col(tab, i) < 0)
    goto error;
  tab->n_eq++;
 
  return tab;
error:
  isl_tab_free(tab);
  return NULL;
}
 
/* Does the sample value of row "row" of "tab" involve the big parameter,
 * if any?
 */
static int row_is_big(struct isl_tab *tab, int row)
{
  return tab->M && !isl_int_is_zero(tab->mat->row[row][2]);
}
 
static int row_is_manifestly_zero(struct isl_tab *tab, int row)
{
  unsigned off = 2 + tab->M;
 
  if (!isl_int_is_zero(tab->mat->row[row][1]))
    return 0;
  if (row_is_big(tab, row))
    return 0;
  return !isl_seq_any_non_zero(tab->mat->row[row] + off + tab->n_dead,
          tab->n_col - tab->n_dead);
}
 
/* Add an equality that is known to be valid for the given tableau.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
int isl_tab_add_valid_eq(struct isl_tab *tab, isl_int *eq)
{
  struct isl_tab_var *var;
  int r;
 
  if (!tab)
    return -1;
  r = isl_tab_add_row(tab, eq);
  if (r < 0)
    return -1;
 
  var = &tab->con[r];
  r = var->index;
  if (row_is_manifestly_zero(tab, r)) {
    var->is_zero = 1;
    if (isl_tab_mark_redundant(tab, r) < 0)
      return -1;
    return 0;
  }
 
  if (isl_int_is_neg(tab->mat->row[r][1])) {
    isl_seq_neg(tab->mat->row[r] + 1, tab->mat->row[r] + 1,
          1 + tab->n_col);
    var->negated = 1;
  }
  var->is_nonneg = 1;
  if (to_col(tab, var) < 0)
    return -1;
  var->is_nonneg = 0;
  if (isl_tab_kill_col(tab, var->index) < 0)
    return -1;
 
  return 0;
}
 
/* Add a zero row to "tab" and return the corresponding index
 * in the constraint array.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
static int add_zero_row(struct isl_tab *tab)
{
  int r;
  isl_int *row;
 
  r = isl_tab_allocate_con(tab);
  if (r < 0)
    return -1;
 
  row = tab->mat->row[tab->con[r].index];
  isl_seq_clr(row + 1, 1 + tab->M + tab->n_col);
  isl_int_set_si(row[0], 1);
 
  return r;
}
 
/* Add equality "eq" and check if it conflicts with the
 * previously added constraints or if it is obviously redundant.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 * If tab->bmap is set, then two rows are needed instead of one.
 */
isl_stat isl_tab_add_eq(struct isl_tab *tab, isl_int *eq)
{
  struct isl_tab_undo *snap = NULL;
  struct isl_tab_var *var;
  int r;
  int row;
  int sgn;
  isl_int cst;
 
  if (!tab)
    return isl_stat_error;
  isl_assert(tab->mat->ctx, !tab->M, return isl_stat_error);
 
  if (tab->need_undo)
    snap = isl_tab_snap(tab);
 
  if (tab->cone) {
    isl_int_init(cst);
    isl_int_set_si(cst, 0);
    isl_int_swap(eq[0], cst);
  }
  r = isl_tab_add_row(tab, eq);
  if (tab->cone) {
    isl_int_swap(eq[0], cst);
    isl_int_clear(cst);
  }
  if (r < 0)
    return isl_stat_error;
 
  var = &tab->con[r];
  row = var->index;
  if (row_is_manifestly_zero(tab, row)) {
    if (snap)
      return isl_tab_rollback(tab, snap);
    return drop_row(tab, row);
  }
 
  if (tab->bmap) {
    tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
    if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
      return isl_stat_error;
    isl_seq_neg(eq, eq, 1 + tab->n_var);
    tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
    isl_seq_neg(eq, eq, 1 + tab->n_var);
    if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
      return isl_stat_error;
    if (!tab->bmap)
      return isl_stat_error;
    if (add_zero_row(tab) < 0)
      return isl_stat_error;
  }
 
  sgn = isl_int_sgn(tab->mat->row[row][1]);
 
  if (sgn > 0) {
    isl_seq_neg(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
          1 + tab->n_col);
    var->negated = 1;
    sgn = -1;
  }
 
  if (sgn < 0) {
    sgn = sign_of_max(tab, var);
    if (sgn < -1)
      return isl_stat_error;
    if (sgn < 0) {
      if (isl_tab_mark_empty(tab) < 0)
        return isl_stat_error;
      return isl_stat_ok;
    }
  }
 
  var->is_nonneg = 1;
  if (to_col(tab, var) < 0)
    return isl_stat_error;
  var->is_nonneg = 0;
  if (isl_tab_kill_col(tab, var->index) < 0)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Construct and return an inequality that expresses an upper bound
 * on the given div.
 * In particular, if the div is given by
 *
 *  d = floor(e/m)
 *
 * then the inequality expresses
 *
 *  m d <= e
 */
static __isl_give isl_vec *ineq_for_div(__isl_keep isl_basic_map *bmap,
  unsigned div)
{
  isl_size total;
  unsigned div_pos;
  struct isl_vec *ineq;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return NULL;
 
  div_pos = 1 + total - bmap->n_div + div;
 
  ineq = isl_vec_alloc(bmap->ctx, 1 + total);
  if (!ineq)
    return NULL;
 
  isl_seq_cpy(ineq->el, bmap->div[div] + 1, 1 + total);
  isl_int_neg(ineq->el[div_pos], bmap->div[div][0]);
  return ineq;
}
 
/* For a div d = floor(f/m), add the constraints
 *
 *    f - m d >= 0
 *    -(f-(m-1)) + m d >= 0
 *
 * Note that the second constraint is the negation of
 *
 *    f - m d >= m
 *
 * If add_ineq is not NULL, then this function is used
 * instead of isl_tab_add_ineq to effectively add the inequalities.
 *
 * This function assumes that at least two more rows and at least
 * two more elements in the constraint array are available in the tableau.
 */
static isl_stat add_div_constraints(struct isl_tab *tab, unsigned div,
  isl_stat (*add_ineq)(void *user, isl_int *), void *user)
{
  isl_size total;
  unsigned div_pos;
  struct isl_vec *ineq;
 
  total = isl_basic_map_dim(tab->bmap, isl_dim_all);
  if (total < 0)
    return isl_stat_error;
  div_pos = 1 + total - tab->bmap->n_div + div;
 
  ineq = ineq_for_div(tab->bmap, div);
  if (!ineq)
    goto error;
 
  if (add_ineq) {
    if (add_ineq(user, ineq->el) < 0)
      goto error;
  } else {
    if (isl_tab_add_ineq(tab, ineq->el) < 0)
      goto error;
  }
 
  isl_seq_neg(ineq->el, tab->bmap->div[div] + 1, 1 + total);
  isl_int_set(ineq->el[div_pos], tab->bmap->div[div][0]);
  isl_int_add(ineq->el[0], ineq->el[0], ineq->el[div_pos]);
  isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
 
  if (add_ineq) {
    if (add_ineq(user, ineq->el) < 0)
      goto error;
  } else {
    if (isl_tab_add_ineq(tab, ineq->el) < 0)
      goto error;
  }
 
  isl_vec_free(ineq);
 
  return isl_stat_ok;
error:
  isl_vec_free(ineq);
  return isl_stat_error;
}
 
/* Check whether the div described by "div" is obviously non-negative.
 * If we are using a big parameter, then we will encode the div
 * as div' = M + div, which is always non-negative.
 * Otherwise, we check whether div is a non-negative affine combination
 * of non-negative variables.
 */
static int div_is_nonneg(struct isl_tab *tab, __isl_keep isl_vec *div)
{
  int i;
 
  if (tab->M)
    return 1;
 
  if (isl_int_is_neg(div->el[1]))
    return 0;
 
  for (i = 0; i < tab->n_var; ++i) {
    if (isl_int_is_neg(div->el[2 + i]))
      return 0;
    if (isl_int_is_zero(div->el[2 + i]))
      continue;
    if (!tab->var[i].is_nonneg)
      return 0;
  }
 
  return 1;
}
 
/* Insert an extra div, prescribed by "div", to the tableau and
 * the associated bmap (which is assumed to be non-NULL).
 * The extra integer division is inserted at (tableau) position "pos".
 * Return "pos" or -1 if an error occurred.
 *
 * If add_ineq is not NULL, then this function is used instead
 * of isl_tab_add_ineq to add the div constraints.
 * This complication is needed because the code in isl_tab_pip
 * wants to perform some extra processing when an inequality
 * is added to the tableau.
 */
int isl_tab_insert_div(struct isl_tab *tab, int pos, __isl_keep isl_vec *div,
  isl_stat (*add_ineq)(void *user, isl_int *), void *user)
{
  int r;
  int nonneg;
  isl_size n_div;
  int o_div;
 
  if (!tab || !div)
    return -1;
 
  if (div->size != 1 + 1 + tab->n_var)
    isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
      "unexpected size", return -1);
 
  n_div = isl_basic_map_dim(tab->bmap, isl_dim_div);
  if (n_div < 0)
    return -1;
  o_div = tab->n_var - n_div;
  if (pos < o_div || pos > tab->n_var)
    isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
      "invalid position", return -1);
 
  nonneg = div_is_nonneg(tab, div);
 
  if (isl_tab_extend_cons(tab, 3) < 0)
    return -1;
  if (isl_tab_extend_vars(tab, 1) < 0)
    return -1;
  r = isl_tab_insert_var(tab, pos);
  if (r < 0)
    return -1;
 
  if (nonneg)
    tab->var[r].is_nonneg = 1;
 
  tab->bmap = isl_basic_map_insert_div(tab->bmap, pos - o_div, div);
  if (!tab->bmap)
    return -1;
  if (isl_tab_push_var(tab, isl_tab_undo_bmap_div, &tab->var[r]) < 0)
    return -1;
 
  if (add_div_constraints(tab, pos - o_div, add_ineq, user) < 0)
    return -1;
 
  return r;
}
 
/* Add an extra div, prescribed by "div", to the tableau and
 * the associated bmap (which is assumed to be non-NULL).
 */
int isl_tab_add_div(struct isl_tab *tab, __isl_keep isl_vec *div)
{
  if (!tab)
    return -1;
  return isl_tab_insert_div(tab, tab->n_var, div, NULL, NULL);
}
 
