isl_tab_basic_map_partial_lexopt: split on parametric rows first

[isl.git] / isl_tab_pip.c

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 * Copyright 2016-2017 Sven Verdoolaege
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France 
 */

#include <isl_ctx_private.h>
#include "isl_map_private.h"
#include <isl_seq.h>
#include "isl_tab.h"
#include "isl_sample.h"
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl_aff_private.h>
#include <isl_constraint_private.h>
#include <isl_options_private.h>
#include <isl_config.h>

#include <bset_to_bmap.c>

/*
 * The implementation of parametric integer linear programming in this file
 * was inspired by the paper "Parametric Integer Programming" and the
 * report "Solving systems of affine (in)equalities" by Paul Feautrier
 * (and others).
 *
 * The strategy used for obtaining a feasible solution is different
 * from the one used in isl_tab.c.  In particular, in isl_tab.c,
 * upon finding a constraint that is not yet satisfied, we pivot
 * in a row that increases the constant term of the row holding the
 * constraint, making sure the sample solution remains feasible
 * for all the constraints it already satisfied.
 * Here, we always pivot in the row holding the constraint,
 * choosing a column that induces the lexicographically smallest
 * increment to the sample solution.
 *
 * By starting out from a sample value that is lexicographically
 * smaller than any integer point in the problem space, the first
 * feasible integer sample point we find will also be the lexicographically
 * smallest.  If all variables can be assumed to be non-negative,
 * then the initial sample value may be chosen equal to zero.
 * However, we will not make this assumption.  Instead, we apply
 * the "big parameter" trick.  Any variable x is then not directly
 * used in the tableau, but instead it is represented by another
 * variable x' = M + x, where M is an arbitrarily large (positive)
 * value.  x' is therefore always non-negative, whatever the value of x.
 * Taking as initial sample value x' = 0 corresponds to x = -M,
 * which is always smaller than any possible value of x.
 *
 * The big parameter trick is used in the main tableau and
 * also in the context tableau if isl_context_lex is used.
 * In this case, each tableaus has its own big parameter.
 * Before doing any real work, we check if all the parameters
 * happen to be non-negative.  If so, we drop the column corresponding
 * to M from the initial context tableau.
 * If isl_context_gbr is used, then the big parameter trick is only
 * used in the main tableau.
 */

struct isl_context;
struct isl_context_op {
        /* detect nonnegative parameters in context and mark them in tab */
        struct isl_tab *(*detect_nonnegative_parameters)(
                        struct isl_context *context, struct isl_tab *tab);
        /* return temporary reference to basic set representation of context */
        struct isl_basic_set *(*peek_basic_set)(struct isl_context *context);
        /* return temporary reference to tableau representation of context */
        struct isl_tab *(*peek_tab)(struct isl_context *context);
        /* add equality; check is 1 if eq may not be valid;
         * update is 1 if we may want to call ineq_sign on context later.
         */
        void (*add_eq)(struct isl_context *context, isl_int *eq,
                        int check, int update);
        /* add inequality; check is 1 if ineq may not be valid;
         * update is 1 if we may want to call ineq_sign on context later.
         */
        void (*add_ineq)(struct isl_context *context, isl_int *ineq,
                        int check, int update);
        /* check sign of ineq based on previous information.
         * strict is 1 if saturation should be treated as a positive sign.
         */
        enum isl_tab_row_sign (*ineq_sign)(struct isl_context *context,
                        isl_int *ineq, int strict);
        /* check if inequality maintains feasibility */
        int (*test_ineq)(struct isl_context *context, isl_int *ineq);
        /* return index of a div that corresponds to "div" */
        int (*get_div)(struct isl_context *context, struct isl_tab *tab,
                        struct isl_vec *div);
        /* insert div "div" to context at "pos" and return non-negativity */
        isl_bool (*insert_div)(struct isl_context *context, int pos,
                __isl_keep isl_vec *div);
        int (*detect_equalities)(struct isl_context *context,
                        struct isl_tab *tab);
        /* return row index of "best" split */
        int (*best_split)(struct isl_context *context, struct isl_tab *tab);
        /* check if context has already been determined to be empty */
        int (*is_empty)(struct isl_context *context);
        /* check if context is still usable */
        int (*is_ok)(struct isl_context *context);
        /* save a copy/snapshot of context */
        void *(*save)(struct isl_context *context);
        /* restore saved context */
        void (*restore)(struct isl_context *context, void *);
        /* discard saved context */
        void (*discard)(void *);
        /* invalidate context */
        void (*invalidate)(struct isl_context *context);
        /* free context */
        __isl_null struct isl_context *(*free)(struct isl_context *context);
};

/* Shared parts of context representation.
 *
 * "n_unknown" is the number of final unknown integer divisions
 * in the input domain.
 */
struct isl_context {
        struct isl_context_op *op;
        int n_unknown;
};

struct isl_context_lex {
        struct isl_context context;
        struct isl_tab *tab;
};

/* A stack (linked list) of solutions of subtrees of the search space.
 *
 * "ma" describes the solution as a function of "dom".
 * In particular, the domain space of "ma" is equal to the space of "dom".
 *
 * If "ma" is NULL, then there is no solution on "dom".
 */
struct isl_partial_sol {
        int level;
        struct isl_basic_set *dom;
        isl_multi_aff *ma;

        struct isl_partial_sol *next;
};

struct isl_sol;
struct isl_sol_callback {
        struct isl_tab_callback callback;
        struct isl_sol *sol;
};

/* isl_sol is an interface for constructing a solution to
 * a parametric integer linear programming problem.
 * Every time the algorithm reaches a state where a solution
 * can be read off from the tableau, the function "add" is called
 * on the isl_sol passed to find_solutions_main.  In a state where
 * the tableau is empty, "add_empty" is called instead.
 * "free" is called to free the implementation specific fields, if any.
 *
 * "error" is set if some error has occurred.  This flag invalidates
 * the remainder of the data structure.
 * If "rational" is set, then a rational optimization is being performed.
 * "level" is the current level in the tree with nodes for each
 * split in the context.
 * If "max" is set, then a maximization problem is being solved, rather than
 * a minimization problem, which means that the variables in the
 * tableau have value "M - x" rather than "M + x".
 * "n_out" is the number of output dimensions in the input.
 * "space" is the space in which the solution (and also the input) lives.
 *
 * The context tableau is owned by isl_sol and is updated incrementally.
 *
 * There are currently three implementations of this interface,
 * isl_sol_map, which simply collects the solutions in an isl_map
 * and (optionally) the parts of the context where there is no solution
 * in an isl_set,
 * isl_sol_pma, which collects an isl_pw_multi_aff instead, and
 * isl_sol_for, which calls a user-defined function for each part of
 * the solution.
 */
struct isl_sol {
        int error;
        int rational;
        int level;
        int max;
        int n_out;
        isl_space *space;
        struct isl_context *context;
        struct isl_partial_sol *partial;
        void (*add)(struct isl_sol *sol,
                __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma);
        void (*add_empty)(struct isl_sol *sol, struct isl_basic_set *bset);
        void (*free)(struct isl_sol *sol);
        struct isl_sol_callback dec_level;
};

static void sol_free(struct isl_sol *sol)
{
        struct isl_partial_sol *partial, *next;
        if (!sol)
                return;
        for (partial = sol->partial; partial; partial = next) {
                next = partial->next;
                isl_basic_set_free(partial->dom);
                isl_multi_aff_free(partial->ma);
                free(partial);
        }
        isl_space_free(sol->space);
        if (sol->context)
                sol->context->op->free(sol->context);
        sol->free(sol);
        free(sol);
}

/* Push a partial solution represented by a domain and function "ma"
 * onto the stack of partial solutions.
 * If "ma" is NULL, then "dom" represents a part of the domain
 * with no solution.
 */
static void sol_push_sol(struct isl_sol *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma)
{
        struct isl_partial_sol *partial;

        if (sol->error || !dom)
                goto error;

        partial = isl_alloc_type(dom->ctx, struct isl_partial_sol);
        if (!partial)
                goto error;

        partial->level = sol->level;
        partial->dom = dom;
        partial->ma = ma;
        partial->next = sol->partial;

        sol->partial = partial;

        return;
error:
        isl_basic_set_free(dom);
        isl_multi_aff_free(ma);
        sol->error = 1;
}

/* Check that the final columns of "M", starting at "first", are zero.
 */
static isl_stat check_final_columns_are_zero(__isl_keep isl_mat *M,
        unsigned first)
{
        int i;
        unsigned rows, cols, n;

        if (!M)
                return isl_stat_error;
        rows = isl_mat_rows(M);
        cols = isl_mat_cols(M);
        n = cols - first;
        for (i = 0; i < rows; ++i)
                if (isl_seq_first_non_zero(M->row[i] + first, n) != -1)
                        isl_die(isl_mat_get_ctx(M), isl_error_internal,
                                "final columns should be zero",
                                return isl_stat_error);
        return isl_stat_ok;
}

/* Set the affine expressions in "ma" according to the rows in "M", which
 * are defined over the local space "ls".
 * The matrix "M" may have extra (zero) columns beyond the number
 * of variables in "ls".
 */
static __isl_give isl_multi_aff *set_from_affine_matrix(
        __isl_take isl_multi_aff *ma, __isl_take isl_local_space *ls,
        __isl_take isl_mat *M)
{
        int i, dim;
        isl_aff *aff;

        if (!ma || !ls || !M)
                goto error;

        dim = isl_local_space_dim(ls, isl_dim_all);
        if (check_final_columns_are_zero(M, 1 + dim) < 0)
                goto error;
        for (i = 1; i < M->n_row; ++i) {
                aff = isl_aff_alloc(isl_local_space_copy(ls));
                if (aff) {
                        isl_int_set(aff->v->el[0], M->row[0][0]);
                        isl_seq_cpy(aff->v->el + 1, M->row[i], 1 + dim);
                }
                aff = isl_aff_normalize(aff);
                ma = isl_multi_aff_set_aff(ma, i - 1, aff);
        }
        isl_local_space_free(ls);
        isl_mat_free(M);

        return ma;
error:
        isl_local_space_free(ls);
        isl_mat_free(M);
        isl_multi_aff_free(ma);
        return NULL;
}

/* Push a partial solution represented by a domain and mapping M
 * onto the stack of partial solutions.
 *
 * The affine matrix "M" maps the dimensions of the context
 * to the output variables.  Convert it into an isl_multi_aff and
 * then call sol_push_sol.
 *
 * Note that the description of the initial context may have involved
 * existentially quantified variables, in which case they also appear
 * in "dom".  These need to be removed before creating the affine
 * expression because an affine expression cannot be defined in terms
 * of existentially quantified variables without a known representation.
 * Since newly added integer divisions are inserted before these
 * existentially quantified variables, they are still in the final
 * positions and the corresponding final columns of "M" are zero
 * because align_context_divs adds the existentially quantified
 * variables of the context to the main tableau without any constraints and
 * any equality constraints that are added later on can only serve
 * to eliminate these existentially quantified variables.
 */
static void sol_push_sol_mat(struct isl_sol *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_mat *M)
{
        isl_local_space *ls;
        isl_multi_aff *ma;
        int n_div, n_known;

        n_div = isl_basic_set_dim(dom, isl_dim_div);
        n_known = n_div - sol->context->n_unknown;

        ma = isl_multi_aff_alloc(isl_space_copy(sol->space));
        ls = isl_basic_set_get_local_space(dom);
        ls = isl_local_space_drop_dims(ls, isl_dim_div,
                                        n_known, n_div - n_known);
        ma = set_from_affine_matrix(ma, ls, M);

        if (!ma)
                dom = isl_basic_set_free(dom);
        sol_push_sol(sol, dom, ma);
}

/* Pop one partial solution from the partial solution stack and
 * pass it on to sol->add or sol->add_empty.
 */
static void sol_pop_one(struct isl_sol *sol)
{
        struct isl_partial_sol *partial;

        partial = sol->partial;
        sol->partial = partial->next;

        if (partial->ma)
                sol->add(sol, partial->dom, partial->ma);
        else
                sol->add_empty(sol, partial->dom);
        free(partial);
}

/* Return a fresh copy of the domain represented by the context tableau.
 */
static struct isl_basic_set *sol_domain(struct isl_sol *sol)
{
        struct isl_basic_set *bset;

        if (sol->error)
                return NULL;

        bset = isl_basic_set_dup(sol->context->op->peek_basic_set(sol->context));
        bset = isl_basic_set_update_from_tab(bset,
                        sol->context->op->peek_tab(sol->context));

        return bset;
}

/* Check whether two partial solutions have the same affine expressions.
 */
static isl_bool same_solution(struct isl_partial_sol *s1,
        struct isl_partial_sol *s2)
{
        if (!s1->ma != !s2->ma)
                return isl_bool_false;
        if (!s1->ma)
                return isl_bool_true;

        return isl_multi_aff_plain_is_equal(s1->ma, s2->ma);
}

/* Swap the initial two partial solutions in "sol".
 *
 * That is, go from
 *
 *      sol->partial = p1; p1->next = p2; p2->next = p3
 *
 * to
 *
 *      sol->partial = p2; p2->next = p1; p1->next = p3
 */
static void swap_initial(struct isl_sol *sol)
{
        struct isl_partial_sol *partial;

        partial = sol->partial;
        sol->partial = partial->next;
        partial->next = partial->next->next;
        sol->partial->next = partial;
}

