isl_map_coalesce: don't drop rational flag from result if set in input

[isl.git] / isl_tab.c

#include "isl_map_private.h"
#include "isl_tab.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)
{
        int i;
        struct isl_tab *tab;

        tab = isl_calloc_type(ctx, struct isl_tab);
        if (!tab)
                return NULL;
        tab->mat = isl_mat_alloc(ctx, n_row, 2 + n_var);
        if (!tab->mat)
                goto error;
        tab->var = isl_alloc_array(ctx, struct isl_tab_var, n_var);
        if (!tab->var)
                goto error;
        tab->con = isl_alloc_array(ctx, struct isl_tab_var, n_row);
        if (!tab->con)
                goto error;
        tab->col_var = isl_alloc_array(ctx, int, n_var);
        if (!tab->col_var)
                goto error;
        tab->row_var = isl_alloc_array(ctx, int, n_row);
        if (!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->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->n_dead = 0;
        tab->n_redundant = 0;
        tab->need_undo = 0;
        tab->rational = 0;
        tab->empty = 0;
        tab->in_undo = 0;
        tab->bottom.type = isl_tab_undo_bottom;
        tab->bottom.next = NULL;
        tab->top = &tab->bottom;
        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

static int extend_cons(struct isl_tab *tab, unsigned n_new)
{
        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, 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;
        }
        return 0;
}

struct isl_tab *isl_tab_extend(struct isl_tab *tab, unsigned n_new)
{
        if (extend_cons(tab, n_new) >= 0)
                return tab;

        isl_tab_free(tab);
        return NULL;
}

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);
        }
        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);
        free(tab->var);
        free(tab->con);
        free(tab->row_var);
        free(tab->col_var);
        free(tab);
}

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];
}

static struct isl_tab_var *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;

        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][2 + var->index]))
                        continue;
                if (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;

        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][2 + var->index]))
                        continue;
                if (var_from_row(tab, i)->is_nonneg)
                        return 0;
        }
        return 1;
}

/* 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;

        isl_int_init(t);
        r = -1;
        for (j = tab->n_redundant; j < tab->n_row; ++j) {
                if (var && j == var->index)
                        continue;
                if (!var_from_row(tab, j)->is_nonneg)
                        continue;
                if (sgn * isl_int_sgn(tab->mat->row[j][2 + c]) >= 0)
                        continue;
                if (r < 0) {
                        r = j;
                        continue;
                }
                isl_int_mul(t, tab->mat->row[r][1], tab->mat->row[j][2 + c]);
                isl_int_submul(t, tab->mat->row[j][1], tab->mat->row[r][2 + c]);
                tsgn = sgn * isl_int_sgn(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];

        c = -1;
        for (j = tab->n_dead; j < tab->n_col; ++j) {
                if (isl_int_is_zero(tr[2 + j]))
                        continue;
                if (isl_int_sgn(tr[2 + 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[2 + 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 variables.
 */
static int is_redundant(struct isl_tab *tab, int row)
{
        int i;

        if (tab->row_var[row] < 0 && !var_from_row(tab, row)->is_nonneg)
                return 0;

        if (isl_int_is_neg(tab->mat->row[row][1]))
                return 0;

        for (i = tab->n_dead; i < tab->n_col; ++i) {
                if (isl_int_is_zero(tab->mat->row[row][2 + i]))
                        continue;
                if (isl_int_is_neg(tab->mat->row[row][2 + 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;
        t = tab->row_var[row1];
        tab->row_var[row1] = tab->row_var[row2];
        tab->row_var[row2] = t;
        var_from_row(tab, row1)->index = row1;
        var_from_row(tab, row2)->index = row2;
        tab->mat = isl_mat_swap_rows(tab->mat, row1, row2);
}

static void push(struct isl_tab *tab,
        enum isl_tab_undo_type type, struct isl_tab_var *var)
{
        struct isl_tab_undo *undo;

        if (!tab->need_undo)
                return;

        undo = isl_alloc_type(tab->mat->ctx, struct isl_tab_undo);
        if (!undo) {
                free_undo(tab);
                tab->top = NULL;
                return;
        }
        undo->type = type;
        undo->var = var;
        undo->next = tab->top;
        tab->top = undo;
}