/* If "track" is set, then we want to keep track of all constraints in tab
 * in its bmap field.  This field is initialized from a copy of "bmap",
 * so we need to make sure that all constraints in "bmap" also appear
 * in the constructed tab.
 */
__isl_give struct isl_tab *isl_tab_from_basic_map(
  __isl_keep isl_basic_map *bmap, int track)
{
  int i;
  struct isl_tab *tab;
  isl_size total;
 
  total = isl_basic_map_dim(bmap, isl_dim_all);
  if (total < 0)
    return NULL;
  tab = isl_tab_alloc(bmap->ctx, total + bmap->n_ineq + 1, total, 0);
  if (!tab)
    return NULL;
  tab->preserve = track;
  tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
  if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
    if (isl_tab_mark_empty(tab) < 0)
      goto error;
    goto done;
  }
  for (i = 0; i < bmap->n_eq; ++i) {
    tab = add_eq(tab, bmap->eq[i]);
    if (!tab)
      return tab;
  }
  for (i = 0; i < bmap->n_ineq; ++i) {
    if (isl_tab_add_ineq(tab, bmap->ineq[i]) < 0)
      goto error;
    if (tab->empty)
      goto done;
  }
done:
  if (track && isl_tab_track_bmap(tab, isl_basic_map_copy(bmap)) < 0)
    goto error;
  return tab;
error:
  isl_tab_free(tab);
  return NULL;
}
 
__isl_give struct isl_tab *isl_tab_from_basic_set(
  __isl_keep isl_basic_set *bset, int track)
{
  return isl_tab_from_basic_map(bset, track);
}
 
/* Construct a tableau corresponding to the recession cone of "bset".
 */
struct isl_tab *isl_tab_from_recession_cone(__isl_keep isl_basic_set *bset,
  int parametric)
{
  isl_int cst;
  int i;
  struct isl_tab *tab;
  isl_size offset = 0;
  isl_size total;
 
  total = isl_basic_set_dim(bset, isl_dim_all);
  if (parametric)
    offset = isl_basic_set_dim(bset, isl_dim_param);
  if (total < 0 || offset < 0)
    return NULL;
  tab = isl_tab_alloc(bset->ctx, bset->n_eq + bset->n_ineq,
        total - offset, 0);
  if (!tab)
    return NULL;
  tab->rational = ISL_F_ISSET(bset, ISL_BASIC_SET_RATIONAL);
  tab->cone = 1;
 
  isl_int_init(cst);
  isl_int_set_si(cst, 0);
  for (i = 0; i < bset->n_eq; ++i) {
    isl_int_swap(bset->eq[i][offset], cst);
    if (offset > 0) {
      if (isl_tab_add_eq(tab, bset->eq[i] + offset) < 0)
        goto error;
    } else
      tab = add_eq(tab, bset->eq[i]);
    isl_int_swap(bset->eq[i][offset], cst);
    if (!tab)
      goto done;
  }
  for (i = 0; i < bset->n_ineq; ++i) {
    int r;
    isl_int_swap(bset->ineq[i][offset], cst);
    r = isl_tab_add_row(tab, bset->ineq[i] + offset);
    isl_int_swap(bset->ineq[i][offset], cst);
    if (r < 0)
      goto error;
    tab->con[r].is_nonneg = 1;
    if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
      goto error;
  }
done:
  isl_int_clear(cst);
  return tab;
error:
  isl_int_clear(cst);
  isl_tab_free(tab);
  return NULL;
}
 
/* Assuming "tab" is the tableau of a cone, check if the cone is
 * bounded, i.e., if it is empty or only contains the origin.
 */
isl_bool isl_tab_cone_is_bounded(struct isl_tab *tab)
{
  int i;
 
  if (!tab)
    return isl_bool_error;
  if (tab->empty)
    return isl_bool_true;
  if (tab->n_dead == tab->n_col)
    return isl_bool_true;
 
  for (;;) {
    for (i = tab->n_redundant; i < tab->n_row; ++i) {
      struct isl_tab_var *var;
      int sgn;
      var = isl_tab_var_from_row(tab, i);
      if (!var->is_nonneg)
        continue;
      sgn = sign_of_max(tab, var);
      if (sgn < -1)
        return isl_bool_error;
      if (sgn != 0)
        return isl_bool_false;
      if (close_row(tab, var, 0) < 0)
        return isl_bool_error;
      break;
    }
    if (tab->n_dead == tab->n_col)
      return isl_bool_true;
    if (i == tab->n_row)
      return isl_bool_false;
  }
}
 
int isl_tab_sample_is_integer(struct isl_tab *tab)
{
  int i;
 
  if (!tab)
    return -1;
 
  for (i = 0; i < tab->n_var; ++i) {
    int row;
    if (!tab->var[i].is_row)
      continue;
    row = tab->var[i].index;
    if (!isl_int_is_divisible_by(tab->mat->row[row][1],
            tab->mat->row[row][0]))
      return 0;
  }
  return 1;
}
 
static struct isl_vec *extract_integer_sample(struct isl_tab *tab)
{
  int i;
  struct isl_vec *vec;
 
  vec = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
  if (!vec)
    return NULL;
 
  isl_int_set_si(vec->block.data[0], 1);
  for (i = 0; i < tab->n_var; ++i) {
    if (!tab->var[i].is_row)
      isl_int_set_si(vec->block.data[1 + i], 0);
    else {
      int row = tab->var[i].index;
      isl_int_divexact(vec->block.data[1 + i],
        tab->mat->row[row][1], tab->mat->row[row][0]);
    }
  }
 
  return vec;
}
 
__isl_give isl_vec *isl_tab_get_sample_value(struct isl_tab *tab)
{
  int i;
  struct isl_vec *vec;
  isl_int m;
 
  if (!tab)
    return NULL;
 
  vec = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
  if (!vec)
    return NULL;
 
  isl_int_init(m);
 
  isl_int_set_si(vec->block.data[0], 1);
  for (i = 0; i < tab->n_var; ++i) {
    int row;
    if (!tab->var[i].is_row) {
      isl_int_set_si(vec->block.data[1 + i], 0);
      continue;
    }
    row = tab->var[i].index;
    isl_int_gcd(m, vec->block.data[0], tab->mat->row[row][0]);
    isl_int_divexact(m, tab->mat->row[row][0], m);
    isl_seq_scale(vec->block.data, vec->block.data, m, 1 + i);
    isl_int_divexact(m, vec->block.data[0], tab->mat->row[row][0]);
    isl_int_mul(vec->block.data[1 + i], m, tab->mat->row[row][1]);
  }
  vec = isl_vec_normalize(vec);
 
  isl_int_clear(m);
  return vec;
}
 
/* Store the sample value of "var" of "tab" rounded up (if sgn > 0)
 * or down (if sgn < 0) to the nearest integer in *v.
 */
static void get_rounded_sample_value(struct isl_tab *tab,
  struct isl_tab_var *var, int sgn, isl_int *v)
{
  if (!var->is_row)
    isl_int_set_si(*v, 0);
  else if (sgn > 0)
    isl_int_cdiv_q(*v, tab->mat->row[var->index][1],
           tab->mat->row[var->index][0]);
  else
    isl_int_fdiv_q(*v, tab->mat->row[var->index][1],
           tab->mat->row[var->index][0]);
}
 
/* Update "bmap" based on the results of the tableau "tab".
 * In particular, implicit equalities are made explicit, redundant constraints
 * are removed and if the sample value happens to be integer, it is stored
 * in "bmap" (unless "bmap" already had an integer sample).
 *
 * The tableau is assumed to have been created from "bmap" using
 * isl_tab_from_basic_map.
 */
__isl_give isl_basic_map *isl_basic_map_update_from_tab(
  __isl_take isl_basic_map *bmap, struct isl_tab *tab)
{
  int i;
  unsigned n_eq;
 
  if (!bmap)
    return NULL;
  if (!tab)
    return bmap;
 
  n_eq = tab->n_eq;
  if (tab->empty)
    bmap = isl_basic_map_set_to_empty(bmap);
  else
    for (i = bmap->n_ineq - 1; i >= 0; --i) {
      if (isl_tab_is_equality(tab, n_eq + i))
        isl_basic_map_inequality_to_equality(bmap, i);
      else if (isl_tab_is_redundant(tab, n_eq + i))
        isl_basic_map_drop_inequality(bmap, i);
    }
  if (bmap->n_eq != n_eq)
    bmap = isl_basic_map_gauss(bmap, NULL);
  if (!tab->rational &&
      bmap && !bmap->sample && isl_tab_sample_is_integer(tab))
    bmap->sample = extract_integer_sample(tab);
  return bmap;
}
 