/* Combine the initial two partial solution of "sol" into
 * a partial solution with the current context domain of "sol" and
 * the function description of the second partial solution in the list.
 * The level of the new partial solution is set to the current level.
 *
 * That is, the first two partial solutions (D1,M1) and (D2,M2) are
 * replaced by (D,M2), where D is the domain of "sol", which is assumed
 * to be the union of D1 and D2, while M1 is assumed to be equal to M2
 * (at least on D1).
 */
static isl_stat combine_initial_into_second(struct isl_sol *sol)
{
        struct isl_partial_sol *partial;
        isl_basic_set *bset;

        partial = sol->partial;

        bset = sol_domain(sol);
        isl_basic_set_free(partial->next->dom);
        partial->next->dom = bset;
        partial->next->level = sol->level;

        if (!bset)
                return isl_stat_error;

        sol->partial = partial->next;
        isl_basic_set_free(partial->dom);
        isl_multi_aff_free(partial->ma);
        free(partial);

        return isl_stat_ok;
}

/* Are "ma1" and "ma2" equal to each other on "dom"?
 *
 * Combine "ma1" and "ma2" with "dom" and check if the results are the same.
 * "dom" may have existentially quantified variables.  Eliminate them first
 * as otherwise they would have to be eliminated twice, in a more complicated
 * context.
 */
static isl_bool equal_on_domain(__isl_keep isl_multi_aff *ma1,
        __isl_keep isl_multi_aff *ma2, __isl_keep isl_basic_set *dom)
{
        isl_set *set;
        isl_pw_multi_aff *pma1, *pma2;
        isl_bool equal;

        set = isl_basic_set_compute_divs(isl_basic_set_copy(dom));
        pma1 = isl_pw_multi_aff_alloc(isl_set_copy(set),
                                        isl_multi_aff_copy(ma1));
        pma2 = isl_pw_multi_aff_alloc(set, isl_multi_aff_copy(ma2));
        equal = isl_pw_multi_aff_is_equal(pma1, pma2);
        isl_pw_multi_aff_free(pma1);
        isl_pw_multi_aff_free(pma2);

        return equal;
}

/* The initial two partial solutions of "sol" are known to be at
 * the same level.
 * If they represent the same solution (on different parts of the domain),
 * then combine them into a single solution at the current level.
 * Otherwise, pop them both.
 *
 * Even if the two partial solution are not obviously the same,
 * one may still be a simplification of the other over its own domain.
 * Also check if the two sets of affine functions are equal when
 * restricted to one of the domains.  If so, combine the two
 * using the set of affine functions on the other domain.
 * That is, for two partial solutions (D1,M1) and (D2,M2),
 * if M1 = M2 on D1, then the pair of partial solutions can
 * be replaced by (D1+D2,M2) and similarly when M1 = M2 on D2.
 */
static isl_stat combine_initial_if_equal(struct isl_sol *sol)
{
        struct isl_partial_sol *partial;
        isl_bool same;

        partial = sol->partial;

        same = same_solution(partial, partial->next);
        if (same < 0)
                return isl_stat_error;
        if (same)
                return combine_initial_into_second(sol);
        if (partial->ma && partial->next->ma) {
                same = equal_on_domain(partial->ma, partial->next->ma,
                                        partial->dom);
                if (same < 0)
                        return isl_stat_error;
                if (same)
                        return combine_initial_into_second(sol);
                same = equal_on_domain(partial->ma, partial->next->ma,
                                        partial->next->dom);
                if (same) {
                        swap_initial(sol);
                        return combine_initial_into_second(sol);
                }
        }

        sol_pop_one(sol);
        sol_pop_one(sol);

        return isl_stat_ok;
}

/* Pop all solutions from the partial solution stack that were pushed onto
 * the stack at levels that are deeper than the current level.
 * If the two topmost elements on the stack have the same level
 * and represent the same solution, then their domains are combined.
 * This combined domain is the same as the current context domain
 * as sol_pop is called each time we move back to a higher level.
 * If the outer level (0) has been reached, then all partial solutions
 * at the current level are also popped off.
 */
static void sol_pop(struct isl_sol *sol)
{
        struct isl_partial_sol *partial;

        if (sol->error)
                return;

        partial = sol->partial;
        if (!partial)
                return;

        if (partial->level == 0 && sol->level == 0) {
                for (partial = sol->partial; partial; partial = sol->partial)
                        sol_pop_one(sol);
                return;
        }

        if (partial->level <= sol->level)
                return;

        if (partial->next && partial->next->level == partial->level) {
                if (combine_initial_if_equal(sol) < 0)
                        goto error;
        } else
                sol_pop_one(sol);

        if (sol->level == 0) {
                for (partial = sol->partial; partial; partial = sol->partial)
                        sol_pop_one(sol);
                return;
        }

        if (0)
error:          sol->error = 1;
}

static void sol_dec_level(struct isl_sol *sol)
{
        if (sol->error)
                return;

        sol->level--;

        sol_pop(sol);
}

static isl_stat sol_dec_level_wrap(struct isl_tab_callback *cb)
{
        struct isl_sol_callback *callback = (struct isl_sol_callback *)cb;

        sol_dec_level(callback->sol);

        return callback->sol->error ? isl_stat_error : isl_stat_ok;
}

/* Move down to next level and push callback onto context tableau
 * to decrease the level again when it gets rolled back across
 * the current state.  That is, dec_level will be called with
 * the context tableau in the same state as it is when inc_level
 * is called.
 */
static void sol_inc_level(struct isl_sol *sol)
{
        struct isl_tab *tab;

        if (sol->error)
                return;

        sol->level++;
        tab = sol->context->op->peek_tab(sol->context);
        if (isl_tab_push_callback(tab, &sol->dec_level.callback) < 0)
                sol->error = 1;
}

static void scale_rows(struct isl_mat *mat, isl_int m, int n_row)
{
        int i;

        if (isl_int_is_one(m))
                return;

        for (i = 0; i < n_row; ++i)
                isl_seq_scale(mat->row[i], mat->row[i], m, mat->n_col);
}

/* Add the solution identified by the tableau and the context tableau.
 *
 * The layout of the variables is as follows.
 *      tab->n_var is equal to the total number of variables in the input
 *                      map (including divs that were copied from the context)
 *                      + the number of extra divs constructed
 *      Of these, the first tab->n_param and the last tab->n_div variables
 *      correspond to the variables in the context, i.e.,
 *              tab->n_param + tab->n_div = context_tab->n_var
 *      tab->n_param is equal to the number of parameters and input
 *                      dimensions in the input map
 *      tab->n_div is equal to the number of divs in the context
 *
 * If there is no solution, then call add_empty with a basic set
 * that corresponds to the context tableau.  (If add_empty is NULL,
 * then do nothing).
 *
 * If there is a solution, then first construct a matrix that maps
 * all dimensions of the context to the output variables, i.e.,
 * the output dimensions in the input map.
 * The divs in the input map (if any) that do not correspond to any
 * div in the context do not appear in the solution.
 * The algorithm will make sure that they have an integer value,
 * but these values themselves are of no interest.
 * We have to be careful not to drop or rearrange any divs in the
 * context because that would change the meaning of the matrix.
 *
 * To extract the value of the output variables, it should be noted
 * that we always use a big parameter M in the main tableau and so
 * the variable stored in this tableau is not an output variable x itself, but
 *      x' = M + x (in case of minimization)
 * or
 *      x' = M - x (in case of maximization)
 * If x' appears in a column, then its optimal value is zero,
 * which means that the optimal value of x is an unbounded number
 * (-M for minimization and M for maximization).
 * We currently assume that the output dimensions in the original map
 * are bounded, so this cannot occur.
 * Similarly, when x' appears in a row, then the coefficient of M in that
 * row is necessarily 1.
 * If the row in the tableau represents
 *      d x' = c + d M + e(y)
 * then, in case of minimization, the corresponding row in the matrix
 * will be
 *      a c + a e(y)
 * with a d = m, the (updated) common denominator of the matrix.
 * In case of maximization, the row will be
 *      -a c - a e(y)
 */
static void sol_add(struct isl_sol *sol, struct isl_tab *tab)
{
        struct isl_basic_set *bset = NULL;
        struct isl_mat *mat = NULL;
        unsigned off;
        int row;
        isl_int m;

        if (sol->error || !tab)
                goto error;

        if (tab->empty && !sol->add_empty)
                return;
        if (sol->context->op->is_empty(sol->context))
                return;

        bset = sol_domain(sol);

        if (tab->empty) {
                sol_push_sol(sol, bset, NULL);
                return;
        }

        off = 2 + tab->M;

        mat = isl_mat_alloc(tab->mat->ctx, 1 + sol->n_out,
                                            1 + tab->n_param + tab->n_div);
        if (!mat)
                goto error;

        isl_int_init(m);

        isl_seq_clr(mat->row[0] + 1, mat->n_col - 1);
        isl_int_set_si(mat->row[0][0], 1);
        for (row = 0; row < sol->n_out; ++row) {
                int i = tab->n_param + row;
                int r, j;

                isl_seq_clr(mat->row[1 + row], mat->n_col);
                if (!tab->var[i].is_row) {
                        if (tab->M)
                                isl_die(mat->ctx, isl_error_invalid,
                                        "unbounded optimum", goto error2);
                        continue;
                }

                r = tab->var[i].index;
                if (tab->M &&
                    isl_int_ne(tab->mat->row[r][2], tab->mat->row[r][0]))
                        isl_die(mat->ctx, isl_error_invalid,
                                "unbounded optimum", goto error2);
                isl_int_gcd(m, mat->row[0][0], tab->mat->row[r][0]);
                isl_int_divexact(m, tab->mat->row[r][0], m);
                scale_rows(mat, m, 1 + row);
                isl_int_divexact(m, mat->row[0][0], tab->mat->row[r][0]);
                isl_int_mul(mat->row[1 + row][0], m, tab->mat->row[r][1]);
                for (j = 0; j < tab->n_param; ++j) {
                        int col;
                        if (tab->var[j].is_row)
                                continue;
                        col = tab->var[j].index;
                        isl_int_mul(mat->row[1 + row][1 + j], m,
                                    tab->mat->row[r][off + col]);
                }
                for (j = 0; j < tab->n_div; ++j) {
                        int col;
                        if (tab->var[tab->n_var - tab->n_div+j].is_row)
                                continue;
                        col = tab->var[tab->n_var - tab->n_div+j].index;
                        isl_int_mul(mat->row[1 + row][1 + tab->n_param + j], m,
                                    tab->mat->row[r][off + col]);
                }
                if (sol->max)
                        isl_seq_neg(mat->row[1 + row], mat->row[1 + row],
                                    mat->n_col);
        }

        isl_int_clear(m);

        sol_push_sol_mat(sol, bset, mat);
        return;
error2:
        isl_int_clear(m);
error:
        isl_basic_set_free(bset);
        isl_mat_free(mat);
        sol->error = 1;
}

struct isl_sol_map {
        struct isl_sol  sol;
        struct isl_map  *map;
        struct isl_set  *empty;
};

static void sol_map_free(struct isl_sol *sol)
{
        struct isl_sol_map *sol_map = (struct isl_sol_map *) sol;
        isl_map_free(sol_map->map);
        isl_set_free(sol_map->empty);
}

/* This function is called for parts of the context where there is
 * no solution, with "bset" corresponding to the context tableau.
 * Simply add the basic set to the set "empty".
 */
static void sol_map_add_empty(struct isl_sol_map *sol,
        struct isl_basic_set *bset)
{
        if (!bset || !sol->empty)
                goto error;

        sol->empty = isl_set_grow(sol->empty, 1);
        bset = isl_basic_set_simplify(bset);
        bset = isl_basic_set_finalize(bset);
        sol->empty = isl_set_add_basic_set(sol->empty, isl_basic_set_copy(bset));
        if (!sol->empty)
                goto error;
        isl_basic_set_free(bset);
        return;
error:
        isl_basic_set_free(bset);
        sol->sol.error = 1;
}

static void sol_map_add_empty_wrap(struct isl_sol *sol,
        struct isl_basic_set *bset)
{
        sol_map_add_empty((struct isl_sol_map *)sol, bset);
}

/* Given a basic set "dom" that represents the context and a tuple of
 * affine expressions "ma" defined over this domain, construct a basic map
 * that expresses this function on the domain.
 */
static void sol_map_add(struct isl_sol_map *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma)
{
        isl_basic_map *bmap;

        if (sol->sol.error || !dom || !ma)
                goto error;

        bmap = isl_basic_map_from_multi_aff2(ma, sol->sol.rational);
        bmap = isl_basic_map_intersect_domain(bmap, dom);
        sol->map = isl_map_grow(sol->map, 1);
        sol->map = isl_map_add_basic_map(sol->map, bmap);
        if (!sol->map)
                sol->sol.error = 1;
        return;
error:
        isl_basic_set_free(dom);
        isl_multi_aff_free(ma);
        sol->sol.error = 1;
}

static void sol_map_add_wrap(struct isl_sol *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma)
{
        sol_map_add((struct isl_sol_map *)sol, dom, ma);
}


/* Store the "parametric constant" of row "row" of tableau "tab" in "line",
 * i.e., the constant term and the coefficients of all variables that
 * appear in the context tableau.
 * Note that the coefficient of the big parameter M is NOT copied.
 * The context tableau may not have a big parameter and even when it
 * does, it is a different big parameter.
 */
static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
{
        int i;
        unsigned off = 2 + tab->M;

        isl_int_set(line[0], tab->mat->row[row][1]);
        for (i = 0; i < tab->n_param; ++i) {
                if (tab->var[i].is_row)
                        isl_int_set_si(line[1 + i], 0);
                else {
                        int col = tab->var[i].index;
                        isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
                }
        }
        for (i = 0; i < tab->n_div; ++i) {
                if (tab->var[tab->n_var - tab->n_div + i].is_row)
                        isl_int_set_si(line[1 + tab->n_param + i], 0);
                else {
                        int col = tab->var[tab->n_var - tab->n_div + i].index;
                        isl_int_set(line[1 + tab->n_param + i],
                                    tab->mat->row[row][off + col]);
                }
        }
}

/* Check if rows "row1" and "row2" have identical "parametric constants",
 * as explained above.
 * In this case, we also insist that the coefficients of the big parameter
 * be the same as the values of the constants will only be the same
 * if these coefficients are also the same.
 */
static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
{
        int i;
        unsigned off = 2 + tab->M;

        if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
                return 0;

        if (tab->M && isl_int_ne(tab->mat->row[row1][2],
                                 tab->mat->row[row2][2]))
                return 0;

        for (i = 0; i < tab->n_param + tab->n_div; ++i) {
                int pos = i < tab->n_param ? i :
                        tab->n_var - tab->n_div + i - tab->n_param;
                int col;

                if (tab->var[pos].is_row)
                        continue;
                col = tab->var[pos].index;
                if (isl_int_ne(tab->mat->row[row1][off + col],
                               tab->mat->row[row2][off + col]))
                        return 0;
        }
        return 1;
}