/* 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.
 */
static int mark_redundant(struct isl_tab *tab, int row)
{
        struct isl_tab_var *var = var_from_row(tab, row);
        var->is_redundant = 1;
        isl_assert(tab->mat->ctx, row >= tab->n_redundant, return);
        if (tab->need_undo || tab->row_var[row] >= 0) {
                if (tab->row_var[row] >= 0) {
                        var->is_nonneg = 1;
                        push(tab, isl_tab_undo_nonneg, var);
                }
                if (row != tab->n_redundant)
                        swap_rows(tab, row, tab->n_redundant);
                push(tab, isl_tab_undo_redundant, var);
                tab->n_redundant++;
                return 0;
        } else {
                if (row != tab->n_row - 1)
                        swap_rows(tab, row, tab->n_row - 1);
                var_from_row(tab, tab->n_row - 1)->index = -1;
                tab->n_row--;
                return 1;
        }
}

static void mark_empty(struct isl_tab *tab)
{
        if (!tab->empty && tab->need_undo)
                push(tab, isl_tab_undo_empty, NULL);
        tab->empty = 1;
}

/* 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)
 *
 */
static void pivot(struct isl_tab *tab, int row, int col)
{
        int i, j;
        int sgn;
        int t;
        struct isl_mat *mat = tab->mat;
        struct isl_tab_var *var;

        isl_int_swap(mat->row[row][0], mat->row[row][2 + 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][2 + col], mat->row[row][2 + col]);
        } else
                for (j = 0; j < 1 + tab->n_col; ++j) {
                        if (j == 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->row[row], 2 + tab->n_col);
        for (i = 0; i < tab->n_row; ++i) {
                if (i == row)
                        continue;
                if (isl_int_is_zero(mat->row[i][2 + col]))
                        continue;
                isl_int_mul(mat->row[i][0], mat->row[i][0], mat->row[row][0]);
                for (j = 0; j < 1 + tab->n_col; ++j) {
                        if (j == 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][2 + col], mat->row[row][1 + j]);
                }
                isl_int_mul(mat->row[i][2 + col],
                            mat->row[i][2 + col], mat->row[row][2 + col]);
                if (!isl_int_is_one(mat->row[row][0]))
                        isl_seq_normalize(mat->row[i], 2 + tab->n_col);
        }
        t = tab->row_var[row];
        tab->row_var[row] = tab->col_var[col];
        tab->col_var[col] = t;
        var = 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;
        if (tab->in_undo)
                return;
        for (i = tab->n_redundant; i < tab->n_row; ++i) {
                if (isl_int_is_zero(mat->row[i][2 + col]))
                        continue;
                if (!var_from_row(tab, i)->frozen &&
                    is_redundant(tab, i))
                        if (mark_redundant(tab, i))
                                --i;
        }
}

/* 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.
 */
static void to_row(struct isl_tab *tab, struct isl_tab_var *var, int sign)
{
        int r;

        if (var->is_row)
                return;

        r = pivot_row(tab, NULL, sign, var->index);
        isl_assert(tab->mat->ctx, r >= 0, return);
        pivot(tab, r, var->index);
}

static void check_table(struct isl_tab *tab)
{
        int i;

        if (tab->empty)
                return;
        for (i = 0; i < tab->n_row; ++i) {
                if (!var_from_row(tab, i)->is_nonneg)
                        continue;
                assert(!isl_int_is_neg(tab->mat->row[i][1]));
        }
}

/* 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;
        to_row(tab, var, 1);
        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]);
                pivot(tab, row, col);
                if (!var->is_row) /* manifestly unbounded */
                        return 1;
        }
        return 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 (isl_int_is_neg(tab->mat->row[var->index][1])) {
                find_pivot(tab, var, var, 1, &row, &col);
                if (row == -1)
                        break;
                pivot(tab, row, col);
                if (!var->is_row) /* manifestly unbounded */
                        return 1;
        }
        return isl_int_sgn(tab->mat->row[var->index][1]);
}