__isl_give isl_basic_set *isl_basic_set_update_from_tab(
  __isl_take isl_basic_set *bset, struct isl_tab *tab)
{
  return bset_from_bmap(isl_basic_map_update_from_tab(bset_to_bmap(bset),
                tab));
}
 
/* Drop the last constraint added to "tab" in position "r".
 * The constraint is expected to have remained in a row.
 */
static isl_stat drop_last_con_in_row(struct isl_tab *tab, int r)
{
  if (!tab->con[r].is_row)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "row unexpectedly moved to column",
      return isl_stat_error);
  if (r + 1 != tab->n_con)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "additional constraints added", return isl_stat_error);
  if (drop_row(tab, tab->con[r].index) < 0)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Given a non-negative variable "var", temporarily add a new non-negative
 * variable that is the opposite of "var", ensuring that "var" can only attain
 * the value zero.  The new variable is removed again before this function
 * returns.  However, the effect of forcing "var" to be zero remains.
 * If var = n/d is a row variable, then the new variable = -n/d.
 * If var is a column variables, then the new variable = -var.
 * If the new variable cannot attain non-negative values, then
 * the resulting tableau is empty.
 * Otherwise, we know the value will be zero and we close the row.
 */
static isl_stat cut_to_hyperplane(struct isl_tab *tab, struct isl_tab_var *var)
{
  unsigned r;
  isl_int *row;
  int sgn;
  unsigned off = 2 + tab->M;
 
  if (var->is_zero)
    return isl_stat_ok;
  if (var->is_redundant || !var->is_nonneg)
    isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
      "expecting non-redundant non-negative variable",
      return isl_stat_error);
 
  if (isl_tab_extend_cons(tab, 1) < 0)
    return isl_stat_error;
 
  r = tab->n_con;
  tab->con[r].index = tab->n_row;
  tab->con[r].is_row = 1;
  tab->con[r].is_nonneg = 0;
  tab->con[r].is_zero = 0;
  tab->con[r].is_redundant = 0;
  tab->con[r].frozen = 0;
  tab->con[r].negated = 0;
  tab->row_var[tab->n_row] = ~r;
  row = tab->mat->row[tab->n_row];
 
  if (var->is_row) {
    isl_int_set(row[0], tab->mat->row[var->index][0]);
    isl_seq_neg(row + 1,
          tab->mat->row[var->index] + 1, 1 + tab->n_col);
  } else {
    isl_int_set_si(row[0], 1);
    isl_seq_clr(row + 1, 1 + tab->n_col);
    isl_int_set_si(row[off + var->index], -1);
  }
 
  tab->n_row++;
  tab->n_con++;
 
  sgn = sign_of_max(tab, &tab->con[r]);
  if (sgn < -1)
    return isl_stat_error;
  if (sgn < 0) {
    if (drop_last_con_in_row(tab, r) < 0)
      return isl_stat_error;
    if (isl_tab_mark_empty(tab) < 0)
      return isl_stat_error;
    return isl_stat_ok;
  }
  tab->con[r].is_nonneg = 1;
  /* sgn == 0 */
  if (close_row(tab, &tab->con[r], 1) < 0)
    return isl_stat_error;
  if (drop_last_con_in_row(tab, r) < 0)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Check that "con" is a valid constraint position for "tab".
 */
static isl_stat isl_tab_check_con(struct isl_tab *tab, int con)
{
  if (!tab)
    return isl_stat_error;
  if (con < 0 || con >= tab->n_con)
    isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
      "position out of bounds", return isl_stat_error);
  return isl_stat_ok;
}
 
/* Given a tableau "tab" and an inequality constraint "con" of the tableau,
 * relax the inequality by one.  That is, the inequality r >= 0 is replaced
 * by r' = r + 1 >= 0.
 * If r is a row variable, we simply increase the constant term by one
 * (taking into account the denominator).
 * If r is a column variable, then we need to modify each row that
 * refers to r = r' - 1 by substituting this equality, effectively
 * subtracting the coefficient of the column from the constant.
 * We should only do this if the minimum is manifestly unbounded,
 * however.  Otherwise, we may end up with negative sample values
 * for non-negative variables.
 * So, if r is a column variable with a minimum that is not
 * manifestly unbounded, then we need to move it to a row.
 * However, the sample value of this row may be negative,
 * even after the relaxation, so we need to restore it.
 * We therefore prefer to pivot a column up to a row, if possible.
 */
int isl_tab_relax(struct isl_tab *tab, int con)
{
  struct isl_tab_var *var;
 
  if (!tab)
    return -1;
 
  var = &tab->con[con];
 
  if (var->is_row && (var->index < 0 || var->index < tab->n_redundant))
    isl_die(tab->mat->ctx, isl_error_invalid,
      "cannot relax redundant constraint", return -1);
  if (!var->is_row && (var->index < 0 || var->index < tab->n_dead))
    isl_die(tab->mat->ctx, isl_error_invalid,
      "cannot relax dead constraint", return -1);
 
  if (!var->is_row && !max_is_manifestly_unbounded(tab, var))
    if (to_row(tab, var, 1) < 0)
      return -1;
  if (!var->is_row && !min_is_manifestly_unbounded(tab, var))
    if (to_row(tab, var, -1) < 0)
      return -1;
 
  if (var->is_row) {
    isl_int_add(tab->mat->row[var->index][1],
        tab->mat->row[var->index][1], tab->mat->row[var->index][0]);
    if (restore_row(tab, var) < 0)
      return -1;
  } else {
    int i;
    unsigned off = 2 + tab->M;
 
    for (i = 0; i < tab->n_row; ++i) {
      if (isl_int_is_zero(tab->mat->row[i][off + var->index]))
        continue;
      isl_int_sub(tab->mat->row[i][1], tab->mat->row[i][1],
          tab->mat->row[i][off + var->index]);
    }
 
  }
 
  if (isl_tab_push_var(tab, isl_tab_undo_relax, var) < 0)
    return -1;
 
  return 0;
}
 
/* Replace the variable v at position "pos" in the tableau "tab"
 * by v' = v + shift.
 *
 * If the variable is in a column, then we first check if we can
 * simply plug in v = v' - shift.  The effect on a row with
 * coefficient f/d for variable v is that the constant term c/d
 * is replaced by (c - f * shift)/d.  If shift is positive and
 * f is negative for each row that needs to remain non-negative,
 * then this is clearly safe.  In other words, if the minimum of v
 * is manifestly unbounded, then we can keep v in a column position.
 * Otherwise, we can pivot it down to a row.
 * Similarly, if shift is negative, we need to check if the maximum
 * of is manifestly unbounded.
 *
 * If the variable is in a row (from the start or after pivoting),
 * then the constant term c/d is replaced by (c + d * shift)/d.
 */
int isl_tab_shift_var(struct isl_tab *tab, int pos, isl_int shift)
{
  struct isl_tab_var *var;
 
  if (!tab)
    return -1;
  if (isl_int_is_zero(shift))
    return 0;
 
  var = &tab->var[pos];
  if (!var->is_row) {
    if (isl_int_is_neg(shift)) {
      if (!max_is_manifestly_unbounded(tab, var))
        if (to_row(tab, var, 1) < 0)
          return -1;
    } else {
      if (!min_is_manifestly_unbounded(tab, var))
        if (to_row(tab, var, -1) < 0)
          return -1;
    }
  }
 
  if (var->is_row) {
    isl_int_addmul(tab->mat->row[var->index][1],
        shift, tab->mat->row[var->index][0]);
  } else {
    int i;
    unsigned off = 2 + tab->M;
 
    for (i = 0; i < tab->n_row; ++i) {
      if (isl_int_is_zero(tab->mat->row[i][off + var->index]))
        continue;
      isl_int_submul(tab->mat->row[i][1],
            shift, tab->mat->row[i][off + var->index]);
    }
 
  }
 
  return 0;
}
 
/* Remove the sign constraint from constraint "con".
 *
 * If the constraint variable was originally marked non-negative,
 * then we make sure we mark it non-negative again during rollback.
 */
int isl_tab_unrestrict(struct isl_tab *tab, int con)
{
  struct isl_tab_var *var;
 
  if (!tab)
    return -1;
 
  var = &tab->con[con];
  if (!var->is_nonneg)
    return 0;
 
  var->is_nonneg = 0;
  if (isl_tab_push_var(tab, isl_tab_undo_unrestrict, var) < 0)
    return -1;
 
  return 0;
}
 
int isl_tab_select_facet(struct isl_tab *tab, int con)
{
  if (!tab)
    return -1;
 
  return cut_to_hyperplane(tab, &tab->con[con]);
}
 
static int may_be_equality(struct isl_tab *tab, int row)
{
  return tab->rational ? isl_int_is_zero(tab->mat->row[row][1])
           : isl_int_lt(tab->mat->row[row][1],
              tab->mat->row[row][0]);
}
 