/* Return an inequality that expresses that the "parametric constant"
 * should be non-negative.
 * This function is only called when the coefficient of the big parameter
 * is equal to zero.
 */
static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
{
        struct isl_vec *ineq;

        ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
        if (!ineq)
                return NULL;

        get_row_parameter_line(tab, row, ineq->el);
        if (ineq)
                ineq = isl_vec_normalize(ineq);

        return ineq;
}

/* Normalize a div expression of the form
 *
 *      [(g*f(x) + c)/(g * m)]
 *
 * with c the constant term and f(x) the remaining coefficients, to
 *
 *      [(f(x) + [c/g])/m]
 */
static void normalize_div(__isl_keep isl_vec *div)
{
        isl_ctx *ctx = isl_vec_get_ctx(div);
        int len = div->size - 2;

        isl_seq_gcd(div->el + 2, len, &ctx->normalize_gcd);
        isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, div->el[0]);

        if (isl_int_is_one(ctx->normalize_gcd))
                return;

        isl_int_divexact(div->el[0], div->el[0], ctx->normalize_gcd);
        isl_int_fdiv_q(div->el[1], div->el[1], ctx->normalize_gcd);
        isl_seq_scale_down(div->el + 2, div->el + 2, ctx->normalize_gcd, len);
}

/* Return an integer division for use in a parametric cut based
 * on the given row.
 * In particular, let the parametric constant of the row be
 *
 *              \sum_i a_i y_i
 *
 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
 * The div returned is equal to
 *
 *              floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
 */
static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
{
        struct isl_vec *div;

        div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
        if (!div)
                return NULL;

        isl_int_set(div->el[0], tab->mat->row[row][0]);
        get_row_parameter_line(tab, row, div->el + 1);
        isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
        normalize_div(div);
        isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);

        return div;
}

/* Return an integer division for use in transferring an integrality constraint
 * to the context.
 * In particular, let the parametric constant of the row be
 *
 *              \sum_i a_i y_i
 *
 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
 * The the returned div is equal to
 *
 *              floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
 */
static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
{
        struct isl_vec *div;

        div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
        if (!div)
                return NULL;

        isl_int_set(div->el[0], tab->mat->row[row][0]);
        get_row_parameter_line(tab, row, div->el + 1);
        normalize_div(div);
        isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);

        return div;
}

/* Construct and return an inequality that expresses an upper bound
 * on the given div.
 * In particular, if the div is given by
 *
 *      d = floor(e/m)
 *
 * then the inequality expresses
 *
 *      m d <= e
 */
static __isl_give isl_vec *ineq_for_div(__isl_keep isl_basic_set *bset,
        unsigned div)
{
        unsigned total;
        unsigned div_pos;
        struct isl_vec *ineq;

        if (!bset)
                return NULL;

        total = isl_basic_set_total_dim(bset);
        div_pos = 1 + total - bset->n_div + div;

        ineq = isl_vec_alloc(bset->ctx, 1 + total);
        if (!ineq)
                return NULL;

        isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
        isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
        return ineq;
}

/* Given a row in the tableau and a div that was created
 * using get_row_split_div and that has been constrained to equality, i.e.,
 *
 *              d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
 *
 * replace the expression "\sum_i {a_i} y_i" in the row by d,
 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
 * The coefficients of the non-parameters in the tableau have been
 * verified to be integral.  We can therefore simply replace coefficient b
 * by floor(b).  For the coefficients of the parameters we have
 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
 * floor(b) = b.
 */
static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
{
        isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
                        tab->mat->row[row][0], 1 + tab->M + tab->n_col);

        isl_int_set_si(tab->mat->row[row][0], 1);

        if (tab->var[tab->n_var - tab->n_div + div].is_row) {
                int drow = tab->var[tab->n_var - tab->n_div + div].index;

                isl_assert(tab->mat->ctx,
                        isl_int_is_one(tab->mat->row[drow][0]), goto error);
                isl_seq_combine(tab->mat->row[row] + 1,
                        tab->mat->ctx->one, tab->mat->row[row] + 1,
                        tab->mat->ctx->one, tab->mat->row[drow] + 1,
                        1 + tab->M + tab->n_col);
        } else {
                int dcol = tab->var[tab->n_var - tab->n_div + div].index;

                isl_int_add_ui(tab->mat->row[row][2 + tab->M + dcol],
                                tab->mat->row[row][2 + tab->M + dcol], 1);
        }

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

/* Check if the (parametric) constant of the given row is obviously
 * negative, meaning that we don't need to consult the context tableau.
 * If there is a big parameter and its coefficient is non-zero,
 * then this coefficient determines the outcome.
 * Otherwise, we check whether the constant is negative and
 * all non-zero coefficients of parameters are negative and
 * belong to non-negative parameters.
 */
static int is_obviously_neg(struct isl_tab *tab, int row)
{
        int i;
        int col;
        unsigned off = 2 + tab->M;

        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;
        }

        if (isl_int_is_nonneg(tab->mat->row[row][1]))
                return 0;
        for (i = 0; i < tab->n_param; ++i) {
                /* Eliminated parameter */
                if (tab->var[i].is_row)
                        continue;
                col = tab->var[i].index;
                if (isl_int_is_zero(tab->mat->row[row][off + col]))
                        continue;
                if (!tab->var[i].is_nonneg)
                        return 0;
                if (isl_int_is_pos(tab->mat->row[row][off + col]))
                        return 0;
        }
        for (i = 0; i < tab->n_div; ++i) {
                if (tab->var[tab->n_var - tab->n_div + i].is_row)
                        continue;
                col = tab->var[tab->n_var - tab->n_div + i].index;
                if (isl_int_is_zero(tab->mat->row[row][off + col]))
                        continue;
                if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
                        return 0;
                if (isl_int_is_pos(tab->mat->row[row][off + col]))
                        return 0;
        }
        return 1;
}

/* Check if the (parametric) constant of the given row is obviously
 * non-negative, meaning that we don't need to consult the context tableau.
 * If there is a big parameter and its coefficient is non-zero,
 * then this coefficient determines the outcome.
 * Otherwise, we check whether the constant is non-negative and
 * all non-zero coefficients of parameters are positive and
 * belong to non-negative parameters.
 */
static int is_obviously_nonneg(struct isl_tab *tab, int row)
{
        int i;
        int col;
        unsigned off = 2 + tab->M;

        if (tab->M) {
                if (isl_int_is_pos(tab->mat->row[row][2]))
                        return 1;
                if (isl_int_is_neg(tab->mat->row[row][2]))
                        return 0;
        }

        if (isl_int_is_neg(tab->mat->row[row][1]))
                return 0;
        for (i = 0; i < tab->n_param; ++i) {
                /* Eliminated parameter */
                if (tab->var[i].is_row)
                        continue;
                col = tab->var[i].index;
                if (isl_int_is_zero(tab->mat->row[row][off + col]))
                        continue;
                if (!tab->var[i].is_nonneg)
                        return 0;
                if (isl_int_is_neg(tab->mat->row[row][off + col]))
                        return 0;
        }
        for (i = 0; i < tab->n_div; ++i) {
                if (tab->var[tab->n_var - tab->n_div + i].is_row)
                        continue;
                col = tab->var[tab->n_var - tab->n_div + i].index;
                if (isl_int_is_zero(tab->mat->row[row][off + col]))
                        continue;
                if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
                        return 0;
                if (isl_int_is_neg(tab->mat->row[row][off + col]))
                        return 0;
        }
        return 1;
}

/* Given a row r and two columns, return the column that would
 * lead to the lexicographically smallest increment in the sample
 * solution when leaving the basis in favor of the row.
 * Pivoting with column c will increment the sample value by a non-negative
 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
 * corresponding to the non-parametric variables.
 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
 * with all other entries in this virtual row equal to zero.
 * If variable v appears in a row, then a_{v,c} is the element in column c
 * of that row.
 *
 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
 * increment.  Otherwise, it's c2.
 */
static int lexmin_col_pair(struct isl_tab *tab,
        int row, int col1, int col2, isl_int tmp)
{
        int i;
        isl_int *tr;

        tr = tab->mat->row[row] + 2 + tab->M;

        for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
                int s1, s2;
                isl_int *r;

                if (!tab->var[i].is_row) {
                        if (tab->var[i].index == col1)
                                return col2;
                        if (tab->var[i].index == col2)
                                return col1;
                        continue;
                }

                if (tab->var[i].index == row)
                        continue;

                r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
                s1 = isl_int_sgn(r[col1]);
                s2 = isl_int_sgn(r[col2]);
                if (s1 == 0 && s2 == 0)
                        continue;
                if (s1 < s2)
                        return col1;
                if (s2 < s1)
                        return col2;

                isl_int_mul(tmp, r[col2], tr[col1]);
                isl_int_submul(tmp, r[col1], tr[col2]);
                if (isl_int_is_pos(tmp))
                        return col1;
                if (isl_int_is_neg(tmp))
                        return col2;
        }
        return -1;
}

/* Does the index into the tab->var or tab->con array "index"
 * correspond to a variable in the context tableau?
 * In particular, it needs to be an index into the tab->var array and
 * it needs to refer to either one of the first tab->n_param variables or
 * one of the last tab->n_div variables.
 */
static int is_parameter_var(struct isl_tab *tab, int index)
{
        if (index < 0)
                return 0;
        if (index < tab->n_param)
                return 1;
        if (index >= tab->n_var - tab->n_div)
                return 1;
        return 0;
}

/* Does column "col" of "tab" refer to a variable in the context tableau?
 */
static int col_is_parameter_var(struct isl_tab *tab, int col)
{
        return is_parameter_var(tab, tab->col_var[col]);
}

/* Does row "row" of "tab" refer to a variable in the context tableau?
 */
static int row_is_parameter_var(struct isl_tab *tab, int row)
{
        return is_parameter_var(tab, tab->row_var[row]);
}

/* Given a row in the tableau, find and return the column that would
 * result in the lexicographically smallest, but positive, increment
 * in the sample point.
 * If there is no such column, then return tab->n_col.
 * If anything goes wrong, return -1.
 */
static int lexmin_pivot_col(struct isl_tab *tab, int row)
{
        int j;
        int col = tab->n_col;
        isl_int *tr;
        isl_int tmp;

        tr = tab->mat->row[row] + 2 + tab->M;

        isl_int_init(tmp);

        for (j = tab->n_dead; j < tab->n_col; ++j) {
                if (col_is_parameter_var(tab, j))
                        continue;

                if (!isl_int_is_pos(tr[j]))
                        continue;

                if (col == tab->n_col)
                        col = j;
                else
                        col = lexmin_col_pair(tab, row, col, j, tmp);
                isl_assert(tab->mat->ctx, col >= 0, goto error);
        }

        isl_int_clear(tmp);
        return col;
error:
        isl_int_clear(tmp);
        return -1;
}

/* Return the first known violated constraint, i.e., a non-negative
 * constraint that currently has an either obviously negative value
 * or a previously determined to be negative value.
 *
 * If any constraint has a negative coefficient for the big parameter,
 * if any, then we return one of these first.
 */
static int first_neg(struct isl_tab *tab)
{
        int row;

        if (tab->M)
                for (row = tab->n_redundant; row < tab->n_row; ++row) {
                        if (!isl_tab_var_from_row(tab, row)->is_nonneg)
                                continue;
                        if (!isl_int_is_neg(tab->mat->row[row][2]))
                                continue;
                        if (tab->row_sign)
                                tab->row_sign[row] = isl_tab_row_neg;
                        return row;
                }
        for (row = tab->n_redundant; row < tab->n_row; ++row) {
                if (!isl_tab_var_from_row(tab, row)->is_nonneg)
                        continue;
                if (tab->row_sign) {
                        if (tab->row_sign[row] == 0 &&
                            is_obviously_neg(tab, row))
                                tab->row_sign[row] = isl_tab_row_neg;
                        if (tab->row_sign[row] != isl_tab_row_neg)
                                continue;
                } else if (!is_obviously_neg(tab, row))
                        continue;
                return row;
        }
        return -1;
}

/* Check whether the invariant that all columns are lexico-positive
 * is satisfied.  This function is not called from the current code
 * but is useful during debugging.
 */
static void check_lexpos(struct isl_tab *tab) __attribute__ ((unused));
static void check_lexpos(struct isl_tab *tab)
{
        unsigned off = 2 + tab->M;
        int col;
        int var;
        int row;

        for (col = tab->n_dead; col < tab->n_col; ++col) {
                if (col_is_parameter_var(tab, col))
                        continue;
                for (var = tab->n_param; var < tab->n_var - tab->n_div; ++var) {
                        if (!tab->var[var].is_row) {
                                if (tab->var[var].index == col)
                                        break;
                                else
                                        continue;
                        }
                        row = tab->var[var].index;
                        if (isl_int_is_zero(tab->mat->row[row][off + col]))
                                continue;
                        if (isl_int_is_pos(tab->mat->row[row][off + col]))
                                break;
                        fprintf(stderr, "lexneg column %d (row %d)\n",
                                col, row);
                }
                if (var >= tab->n_var - tab->n_div)
                        fprintf(stderr, "zero column %d\n", col);
        }
}

/* Report to the caller that the given constraint is part of an encountered
 * conflict.
 */
static int report_conflicting_constraint(struct isl_tab *tab, int con)
{
        return tab->conflict(con, tab->conflict_user);
}

