isl_mat.c: extract out shared check_col

[isl.git] / isl_tab.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2013      Ecole Normale Superieure
 * Copyright 2014      INRIA Rocquencourt
 *
 * 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>

/*
 * 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;
        tab->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 struct isl_mat *tab_mat_product(struct isl_mat *mat1,
        struct 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 int push_union(struct isl_tab *tab,
        enum isl_tab_undo_type type, union isl_tab_undo_val u) WARN_UNUSED;
static int 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 -1;
        if (!tab->need_undo)
                return 0;

        undo = isl_alloc_type(tab->mat->ctx, struct isl_tab_undo);
        if (!undo)
                return -1;
        undo->type = type;
        undo->u = u;
        undo->next = tab->top;
        tab->top = undo;

        return 0;
}

int 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);
}

int 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.
 */
int 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 -1;
        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);
}

int 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);
}

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.
 */
int isl_tab_save_samples(struct isl_tab *tab)
{
        union isl_tab_undo_val u;

        if (!tab)
                return -1;

        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;
}

int isl_tab_mark_empty(struct isl_tab *tab)
{
        if (!tab)
                return -1;
        if (!tab->empty && tab->need_undo)
                if (isl_tab_push(tab, isl_tab_undo_empty) < 0)
                        return -1;
        tab->empty = 1;
        return 0;
}

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 down until either
 *      - no more down 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_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
                                    tab->n_col - tab->n_dead) != -1)
                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.
 */
static int close_row(struct isl_tab *tab, struct isl_tab_var *var) WARN_UNUSED;
static int close_row(struct isl_tab *tab, struct isl_tab_var *var)
{
        int j;
        struct isl_mat *mat = tab->mat;
        unsigned off = 2 + tab->M;

        isl_assert(tab->mat->ctx, var->is_nonneg, return -1);
        var->is_zero = 1;
        if (tab->need_undo)
                if (isl_tab_push_var(tab, isl_tab_undo_zero, var) < 0)
                        return -1;
        for (j = tab->n_dead; j < tab->n_col; ++j) {
                int recheck;
                if (isl_int_is_zero(mat->row[var->index][off + j]))
                        continue;
                isl_assert(tab->mat->ctx,
                    isl_int_is_neg(mat->row[var->index][off + j]), return -1);
                recheck = isl_tab_kill_col(tab, j);
                if (recheck < 0)
                        return -1;
                if (recheck)
                        --j;
        }
        if (isl_tab_mark_redundant(tab, var->index) < 0)
                return -1;
        if (tab_is_manifestly_empty(tab) && isl_tab_mark_empty(tab) < 0)
                return -1;
        return 0;
}

/* Add a constraint to the tableau and allocate a row for it.
 * Return the index into the constraint array "con".
 */
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 variable to the tableau and allocate a column for it.
 * Return the index into the variable array "var".
 */
int isl_tab_allocate_var(struct isl_tab *tab)
{
        if (!tab)
                return -1;

        return isl_tab_insert_var(tab, tab->n_var);
}

/* 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.
 *
 * 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 int drop_row(struct isl_tab *tab, int row)
{
        isl_assert(tab->mat->ctx, ~tab->row_var[row] == tab->n_con - 1, return -1);
        if (row != tab->n_row - 1)
                swap_rows(tab, row, tab->n_row - 1);
        tab->n_row--;
        tab->n_con--;
        return 0;
}

/* 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 int 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 -1;
        return 0;
}

/* Add inequality "ineq" and check if it conflicts with the
 * previously added constraints or if it is obviously redundant.
 */
int isl_tab_add_ineq(struct isl_tab *tab, isl_int *ineq)
{
        int r;
        int sgn;
        isl_int cst;

        if (!tab)
                return -1;
        if (tab->bmap) {
                struct isl_basic_map *bmap = tab->bmap;

                isl_assert(tab->mat->ctx, tab->n_eq == bmap->n_eq, return -1);
                isl_assert(tab->mat->ctx,
                            tab->n_con == bmap->n_eq + bmap->n_ineq, return -1);
                tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq);
                if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
                        return -1;
                if (!tab->bmap)
                        return -1;
        }
        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 -1;
        tab->con[r].is_nonneg = 1;
        if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
                return -1;
        if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
                if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
                        return -1;
                return 0;
        }

        sgn = restore_row(tab, &tab->con[r]);
        if (sgn < -1)
                return -1;
        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 -1;
        return 0;
}

/* 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.
 */
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;
}

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 (tab->M && !isl_int_is_zero(tab->mat->row[row][2]))
                return 0;
        return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
                                        tab->n_col - tab->n_dead) == -1;
}

/* Add an equality that is known to be valid for the given 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;
}

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.
 */
int 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 -1;
        isl_assert(tab->mat->ctx, !tab->M, return -1);

        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 -1;

        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 -1;
                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 -1;
                if (!tab->bmap)
                        return -1;
                if (add_zero_row(tab) < 0)
                        return -1;
        }

        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 -1;
                if (sgn < 0) {
                        if (isl_tab_mark_empty(tab) < 0)
                                return -1;
                        return 0;
                }
        }