/* Return an isl_tab_var that has been marked or NULL if no such
 * variable can be found.
 * The marked field has only been set for variables that
 * appear in non-redundant rows or non-dead columns.
 *
 * Pick the last constraint variable that is marked and
 * that appears in either a non-redundant row or a non-dead columns.
 * Since the returned variable is tested for being a redundant constraint or
 * an implicit equality, there is no need to return any tab variable that
 * corresponds to a variable.
 */
static struct isl_tab_var *select_marked(struct isl_tab *tab)
{
  int i;
  struct isl_tab_var *var;
 
  for (i = tab->n_con - 1; i >= 0; --i) {
    var = &tab->con[i];
    if (var->index < 0)
      continue;
    if (var->is_row && var->index < tab->n_redundant)
      continue;
    if (!var->is_row && var->index < tab->n_dead)
      continue;
    if (var->marked)
      return var;
  }
 
  return NULL;
}
 
/* Check for (near) equalities among the constraints.
 * A constraint is an equality if it is non-negative and if
 * its maximal value is either
 *  - zero (in case of rational tableaus), or
 *  - strictly less than 1 (in case of integer tableaus)
 *
 * We first mark all non-redundant and non-dead variables that
 * are not frozen and not obviously not an equality.
 * Then we iterate over all marked variables if they can attain
 * any values larger than zero or at least one.
 * If the maximal value is zero, we mark any column variables
 * that appear in the row as being zero and mark the row as being redundant.
 * Otherwise, if the maximal value is strictly less than one (and the
 * tableau is integer), then we restrict the value to being zero
 * by adding an opposite non-negative variable.
 * The order in which the variables are considered is not important.
 */
int isl_tab_detect_implicit_equalities(struct isl_tab *tab)
{
  int i;
  unsigned n_marked;
 
  if (!tab)
    return -1;
  if (tab->empty)
    return 0;
  if (tab->n_dead == tab->n_col)
    return 0;
 
  n_marked = 0;
  for (i = tab->n_redundant; i < tab->n_row; ++i) {
    struct isl_tab_var *var = isl_tab_var_from_row(tab, i);
    var->marked = !var->frozen && var->is_nonneg &&
      may_be_equality(tab, i);
    if (var->marked)
      n_marked++;
  }
  for (i = tab->n_dead; i < tab->n_col; ++i) {
    struct isl_tab_var *var = var_from_col(tab, i);
    var->marked = !var->frozen && var->is_nonneg;
    if (var->marked)
      n_marked++;
  }
  while (n_marked) {
    struct isl_tab_var *var;
    int sgn;
    var = select_marked(tab);
    if (!var)
      break;
    var->marked = 0;
    n_marked--;
    sgn = sign_of_max(tab, var);
    if (sgn < 0)
      return -1;
    if (sgn == 0) {
      if (close_row(tab, var, 0) < 0)
        return -1;
    } else if (!tab->rational && !at_least_one(tab, var)) {
      if (cut_to_hyperplane(tab, var) < 0)
        return -1;
      return isl_tab_detect_implicit_equalities(tab);
    }
    for (i = tab->n_redundant; i < tab->n_row; ++i) {
      var = isl_tab_var_from_row(tab, i);
      if (!var->marked)
        continue;
      if (may_be_equality(tab, i))
        continue;
      var->marked = 0;
      n_marked--;
    }
  }
 
  return 0;
}
 
/* Update the element of row_var or col_var that corresponds to
 * constraint tab->con[i] to a move from position "old" to position "i".
 */
static int update_con_after_move(struct isl_tab *tab, int i, int old)
{
  int *p;
  int index;
 
  index = tab->con[i].index;
  if (index == -1)
    return 0;
  p = tab->con[i].is_row ? tab->row_var : tab->col_var;
  if (p[index] != ~old)
    isl_die(tab->mat->ctx, isl_error_internal,
      "broken internal state", return -1);
  p[index] = ~i;
 
  return 0;
}
 
/* Interchange constraints "con1" and "con2" in "tab".
 * In particular, interchange the contents of these entries in tab->con.
 * Since tab->col_var and tab->row_var point back into this array,
 * they need to be updated accordingly.
 */
isl_stat isl_tab_swap_constraints(struct isl_tab *tab, int con1, int con2)
{
  struct isl_tab_var var;
 
  if (isl_tab_check_con(tab, con1) < 0 ||
      isl_tab_check_con(tab, con2) < 0)
    return isl_stat_error;
 
  var = tab->con[con1];
  tab->con[con1] = tab->con[con2];
  if (update_con_after_move(tab, con1, con2) < 0)
    return isl_stat_error;
  tab->con[con2] = var;
  if (update_con_after_move(tab, con2, con1) < 0)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Rotate the "n" constraints starting at "first" to the right,
 * putting the last constraint in the position of the first constraint.
 */
static isl_stat rotate_constraints_right(struct isl_tab *tab, int first, int n)
{
  int i, last;
  struct isl_tab_var var;
 
  if (n <= 1)
    return isl_stat_ok;
 
  last = first + n - 1;
  var = tab->con[last];
  for (i = last; i > first; --i) {
    tab->con[i] = tab->con[i - 1];
    if (update_con_after_move(tab, i, i - 1) < 0)
      return isl_stat_error;
  }
  tab->con[first] = var;
  if (update_con_after_move(tab, first, last) < 0)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Rotate the "n" constraints starting at "first" to the left,
 * putting the first constraint in the position of the last constraint.
 */
static isl_stat rotate_constraints_left(struct isl_tab *tab, int first, int n)
{
  int i, last;
  struct isl_tab_var var;
 
  if (n <= 1)
    return isl_stat_ok;
 
  last = first + n - 1;
  var = tab->con[first];
  for (i = first; i < last; ++i) {
    tab->con[i] = tab->con[i + 1];
    if (update_con_after_move(tab, i, i + 1) < 0)
      return isl_stat_error;
  }
  tab->con[last] = var;
  if (update_con_after_move(tab, last, first) < 0)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Drop the "n" entries starting at position "first" in tab->con, moving all
 * subsequent entries down.
 * Since some of the entries of tab->row_var and tab->col_var contain
 * indices into this array, they have to be updated accordingly.
 */
static isl_stat con_drop_entries(struct isl_tab *tab,
  unsigned first, unsigned n)
{
  int i;
 
  if (first + n > tab->n_con || first + n < first)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "invalid range", return isl_stat_error);
 
  tab->n_con -= n;
 
  for (i = first; i < tab->n_con; ++i) {
    tab->con[i] = tab->con[i + n];
    if (update_con_after_move(tab, i, i + n) < 0)
      return isl_stat_error;
  }
 
  return isl_stat_ok;
}
 
/* isl_basic_map_gauss5 callback that gets called when
 * two (equality) constraints "a" and "b" get interchanged
 * in the basic map.  Perform the same interchange in "tab".
 */
static isl_stat swap_eq(unsigned a, unsigned b, void *user)
{
  struct isl_tab *tab = user;
 
  return isl_tab_swap_constraints(tab, a, b);
}
 
/* isl_basic_map_gauss5 callback that gets called when
 * the final "n" equality constraints get removed.
 * As a special case, if "n" is equal to the total number
 * of equality constraints, then this means the basic map
 * turned out to be empty.
 * Drop the same number of equality constraints from "tab" or
 * mark it empty in the special case.
 */
static isl_stat drop_eq(unsigned n, void *user)
{
  struct isl_tab *tab = user;
 
  if (tab->n_eq == n)
    return isl_tab_mark_empty(tab);
 
  tab->n_eq -= n;
  return con_drop_entries(tab, tab->n_eq, n);
}
 
/* If "bmap" has more than a single reference, then call
 * isl_basic_map_gauss on it, updating "tab" accordingly.
 */
static __isl_give isl_basic_map *gauss_if_shared(__isl_take isl_basic_map *bmap,
  struct isl_tab *tab)
{
  isl_bool single;
 
  single = isl_basic_map_has_single_reference(bmap);
  if (single < 0)
    return isl_basic_map_free(bmap);
  if (single)
    return bmap;
  return isl_basic_map_gauss5(bmap, NULL, &swap_eq, &drop_eq, tab);
}
 