/* Given a conflicting row in the tableau, report all constraints
 * involved in the row to the caller.  That is, the row itself
 * (if it represents a constraint) and all constraint columns with
 * non-zero (and therefore negative) coefficients.
 */
static int report_conflict(struct isl_tab *tab, int row)
{
        int j;
        isl_int *tr;

        if (!tab->conflict)
                return 0;

        if (tab->row_var[row] < 0 &&
            report_conflicting_constraint(tab, ~tab->row_var[row]) < 0)
                return -1;

        tr = tab->mat->row[row] + 2 + tab->M;

        for (j = tab->n_dead; j < tab->n_col; ++j) {
                if (col_is_parameter_var(tab, j))
                        continue;

                if (!isl_int_is_neg(tr[j]))
                        continue;

                if (tab->col_var[j] < 0 &&
                    report_conflicting_constraint(tab, ~tab->col_var[j]) < 0)
                        return -1;
        }

        return 0;
}

/* Resolve all known or obviously violated constraints through pivoting.
 * In particular, as long as we can find any violated constraint, we
 * look for a pivoting column that would result in the lexicographically
 * smallest increment in the sample point.  If there is no such column
 * then the tableau is infeasible.
 */
static int restore_lexmin(struct isl_tab *tab) WARN_UNUSED;
static int restore_lexmin(struct isl_tab *tab)
{
        int row, col;

        if (!tab)
                return -1;
        if (tab->empty)
                return 0;
        while ((row = first_neg(tab)) != -1) {
                col = lexmin_pivot_col(tab, row);
                if (col >= tab->n_col) {
                        if (report_conflict(tab, row) < 0)
                                return -1;
                        if (isl_tab_mark_empty(tab) < 0)
                                return -1;
                        return 0;
                }
                if (col < 0)
                        return -1;
                if (isl_tab_pivot(tab, row, col) < 0)
                        return -1;
        }
        return 0;
}

/* Given a row that represents an equality, look for an appropriate
 * pivoting column.
 * In particular, if there are any non-zero coefficients among
 * the non-parameter variables, then we take the last of these
 * variables.  Eliminating this variable in terms of the other
 * variables and/or parameters does not influence the property
 * that all column in the initial tableau are lexicographically
 * positive.  The row corresponding to the eliminated variable
 * will only have non-zero entries below the diagonal of the
 * initial tableau.  That is, we transform
 *
 *              I                               I
 *                1             into            a
 *                  I                             I
 *
 * If there is no such non-parameter variable, then we are dealing with
 * pure parameter equality and we pick any parameter with coefficient 1 or -1
 * for elimination.  This will ensure that the eliminated parameter
 * always has an integer value whenever all the other parameters are integral.
 * If there is no such parameter then we return -1.
 */
static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
{
        unsigned off = 2 + tab->M;
        int i;

        for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
                int col;
                if (tab->var[i].is_row)
                        continue;
                col = tab->var[i].index;
                if (col <= tab->n_dead)
                        continue;
                if (!isl_int_is_zero(tab->mat->row[row][off + col]))
                        return col;
        }
        for (i = tab->n_dead; i < tab->n_col; ++i) {
                if (isl_int_is_one(tab->mat->row[row][off + i]))
                        return i;
                if (isl_int_is_negone(tab->mat->row[row][off + i]))
                        return i;
        }
        return -1;
}

/* Add an equality that is known to be valid to the tableau.
 * We first check if we can eliminate a variable or a parameter.
 * If not, we add the equality as two inequalities.
 * In this case, the equality was a pure parameter equality and there
 * is no need to resolve any constraint violations.
 *
 * This function assumes that at least two more rows and at least
 * two more elements in the constraint array are available in the tableau.
 */
static struct isl_tab *add_lexmin_valid_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 = last_var_col_or_int_par_col(tab, r);
        if (i < 0) {
                tab->con[r].is_nonneg = 1;
                if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
                        goto error;
                isl_seq_neg(eq, eq, 1 + tab->n_var);
                r = isl_tab_add_row(tab, eq);
                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;
        } else {
                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;
}

/* Check if the given row is a pure constant.
 */
static int is_constant(struct isl_tab *tab, int row)
{
        unsigned off = 2 + tab->M;

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

/* Is the given row a parametric constant?
 * That is, does it only involve variables that also appear in the context?
 */
static int is_parametric_constant(struct isl_tab *tab, int row)
{
        unsigned off = 2 + tab->M;
        int col;

        for (col = tab->n_dead; col < tab->n_col; ++col) {
                if (col_is_parameter_var(tab, col))
                        continue;
                if (isl_int_is_zero(tab->mat->row[row][off + col]))
                        continue;
                return 0;
        }

        return 1;
}

/* Add an equality that may or may not be valid to the tableau.
 * If the resulting row is a pure constant, then it must be zero.
 * Otherwise, the resulting tableau is empty.
 *
 * If the row is not a pure constant, then we add two inequalities,
 * each time checking that they can be satisfied.
 * In the end we try to use one of the two constraints to eliminate
 * a column.
 *
 * This function assumes that at least two more rows and at least
 * two more elements in the constraint array are available in the tableau.
 */
static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED;
static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
{
        int r1, r2;
        int row;
        struct isl_tab_undo *snap;

        if (!tab)
                return -1;
        snap = isl_tab_snap(tab);
        r1 = isl_tab_add_row(tab, eq);
        if (r1 < 0)
                return -1;
        tab->con[r1].is_nonneg = 1;
        if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]) < 0)
                return -1;

        row = tab->con[r1].index;
        if (is_constant(tab, row)) {
                if (!isl_int_is_zero(tab->mat->row[row][1]) ||
                    (tab->M && !isl_int_is_zero(tab->mat->row[row][2]))) {
                        if (isl_tab_mark_empty(tab) < 0)
                                return -1;
                        return 0;
                }
                if (isl_tab_rollback(tab, snap) < 0)
                        return -1;
                return 0;
        }

        if (restore_lexmin(tab) < 0)
                return -1;
        if (tab->empty)
                return 0;

        isl_seq_neg(eq, eq, 1 + tab->n_var);

        r2 = isl_tab_add_row(tab, eq);
        if (r2 < 0)
                return -1;
        tab->con[r2].is_nonneg = 1;
        if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]) < 0)
                return -1;

        if (restore_lexmin(tab) < 0)
                return -1;
        if (tab->empty)
                return 0;

        if (!tab->con[r1].is_row) {
                if (isl_tab_kill_col(tab, tab->con[r1].index) < 0)
                        return -1;
        } else if (!tab->con[r2].is_row) {
                if (isl_tab_kill_col(tab, tab->con[r2].index) < 0)
                        return -1;
        }

        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;
        }

        return 0;
}

/* Add an inequality to the tableau, resolving violations using
 * restore_lexmin.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
{
        int r;

        if (!tab)
                return NULL;
        if (tab->bmap) {
                tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq);
                if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
                        goto error;
                if (!tab->bmap)
                        goto error;
        }
        r = isl_tab_add_row(tab, ineq);
        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;
        if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
                if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
                        goto error;
                return tab;
        }

        if (restore_lexmin(tab) < 0)
                goto error;
        if (!tab->empty && 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)
                        goto error;
        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

/* Check if the coefficients of the parameters are all integral.
 */
static int integer_parameter(struct isl_tab *tab, int row)
{
        int i;
        int col;
        unsigned off = 2 + tab->M;

        for (i = 0; i < tab->n_param; ++i) {
                /* Eliminated parameter */
                if (tab->var[i].is_row)
                        continue;
                col = tab->var[i].index;
                if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
                                                tab->mat->row[row][0]))
                        return 0;
        }
        for (i = 0; i < tab->n_div; ++i) {
                if (tab->var[tab->n_var - tab->n_div + i].is_row)
                        continue;
                col = tab->var[tab->n_var - tab->n_div + i].index;
                if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
                                                tab->mat->row[row][0]))
                        return 0;
        }
        return 1;
}

/* Check if the coefficients of the non-parameter variables are all integral.
 */
static int integer_variable(struct isl_tab *tab, int row)
{
        int i;
        unsigned off = 2 + tab->M;

        for (i = tab->n_dead; i < tab->n_col; ++i) {
                if (col_is_parameter_var(tab, i))
                        continue;
                if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
                                                tab->mat->row[row][0]))
                        return 0;
        }
        return 1;
}

/* Check if the constant term is integral.
 */
static int integer_constant(struct isl_tab *tab, int row)
{
        return isl_int_is_divisible_by(tab->mat->row[row][1],
                                        tab->mat->row[row][0]);
}

#define I_CST   1 << 0
#define I_PAR   1 << 1
#define I_VAR   1 << 2

/* Check for next (non-parameter) variable after "var" (first if var == -1)
 * that is non-integer and therefore requires a cut and return
 * the index of the variable.
 * For parametric tableaus, there are three parts in a row,
 * the constant, the coefficients of the parameters and the rest.
 * For each part, we check whether the coefficients in that part
 * are all integral and if so, set the corresponding flag in *f.
 * If the constant and the parameter part are integral, then the
 * current sample value is integral and no cut is required
 * (irrespective of whether the variable part is integral).
 */
static int next_non_integer_var(struct isl_tab *tab, int var, int *f)
{
        var = var < 0 ? tab->n_param : var + 1;

        for (; var < tab->n_var - tab->n_div; ++var) {
                int flags = 0;
                int row;
                if (!tab->var[var].is_row)
                        continue;
                row = tab->var[var].index;
                if (integer_constant(tab, row))
                        ISL_FL_SET(flags, I_CST);
                if (integer_parameter(tab, row))
                        ISL_FL_SET(flags, I_PAR);
                if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
                        continue;
                if (integer_variable(tab, row))
                        ISL_FL_SET(flags, I_VAR);
                *f = flags;
                return var;
        }
        return -1;
}

/* Check for first (non-parameter) variable that is non-integer and
 * therefore requires a cut and return the corresponding row.
 * For parametric tableaus, there are three parts in a row,
 * the constant, the coefficients of the parameters and the rest.
 * For each part, we check whether the coefficients in that part
 * are all integral and if so, set the corresponding flag in *f.
 * If the constant and the parameter part are integral, then the
 * current sample value is integral and no cut is required
 * (irrespective of whether the variable part is integral).
 */
static int first_non_integer_row(struct isl_tab *tab, int *f)
{
        int var = next_non_integer_var(tab, -1, f);

        return var < 0 ? -1 : tab->var[var].index;
}

/* Add a (non-parametric) cut to cut away the non-integral sample
 * value of the given row.
 *
 * If the row is given by
 *
 *      m r = f + \sum_i a_i y_i
 *
 * then the cut is
 *
 *      c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
 *
 * The big parameter, if any, is ignored, since it is assumed to be big
 * enough to be divisible by any integer.
 * If the tableau is actually a parametric tableau, then this function
 * is only called when all coefficients of the parameters are integral.
 * The cut therefore has zero coefficients for the parameters.
 *
 * The current value is known to be negative, so row_sign, if it
 * exists, is set accordingly.
 *
 * Return the row of the cut or -1.
 */
static int add_cut(struct isl_tab *tab, int row)
{
        int i;
        int r;
        isl_int *r_row;
        unsigned off = 2 + tab->M;

        if (isl_tab_extend_cons(tab, 1) < 0)
                return -1;
        r = isl_tab_allocate_con(tab);
        if (r < 0)
                return -1;

        r_row = tab->mat->row[tab->con[r].index];
        isl_int_set(r_row[0], tab->mat->row[row][0]);
        isl_int_neg(r_row[1], tab->mat->row[row][1]);
        isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
        isl_int_neg(r_row[1], r_row[1]);
        if (tab->M)
                isl_int_set_si(r_row[2], 0);
        for (i = 0; i < tab->n_col; ++i)
                isl_int_fdiv_r(r_row[off + i],
                        tab->mat->row[row][off + i], tab->mat->row[row][0]);

        tab->con[r].is_nonneg = 1;
        if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
                return -1;
        if (tab->row_sign)
                tab->row_sign[tab->con[r].index] = isl_tab_row_neg;

        return tab->con[r].index;
}

#define CUT_ALL 1
#define CUT_ONE 0

/* Given a non-parametric tableau, add cuts until an integer
 * sample point is obtained or until the tableau is determined
 * to be integer infeasible.
 * As long as there is any non-integer value in the sample point,
 * we add appropriate cuts, if possible, for each of these
 * non-integer values and then resolve the violated
 * cut constraints using restore_lexmin.
 * If one of the corresponding rows is equal to an integral
 * combination of variables/constraints plus a non-integral constant,
 * then there is no way to obtain an integer point and we return
 * a tableau that is marked empty.
 * The parameter cutting_strategy controls the strategy used when adding cuts
 * to remove non-integer points. CUT_ALL adds all possible cuts
 * before continuing the search. CUT_ONE adds only one cut at a time.
 */
static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab,
        int cutting_strategy)
{
        int var;
        int row;
        int flags;

        if (!tab)
                return NULL;
        if (tab->empty)
                return tab;

        while ((var = next_non_integer_var(tab, -1, &flags)) != -1) {
                do {
                        if (ISL_FL_ISSET(flags, I_VAR)) {
                                if (isl_tab_mark_empty(tab) < 0)
                                        goto error;
                                return tab;
                        }
                        row = tab->var[var].index;
                        row = add_cut(tab, row);
                        if (row < 0)
                                goto error;
                        if (cutting_strategy == CUT_ONE)
                                break;
                } while ((var = next_non_integer_var(tab, var, &flags)) != -1);
                if (restore_lexmin(tab) < 0)
                        goto error;
                if (tab->empty)
                        break;
        }
        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

/* Check whether all the currently active samples also satisfy the inequality
 * "ineq" (treated as an equality if eq is set).
 * Remove those samples that do not.
 */
static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
{
        int i;
        isl_int v;

        if (!tab)
                return NULL;

        isl_assert(tab->mat->ctx, tab->bmap, goto error);
        isl_assert(tab->mat->ctx, tab->samples, goto error);
        isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);

        isl_int_init(v);
        for (i = tab->n_outside; i < tab->n_sample; ++i) {
                int sgn;
                isl_seq_inner_product(ineq, tab->samples->row[i],
                                        1 + tab->n_var, &v);
                sgn = isl_int_sgn(v);
                if (eq ? (sgn == 0) : (sgn >= 0))
                        continue;
                tab = isl_tab_drop_sample(tab, i);
                if (!tab)
                        break;
        }
        isl_int_clear(v);