/* 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;
                pivot(tab, row, col);
        }
        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;

        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);
                pivot(tab, row, col);
                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)
                                        pivot(tab, row, col);
                                else
                                        restore_row(tab, var);
                        }
                        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);
                pivot(tab, row, col);
                if (var->is_redundant)
                        return 0;
        }
        if (var->is_nonneg) {
                /* pivot back to non-negative value */
                if (!pivot_var->is_redundant && pivot_var->index == row)
                        pivot(tab, row, col);
                else
                        restore_row(tab, var);
        }
        return -1;
}

/* Return 1 if "var" can attain values <= -1.
 * Return 0 otherwise.
 *
 * The sample value of "var" is assumed to be non-negative when the
 * the function is called and will be made non-negative again before
 * the function returns.
 */
static int 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);
                pivot(tab, row, col);
                if (var->is_redundant)
                        return 0;
                if (isl_int_is_neg(tab->mat->row[var->index][1]) &&
                    isl_int_abs_ge(tab->mat->row[var->index][1],
                                   tab->mat->row[var->index][0])) {
                        if (var->is_nonneg) {
                                if (!pivot_var->is_redundant &&
                                    pivot_var->index == row)
                                        pivot(tab, row, col);
                                else
                                        restore_row(tab, var);
                        }
                        return 1;
                }
        }
        if (var->is_redundant)
                return 0;
        do {
                find_pivot(tab, var, var, -1, &row, &col);
                if (row == var->index)
                        return 1;
                if (row == -1)
                        return 0;
                pivot_var = var_from_col(tab, col);
                pivot(tab, row, col);
                if (var->is_redundant)
                        return 0;
        } while (!isl_int_is_neg(tab->mat->row[var->index][1]) ||
                 isl_int_abs_lt(tab->mat->row[var->index][1],
                                tab->mat->row[var->index][0]));
        if (var->is_nonneg) {
                /* pivot back to non-negative value */
                if (!pivot_var->is_redundant && pivot_var->index == row)
                        pivot(tab, row, col);
                restore_row(tab, var);
        }
        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;
        to_row(tab, var, 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;
                pivot(tab, row, col);
        }
        return 1;
}

static void swap_cols(struct isl_tab *tab, int col1, int col2)
{
        int t;
        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, 2 + col1, 2 + 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.
 */
static int kill_col(struct isl_tab *tab, int col)
{
        var_from_col(tab, col)->is_zero = 1;
        if (tab->need_undo) {
                push(tab, isl_tab_undo_zero, var_from_col(tab, col));
                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;
        }
}

/* 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 void close_row(struct isl_tab *tab, struct isl_tab_var *var)
{
        int j;
        struct isl_mat *mat = tab->mat;

        isl_assert(tab->mat->ctx, var->is_nonneg, return);
        var->is_zero = 1;
        for (j = tab->n_dead; j < tab->n_col; ++j) {
                if (isl_int_is_zero(mat->row[var->index][2 + j]))
                        continue;
                isl_assert(tab->mat->ctx,
                        isl_int_is_neg(mat->row[var->index][2 + j]), return);
                if (kill_col(tab, j))
                        --j;
        }
        mark_redundant(tab, var->index);
}

/* 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.
 */
static int add_row(struct isl_tab *tab, isl_int *line)
{
        int i;
        unsigned r;
        isl_int *row;
        isl_int a, b;

        isl_assert(tab->mat->ctx, tab->n_row < tab->mat->n_row, return -1);

        isl_int_init(a);
        isl_int_init(b);
        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->row_var[tab->n_row] = ~r;
        row = tab->mat->row[tab->n_row];
        isl_int_set_si(row[0], 1);
        isl_int_set(row[1], line[0]);
        isl_seq_clr(row + 2, 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->n_col);
                } else
                        isl_int_addmul(row[2 + tab->var[i].index],
                                                        line[1 + i], row[0]);
        }
        isl_seq_normalize(row, 2 + tab->n_col);
        tab->n_row++;
        tab->n_con++;
        push(tab, isl_tab_undo_allocate, &tab->con[r]);
        isl_int_clear(a);
        isl_int_clear(b);