/* Make the equalities that are implicit in "bmap" but that have been
 * detected in the corresponding "tab" explicit in "bmap" and update
 * "tab" to reflect the new order of the constraints.
 *
 * In particular, if inequality i is an implicit equality then
 * isl_basic_map_inequality_to_equality will move the inequality
 * in front of the other equality and it will move the last inequality
 * in the position of inequality i.
 * In the tableau, the inequalities of "bmap" are stored after the equalities
 * and so the original order
 *
 *    E E E E E A A A I B B B B L
 *
 * is changed into
 *
 *    I E E E E E A A A L B B B B
 *
 * where I is the implicit equality, the E are equalities,
 * the A inequalities before I, the B inequalities after I and
 * L the last inequality.
 * We therefore need to rotate to the right two sets of constraints,
 * those up to and including I and those after I.
 *
 * If "tab" contains any constraints that are not in "bmap" then they
 * appear after those in "bmap" and they should be left untouched.
 *
 * If the operation may need to be undone, then keep track
 * of the inequality constraints that have been turned
 * into equality constraints.
 *
 * Note that this function only calls isl_basic_map_gauss
 * (in case some equality constraints got detected)
 * if "bmap" has more than one reference and if the operation
 * does not need to be undone.
 * If it only has a single reference, then it is left in a temporary state,
 * because the caller may require this state.
 * Calling isl_basic_map_gauss is then the responsibility of the caller.
 * This is also the case if the operation may need to be undone.
 */
__isl_give isl_basic_map *isl_tab_make_equalities_explicit(struct isl_tab *tab,
  __isl_take isl_basic_map *bmap)
{
  int i;
  unsigned n_eq;
 
  if (!tab || !bmap)
    return isl_basic_map_free(bmap);
  if (tab->empty)
    return bmap;
 
  n_eq = tab->n_eq;
  for (i = bmap->n_ineq - 1; i >= 0; --i) {
    if (!isl_tab_is_equality(tab, bmap->n_eq + i))
      continue;
    isl_basic_map_inequality_to_equality(bmap, i);
    if (rotate_constraints_right(tab, 0, tab->n_eq + i + 1) < 0)
      return isl_basic_map_free(bmap);
    if (rotate_constraints_right(tab, tab->n_eq + i + 1,
          bmap->n_ineq - i) < 0)
      return isl_basic_map_free(bmap);
    tab->n_eq++;
    if (tab->need_undo)
      isl_tab_push_ineq_to_eq(tab, i);
  }
 
  if (!tab->need_undo && n_eq != tab->n_eq)
    bmap = gauss_if_shared(bmap, tab);
 
  return bmap;
}
 
/* Undo the effect of turning an inequality constraint
 * into an equality constraint in isl_tab_make_equalities_explicit.
 * "ineq" is the original position of the inequality constraint that
 * now appears as the first equality constraint.
 *
 * That is, the order
 *
 *    I E E E E E A A A L B B B B
 *
 * needs to be changed back into
 *
 *    E E E E E A A A I B B B B L
 *
 * where I is the inequality turned equality, the E are the original equalities,
 * the A inequalities originally before I,
 * the B inequalities originally after I and
 * L the originally last inequality.
 *
 * Two groups of constraints therefore need to be rotated left,
 * those up to and including the original position of I and
 * those after this position.
 */
static isl_stat first_eq_to_ineq(struct isl_tab *tab, int ineq)
{
  unsigned n_ineq, n_eq;
 
  if (!tab)
    return isl_stat_error;
 
  n_ineq = tab->n_con - tab->n_eq;
  tab->n_eq--;
  n_eq = tab->n_eq;
  if (rotate_constraints_left(tab, 0, n_eq + ineq + 1) < 0)
    return isl_stat_error;
  if (rotate_constraints_left(tab, n_eq + ineq + 1, n_ineq - ineq) < 0)
    return isl_stat_error;
  return isl_stat_ok;
}
 
static int con_is_redundant(struct isl_tab *tab, struct isl_tab_var *var)
{
  if (!tab)
    return -1;
  if (tab->rational) {
    int sgn = sign_of_min(tab, var);
    if (sgn < -1)
      return -1;
    return sgn >= 0;
  } else {
    int irred = isl_tab_min_at_most_neg_one(tab, var);
    if (irred < 0)
      return -1;
    return !irred;
  }
}
 
/* Check for (near) redundant constraints.
 * A constraint is redundant if it is non-negative and if
 * its minimal value (temporarily ignoring the non-negativity) is either
 *  - zero (in case of rational tableaus), or
 *  - strictly larger than -1 (in case of integer tableaus)
 *
 * We first mark all non-redundant and non-dead variables that
 * are not frozen and not obviously negatively unbounded.
 * Then we iterate over all marked variables if they can attain
 * any values smaller than zero or at most negative one.
 * If not, we mark the row as being redundant (assuming it hasn't
 * been detected as being obviously redundant in the mean time).
 */
int isl_tab_detect_redundant(struct isl_tab *tab)
{
  int i;
  unsigned n_marked;
 
  if (!tab)
    return -1;
  if (tab->empty)
    return 0;
  if (tab->n_redundant == tab->n_row)
    return 0;
 
  n_marked = 0;
  for (i = tab->n_redundant; i < tab->n_row; ++i) {
    struct isl_tab_var *var = isl_tab_var_from_row(tab, i);
    var->marked = !var->frozen && var->is_nonneg;
    if (var->marked)
      n_marked++;
  }
  for (i = tab->n_dead; i < tab->n_col; ++i) {
    struct isl_tab_var *var = var_from_col(tab, i);
    var->marked = !var->frozen && var->is_nonneg &&
      !min_is_manifestly_unbounded(tab, var);
    if (var->marked)
      n_marked++;
  }
  while (n_marked) {
    struct isl_tab_var *var;
    int red;
    var = select_marked(tab);
    if (!var)
      break;
    var->marked = 0;
    n_marked--;
    red = con_is_redundant(tab, var);
    if (red < 0)
      return -1;
    if (red && !var->is_redundant)
      if (isl_tab_mark_redundant(tab, var->index) < 0)
        return -1;
    for (i = tab->n_dead; i < tab->n_col; ++i) {
      var = var_from_col(tab, i);
      if (!var->marked)
        continue;
      if (!min_is_manifestly_unbounded(tab, var))
        continue;
      var->marked = 0;
      n_marked--;
    }
  }
 
  return 0;
}
 
int isl_tab_is_equality(struct isl_tab *tab, int con)
{
  int row;
  unsigned off;
 
  if (!tab)
    return -1;
  if (tab->con[con].is_zero)
    return 1;
  if (tab->con[con].is_redundant)
    return 0;
  if (!tab->con[con].is_row)
    return tab->con[con].index < tab->n_dead;
 
  row = tab->con[con].index;
 
  off = 2 + tab->M;
  return isl_int_is_zero(tab->mat->row[row][1]) &&
    !row_is_big(tab, row) &&
    !isl_seq_any_non_zero(tab->mat->row[row] + off + tab->n_dead,
          tab->n_col - tab->n_dead);
}
 
/* Return the minimal value of the affine expression "f" with denominator
 * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
 * the expression cannot attain arbitrarily small values.
 * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
 * The return value reflects the nature of the result (empty, unbounded,
 * minimal value returned in *opt).
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
enum isl_lp_result isl_tab_min(struct isl_tab *tab,
  isl_int *f, isl_int denom, isl_int *opt, isl_int *opt_denom,
  unsigned flags)
{
  int r;
  enum isl_lp_result res = isl_lp_ok;
  struct isl_tab_var *var;
  struct isl_tab_undo *snap;
 
  if (!tab)
    return isl_lp_error;
 
  if (tab->empty)
    return isl_lp_empty;
 
  snap = isl_tab_snap(tab);
  r = isl_tab_add_row(tab, f);
  if (r < 0)
    return isl_lp_error;
  var = &tab->con[r];
  for (;;) {
    int row, col;
    find_pivot(tab, var, var, -1, &row, &col);
    if (row == var->index) {
      res = isl_lp_unbounded;
      break;
    }
    if (row == -1)
      break;
    if (isl_tab_pivot(tab, row, col) < 0)
      return isl_lp_error;
  }
  isl_int_mul(tab->mat->row[var->index][0],
        tab->mat->row[var->index][0], denom);
  if (ISL_FL_ISSET(flags, ISL_TAB_SAVE_DUAL)) {
    int i;
 
    isl_vec_free(tab->dual);
    tab->dual = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_con);
    if (!tab->dual)
      return isl_lp_error;
    isl_int_set(tab->dual->el[0], tab->mat->row[var->index][0]);
    for (i = 0; i < tab->n_con; ++i) {
      int pos;
      if (tab->con[i].is_row) {
        isl_int_set_si(tab->dual->el[1 + i], 0);
        continue;
      }
      pos = 2 + tab->M + tab->con[i].index;
      if (tab->con[i].negated)
        isl_int_neg(tab->dual->el[1 + i],
              tab->mat->row[var->index][pos]);
      else
        isl_int_set(tab->dual->el[1 + i],
              tab->mat->row[var->index][pos]);
    }
  }
  if (opt && res == isl_lp_ok) {
    if (opt_denom) {
      isl_int_set(*opt, tab->mat->row[var->index][1]);
      isl_int_set(*opt_denom, tab->mat->row[var->index][0]);
    } else
      get_rounded_sample_value(tab, var, 1, opt);
  }
  if (isl_tab_rollback(tab, snap) < 0)
    return isl_lp_error;
  return res;
}
 
/* Is the constraint at position "con" marked as being redundant?
 * If it is marked as representing an equality, then it is not
 * considered to be redundant.
 * Note that isl_tab_mark_redundant marks both the isl_tab_var as
 * redundant and moves the corresponding row into the first
 * tab->n_redundant positions (or removes the row, assigning it index -1),
 * so the final test is actually redundant itself.
 */
int isl_tab_is_redundant(struct isl_tab *tab, int con)
{
  if (isl_tab_check_con(tab, con) < 0)
    return -1;
  if (tab->con[con].is_zero)
    return 0;
  if (tab->con[con].is_redundant)
    return 1;
  return tab->con[con].is_row && tab->con[con].index < tab->n_redundant;
}
 