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

/* Check whether the sample value of the tableau is finite,
 * i.e., either the tableau does not use a big parameter, or
 * all values of the variables are equal to the big parameter plus
 * some constant.  This constant is the actual sample value.
 */
static int sample_is_finite(struct isl_tab *tab)
{
        int i;

        if (!tab->M)
                return 1;

        for (i = 0; i < tab->n_var; ++i) {
                int row;
                if (!tab->var[i].is_row)
                        return 0;
                row = tab->var[i].index;
                if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
                        return 0;
        }
        return 1;
}

/* Check if the context tableau of sol has any integer points.
 * Leave tab in empty state if no integer point can be found.
 * If an integer point can be found and if moreover it is finite,
 * then it is added to the list of sample values.
 *
 * This function is only called when none of the currently active sample
 * values satisfies the most recently added constraint.
 */
static struct isl_tab *check_integer_feasible(struct isl_tab *tab)
{
        struct isl_tab_undo *snap;

        if (!tab)
                return NULL;

        snap = isl_tab_snap(tab);
        if (isl_tab_push_basis(tab) < 0)
                goto error;

        tab = cut_to_integer_lexmin(tab, CUT_ALL);
        if (!tab)
                goto error;

        if (!tab->empty && sample_is_finite(tab)) {
                struct isl_vec *sample;

                sample = isl_tab_get_sample_value(tab);

                if (isl_tab_add_sample(tab, sample) < 0)
                        goto error;
        }

        if (!tab->empty && isl_tab_rollback(tab, snap) < 0)
                goto error;

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

/* Check if any of the currently active sample values satisfies
 * the inequality "ineq" (an equality if eq is set).
 */
static int tab_has_valid_sample(struct isl_tab *tab, isl_int *ineq, int eq)
{
        int i;
        isl_int v;

        if (!tab)
                return -1;

        isl_assert(tab->mat->ctx, tab->bmap, return -1);
        isl_assert(tab->mat->ctx, tab->samples, return -1);
        isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1);

        isl_int_init(v);
        for (i = tab->n_outside; i < tab->n_sample; ++i) {
                int sgn;
                isl_seq_inner_product(ineq, tab->samples->row[i],
                                        1 + tab->n_var, &v);
                sgn = isl_int_sgn(v);
                if (eq ? (sgn == 0) : (sgn >= 0))
                        break;
        }
        isl_int_clear(v);

        return i < tab->n_sample;
}

/* Insert a div specified by "div" to the tableau "tab" at position "pos" and
 * return isl_bool_true if the div is obviously non-negative.
 */
static isl_bool context_tab_insert_div(struct isl_tab *tab, int pos,
        __isl_keep isl_vec *div,
        isl_stat (*add_ineq)(void *user, isl_int *), void *user)
{
        int i;
        int r;
        struct isl_mat *samples;
        int nonneg;

        r = isl_tab_insert_div(tab, pos, div, add_ineq, user);
        if (r < 0)
                return isl_bool_error;
        nonneg = tab->var[r].is_nonneg;
        tab->var[r].frozen = 1;

        samples = isl_mat_extend(tab->samples,
                        tab->n_sample, 1 + tab->n_var);
        tab->samples = samples;
        if (!samples)
                return isl_bool_error;
        for (i = tab->n_outside; i < samples->n_row; ++i) {
                isl_seq_inner_product(div->el + 1, samples->row[i],
                        div->size - 1, &samples->row[i][samples->n_col - 1]);
                isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
                               samples->row[i][samples->n_col - 1], div->el[0]);
        }
        tab->samples = isl_mat_move_cols(tab->samples, 1 + pos,
                                        1 + tab->n_var - 1, 1);
        if (!tab->samples)
                return isl_bool_error;

        return nonneg;
}

/* Add a div specified by "div" to both the main tableau and
 * the context tableau.  In case of the main tableau, we only
 * need to add an extra div.  In the context tableau, we also
 * need to express the meaning of the div.
 * Return the index of the div or -1 if anything went wrong.
 *
 * The new integer division is added before any unknown integer
 * divisions in the context to ensure that it does not get
 * equated to some linear combination involving unknown integer
 * divisions.
 */
static int add_div(struct isl_tab *tab, struct isl_context *context,
        __isl_keep isl_vec *div)
{
        int r;
        int pos;
        isl_bool nonneg;
        struct isl_tab *context_tab = context->op->peek_tab(context);

        if (!tab || !context_tab)
                goto error;

        pos = context_tab->n_var - context->n_unknown;
        if ((nonneg = context->op->insert_div(context, pos, div)) < 0)
                goto error;

        if (!context->op->is_ok(context))
                goto error;

        pos = tab->n_var - context->n_unknown;
        if (isl_tab_extend_vars(tab, 1) < 0)
                goto error;
        r = isl_tab_insert_var(tab, pos);
        if (r < 0)
                goto error;
        if (nonneg)
                tab->var[r].is_nonneg = 1;
        tab->var[r].frozen = 1;
        tab->n_div++;

        return tab->n_div - 1 - context->n_unknown;
error:
        context->op->invalidate(context);
        return -1;
}

static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
{
        int i;
        unsigned total = isl_basic_map_total_dim(tab->bmap);

        for (i = 0; i < tab->bmap->n_div; ++i) {
                if (isl_int_ne(tab->bmap->div[i][0], denom))
                        continue;
                if (!isl_seq_eq(tab->bmap->div[i] + 1, div, 1 + total))
                        continue;
                return i;
        }
        return -1;
}

/* Return the index of a div that corresponds to "div".
 * We first check if we already have such a div and if not, we create one.
 */
static int get_div(struct isl_tab *tab, struct isl_context *context,
        struct isl_vec *div)
{
        int d;
        struct isl_tab *context_tab = context->op->peek_tab(context);

        if (!context_tab)
                return -1;

        d = find_div(context_tab, div->el + 1, div->el[0]);
        if (d != -1)
                return d;

        return add_div(tab, context, div);
}

/* Add a parametric cut to cut away the non-integral sample value
 * of the give row.
 * Let a_i be the coefficients of the constant term and the parameters
 * and let b_i be the coefficients of the variables or constraints
 * in basis of the tableau.
 * Let q be the div q = floor(\sum_i {-a_i} y_i).
 *
 * The cut is expressed as
 *
 *      c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
 *
 * If q did not already exist in the context tableau, then it is added first.
 * If q is in a column of the main tableau then the "+ q" can be accomplished
 * by setting the corresponding entry to the denominator of the constraint.
 * If q happens to be in a row of the main tableau, then the corresponding
 * row needs to be added instead (taking care of the denominators).
 * Note that this is very unlikely, but perhaps not entirely impossible.
 *
 * The current value of the cut is known to be negative (or at least
 * non-positive), so row_sign is set accordingly.
 *
 * Return the row of the cut or -1.
 */
static int add_parametric_cut(struct isl_tab *tab, int row,
        struct isl_context *context)
{
        struct isl_vec *div;
        int d;
        int i;
        int r;
        isl_int *r_row;
        int col;
        int n;
        unsigned off = 2 + tab->M;

        if (!context)
                return -1;

        div = get_row_parameter_div(tab, row);
        if (!div)
                return -1;

        n = tab->n_div - context->n_unknown;
        d = context->op->get_div(context, tab, div);
        isl_vec_free(div);
        if (d < 0)
                return -1;

        if (isl_tab_extend_cons(tab, 1) < 0)
                return -1;
        r = isl_tab_allocate_con(tab);
        if (r < 0)
                return -1;

        r_row = tab->mat->row[tab->con[r].index];
        isl_int_set(r_row[0], tab->mat->row[row][0]);
        isl_int_neg(r_row[1], tab->mat->row[row][1]);
        isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
        isl_int_neg(r_row[1], r_row[1]);
        if (tab->M)
                isl_int_set_si(r_row[2], 0);
        for (i = 0; i < tab->n_param; ++i) {
                if (tab->var[i].is_row)
                        continue;
                col = tab->var[i].index;
                isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
                isl_int_fdiv_r(r_row[off + col], r_row[off + col],
                                tab->mat->row[row][0]);
                isl_int_neg(r_row[off + col], r_row[off + col]);
        }
        for (i = 0; i < tab->n_div; ++i) {
                if (tab->var[tab->n_var - tab->n_div + i].is_row)
                        continue;
                col = tab->var[tab->n_var - tab->n_div + i].index;
                isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
                isl_int_fdiv_r(r_row[off + col], r_row[off + col],
                                tab->mat->row[row][0]);
                isl_int_neg(r_row[off + col], r_row[off + col]);
        }
        for (i = 0; i < tab->n_col; ++i) {
                if (tab->col_var[i] >= 0 &&
                    (tab->col_var[i] < tab->n_param ||
                     tab->col_var[i] >= tab->n_var - tab->n_div))
                        continue;
                isl_int_fdiv_r(r_row[off + i],
                        tab->mat->row[row][off + i], tab->mat->row[row][0]);
        }
        if (tab->var[tab->n_var - tab->n_div + d].is_row) {
                isl_int gcd;
                int d_row = tab->var[tab->n_var - tab->n_div + d].index;
                isl_int_init(gcd);
                isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
                isl_int_divexact(r_row[0], r_row[0], gcd);
                isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
                isl_seq_combine(r_row + 1, gcd, r_row + 1,
                                r_row[0], tab->mat->row[d_row] + 1,
                                off - 1 + tab->n_col);
                isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
                isl_int_clear(gcd);
        } else {
                col = tab->var[tab->n_var - tab->n_div + d].index;
                isl_int_set(r_row[off + col], tab->mat->row[row][0]);
        }

        tab->con[r].is_nonneg = 1;
        if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
                return -1;
        if (tab->row_sign)
                tab->row_sign[tab->con[r].index] = isl_tab_row_neg;

        row = tab->con[r].index;

        if (d >= n && context->op->detect_equalities(context, tab) < 0)
                return -1;

        return row;
}

/* Construct a tableau for bmap that can be used for computing
 * the lexicographic minimum (or maximum) of bmap.
 * If not NULL, then dom is the domain where the minimum
 * should be computed.  In this case, we set up a parametric
 * tableau with row signs (initialized to "unknown").
 * If M is set, then the tableau will use a big parameter.
 * If max is set, then a maximum should be computed instead of a minimum.
 * This means that for each variable x, the tableau will contain the variable
 * x' = M - x, rather than x' = M + x.  This in turn means that the coefficient
 * of the variables in all constraints are negated prior to adding them
 * to the tableau.
 */
static __isl_give struct isl_tab *tab_for_lexmin(__isl_keep isl_basic_map *bmap,
        __isl_keep isl_basic_set *dom, unsigned M, int max)
{
        int i;
        struct isl_tab *tab;
        unsigned n_var;
        unsigned o_var;

        tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
                            isl_basic_map_total_dim(bmap), M);
        if (!tab)
                return NULL;

        tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
        if (dom) {
                tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
                tab->n_div = dom->n_div;
                tab->row_sign = isl_calloc_array(bmap->ctx,
                                        enum isl_tab_row_sign, tab->mat->n_row);
                if (tab->mat->n_row && !tab->row_sign)
                        goto error;
        }
        if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
                if (isl_tab_mark_empty(tab) < 0)
                        goto error;
                return tab;
        }

        for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
                tab->var[i].is_nonneg = 1;
                tab->var[i].frozen = 1;
        }
        o_var = 1 + tab->n_param;
        n_var = tab->n_var - tab->n_param - tab->n_div;
        for (i = 0; i < bmap->n_eq; ++i) {
                if (max)
                        isl_seq_neg(bmap->eq[i] + o_var,
                                    bmap->eq[i] + o_var, n_var);
                tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
                if (max)
                        isl_seq_neg(bmap->eq[i] + o_var,
                                    bmap->eq[i] + o_var, n_var);
                if (!tab || tab->empty)
                        return tab;
        }
        if (bmap->n_eq && restore_lexmin(tab) < 0)
                goto error;
        for (i = 0; i < bmap->n_ineq; ++i) {
                if (max)
                        isl_seq_neg(bmap->ineq[i] + o_var,
                                    bmap->ineq[i] + o_var, n_var);
                tab = add_lexmin_ineq(tab, bmap->ineq[i]);
                if (max)
                        isl_seq_neg(bmap->ineq[i] + o_var,
                                    bmap->ineq[i] + o_var, n_var);
                if (!tab || tab->empty)
                        return tab;
        }
        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

/* Given a main tableau where more than one row requires a split,
 * determine and return the "best" row to split on.
 *
 * If any of the rows requiring a split only involves
 * variables that also appear in the context tableau,
 * then the negative part is guaranteed not to have a solution.
 * It is therefore best to split on any of these rows first.
 *
 * Otherwise,
 * given two rows in the main tableau, if the inequality corresponding
 * to the first row is redundant with respect to that of the second row
 * in the current tableau, then it is better to split on the second row,
 * since in the positive part, both rows will be positive.
 * (In the negative part a pivot will have to be performed and just about
 * anything can happen to the sign of the other row.)
 *
 * As a simple heuristic, we therefore select the row that makes the most
 * of the other rows redundant.
 *
 * Perhaps it would also be useful to look at the number of constraints
 * that conflict with any given constraint.
 *
 * best is the best row so far (-1 when we have not found any row yet).
 * best_r is the number of other rows made redundant by row best.
 * When best is still -1, bset_r is meaningless, but it is initialized
 * to some arbitrary value (0) anyway.  Without this redundant initialization
 * valgrind may warn about uninitialized memory accesses when isl
 * is compiled with some versions of gcc.
 */
static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
{
        struct isl_tab_undo *snap;
        int split;
        int row;
        int best = -1;
        int best_r = 0;

        if (isl_tab_extend_cons(context_tab, 2) < 0)
                return -1;

        snap = isl_tab_snap(context_tab);

        for (split = tab->n_redundant; split < tab->n_row; ++split) {
                struct isl_tab_undo *snap2;
                struct isl_vec *ineq = NULL;
                int r = 0;
                int ok;

                if (!isl_tab_var_from_row(tab, split)->is_nonneg)
                        continue;
                if (tab->row_sign[split] != isl_tab_row_any)
                        continue;

                if (is_parametric_constant(tab, split))
                        return split;

                ineq = get_row_parameter_ineq(tab, split);
                if (!ineq)
                        return -1;
                ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0;
                isl_vec_free(ineq);
                if (!ok)
                        return -1;

                snap2 = isl_tab_snap(context_tab);

                for (row = tab->n_redundant; row < tab->n_row; ++row) {
                        struct isl_tab_var *var;