        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;
}

/* 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 struct isl_vec *ineq_for_div(struct isl_basic_map *bmap, unsigned div)
{
        unsigned total;
        unsigned div_pos;
        struct isl_vec *ineq;

        if (!bmap)
                return NULL;

        total = isl_basic_map_total_dim(bmap);
        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.
 */
static int add_div_constraints(struct isl_tab *tab, unsigned div,
        int (*add_ineq)(void *user, isl_int *), void *user)
{
        unsigned total;
        unsigned div_pos;
        struct isl_vec *ineq;

        total = isl_basic_map_total_dim(tab->bmap);
        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 0;
error:
        isl_vec_free(ineq);
        return -1;
}

/* 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;
}

/* Add an extra div, prescribed by "div" to the tableau and
 * the associated bmap (which is assumed to be non-NULL).
 *
 * 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_add_div(struct isl_tab *tab, __isl_keep isl_vec *div,
        int (*add_ineq)(void *user, isl_int *), void *user)
{
        int r;
        int k;
        int nonneg;

        if (!tab || !div)
                return -1;

        isl_assert(tab->mat->ctx, tab->bmap, 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_allocate_var(tab);
        if (r < 0)
                return -1;

        if (nonneg)
                tab->var[r].is_nonneg = 1;

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

        if (add_div_constraints(tab, k, add_ineq, user) < 0)
                return -1;

        return r;
}

/* 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;

        if (!bmap)
                return NULL;
        tab = isl_tab_alloc(bmap->ctx,
                            isl_basic_map_total_dim(bmap) + bmap->n_ineq + 1,
                            isl_basic_map_total_dim(bmap), 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;
        unsigned offset = 0;

        if (!bset)
                return NULL;
        if (parametric)
                offset = isl_basic_set_dim(bset, isl_dim_param);
        tab = isl_tab_alloc(bset->ctx, bset->n_eq + bset->n_ineq,
                                isl_basic_set_total_dim(bset) - 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.
 */
int isl_tab_cone_is_bounded(struct isl_tab *tab)
{
        int i;

        if (!tab)
                return -1;
        if (tab->empty)
                return 1;
        if (tab->n_dead == tab->n_col)
                return 1;

        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 -1;
                        if (sgn != 0)
                                return 0;
                        if (close_row(tab, var) < 0)
                                return -1;
                        break;
                }
                if (tab->n_dead == tab->n_col)
                        return 1;
                if (i == tab->n_row)
                        return 0;
        }
}

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;
}

struct 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;
}

/* 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.
 */
struct isl_basic_map *isl_basic_map_update_from_tab(struct 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;
}

struct isl_basic_set *isl_basic_set_update_from_tab(struct isl_basic_set *bset,
        struct isl_tab *tab)
{
        return (struct isl_basic_set *)isl_basic_map_update_from_tab(
                (struct isl_basic_map *)bset, tab);
}

/* Given a non-negative variable "var", add a new non-negative variable
 * that is the opposite of "var", ensuring that var can only attain the
 * value zero.
 * 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 int 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 0;
        isl_assert(tab->mat->ctx, !var->is_redundant, return -1);
        isl_assert(tab->mat->ctx, var->is_nonneg, return -1);

        if (isl_tab_extend_cons(tab, 1) < 0)
                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;
        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++;
        if (isl_tab_push_var(tab, isl_tab_undo_allocate, &tab->con[r]) < 0)
                return -1;

        sgn = sign_of_max(tab, &tab->con[r]);
        if (sgn < -1)
                return -1;
        if (sgn < 0) {
                if (isl_tab_mark_empty(tab) < 0)
                        return -1;
                return 0;
        }
        tab->con[r].is_nonneg = 1;
        if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
                return -1;
        /* sgn == 0 */
        if (close_row(tab, &tab->con[r]) < 0)
                return -1;

        return 0;
}

/* 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]);
}

/* 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.
 */
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;
                for (i = tab->n_redundant; i < tab->n_row; ++i) {
                        var = isl_tab_var_from_row(tab, i);
                        if (var->marked)
                                break;
                }
                if (i == tab->n_row) {
                        for (i = tab->n_dead; i < tab->n_col; ++i) {
                                var = var_from_col(tab, i);
                                if (var->marked)
                                        break;
                        }
                        if (i == tab->n_col)
                                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)
                                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;
}

/* Rotate the "n" constraints starting at "first" to the right,
 * putting the last constraint in the position of the first constraint.
 */
static int rotate_constraints(struct isl_tab *tab, int first, int n)
{
        int i, last;
        struct isl_tab_var var;

        if (n <= 1)
                return 0;

        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 -1;
        }
        tab->con[first] = var;
        if (update_con_after_move(tab, first, last) < 0)
                return -1;

        return 0;
}

/* 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.
 *
 * Note that this function leaves "bmap" in a temporary state
 * as it does not call isl_basic_map_gauss.  Calling this function
 * is the responsibility of the caller.
 */
__isl_give isl_basic_map *isl_tab_make_equalities_explicit(struct isl_tab *tab,
        __isl_take isl_basic_map *bmap)
{
        int i;

        if (!tab || !bmap)
                return isl_basic_map_free(bmap);
        if (tab->empty)
                return bmap;