/* Is variable "var" of "tab" fixed to a constant value by its row
 * in the tableau?
 * If so and if "value" is not NULL, then store this constant value
 * in "value".
 *
 * That is, is it a row variable that only has non-zero coefficients
 * for dead columns?
 */
static isl_bool is_constant(struct isl_tab *tab, struct isl_tab_var *var,
  isl_int *value)
{
  unsigned off = 2 + tab->M;
  isl_mat *mat = tab->mat;
  int n;
  int row;
 
  if (!var->is_row)
    return isl_bool_false;
  row = var->index;
  if (row_is_big(tab, row))
    return isl_bool_false;
  n = tab->n_col - tab->n_dead;
  if (isl_seq_any_non_zero(mat->row[row] + off + tab->n_dead, n))
    return isl_bool_false;
  if (value)
    isl_int_divexact(*value, mat->row[row][1], mat->row[row][0]);
  return isl_bool_true;
}
 
/* Has the variable "var' of "tab" reached a value that is greater than
 * or equal (if sgn > 0) or smaller than or equal (if sgn < 0) to "target"?
 * "tmp" has been initialized by the caller and can be used
 * to perform local computations.
 *
 * If the sample value involves the big parameter, then any value
 * is reached.
 * Otherwise check if n/d >= t, i.e., n >= d * t (if sgn > 0)
 * or n/d <= t, i.e., n <= d * t (if sgn < 0).
 */
static int reached(struct isl_tab *tab, struct isl_tab_var *var, int sgn,
  isl_int target, isl_int *tmp)
{
  if (row_is_big(tab, var->index))
    return 1;
  isl_int_mul(*tmp, tab->mat->row[var->index][0], target);
  if (sgn > 0)
    return isl_int_ge(tab->mat->row[var->index][1], *tmp);
  else
    return isl_int_le(tab->mat->row[var->index][1], *tmp);
}
 
/* Can variable "var" of "tab" attain the value "target" by
 * pivoting up (if sgn > 0) or down (if sgn < 0)?
 * If not, then pivot up [down] to the greatest [smallest]
 * rational value.
 * "tmp" has been initialized by the caller and can be used
 * to perform local computations.
 *
 * If the variable is manifestly unbounded in the desired direction,
 * then it can attain any value.
 * Otherwise, it can be moved to a row.
 * Continue pivoting until the target is reached.
 * If no more pivoting can be performed, the maximal [minimal]
 * rational value has been reached and the target cannot be reached.
 * If the variable would be pivoted into a manifestly unbounded column,
 * then the target can be reached.
 */
static isl_bool var_reaches(struct isl_tab *tab, struct isl_tab_var *var,
  int sgn, isl_int target, isl_int *tmp)
{
  int row, col;
 
  if (sgn < 0 && min_is_manifestly_unbounded(tab, var))
    return isl_bool_true;
  if (sgn > 0 && max_is_manifestly_unbounded(tab, var))
    return isl_bool_true;
  if (to_row(tab, var, sgn) < 0)
    return isl_bool_error;
  while (!reached(tab, var, sgn, target, tmp)) {
    find_pivot(tab, var, var, sgn, &row, &col);
    if (row == -1)
      return isl_bool_false;
    if (row == var->index)
      return isl_bool_true;
    if (isl_tab_pivot(tab, row, col) < 0)
      return isl_bool_error;
  }
 
  return isl_bool_true;
}
 
/* Check if variable "var" of "tab" can only attain a single (integer)
 * value, and, if so, add an equality constraint to fix the variable
 * to this single value and store the result in "target".
 * "target" and "tmp" have been initialized by the caller.
 *
 * Given the current sample value, round it down and check
 * whether it is possible to attain a strictly smaller integer value.
 * If so, the variable is not restricted to a single integer value.
 * Otherwise, the search stops at the smallest rational value.
 * Round up this value and check whether it is possible to attain
 * a strictly greater integer value.
 * If so, the variable is not restricted to a single integer value.
 * Otherwise, the search stops at the greatest rational value.
 * If rounding down this value yields a value that is different
 * from rounding up the smallest rational value, then the variable
 * cannot attain any integer value.  Mark the tableau empty.
 * Otherwise, add an equality constraint that fixes the variable
 * to the single integer value found.
 */
static isl_bool detect_constant_with_tmp(struct isl_tab *tab,
  struct isl_tab_var *var, isl_int *target, isl_int *tmp)
{
  isl_bool reached;
  isl_vec *eq;
  int pos;
  isl_stat r;
 
  get_rounded_sample_value(tab, var, -1, target);
  isl_int_sub_ui(*target, *target, 1);
  reached = var_reaches(tab, var, -1, *target, tmp);
  if (reached < 0 || reached)
    return isl_bool_not(reached);
  get_rounded_sample_value(tab, var, 1, target);
  isl_int_add_ui(*target, *target, 1);
  reached = var_reaches(tab, var, 1, *target, tmp);
  if (reached < 0 || reached)
    return isl_bool_not(reached);
  get_rounded_sample_value(tab, var, -1, tmp);
  isl_int_sub_ui(*target, *target, 1);
  if (isl_int_ne(*target, *tmp)) {
    if (isl_tab_mark_empty(tab) < 0)
      return isl_bool_error;
    return isl_bool_false;
  }
 
  if (isl_tab_extend_cons(tab, 1) < 0)
    return isl_bool_error;
  eq = isl_vec_alloc(isl_tab_get_ctx(tab), 1 + tab->n_var);
  if (!eq)
    return isl_bool_error;
  pos = var - tab->var;
  isl_seq_clr(eq->el + 1, tab->n_var);
  isl_int_set_si(eq->el[1 + pos], -1);
  isl_int_set(eq->el[0], *target);
  r = isl_tab_add_eq(tab, eq->el);
  isl_vec_free(eq);
 
  return r < 0 ? isl_bool_error : isl_bool_true;
}
 
/* Check if variable "var" of "tab" can only attain a single (integer)
 * value, and, if so, add an equality constraint to fix the variable
 * to this single value and store the result in "value" (if "value"
 * is not NULL).
 *
 * If the current sample value involves the big parameter,
 * then the variable cannot have a fixed integer value.
 * If the variable is already fixed to a single value by its row, then
 * there is no need to add another equality constraint.
 *
 * Otherwise, allocate some temporary variables and continue
 * with detect_constant_with_tmp.
 */
static isl_bool get_constant(struct isl_tab *tab, struct isl_tab_var *var,
  isl_int *value)
{
  isl_int target, tmp;
  isl_bool is_cst;
 
  if (var->is_row && row_is_big(tab, var->index))
    return isl_bool_false;
  is_cst = is_constant(tab, var, value);
  if (is_cst < 0 || is_cst)
    return is_cst;
 
  if (!value)
    isl_int_init(target);
  isl_int_init(tmp);
 
  is_cst = detect_constant_with_tmp(tab, var,
              value ? value : &target, &tmp);
 
  isl_int_clear(tmp);
  if (!value)
    isl_int_clear(target);
 
  return is_cst;
}
 
/* Check if variable "var" of "tab" can only attain a single (integer)
 * value, and, if so, add an equality constraint to fix the variable
 * to this single value and store the result in "value" (if "value"
 * is not NULL).
 *
 * For rational tableaus, nothing needs to be done.
 */
isl_bool isl_tab_is_constant(struct isl_tab *tab, int var, isl_int *value)
{
  if (!tab)
    return isl_bool_error;
  if (var < 0 || var >= tab->n_var)
    isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
      "position out of bounds", return isl_bool_error);
  if (tab->rational)
    return isl_bool_false;
 
  return get_constant(tab, &tab->var[var], value);
}
 
/* Check if any of the variables of "tab" can only attain a single (integer)
 * value, and, if so, add equality constraints to fix those variables
 * to these single values.
 *
 * For rational tableaus, nothing needs to be done.
 */
isl_stat isl_tab_detect_constants(struct isl_tab *tab)
{
  int i;
 
  if (!tab)
    return isl_stat_error;
  if (tab->rational)
    return isl_stat_ok;
 
  for (i = 0; i < tab->n_var; ++i) {
    if (get_constant(tab, &tab->var[i], NULL) < 0)
      return isl_stat_error;
  }
 
  return isl_stat_ok;
}
 
/* Take a snapshot of the tableau that can be restored by a call to
 * isl_tab_rollback.
 */
struct isl_tab_undo *isl_tab_snap(struct isl_tab *tab)
{
  if (!tab)
    return NULL;
  tab->need_undo = 1;
  return tab->top;
}
 
/* Does "tab" need to keep track of undo information?
 * That is, was a snapshot taken that may need to be restored?
 */
isl_bool isl_tab_need_undo(struct isl_tab *tab)
{
  if (!tab)
    return isl_bool_error;
 
  return isl_bool_ok(tab->need_undo);
}
 
/* Remove all tracking of undo information from "tab", invalidating
 * any snapshots that may have been taken of the tableau.
 * Since all snapshots have been invalidated, there is also
 * no need to start keeping track of undo information again.
 */
void isl_tab_clear_undo(struct isl_tab *tab)
{
  if (!tab)
    return;
 
  free_undo(tab);
  tab->need_undo = 0;
}
 
/* Undo the operation performed by isl_tab_relax.
 */
static isl_stat unrelax(struct isl_tab *tab, struct isl_tab_var *var)
  WARN_UNUSED;
static isl_stat unrelax(struct isl_tab *tab, struct isl_tab_var *var)
{
  unsigned off = 2 + tab->M;
 
  if (!var->is_row && !max_is_manifestly_unbounded(tab, var))
    if (to_row(tab, var, 1) < 0)
      return isl_stat_error;
 
  if (var->is_row) {
    isl_int_sub(tab->mat->row[var->index][1],
        tab->mat->row[var->index][1], tab->mat->row[var->index][0]);
    if (var->is_nonneg) {
      int sgn = restore_row(tab, var);
      isl_assert(tab->mat->ctx, sgn >= 0,
        return isl_stat_error);
    }
  } else {
    int i;
 
    for (i = 0; i < tab->n_row; ++i) {
      if (isl_int_is_zero(tab->mat->row[i][off + var->index]))
        continue;
      isl_int_add(tab->mat->row[i][1], tab->mat->row[i][1],
          tab->mat->row[i][off + var->index]);
    }
 