                        if (row == split)
                                continue;
                        if (!isl_tab_var_from_row(tab, row)->is_nonneg)
                                continue;
                        if (tab->row_sign[row] != isl_tab_row_any)
                                continue;

                        ineq = get_row_parameter_ineq(tab, row);
                        if (!ineq)
                                return -1;
                        ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0;
                        isl_vec_free(ineq);
                        if (!ok)
                                return -1;
                        var = &context_tab->con[context_tab->n_con - 1];
                        if (!context_tab->empty &&
                            !isl_tab_min_at_most_neg_one(context_tab, var))
                                r++;
                        if (isl_tab_rollback(context_tab, snap2) < 0)
                                return -1;
                }
                if (best == -1 || r > best_r) {
                        best = split;
                        best_r = r;
                }
                if (isl_tab_rollback(context_tab, snap) < 0)
                        return -1;
        }

        return best;
}

static struct isl_basic_set *context_lex_peek_basic_set(
        struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        if (!clex->tab)
                return NULL;
        return isl_tab_peek_bset(clex->tab);
}

static struct isl_tab *context_lex_peek_tab(struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        return clex->tab;
}

static void context_lex_add_eq(struct isl_context *context, isl_int *eq,
                int check, int update)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        if (isl_tab_extend_cons(clex->tab, 2) < 0)
                goto error;
        if (add_lexmin_eq(clex->tab, eq) < 0)
                goto error;
        if (check) {
                int v = tab_has_valid_sample(clex->tab, eq, 1);
                if (v < 0)
                        goto error;
                if (!v)
                        clex->tab = check_integer_feasible(clex->tab);
        }
        if (update)
                clex->tab = check_samples(clex->tab, eq, 1);
        return;
error:
        isl_tab_free(clex->tab);
        clex->tab = NULL;
}

static void context_lex_add_ineq(struct isl_context *context, isl_int *ineq,
                int check, int update)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        if (isl_tab_extend_cons(clex->tab, 1) < 0)
                goto error;
        clex->tab = add_lexmin_ineq(clex->tab, ineq);
        if (check) {
                int v = tab_has_valid_sample(clex->tab, ineq, 0);
                if (v < 0)
                        goto error;
                if (!v)
                        clex->tab = check_integer_feasible(clex->tab);
        }
        if (update)
                clex->tab = check_samples(clex->tab, ineq, 0);
        return;
error:
        isl_tab_free(clex->tab);
        clex->tab = NULL;
}

static isl_stat context_lex_add_ineq_wrap(void *user, isl_int *ineq)
{
        struct isl_context *context = (struct isl_context *)user;
        context_lex_add_ineq(context, ineq, 0, 0);
        return context->op->is_ok(context) ? isl_stat_ok : isl_stat_error;
}

/* Check which signs can be obtained by "ineq" on all the currently
 * active sample values.  See row_sign for more information.
 */
static enum isl_tab_row_sign tab_ineq_sign(struct isl_tab *tab, isl_int *ineq,
        int strict)
{
        int i;
        int sgn;
        isl_int tmp;
        enum isl_tab_row_sign res = isl_tab_row_unknown;

        isl_assert(tab->mat->ctx, tab->samples, return isl_tab_row_unknown);
        isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var,
                        return isl_tab_row_unknown);

        isl_int_init(tmp);
        for (i = tab->n_outside; i < tab->n_sample; ++i) {
                isl_seq_inner_product(tab->samples->row[i], ineq,
                                        1 + tab->n_var, &tmp);
                sgn = isl_int_sgn(tmp);
                if (sgn > 0 || (sgn == 0 && strict)) {
                        if (res == isl_tab_row_unknown)
                                res = isl_tab_row_pos;
                        if (res == isl_tab_row_neg)
                                res = isl_tab_row_any;
                }
                if (sgn < 0) {
                        if (res == isl_tab_row_unknown)
                                res = isl_tab_row_neg;
                        if (res == isl_tab_row_pos)
                                res = isl_tab_row_any;
                }
                if (res == isl_tab_row_any)
                        break;
        }
        isl_int_clear(tmp);

        return res;
}

static enum isl_tab_row_sign context_lex_ineq_sign(struct isl_context *context,
                        isl_int *ineq, int strict)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        return tab_ineq_sign(clex->tab, ineq, strict);
}

/* Check whether "ineq" can be added to the tableau without rendering
 * it infeasible.
 */
static int context_lex_test_ineq(struct isl_context *context, isl_int *ineq)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        struct isl_tab_undo *snap;
        int feasible;

        if (!clex->tab)
                return -1;

        if (isl_tab_extend_cons(clex->tab, 1) < 0)
                return -1;

        snap = isl_tab_snap(clex->tab);
        if (isl_tab_push_basis(clex->tab) < 0)
                return -1;
        clex->tab = add_lexmin_ineq(clex->tab, ineq);
        clex->tab = check_integer_feasible(clex->tab);
        if (!clex->tab)
                return -1;
        feasible = !clex->tab->empty;
        if (isl_tab_rollback(clex->tab, snap) < 0)
                return -1;

        return feasible;
}

static int context_lex_get_div(struct isl_context *context, struct isl_tab *tab,
                struct isl_vec *div)
{
        return get_div(tab, context, div);
}

/* Insert a div specified by "div" to the context tableau at position "pos" and
 * return isl_bool_true if the div is obviously non-negative.
 * context_tab_add_div will always return isl_bool_true, because all variables
 * in a isl_context_lex tableau are non-negative.
 * However, if we are using a big parameter in the context, then this only
 * reflects the non-negativity of the variable used to _encode_ the
 * div, i.e., div' = M + div, so we can't draw any conclusions.
 */
static isl_bool context_lex_insert_div(struct isl_context *context, int pos,
        __isl_keep isl_vec *div)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        isl_bool nonneg;
        nonneg = context_tab_insert_div(clex->tab, pos, div,
                                        context_lex_add_ineq_wrap, context);
        if (nonneg < 0)
                return isl_bool_error;
        if (clex->tab->M)
                return isl_bool_false;
        return nonneg;
}

static int context_lex_detect_equalities(struct isl_context *context,
                struct isl_tab *tab)
{
        return 0;
}

static int context_lex_best_split(struct isl_context *context,
                struct isl_tab *tab)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        struct isl_tab_undo *snap;
        int r;

        snap = isl_tab_snap(clex->tab);
        if (isl_tab_push_basis(clex->tab) < 0)
                return -1;
        r = best_split(tab, clex->tab);

        if (r >= 0 && isl_tab_rollback(clex->tab, snap) < 0)
                return -1;

        return r;
}

static int context_lex_is_empty(struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        if (!clex->tab)
                return -1;
        return clex->tab->empty;
}

static void *context_lex_save(struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        struct isl_tab_undo *snap;

        snap = isl_tab_snap(clex->tab);
        if (isl_tab_push_basis(clex->tab) < 0)
                return NULL;
        if (isl_tab_save_samples(clex->tab) < 0)
                return NULL;

        return snap;
}

static void context_lex_restore(struct isl_context *context, void *save)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        if (isl_tab_rollback(clex->tab, (struct isl_tab_undo *)save) < 0) {
                isl_tab_free(clex->tab);
                clex->tab = NULL;
        }
}

static void context_lex_discard(void *save)
{
}

static int context_lex_is_ok(struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        return !!clex->tab;
}

/* For each variable in the context tableau, check if the variable can
 * only attain non-negative values.  If so, mark the parameter as non-negative
 * in the main tableau.  This allows for a more direct identification of some
 * cases of violated constraints.
 */
static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
        struct isl_tab *context_tab)
{
        int i;
        struct isl_tab_undo *snap;
        struct isl_vec *ineq = NULL;
        struct isl_tab_var *var;
        int n;

        if (context_tab->n_var == 0)
                return tab;

        ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
        if (!ineq)
                goto error;

        if (isl_tab_extend_cons(context_tab, 1) < 0)
                goto error;

        snap = isl_tab_snap(context_tab);

        n = 0;
        isl_seq_clr(ineq->el, ineq->size);
        for (i = 0; i < context_tab->n_var; ++i) {
                isl_int_set_si(ineq->el[1 + i], 1);
                if (isl_tab_add_ineq(context_tab, ineq->el) < 0)
                        goto error;
                var = &context_tab->con[context_tab->n_con - 1];
                if (!context_tab->empty &&
                    !isl_tab_min_at_most_neg_one(context_tab, var)) {
                        int j = i;
                        if (i >= tab->n_param)
                                j = i - tab->n_param + tab->n_var - tab->n_div;
                        tab->var[j].is_nonneg = 1;
                        n++;
                }
                isl_int_set_si(ineq->el[1 + i], 0);
                if (isl_tab_rollback(context_tab, snap) < 0)
                        goto error;
        }

        if (context_tab->M && n == context_tab->n_var) {
                context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
                context_tab->M = 0;
        }

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

static struct isl_tab *context_lex_detect_nonnegative_parameters(
        struct isl_context *context, struct isl_tab *tab)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        struct isl_tab_undo *snap;

        if (!tab)
                return NULL;

        snap = isl_tab_snap(clex->tab);
        if (isl_tab_push_basis(clex->tab) < 0)
                goto error;

        tab = tab_detect_nonnegative_parameters(tab, clex->tab);

        if (isl_tab_rollback(clex->tab, snap) < 0)
                goto error;

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

static void context_lex_invalidate(struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        isl_tab_free(clex->tab);
        clex->tab = NULL;
}

static __isl_null struct isl_context *context_lex_free(
        struct isl_context *context)
{
        struct isl_context_lex *clex = (struct isl_context_lex *)context;
        isl_tab_free(clex->tab);
        free(clex);

        return NULL;
}

struct isl_context_op isl_context_lex_op = {
        context_lex_detect_nonnegative_parameters,
        context_lex_peek_basic_set,
        context_lex_peek_tab,
        context_lex_add_eq,
        context_lex_add_ineq,
        context_lex_ineq_sign,
        context_lex_test_ineq,
        context_lex_get_div,
        context_lex_insert_div,
        context_lex_detect_equalities,
        context_lex_best_split,
        context_lex_is_empty,
        context_lex_is_ok,
        context_lex_save,
        context_lex_restore,
        context_lex_discard,
        context_lex_invalidate,
        context_lex_free,
};

static struct isl_tab *context_tab_for_lexmin(__isl_take isl_basic_set *bset)
{
        struct isl_tab *tab;

        if (!bset)
                return NULL;
        tab = tab_for_lexmin(bset_to_bmap(bset), NULL, 1, 0);
        if (isl_tab_track_bset(tab, bset) < 0)
                goto error;
        tab = isl_tab_init_samples(tab);
        return tab;
error:
        isl_tab_free(tab);
        return NULL;
}

static struct isl_context *isl_context_lex_alloc(struct isl_basic_set *dom)
{
        struct isl_context_lex *clex;

        if (!dom)
                return NULL;

        clex = isl_alloc_type(dom->ctx, struct isl_context_lex);
        if (!clex)
                return NULL;

        clex->context.op = &isl_context_lex_op;

        clex->tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
        if (restore_lexmin(clex->tab) < 0)
                goto error;
        clex->tab = check_integer_feasible(clex->tab);
        if (!clex->tab)
                goto error;

        return &clex->context;
error:
        clex->context.op->free(&clex->context);
        return NULL;
}

/* Representation of the context when using generalized basis reduction.
 *
 * "shifted" contains the offsets of the unit hypercubes that lie inside the
 * context.  Any rational point in "shifted" can therefore be rounded
 * up to an integer point in the context.
 * If the context is constrained by any equality, then "shifted" is not used
 * as it would be empty.
 */
struct isl_context_gbr {
        struct isl_context context;
        struct isl_tab *tab;
        struct isl_tab *shifted;
        struct isl_tab *cone;
};

static struct isl_tab *context_gbr_detect_nonnegative_parameters(
        struct isl_context *context, struct isl_tab *tab)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        if (!tab)
                return NULL;
        return tab_detect_nonnegative_parameters(tab, cgbr->tab);
}

static struct isl_basic_set *context_gbr_peek_basic_set(
        struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        if (!cgbr->tab)
                return NULL;
        return isl_tab_peek_bset(cgbr->tab);
}

static struct isl_tab *context_gbr_peek_tab(struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        return cgbr->tab;
}