        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(tab, 0, tab->n_eq + i + 1) < 0)
                        return isl_basic_map_free(bmap);
                if (rotate_constraints(tab, tab->n_eq + i + 1,
                                        bmap->n_ineq - i) < 0)
                        return isl_basic_map_free(bmap);
                tab->n_eq++;
        }

        return bmap;
}

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;
        }
}

/* 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,
 * 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) 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]) &&
                (!tab->M || isl_int_is_zero(tab->mat->row[row][2])) &&
                isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
                                        tab->n_col - tab->n_dead) == -1;
}

/* 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).
 */
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
                        isl_int_cdiv_q(*opt, tab->mat->row[var->index][1],
                                             tab->mat->row[var->index][0]);
        }
        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 (!tab)
                return -1;
        if (con < 0 || con >= tab->n_con)
                isl_die(isl_tab_get_ctx(tab), isl_error_invalid,
                        "position out of bounds", 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;
}

/* Take a snapshot of the tableau that can be restored by s 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;
}

/* Undo the operation performed by isl_tab_relax.
 */
static int unrelax(struct isl_tab *tab, struct isl_tab_var *var) WARN_UNUSED;
static int 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 -1;

        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 -1);
                }
        } 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 0;
}

/* 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 int ununrestrict(struct isl_tab *tab, struct isl_tab_var *var)
{
        var->is_nonneg = 1;

        if (var->is_row && restore_row(tab, var) < -1)
                return -1;

        return 0;
}

static int perform_undo_var(struct isl_tab *tab, struct isl_tab_undo *undo) WARN_UNUSED;
static int 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:
                var->is_redundant = 0;
                tab->n_redundant--;
                restore_row(tab, isl_tab_var_from_row(tab, tab->n_redundant));
                break;
        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 -1);
                        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 -1;
                        } else if (!min_is_manifestly_unbounded(tab, var)) {
                                if (to_row(tab, var, -1) < 0)
                                        return -1;
                        } else
                                if (to_row(tab, var, 0) < 0)
                                        return -1;
                }
                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 -1);
        }

        return 0;
}

/* 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 int perform_undo(struct isl_tab *tab, struct isl_tab_undo *undo) WARN_UNUSED;
static int 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:
                return isl_basic_map_free_equality(tab->bmap, 1);
        case isl_tab_undo_bmap_ineq:
                return isl_basic_map_free_inequality(tab->bmap, 1);
        case isl_tab_undo_bmap_div:
                if (isl_basic_map_free_div(tab->bmap, 1) < 0)
                        return -1;
                if (tab->samples)
                        tab->samples->n_col--;
                break;
        case isl_tab_undo_saved_basis:
                if (restore_basis(tab, undo->u.col_var) < 0)
                        return -1;
                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);
        default:
                isl_assert(tab->mat->ctx, 0, return -1);
        }
        return 0;
}

/* Return the tableau to the state it was in when the snapshot "snap"
 * was taken.
 */
int isl_tab_rollback(struct isl_tab *tab, struct isl_tab_undo *snap)
{
        struct isl_tab_undo *undo, *next;

        if (!tab)
                return -1;

        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 -1;
                }
                free_undo_record(undo);
        }
        tab->in_undo = 0;
        tab->top = undo;
        if (!undo)
                return -1;
        return 0;
}

/* 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;
        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;

        pos = isl_seq_first_non_zero(
                        tab->mat->row[row] + off + tab->n_dead + pos + 1,
                        tab->n_col - tab->n_dead - pos - 1);

        return pos == -1 ? 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;
}

int 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 0;
        }

        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 0;
error:
        isl_basic_map_free(bmap);
        return -1;
}

int isl_tab_track_bset(struct isl_tab *tab, __isl_take isl_basic_set *bset)
{
        return isl_tab_track_bmap(tab, (isl_basic_map *)bset);
}

__isl_keep isl_basic_set *isl_tab_peek_bset(struct isl_tab *tab)
{
        if (!tab)
                return NULL;

        return (isl_basic_set *)tab->bmap;
}

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) ? "; "
                                                                    : ", ");
                fprintf(out, "%c%d%s", tab->var[i].is_row ? 'r' : 'c',
                                        tab->var[i].index,
                                        tab->var[i].is_zero ? " [=0]" :
                                        tab->var[i].is_redundant ? " [R]" : "");
        }
        fprintf(out, "]\n");
        fprintf(out, "%*s[", indent, "");
        for (i = 0; i < tab->n_con; ++i) {
                if (i)
                        fprintf(out, ", ");
                fprintf(out, "%c%d%s", tab->con[i].is_row ? 'r' : 'c',
                                        tab->con[i].index,
                                        tab->con[i].is_zero ? " [=0]" :
                                        tab->con[i].is_redundant ? " [R]" : "");
        }
        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);
}