  }
 
  return isl_stat_ok;
}
 
/* Undo the operation performed by isl_tab_unrestrict.
 *
 * In particular, mark the variable as being non-negative and make
 * sure the sample value respects this constraint.
 */
static isl_stat ununrestrict(struct isl_tab *tab, struct isl_tab_var *var)
{
  var->is_nonneg = 1;
 
  if (var->is_row && restore_row(tab, var) < -1)
    return isl_stat_error;
 
  return isl_stat_ok;
}
 
/* Unmark the last redundant row in "tab" as being redundant.
 * This undoes part of the modifications performed by isl_tab_mark_redundant.
 * In particular, remove the redundant mark and make
 * sure the sample value respects the constraint again.
 * A variable that is marked non-negative by isl_tab_mark_redundant
 * is covered by a separate undo record.
 */
static isl_stat restore_last_redundant(struct isl_tab *tab)
{
  struct isl_tab_var *var;
 
  if (tab->n_redundant < 1)
    isl_die(isl_tab_get_ctx(tab), isl_error_internal,
      "no redundant rows", return isl_stat_error);
 
  var = isl_tab_var_from_row(tab, tab->n_redundant - 1);
  var->is_redundant = 0;
  tab->n_redundant--;
  restore_row(tab, var);
 
  return isl_stat_ok;
}
 
static isl_stat perform_undo_var(struct isl_tab *tab, struct isl_tab_undo *undo)
  WARN_UNUSED;
static isl_stat perform_undo_var(struct isl_tab *tab, struct isl_tab_undo *undo)
{
  struct isl_tab_var *var = var_from_index(tab, undo->u.var_index);
  switch (undo->type) {
  case isl_tab_undo_nonneg:
    var->is_nonneg = 0;
    break;
  case isl_tab_undo_redundant:
    if (!var->is_row || var->index != tab->n_redundant - 1)
      isl_die(isl_tab_get_ctx(tab), isl_error_internal,
        "not undoing last redundant row",
        return isl_stat_error);
    return restore_last_redundant(tab);
  case isl_tab_undo_freeze:
    var->frozen = 0;
    break;
  case isl_tab_undo_zero:
    var->is_zero = 0;
    if (!var->is_row)
      tab->n_dead--;
    break;
  case isl_tab_undo_allocate:
    if (undo->u.var_index >= 0) {
      isl_assert(tab->mat->ctx, !var->is_row,
        return isl_stat_error);
      return drop_col(tab, var->index);
    }
    if (!var->is_row) {
      if (!max_is_manifestly_unbounded(tab, var)) {
        if (to_row(tab, var, 1) < 0)
          return isl_stat_error;
      } else if (!min_is_manifestly_unbounded(tab, var)) {
        if (to_row(tab, var, -1) < 0)
          return isl_stat_error;
      } else
        if (to_row(tab, var, 0) < 0)
          return isl_stat_error;
    }
    return drop_row(tab, var->index);
  case isl_tab_undo_relax:
    return unrelax(tab, var);
  case isl_tab_undo_unrestrict:
    return ununrestrict(tab, var);
  default:
    isl_die(tab->mat->ctx, isl_error_internal,
      "perform_undo_var called on invalid undo record",
      return isl_stat_error);
  }
 
  return isl_stat_ok;
}
 
/* Restore all rows that have been marked redundant by isl_tab_mark_redundant
 * and that have been preserved in the tableau.
 * Note that isl_tab_mark_redundant may also have marked some variables
 * as being non-negative before marking them redundant.  These need
 * to be removed as well as otherwise some constraints could end up
 * getting marked redundant with respect to the variable.
 */
isl_stat isl_tab_restore_redundant(struct isl_tab *tab)
{
  if (!tab)
    return isl_stat_error;
 
  if (tab->need_undo)
    isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
      "manually restoring redundant constraints "
      "interferes with undo history",
      return isl_stat_error);
 
  while (tab->n_redundant > 0) {
    if (tab->row_var[tab->n_redundant - 1] >= 0) {
      struct isl_tab_var *var;
 
      var = isl_tab_var_from_row(tab, tab->n_redundant - 1);
      var->is_nonneg = 0;
    }
    restore_last_redundant(tab);
  }
  return isl_stat_ok;
}
 
/* Undo the addition of an integer division to the basic map representation
 * of "tab" in position "pos".
 */
static isl_stat drop_bmap_div(struct isl_tab *tab, int pos)
{
  int off;
  isl_size n_div;
 
  n_div = isl_basic_map_dim(tab->bmap, isl_dim_div);
  if (n_div < 0)
    return isl_stat_error;
  off = tab->n_var - n_div;
  tab->bmap = isl_basic_map_drop_div(tab->bmap, pos - off);
  if (!tab->bmap)
    return isl_stat_error;
  if (tab->samples) {
    tab->samples = isl_mat_drop_cols(tab->samples, 1 + pos, 1);
    if (!tab->samples)
      return isl_stat_error;
  }
 
  return isl_stat_ok;
}
 
/* Restore the tableau to the state where the basic variables
 * are those in "col_var".
 * We first construct a list of variables that are currently in
 * the basis, but shouldn't.  Then we iterate over all variables
 * that should be in the basis and for each one that is currently
 * not in the basis, we exchange it with one of the elements of the
 * list constructed before.
 * We can always find an appropriate variable to pivot with because
 * the current basis is mapped to the old basis by a non-singular
 * matrix and so we can never end up with a zero row.
 */
static int restore_basis(struct isl_tab *tab, int *col_var)
{
  int i, j;
  int n_extra = 0;
  int *extra = NULL;  /* current columns that contain bad stuff */
  unsigned off = 2 + tab->M;
 
  extra = isl_alloc_array(tab->mat->ctx, int, tab->n_col);
  if (tab->n_col && !extra)
    goto error;
  for (i = 0; i < tab->n_col; ++i) {
    for (j = 0; j < tab->n_col; ++j)
      if (tab->col_var[i] == col_var[j])
        break;
    if (j < tab->n_col)
      continue;
    extra[n_extra++] = i;
  }
  for (i = 0; i < tab->n_col && n_extra > 0; ++i) {
    struct isl_tab_var *var;
    int row;
 
    for (j = 0; j < tab->n_col; ++j)
      if (col_var[i] == tab->col_var[j])
        break;
    if (j < tab->n_col)
      continue;
    var = var_from_index(tab, col_var[i]);
    row = var->index;
    for (j = 0; j < n_extra; ++j)
      if (!isl_int_is_zero(tab->mat->row[row][off+extra[j]]))
        break;
    isl_assert(tab->mat->ctx, j < n_extra, goto error);
    if (isl_tab_pivot(tab, row, extra[j]) < 0)
      goto error;
    extra[j] = extra[--n_extra];
  }
 
  free(extra);
  return 0;
error:
  free(extra);
  return -1;
}
 
/* Remove all samples with index n or greater, i.e., those samples
 * that were added since we saved this number of samples in
 * isl_tab_save_samples.
 */
static void drop_samples_since(struct isl_tab *tab, int n)
{
  int i;
 
  for (i = tab->n_sample - 1; i >= 0 && tab->n_sample > n; --i) {
    if (tab->sample_index[i] < n)
      continue;
 
    if (i != tab->n_sample - 1) {
      int t = tab->sample_index[tab->n_sample-1];
      tab->sample_index[tab->n_sample-1] = tab->sample_index[i];
      tab->sample_index[i] = t;
      isl_mat_swap_rows(tab->samples, tab->n_sample-1, i);
    }
    tab->n_sample--;
  }
}
 
static isl_stat perform_undo(struct isl_tab *tab, struct isl_tab_undo *undo)
  WARN_UNUSED;
static isl_stat perform_undo(struct isl_tab *tab, struct isl_tab_undo *undo)
{
  switch (undo->type) {
  case isl_tab_undo_rational:
    tab->rational = 0;
    break;
  case isl_tab_undo_empty:
    tab->empty = 0;
    break;
  case isl_tab_undo_nonneg:
  case isl_tab_undo_redundant:
  case isl_tab_undo_freeze:
  case isl_tab_undo_zero:
  case isl_tab_undo_allocate:
  case isl_tab_undo_relax:
  case isl_tab_undo_unrestrict:
    return perform_undo_var(tab, undo);
  case isl_tab_undo_bmap_eq:
    tab->bmap = isl_basic_map_free_equality(tab->bmap, 1);
    return tab->bmap ? isl_stat_ok : isl_stat_error;
  case isl_tab_undo_bmap_ineq:
    tab->bmap = isl_basic_map_free_inequality(tab->bmap, 1);
    return tab->bmap ? isl_stat_ok : isl_stat_error;
  case isl_tab_undo_bmap_div:
    return drop_bmap_div(tab, undo->u.var_index);
  case isl_tab_undo_saved_basis:
    if (restore_basis(tab, undo->u.col_var) < 0)
      return isl_stat_error;
    break;
  case isl_tab_undo_drop_sample:
    tab->n_outside--;
    break;
  case isl_tab_undo_saved_samples:
    drop_samples_since(tab, undo->u.n);
    break;
  case isl_tab_undo_callback:
    return undo->u.callback->run(undo->u.callback);
  case isl_tab_undo_ineq_to_eq:
    return first_eq_to_ineq(tab, undo->u.n);
  default:
    isl_assert(tab->mat->ctx, 0, return isl_stat_error);
  }
  return isl_stat_ok;
}
 