/* Initialize the "shifted" tableau of the context, which
 * contains the constraints of the original tableau shifted
 * by the sum of all negative coefficients.  This ensures
 * that any rational point in the shifted tableau can
 * be rounded up to yield an integer point in the original tableau.
 */
static void gbr_init_shifted(struct isl_context_gbr *cgbr)
{
        int i, j;
        struct isl_vec *cst;
        struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab);
        unsigned dim = isl_basic_set_total_dim(bset);

        cst = isl_vec_alloc(cgbr->tab->mat->ctx, bset->n_ineq);
        if (!cst)
                return;

        for (i = 0; i < bset->n_ineq; ++i) {
                isl_int_set(cst->el[i], bset->ineq[i][0]);
                for (j = 0; j < dim; ++j) {
                        if (!isl_int_is_neg(bset->ineq[i][1 + j]))
                                continue;
                        isl_int_add(bset->ineq[i][0], bset->ineq[i][0],
                                    bset->ineq[i][1 + j]);
                }
        }

        cgbr->shifted = isl_tab_from_basic_set(bset, 0);

        for (i = 0; i < bset->n_ineq; ++i)
                isl_int_set(bset->ineq[i][0], cst->el[i]);

        isl_vec_free(cst);
}

/* Check if the shifted tableau is non-empty, and if so
 * use the sample point to construct an integer point
 * of the context tableau.
 */
static struct isl_vec *gbr_get_shifted_sample(struct isl_context_gbr *cgbr)
{
        struct isl_vec *sample;

        if (!cgbr->shifted)
                gbr_init_shifted(cgbr);
        if (!cgbr->shifted)
                return NULL;
        if (cgbr->shifted->empty)
                return isl_vec_alloc(cgbr->tab->mat->ctx, 0);

        sample = isl_tab_get_sample_value(cgbr->shifted);
        sample = isl_vec_ceil(sample);

        return sample;
}

static __isl_give isl_basic_set *drop_constant_terms(
        __isl_take isl_basic_set *bset)
{
        int i;

        if (!bset)
                return NULL;

        for (i = 0; i < bset->n_eq; ++i)
                isl_int_set_si(bset->eq[i][0], 0);

        for (i = 0; i < bset->n_ineq; ++i)
                isl_int_set_si(bset->ineq[i][0], 0);

        return bset;
}

static int use_shifted(struct isl_context_gbr *cgbr)
{
        if (!cgbr->tab)
                return 0;
        return cgbr->tab->bmap->n_eq == 0 && cgbr->tab->bmap->n_div == 0;
}

static struct isl_vec *gbr_get_sample(struct isl_context_gbr *cgbr)
{
        struct isl_basic_set *bset;
        struct isl_basic_set *cone;

        if (isl_tab_sample_is_integer(cgbr->tab))
                return isl_tab_get_sample_value(cgbr->tab);

        if (use_shifted(cgbr)) {
                struct isl_vec *sample;

                sample = gbr_get_shifted_sample(cgbr);
                if (!sample || sample->size > 0)
                        return sample;

                isl_vec_free(sample);
        }

        if (!cgbr->cone) {
                bset = isl_tab_peek_bset(cgbr->tab);
                cgbr->cone = isl_tab_from_recession_cone(bset, 0);
                if (!cgbr->cone)
                        return NULL;
                if (isl_tab_track_bset(cgbr->cone,
                                        isl_basic_set_copy(bset)) < 0)
                        return NULL;
        }
        if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0)
                return NULL;

        if (cgbr->cone->n_dead == cgbr->cone->n_col) {
                struct isl_vec *sample;
                struct isl_tab_undo *snap;

                if (cgbr->tab->basis) {
                        if (cgbr->tab->basis->n_col != 1 + cgbr->tab->n_var) {
                                isl_mat_free(cgbr->tab->basis);
                                cgbr->tab->basis = NULL;
                        }
                        cgbr->tab->n_zero = 0;
                        cgbr->tab->n_unbounded = 0;
                }

                snap = isl_tab_snap(cgbr->tab);

                sample = isl_tab_sample(cgbr->tab);

                if (!sample || isl_tab_rollback(cgbr->tab, snap) < 0) {
                        isl_vec_free(sample);
                        return NULL;
                }

                return sample;
        }

        cone = isl_basic_set_dup(isl_tab_peek_bset(cgbr->cone));
        cone = drop_constant_terms(cone);
        cone = isl_basic_set_update_from_tab(cone, cgbr->cone);
        cone = isl_basic_set_underlying_set(cone);
        cone = isl_basic_set_gauss(cone, NULL);

        bset = isl_basic_set_dup(isl_tab_peek_bset(cgbr->tab));
        bset = isl_basic_set_update_from_tab(bset, cgbr->tab);
        bset = isl_basic_set_underlying_set(bset);
        bset = isl_basic_set_gauss(bset, NULL);

        return isl_basic_set_sample_with_cone(bset, cone);
}

static void check_gbr_integer_feasible(struct isl_context_gbr *cgbr)
{
        struct isl_vec *sample;

        if (!cgbr->tab)
                return;

        if (cgbr->tab->empty)
                return;

        sample = gbr_get_sample(cgbr);
        if (!sample)
                goto error;

        if (sample->size == 0) {
                isl_vec_free(sample);
                if (isl_tab_mark_empty(cgbr->tab) < 0)
                        goto error;
                return;
        }

        if (isl_tab_add_sample(cgbr->tab, sample) < 0)
                goto error;

        return;
error:
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
}

static struct isl_tab *add_gbr_eq(struct isl_tab *tab, isl_int *eq)
{
        if (!tab)
                return NULL;

        if (isl_tab_extend_cons(tab, 2) < 0)
                goto error;

        if (isl_tab_add_eq(tab, eq) < 0)
                goto error;

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

/* Add the equality described by "eq" to the context.
 * If "check" is set, then we check if the context is empty after
 * adding the equality.
 * If "update" is set, then we check if the samples are still valid.
 *
 * We do not explicitly add shifted copies of the equality to
 * cgbr->shifted since they would conflict with each other.
 * Instead, we directly mark cgbr->shifted empty.
 */
static void context_gbr_add_eq(struct isl_context *context, isl_int *eq,
                int check, int update)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;

        cgbr->tab = add_gbr_eq(cgbr->tab, eq);

        if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) {
                if (isl_tab_mark_empty(cgbr->shifted) < 0)
                        goto error;
        }

        if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
                if (isl_tab_extend_cons(cgbr->cone, 2) < 0)
                        goto error;
                if (isl_tab_add_eq(cgbr->cone, eq) < 0)
                        goto error;
        }

        if (check) {
                int v = tab_has_valid_sample(cgbr->tab, eq, 1);
                if (v < 0)
                        goto error;
                if (!v)
                        check_gbr_integer_feasible(cgbr);
        }
        if (update)
                cgbr->tab = check_samples(cgbr->tab, eq, 1);
        return;
error:
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
}

static void add_gbr_ineq(struct isl_context_gbr *cgbr, isl_int *ineq)
{
        if (!cgbr->tab)
                return;

        if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
                goto error;

        if (isl_tab_add_ineq(cgbr->tab, ineq) < 0)
                goto error;

        if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) {
                int i;
                unsigned dim;
                dim = isl_basic_map_total_dim(cgbr->tab->bmap);

                if (isl_tab_extend_cons(cgbr->shifted, 1) < 0)
                        goto error;

                for (i = 0; i < dim; ++i) {
                        if (!isl_int_is_neg(ineq[1 + i]))
                                continue;
                        isl_int_add(ineq[0], ineq[0], ineq[1 + i]);
                }

                if (isl_tab_add_ineq(cgbr->shifted, ineq) < 0)
                        goto error;

                for (i = 0; i < dim; ++i) {
                        if (!isl_int_is_neg(ineq[1 + i]))
                                continue;
                        isl_int_sub(ineq[0], ineq[0], ineq[1 + i]);
                }
        }

        if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
                if (isl_tab_extend_cons(cgbr->cone, 1) < 0)
                        goto error;
                if (isl_tab_add_ineq(cgbr->cone, ineq) < 0)
                        goto error;
        }

        return;
error:
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
}

static void context_gbr_add_ineq(struct isl_context *context, isl_int *ineq,
                int check, int update)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;

        add_gbr_ineq(cgbr, ineq);
        if (!cgbr->tab)
                return;

        if (check) {
                int v = tab_has_valid_sample(cgbr->tab, ineq, 0);
                if (v < 0)
                        goto error;
                if (!v)
                        check_gbr_integer_feasible(cgbr);
        }
        if (update)
                cgbr->tab = check_samples(cgbr->tab, ineq, 0);
        return;
error:
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
}

static isl_stat context_gbr_add_ineq_wrap(void *user, isl_int *ineq)
{
        struct isl_context *context = (struct isl_context *)user;
        context_gbr_add_ineq(context, ineq, 0, 0);
        return context->op->is_ok(context) ? isl_stat_ok : isl_stat_error;
}

static enum isl_tab_row_sign context_gbr_ineq_sign(struct isl_context *context,
                        isl_int *ineq, int strict)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        return tab_ineq_sign(cgbr->tab, ineq, strict);
}

/* Check whether "ineq" can be added to the tableau without rendering
 * it infeasible.
 */
static int context_gbr_test_ineq(struct isl_context *context, isl_int *ineq)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        struct isl_tab_undo *snap;
        struct isl_tab_undo *shifted_snap = NULL;
        struct isl_tab_undo *cone_snap = NULL;
        int feasible;

        if (!cgbr->tab)
                return -1;

        if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
                return -1;

        snap = isl_tab_snap(cgbr->tab);
        if (cgbr->shifted)
                shifted_snap = isl_tab_snap(cgbr->shifted);
        if (cgbr->cone)
                cone_snap = isl_tab_snap(cgbr->cone);
        add_gbr_ineq(cgbr, ineq);
        check_gbr_integer_feasible(cgbr);
        if (!cgbr->tab)
                return -1;
        feasible = !cgbr->tab->empty;
        if (isl_tab_rollback(cgbr->tab, snap) < 0)
                return -1;
        if (shifted_snap) {
                if (isl_tab_rollback(cgbr->shifted, shifted_snap))
                        return -1;
        } else if (cgbr->shifted) {
                isl_tab_free(cgbr->shifted);
                cgbr->shifted = NULL;
        }
        if (cone_snap) {
                if (isl_tab_rollback(cgbr->cone, cone_snap))
                        return -1;
        } else if (cgbr->cone) {
                isl_tab_free(cgbr->cone);
                cgbr->cone = NULL;
        }

        return feasible;
}

/* Return the column of the last of the variables associated to
 * a column that has a non-zero coefficient.
 * This function is called in a context where only coefficients
 * of parameters or divs can be non-zero.
 */
static int last_non_zero_var_col(struct isl_tab *tab, isl_int *p)
{
        int i;
        int col;

        if (tab->n_var == 0)
                return -1;

        for (i = tab->n_var - 1; i >= 0; --i) {
                if (i >= tab->n_param && i < tab->n_var - tab->n_div)
                        continue;
                if (tab->var[i].is_row)
                        continue;
                col = tab->var[i].index;
                if (!isl_int_is_zero(p[col]))
                        return col;
        }

        return -1;
}

/* Look through all the recently added equalities in the context
 * to see if we can propagate any of them to the main tableau.
 *
 * The newly added equalities in the context are encoded as pairs
 * of inequalities starting at inequality "first".
 *
 * We tentatively add each of these equalities to the main tableau
 * and if this happens to result in a row with a final coefficient
 * that is one or negative one, we use it to kill a column
 * in the main tableau.  Otherwise, we discard the tentatively
 * added row.
 * This tentative addition of equality constraints turns
 * on the undo facility of the tableau.  Turn it off again
 * at the end, assuming it was turned off to begin with.
 *
 * Return 0 on success and -1 on failure.
 */
static int propagate_equalities(struct isl_context_gbr *cgbr,
        struct isl_tab *tab, unsigned first)
{
        int i;
        struct isl_vec *eq = NULL;
        isl_bool needs_undo;

        needs_undo = isl_tab_need_undo(tab);
        if (needs_undo < 0)
                goto error;
        eq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
        if (!eq)
                goto error;

        if (isl_tab_extend_cons(tab, (cgbr->tab->bmap->n_ineq - first)/2) < 0)
                goto error;

        isl_seq_clr(eq->el + 1 + tab->n_param,
                    tab->n_var - tab->n_param - tab->n_div);
        for (i = first; i < cgbr->tab->bmap->n_ineq; i += 2) {
                int j;
                int r;
                struct isl_tab_undo *snap;
                snap = isl_tab_snap(tab);

                isl_seq_cpy(eq->el, cgbr->tab->bmap->ineq[i], 1 + tab->n_param);
                isl_seq_cpy(eq->el + 1 + tab->n_var - tab->n_div,
                            cgbr->tab->bmap->ineq[i] + 1 + tab->n_param,
                            tab->n_div);

                r = isl_tab_add_row(tab, eq->el);
                if (r < 0)
                        goto error;
                r = tab->con[r].index;
                j = last_non_zero_var_col(tab, tab->mat->row[r] + 2 + tab->M);
                if (j < 0 || j < tab->n_dead ||
                    !isl_int_is_one(tab->mat->row[r][0]) ||
                    (!isl_int_is_one(tab->mat->row[r][2 + tab->M + j]) &&
                     !isl_int_is_negone(tab->mat->row[r][2 + tab->M + j]))) {
                        if (isl_tab_rollback(tab, snap) < 0)
                                goto error;
                        continue;
                }
                if (isl_tab_pivot(tab, r, j) < 0)
                        goto error;
                if (isl_tab_kill_col(tab, j) < 0)
                        goto error;

                if (restore_lexmin(tab) < 0)
                        goto error;
        }

        if (!needs_undo)
                isl_tab_clear_undo(tab);
        isl_vec_free(eq);

        return 0;
error:
        isl_vec_free(eq);
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
        return -1;
}

static int context_gbr_detect_equalities(struct isl_context *context,
        struct isl_tab *tab)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        unsigned n_ineq;

        if (!cgbr->cone) {
                struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab);
                cgbr->cone = isl_tab_from_recession_cone(bset, 0);
                if (!cgbr->cone)
                        goto error;
                if (isl_tab_track_bset(cgbr->cone,
                                        isl_basic_set_copy(bset)) < 0)
                        goto error;
        }
        if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0)
                goto error;

        n_ineq = cgbr->tab->bmap->n_ineq;
        cgbr->tab = isl_tab_detect_equalities(cgbr->tab, cgbr->cone);
        if (!cgbr->tab)
                return -1;
        if (cgbr->tab->bmap->n_ineq > n_ineq &&
            propagate_equalities(cgbr, tab, n_ineq) < 0)
                return -1;

        return 0;
error:
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
        return -1;
}

static int context_gbr_get_div(struct isl_context *context, struct isl_tab *tab,
                struct isl_vec *div)
{
        return get_div(tab, context, div);
}

static isl_bool context_gbr_insert_div(struct isl_context *context, int pos,
        __isl_keep isl_vec *div)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        if (cgbr->cone) {
                int r, n_div, o_div;

                n_div = isl_basic_map_dim(cgbr->cone->bmap, isl_dim_div);
                o_div = cgbr->cone->n_var - n_div;

                if (isl_tab_extend_cons(cgbr->cone, 3) < 0)
                        return isl_bool_error;
                if (isl_tab_extend_vars(cgbr->cone, 1) < 0)
                        return isl_bool_error;
                if ((r = isl_tab_insert_var(cgbr->cone, pos)) <0)
                        return isl_bool_error;