        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;
}

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

        if (!tab)
                return NULL;
        r = add_row(tab, ineq);
        if (r < 0)
                goto error;
        tab->con[r].is_nonneg = 1;
        push(tab, isl_tab_undo_nonneg, &tab->con[r]);
        if (is_redundant(tab, tab->con[r].index)) {
                mark_redundant(tab, tab->con[r].index);
                return tab;
        }

        sgn = restore_row(tab, &tab->con[r]);
        if (sgn < 0)
                mark_empty(tab);
        else if (tab->con[r].is_row &&
                 is_redundant(tab, tab->con[r].index))
                mark_redundant(tab, tab->con[r].index);
        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

/* 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)
{
        int i;
        int row, col;

        if (!var->is_row)
                return;

        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);
                pivot(tab, row, col);
                if (!var->is_row)
                        return;
        }

        for (i = tab->n_dead; i < tab->n_col; ++i)
                if (!isl_int_is_zero(tab->mat->row[var->index][2 + i]))
                        break;

        isl_assert(tab->mat->ctx, i < tab->n_col, return -1);
        pivot(tab, var->index, i);

        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 = add_row(tab, eq);
        if (r < 0)
                goto error;

        r = tab->con[r].index;
        for (i = tab->n_dead; i < tab->n_col; ++i) {
                if (isl_int_is_zero(tab->mat->row[r][2 + i]))
                        continue;
                pivot(tab, r, i);
                kill_col(tab, i);
                break;
        }
        tab->n_eq++;

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

/* Add an equality that is known to be valid for the given tableau.
 */
struct isl_tab *isl_tab_add_valid_eq(struct isl_tab *tab, isl_int *eq)
{
        struct isl_tab_var *var;
        int i;
        int r;

        if (!tab)
                return NULL;
        r = add_row(tab, eq);
        if (r < 0)
                goto error;

        var = &tab->con[r];
        r = var->index;
        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->is_nonneg = 1;
        if (to_col(tab, var) < 0)
                goto error;
        var->is_nonneg = 0;
        kill_col(tab, var->index);

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

struct isl_tab *isl_tab_from_basic_map(struct isl_basic_map *bmap)
{
        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));
        if (!tab)
                return NULL;
        tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
                mark_empty(tab);
                return tab;
        }
        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) {
                tab = isl_tab_add_ineq(tab, bmap->ineq[i]);
                if (!tab || tab->empty)
                        return tab;
        }
        return tab;
}

struct isl_tab *isl_tab_from_basic_set(struct isl_basic_set *bset)
{
        return isl_tab_from_basic_map((struct isl_basic_map *)bset);
}

/* Construct a tableau corresponding to the recession cone of "bmap".
 */
struct isl_tab *isl_tab_from_recession_cone(struct isl_basic_map *bmap)
{
        isl_int cst;
        int i;
        struct isl_tab *tab;

        if (!bmap)
                return NULL;
        tab = isl_tab_alloc(bmap->ctx, bmap->n_eq + bmap->n_ineq,
                                isl_basic_map_total_dim(bmap));
        if (!tab)
                return NULL;
        tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);

        isl_int_init(cst);
        for (i = 0; i < bmap->n_eq; ++i) {
                isl_int_swap(bmap->eq[i][0], cst);
                tab = add_eq(tab, bmap->eq[i]);
                isl_int_swap(bmap->eq[i][0], cst);
                if (!tab)
                        goto done;
        }
        for (i = 0; i < bmap->n_ineq; ++i) {
                int r;
                isl_int_swap(bmap->ineq[i][0], cst);
                r = add_row(tab, bmap->ineq[i]);
                isl_int_swap(bmap->ineq[i][0], cst);
                if (r < 0)
                        goto error;
                tab->con[r].is_nonneg = 1;
                push(tab, isl_tab_undo_nonneg, &tab->con[r]);
        }
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;
                        var = var_from_row(tab, i);
                        if (!var->is_nonneg)
                                continue;
                        if (sign_of_max(tab, var) != 0)
                                return 0;
                        close_row(tab, var);
                        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]);
        }
        isl_seq_normalize(vec->block.data, vec->size);