/* Return the tableau to the state it was in when the snapshot "snap"
 * was taken.
 */
isl_stat isl_tab_rollback(struct isl_tab *tab, struct isl_tab_undo *snap)
{
  struct isl_tab_undo *undo, *next;
 
  if (!tab)
    return isl_stat_error;
 
  tab->in_undo = 1;
  for (undo = tab->top; undo && undo != &tab->bottom; undo = next) {
    next = undo->next;
    if (undo == snap)
      break;
    if (perform_undo(tab, undo) < 0) {
      tab->top = undo;
      free_undo(tab);
      tab->in_undo = 0;
      return isl_stat_error;
    }
    free_undo_record(undo);
  }
  tab->in_undo = 0;
  tab->top = undo;
  if (!undo)
    return isl_stat_error;
  return isl_stat_ok;
}
 
/* The given row "row" represents an inequality violated by all
 * points in the tableau.  Check for some special cases of such
 * separating constraints.
 * In particular, if the row has been reduced to the constant -1,
 * then we know the inequality is adjacent (but opposite) to
 * an equality in the tableau.
 * If the row has been reduced to r = c*(-1 -r'), with r' an inequality
 * of the tableau and c a positive constant, then the inequality
 * is adjacent (but opposite) to the inequality r'.
 */
static enum isl_ineq_type separation_type(struct isl_tab *tab, unsigned row)
{
  int pos;
  int separate;
  unsigned off = 2 + tab->M;
 
  if (tab->rational)
    return isl_ineq_separate;
 
  if (!isl_int_is_one(tab->mat->row[row][0]))
    return isl_ineq_separate;
 
  pos = isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
          tab->n_col - tab->n_dead);
  if (pos == -1) {
    if (isl_int_is_negone(tab->mat->row[row][1]))
      return isl_ineq_adj_eq;
    else
      return isl_ineq_separate;
  }
 
  if (!isl_int_eq(tab->mat->row[row][1],
      tab->mat->row[row][off + tab->n_dead + pos]))
    return isl_ineq_separate;
 
  separate = isl_seq_any_non_zero(
      tab->mat->row[row] + off + tab->n_dead + pos + 1,
      tab->n_col - tab->n_dead - pos - 1);
 
  return !separate ? isl_ineq_adj_ineq : isl_ineq_separate;
}
 
/* Check the effect of inequality "ineq" on the tableau "tab".
 * The result may be
 *  isl_ineq_redundant: satisfied by all points in the tableau
 *  isl_ineq_separate:  satisfied by no point in the tableau
 *  isl_ineq_cut:   satisfied by some by not all points
 *  isl_ineq_adj_eq:  adjacent to an equality
 *  isl_ineq_adj_ineq:  adjacent to an inequality.
 */
enum isl_ineq_type isl_tab_ineq_type(struct isl_tab *tab, isl_int *ineq)
{
  enum isl_ineq_type type = isl_ineq_error;
  struct isl_tab_undo *snap = NULL;
  int con;
  int row;
 
  if (!tab)
    return isl_ineq_error;
 
  if (isl_tab_extend_cons(tab, 1) < 0)
    return isl_ineq_error;
 
  snap = isl_tab_snap(tab);
 
  con = isl_tab_add_row(tab, ineq);
  if (con < 0)
    goto error;
 
  row = tab->con[con].index;
  if (isl_tab_row_is_redundant(tab, row))
    type = isl_ineq_redundant;
  else if (isl_int_is_neg(tab->mat->row[row][1]) &&
     (tab->rational ||
        isl_int_abs_ge(tab->mat->row[row][1],
           tab->mat->row[row][0]))) {
    int nonneg = at_least_zero(tab, &tab->con[con]);
    if (nonneg < 0)
      goto error;
    if (nonneg)
      type = isl_ineq_cut;
    else
      type = separation_type(tab, row);
  } else {
    int red = con_is_redundant(tab, &tab->con[con]);
    if (red < 0)
      goto error;
    if (!red)
      type = isl_ineq_cut;
    else
      type = isl_ineq_redundant;
  }
 
  if (isl_tab_rollback(tab, snap))
    return isl_ineq_error;
  return type;
error:
  return isl_ineq_error;
}
 
isl_stat isl_tab_track_bmap(struct isl_tab *tab, __isl_take isl_basic_map *bmap)
{
  bmap = isl_basic_map_cow(bmap);
  if (!tab || !bmap)
    goto error;
 
  if (tab->empty) {
    bmap = isl_basic_map_set_to_empty(bmap);
    if (!bmap)
      goto error;
    tab->bmap = bmap;
    return isl_stat_ok;
  }
 
  isl_assert(tab->mat->ctx, tab->n_eq == bmap->n_eq, goto error);
  isl_assert(tab->mat->ctx,
        tab->n_con == bmap->n_eq + bmap->n_ineq, goto error);
 
  tab->bmap = bmap;
 
  return isl_stat_ok;
error:
  isl_basic_map_free(bmap);
  return isl_stat_error;
}
 
isl_stat isl_tab_track_bset(struct isl_tab *tab, __isl_take isl_basic_set *bset)
{
  return isl_tab_track_bmap(tab, bset_to_bmap(bset));
}
 
__isl_keep isl_basic_set *isl_tab_peek_bset(struct isl_tab *tab)
{
  if (!tab)
    return NULL;
 
  return bset_from_bmap(tab->bmap);
}
 
/* Print information about a tab variable representing a variable or
 * a constraint.
 * In particular, print its position (row or column) in the tableau and
 * an indication of whether it is zero, redundant and/or frozen.
 * Note that only constraints can be frozen.
 */
static void print_tab_var(FILE *out, struct isl_tab_var *var)
{
  fprintf(out, "%c%d%s%s", var->is_row ? 'r' : 'c',
        var->index,
        var->is_zero ? " [=0]" :
        var->is_redundant ? " [R]" : "",
        var->frozen ? " [F]" : "");
}
 
static void isl_tab_print_internal(__isl_keep struct isl_tab *tab,
  FILE *out, int indent)
{
  unsigned r, c;
  int i;
 
  if (!tab) {
    fprintf(out, "%*snull tab\n", indent, "");
    return;
  }
  fprintf(out, "%*sn_redundant: %d, n_dead: %d", indent, "",
    tab->n_redundant, tab->n_dead);
  if (tab->rational)
    fprintf(out, ", rational");
  if (tab->empty)
    fprintf(out, ", empty");
  fprintf(out, "\n");
  fprintf(out, "%*s[", indent, "");
  for (i = 0; i < tab->n_var; ++i) {
    if (i)
      fprintf(out, (i == tab->n_param ||
              i == tab->n_var - tab->n_div) ? "; "
                    : ", ");
    print_tab_var(out, &tab->var[i]);
  }
  fprintf(out, "]\n");
  fprintf(out, "%*s[", indent, "");
  for (i = 0; i < tab->n_con; ++i) {
    if (i)
      fprintf(out, ", ");
    print_tab_var(out, &tab->con[i]);
  }
  fprintf(out, "]\n");
  fprintf(out, "%*s[", indent, "");
  for (i = 0; i < tab->n_row; ++i) {
    const char *sign = "";
    if (i)
      fprintf(out, ", ");
    if (tab->row_sign) {
      if (tab->row_sign[i] == isl_tab_row_unknown)
        sign = "?";
      else if (tab->row_sign[i] == isl_tab_row_neg)
        sign = "-";
      else if (tab->row_sign[i] == isl_tab_row_pos)
        sign = "+";
      else
        sign = "+-";
    }
    fprintf(out, "r%d: %d%s%s", i, tab->row_var[i],
        isl_tab_var_from_row(tab, i)->is_nonneg ? " [>=0]" : "", sign);
  }
  fprintf(out, "]\n");
  fprintf(out, "%*s[", indent, "");
  for (i = 0; i < tab->n_col; ++i) {
    if (i)
      fprintf(out, ", ");
    fprintf(out, "c%d: %d%s", i, tab->col_var[i],
        var_from_col(tab, i)->is_nonneg ? " [>=0]" : "");
  }
  fprintf(out, "]\n");
  r = tab->mat->n_row;
  tab->mat->n_row = tab->n_row;
  c = tab->mat->n_col;
  tab->mat->n_col = 2 + tab->M + tab->n_col;
  isl_mat_print_internal(tab->mat, out, indent);
  tab->mat->n_row = r;
  tab->mat->n_col = c;
  if (tab->bmap)
    isl_basic_map_print_internal(tab->bmap, out, indent);
}
 
void isl_tab_dump(__isl_keep struct isl_tab *tab)
{
  isl_tab_print_internal(tab, stderr, 0);
}