        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 (!tab->rational &&
            !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 struct isl_tab *cut_to_hyperplane(struct isl_tab *tab,
        struct isl_tab_var *var)
{
        unsigned r;
        isl_int *row;
        int sgn;

        if (extend_cons(tab, 1) < 0)
                goto error;

        r = tab->n_con;
        tab->con[r].index = tab->n_row;
        tab->con[r].is_row = 1;
        tab->con[r].is_nonneg = 0;
        tab->con[r].is_zero = 0;
        tab->con[r].is_redundant = 0;
        tab->con[r].frozen = 0;
        tab->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[2 + var->index], -1);
        }

        tab->n_row++;
        tab->n_con++;
        push(tab, isl_tab_undo_allocate, &tab->con[r]);

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

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

/* 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.
 */
struct isl_tab *isl_tab_relax(struct isl_tab *tab, int con)
{
        struct isl_tab_var *var;
        if (!tab)
                return NULL;

        var = &tab->con[con];

        if (!var->is_row && !max_is_manifestly_unbounded(tab, var))
                to_row(tab, var, 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]);
        else {
                int i;

                for (i = 0; i < tab->n_row; ++i) {
                        if (isl_int_is_zero(tab->mat->row[i][2 + var->index]))
                                continue;
                        isl_int_sub(tab->mat->row[i][1], tab->mat->row[i][1],
                            tab->mat->row[i][2 + var->index]);
                }

        }

        push(tab, isl_tab_undo_relax, var);

        return tab;
}

struct isl_tab *isl_tab_select_facet(struct isl_tab *tab, int con)
{
        if (!tab)
                return NULL;

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

/* 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.
 */
struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab)
{
        int i;
        unsigned n_marked;

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

        n_marked = 0;
        for (i = tab->n_redundant; i < tab->n_row; ++i) {
                struct isl_tab_var *var = 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;
                for (i = tab->n_redundant; i < tab->n_row; ++i) {
                        var = 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--;
                if (sign_of_max(tab, var) == 0)
                        close_row(tab, var);
                else if (!tab->rational && !at_least_one(tab, var)) {
                        tab = cut_to_hyperplane(tab, var);
                        return isl_tab_detect_equalities(tab);
                }
                for (i = tab->n_redundant; i < tab->n_row; ++i) {
                        var = var_from_row(tab, i);
                        if (!var->marked)
                                continue;
                        if (may_be_equality(tab, i))
                                continue;
                        var->marked = 0;
                        n_marked--;
                }
        }

        return tab;
}

/* 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).
 */
struct isl_tab *isl_tab_detect_redundant(struct isl_tab *tab)
{
        int i;
        unsigned n_marked;

        if (!tab)
                return NULL;
        if (tab->empty)
                return tab;
        if (tab->n_redundant == tab->n_row)
                return tab;

        n_marked = 0;
        for (i = tab->n_redundant; i < tab->n_row; ++i) {
                struct isl_tab_var *var = 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;
                for (i = tab->n_redundant; i < tab->n_row; ++i) {
                        var = 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--;
                if ((tab->rational ? (sign_of_min(tab, var) >= 0)
                                   : !min_at_most_neg_one(tab, var)) &&
                    !var->is_redundant)
                        mark_redundant(tab, var->index);
                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 tab;
}

int isl_tab_is_equality(struct isl_tab *tab, int con)
{
        int row;

        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;

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

/* Return the minimial 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,
 * minmimal 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->empty)
                return isl_lp_empty;

        snap = isl_tab_snap(tab);
        r = add_row(tab, f);
        if (r < 0)
                return isl_lp_error;
        var = &tab->con[r];
        isl_int_mul(tab->mat->row[var->index][0],
                    tab->mat->row[var->index][0], denom);
        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;
                pivot(tab, row, col);
        }
        if (isl_tab_rollback(tab, snap) < 0)
                return isl_lp_error;
        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) {
                        if (tab->con[i].is_row)
                                isl_int_set_si(tab->dual->el[1 + i], 0);
                        else {
                                int pos = 2 + tab->con[i].index;
                                isl_int_set(tab->dual->el[1 + i],
                                            tab->mat->row[var->index][pos]);
                        }
                }
        }
        if (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]);
        }
        return res;
}

int isl_tab_is_redundant(struct isl_tab *tab, int con)
{
        int row;
        unsigned n_col;

        if (!tab)
                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 void unrelax(struct isl_tab *tab, struct isl_tab_var *var)
{
        if (!var->is_row && !max_is_manifestly_unbounded(tab, var))
                to_row(tab, var, 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]);
        else {
                int i;

                for (i = 0; i < tab->n_row; ++i) {
                        if (isl_int_is_zero(tab->mat->row[i][2 + var->index]))
                                continue;
                        isl_int_add(tab->mat->row[i][1], tab->mat->row[i][1],
                            tab->mat->row[i][2 + var->index]);
                }

        }
}

static void perform_undo(struct isl_tab *tab, struct isl_tab_undo *undo)
{
        switch(undo->type) {
        case isl_tab_undo_empty:
                tab->empty = 0;
                break;
        case isl_tab_undo_nonneg:
                undo->var->is_nonneg = 0;
                break;
        case isl_tab_undo_redundant:
                undo->var->is_redundant = 0;
                tab->n_redundant--;
                break;
        case isl_tab_undo_zero:
                undo->var->is_zero = 0;
                tab->n_dead--;
                break;
        case isl_tab_undo_allocate:
                if (!undo->var->is_row) {
                        if (max_is_manifestly_unbounded(tab, undo->var))
                                to_row(tab, undo->var, -1);
                        else
                                to_row(tab, undo->var, 1);
                }
                drop_row(tab, undo->var->index);
                break;
        case isl_tab_undo_relax:
                unrelax(tab, undo->var);
                break;
        }
}

/* 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;
                perform_undo(tab, undo);
                free(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 = -1 -r', with r' an inequality
 * of the tableau, 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;

        if (tab->rational)
                return isl_ineq_separate;

        if (!isl_int_is_one(tab->mat->row[row][0]))
                return isl_ineq_separate;
        if (!isl_int_is_negone(tab->mat->row[row][1]))
                return isl_ineq_separate;

        pos = isl_seq_first_non_zero(tab->mat->row[row] + 2 + tab->n_dead,
                                        tab->n_col - tab->n_dead);
        if (pos == -1)
                return isl_ineq_adj_eq;

        if (!isl_int_is_negone(tab->mat->row[row][2 + tab->n_dead + pos]))
                return isl_ineq_separate;

        pos = isl_seq_first_non_zero(
                        tab->mat->row[row] + 2 + 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 (extend_cons(tab, 1) < 0)
                return isl_ineq_error;

        snap = isl_tab_snap(tab);

        con = add_row(tab, ineq);
        if (con < 0)
                goto error;

        row = tab->con[con].index;
        if (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]))) {
                if (at_least_zero(tab, &tab->con[con]))
                        type = isl_ineq_cut;
                else
                        type = separation_type(tab, row);
        } else if (tab->rational ? (sign_of_min(tab, &tab->con[con]) < 0)
                             : min_at_most_neg_one(tab, &tab->con[con]))
                type = isl_ineq_cut;
        else
                type = isl_ineq_redundant;

        if (isl_tab_rollback(tab, snap))
                return isl_ineq_error;
        return type;
error:
        isl_tab_rollback(tab, snap);
        return isl_ineq_error;
}

void isl_tab_dump(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, ", ");
                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) {
                if (i)
                        fprintf(out, ", ");
                fprintf(out, "r%d: %d%s", i, tab->row_var[i],
                    var_from_row(tab, i)->is_nonneg ? " [>=0]" : "");
        }
        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->n_col;
        isl_mat_dump(tab->mat, out, indent);
        tab->mat->n_row = r;
        tab->mat->n_col = c;
}