                cgbr->cone->bmap = isl_basic_map_insert_div(cgbr->cone->bmap,
                                                    r - o_div, div);
                if (!cgbr->cone->bmap)
                        return isl_bool_error;
                if (isl_tab_push_var(cgbr->cone, isl_tab_undo_bmap_div,
                                    &cgbr->cone->var[r]) < 0)
                        return isl_bool_error;
        }
        return context_tab_insert_div(cgbr->tab, pos, div,
                                        context_gbr_add_ineq_wrap, context);
}

static int context_gbr_best_split(struct isl_context *context,
                struct isl_tab *tab)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        struct isl_tab_undo *snap;
        int r;

        snap = isl_tab_snap(cgbr->tab);
        r = best_split(tab, cgbr->tab);

        if (r >= 0 && isl_tab_rollback(cgbr->tab, snap) < 0)
                return -1;

        return r;
}

static int context_gbr_is_empty(struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        if (!cgbr->tab)
                return -1;
        return cgbr->tab->empty;
}

struct isl_gbr_tab_undo {
        struct isl_tab_undo *tab_snap;
        struct isl_tab_undo *shifted_snap;
        struct isl_tab_undo *cone_snap;
};

static void *context_gbr_save(struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        struct isl_gbr_tab_undo *snap;

        if (!cgbr->tab)
                return NULL;

        snap = isl_alloc_type(cgbr->tab->mat->ctx, struct isl_gbr_tab_undo);
        if (!snap)
                return NULL;

        snap->tab_snap = isl_tab_snap(cgbr->tab);
        if (isl_tab_save_samples(cgbr->tab) < 0)
                goto error;

        if (cgbr->shifted)
                snap->shifted_snap = isl_tab_snap(cgbr->shifted);
        else
                snap->shifted_snap = NULL;

        if (cgbr->cone)
                snap->cone_snap = isl_tab_snap(cgbr->cone);
        else
                snap->cone_snap = NULL;

        return snap;
error:
        free(snap);
        return NULL;
}

static void context_gbr_restore(struct isl_context *context, void *save)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
        if (!snap)
                goto error;
        if (isl_tab_rollback(cgbr->tab, snap->tab_snap) < 0)
                goto error;

        if (snap->shifted_snap) {
                if (isl_tab_rollback(cgbr->shifted, snap->shifted_snap) < 0)
                        goto error;
        } else if (cgbr->shifted) {
                isl_tab_free(cgbr->shifted);
                cgbr->shifted = NULL;
        }

        if (snap->cone_snap) {
                if (isl_tab_rollback(cgbr->cone, snap->cone_snap) < 0)
                        goto error;
        } else if (cgbr->cone) {
                isl_tab_free(cgbr->cone);
                cgbr->cone = NULL;
        }

        free(snap);

        return;
error:
        free(snap);
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
}

static void context_gbr_discard(void *save)
{
        struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
        free(snap);
}

static int context_gbr_is_ok(struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        return !!cgbr->tab;
}

static void context_gbr_invalidate(struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        isl_tab_free(cgbr->tab);
        cgbr->tab = NULL;
}

static __isl_null struct isl_context *context_gbr_free(
        struct isl_context *context)
{
        struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
        isl_tab_free(cgbr->tab);
        isl_tab_free(cgbr->shifted);
        isl_tab_free(cgbr->cone);
        free(cgbr);

        return NULL;
}

struct isl_context_op isl_context_gbr_op = {
        context_gbr_detect_nonnegative_parameters,
        context_gbr_peek_basic_set,
        context_gbr_peek_tab,
        context_gbr_add_eq,
        context_gbr_add_ineq,
        context_gbr_ineq_sign,
        context_gbr_test_ineq,
        context_gbr_get_div,
        context_gbr_insert_div,
        context_gbr_detect_equalities,
        context_gbr_best_split,
        context_gbr_is_empty,
        context_gbr_is_ok,
        context_gbr_save,
        context_gbr_restore,
        context_gbr_discard,
        context_gbr_invalidate,
        context_gbr_free,
};

static struct isl_context *isl_context_gbr_alloc(__isl_keep isl_basic_set *dom)
{
        struct isl_context_gbr *cgbr;

        if (!dom)
                return NULL;

        cgbr = isl_calloc_type(dom->ctx, struct isl_context_gbr);
        if (!cgbr)
                return NULL;

        cgbr->context.op = &isl_context_gbr_op;

        cgbr->shifted = NULL;
        cgbr->cone = NULL;
        cgbr->tab = isl_tab_from_basic_set(dom, 1);
        cgbr->tab = isl_tab_init_samples(cgbr->tab);
        if (!cgbr->tab)
                goto error;
        check_gbr_integer_feasible(cgbr);

        return &cgbr->context;
error:
        cgbr->context.op->free(&cgbr->context);
        return NULL;
}

/* Allocate a context corresponding to "dom".
 * The representation specific fields are initialized by
 * isl_context_lex_alloc or isl_context_gbr_alloc.
 * The shared "n_unknown" field is initialized to the number
 * of final unknown integer divisions in "dom".
 */
static struct isl_context *isl_context_alloc(__isl_keep isl_basic_set *dom)
{
        struct isl_context *context;
        int first;

        if (!dom)
                return NULL;

        if (dom->ctx->opt->context == ISL_CONTEXT_LEXMIN)
                context = isl_context_lex_alloc(dom);
        else
                context = isl_context_gbr_alloc(dom);

        if (!context)
                return NULL;

        first = isl_basic_set_first_unknown_div(dom);
        if (first < 0)
                return context->op->free(context);
        context->n_unknown = isl_basic_set_dim(dom, isl_dim_div) - first;

        return context;
}

/* Initialize some common fields of "sol", which keeps track
 * of the solution of an optimization problem on "bmap" over
 * the domain "dom".
 * If "max" is set, then a maximization problem is being solved, rather than
 * a minimization problem, which means that the variables in the
 * tableau have value "M - x" rather than "M + x".
 */
static isl_stat sol_init(struct isl_sol *sol, __isl_keep isl_basic_map *bmap,
        __isl_keep isl_basic_set *dom, int max)
{
        sol->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
        sol->dec_level.callback.run = &sol_dec_level_wrap;
        sol->dec_level.sol = sol;
        sol->max = max;
        sol->n_out = isl_basic_map_dim(bmap, isl_dim_out);
        sol->space = isl_basic_map_get_space(bmap);

        sol->context = isl_context_alloc(dom);
        if (!sol->space || !sol->context)
                return isl_stat_error;

        return isl_stat_ok;
}

/* Construct an isl_sol_map structure for accumulating the solution.
 * If track_empty is set, then we also keep track of the parts
 * of the context where there is no solution.
 * If max is set, then we are solving a maximization, rather than
 * a minimization problem, which means that the variables in the
 * tableau have value "M - x" rather than "M + x".
 */
static struct isl_sol *sol_map_init(__isl_keep isl_basic_map *bmap,
        __isl_take isl_basic_set *dom, int track_empty, int max)
{
        struct isl_sol_map *sol_map = NULL;
        isl_space *space;

        if (!bmap)
                goto error;

        sol_map = isl_calloc_type(bmap->ctx, struct isl_sol_map);
        if (!sol_map)
                goto error;

        sol_map->sol.free = &sol_map_free;
        if (sol_init(&sol_map->sol, bmap, dom, max) < 0)
                goto error;
        sol_map->sol.add = &sol_map_add_wrap;
        sol_map->sol.add_empty = track_empty ? &sol_map_add_empty_wrap : NULL;
        space = isl_space_copy(sol_map->sol.space);
        sol_map->map = isl_map_alloc_space(space, 1, ISL_MAP_DISJOINT);
        if (!sol_map->map)
                goto error;

        if (track_empty) {
                sol_map->empty = isl_set_alloc_space(isl_basic_set_get_space(dom),
                                                        1, ISL_SET_DISJOINT);
                if (!sol_map->empty)
                        goto error;
        }

        isl_basic_set_free(dom);
        return &sol_map->sol;
error:
        isl_basic_set_free(dom);
        sol_free(&sol_map->sol);
        return NULL;
}

/* Check whether all coefficients of (non-parameter) variables
 * are non-positive, meaning that no pivots can be performed on the row.
 */
static int is_critical(struct isl_tab *tab, int row)
{
        int j;
        unsigned off = 2 + tab->M;

        for (j = tab->n_dead; j < tab->n_col; ++j) {
                if (col_is_parameter_var(tab, j))
                        continue;

                if (isl_int_is_pos(tab->mat->row[row][off + j]))
                        return 0;
        }

        return 1;
}

/* Check whether the inequality represented by vec is strict over the integers,
 * i.e., there are no integer values satisfying the constraint with
 * equality.  This happens if the gcd of the coefficients is not a divisor
 * of the constant term.  If so, scale the constraint down by the gcd
 * of the coefficients.
 */
static int is_strict(struct isl_vec *vec)
{
        isl_int gcd;
        int strict = 0;

        isl_int_init(gcd);
        isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
        if (!isl_int_is_one(gcd)) {
                strict = !isl_int_is_divisible_by(vec->el[0], gcd);
                isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
                isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
        }
        isl_int_clear(gcd);

        return strict;
}

/* Determine the sign of the given row of the main tableau.
 * The result is one of
 *      isl_tab_row_pos: always non-negative; no pivot needed
 *      isl_tab_row_neg: always non-positive; pivot
 *      isl_tab_row_any: can be both positive and negative; split
 *
 * We first handle some simple cases
 *      - the row sign may be known already
 *      - the row may be obviously non-negative
 *      - the parametric constant may be equal to that of another row
 *        for which we know the sign.  This sign will be either "pos" or
 *        "any".  If it had been "neg" then we would have pivoted before.
 *
 * If none of these cases hold, we check the value of the row for each
 * of the currently active samples.  Based on the signs of these values
 * we make an initial determination of the sign of the row.
 *
 *      all zero                        ->      unk(nown)
 *      all non-negative                ->      pos
 *      all non-positive                ->      neg
 *      both negative and positive      ->      all
 *
 * If we end up with "all", we are done.
 * Otherwise, we perform a check for positive and/or negative
 * values as follows.
 *
 *      samples        neg             unk             pos
 *      <0 ?                        Y        N      Y        N
 *                                          pos    any      pos
 *      >0 ?         Y      N    Y     N
 *                  any    neg  any   neg
 *
 * There is no special sign for "zero", because we can usually treat zero
 * as either non-negative or non-positive, whatever works out best.
 * However, if the row is "critical", meaning that pivoting is impossible
 * then we don't want to limp zero with the non-positive case, because
 * then we we would lose the solution for those values of the parameters
 * where the value of the row is zero.  Instead, we treat 0 as non-negative
 * ensuring a split if the row can attain both zero and negative values.
 * The same happens when the original constraint was one that could not
 * be satisfied with equality by any integer values of the parameters.
 * In this case, we normalize the constraint, but then a value of zero
 * for the normalized constraint is actually a positive value for the
 * original constraint, so again we need to treat zero as non-negative.
 * In both these cases, we have the following decision tree instead:
 *
 *      all non-negative                ->      pos
 *      all negative                    ->      neg
 *      both negative and non-negative  ->      all
 *
 *      samples        neg                             pos
 *      <0 ?                                        Y        N
 *                                                 any      pos
 *      >=0 ?        Y      N
 *                  any    neg
 */
static enum isl_tab_row_sign row_sign(struct isl_tab *tab,
        struct isl_sol *sol, int row)
{
        struct isl_vec *ineq = NULL;
        enum isl_tab_row_sign res = isl_tab_row_unknown;
        int critical;
        int strict;
        int row2;

        if (tab->row_sign[row] != isl_tab_row_unknown)
                return tab->row_sign[row];
        if (is_obviously_nonneg(tab, row))
                return isl_tab_row_pos;
        for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
                if (tab->row_sign[row2] == isl_tab_row_unknown)
                        continue;
                if (identical_parameter_line(tab, row, row2))
                        return tab->row_sign[row2];
        }

        critical = is_critical(tab, row);

        ineq = get_row_parameter_ineq(tab, row);
        if (!ineq)
                goto error;

        strict = is_strict(ineq);

        res = sol->context->op->ineq_sign(sol->context, ineq->el,
                                          critical || strict);

        if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
                /* test for negative values */
                int feasible;
                isl_seq_neg(ineq->el, ineq->el, ineq->size);
                isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);

                feasible = sol->context->op->test_ineq(sol->context, ineq->el);
                if (feasible < 0)
                        goto error;
                if (!feasible)
                        res = isl_tab_row_pos;
                else
                        res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
                                                           : isl_tab_row_any;
                if (res == isl_tab_row_neg) {
                        isl_seq_neg(ineq->el, ineq->el, ineq->size);
                        isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
                }
        }

        if (res == isl_tab_row_neg) {
                /* test for positive values */
                int feasible;
                if (!critical && !strict)
                        isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);

                feasible = sol->context->op->test_ineq(sol->context, ineq->el);
                if (feasible < 0)
                        goto error;
                if (feasible)
                        res = isl_tab_row_any;
        }

        isl_vec_free(ineq);
        return res;
error:
        isl_vec_free(ineq);
        return isl_tab_row_unknown;
}

static void find_solutions(struct isl_sol *sol, struct isl_tab *tab);

/* Find solutions for values of the parameters that satisfy the given
 * inequality.
 *
 * We currently take a snapshot of the context tableau that is reset
 * when we return from this function, while we make a copy of the main
 * tableau, leaving the original main tableau untouched.
 * These are fairly arbitrary choices.  Making a copy also of the context
 * tableau would obviate the need to undo any changes made to it later,
 * while taking a snapshot of the main tableau could reduce memory usage.
 * If we were to switch to taking a snapshot of the main tableau,
 * we would have to keep in mind that we need to save the row signs
 * and that we need to do this before saving the current basis
 * such that the basis has been restore before we restore the row signs.
 */
static void find_in_pos(struct isl_sol *sol, struct isl_tab *tab, isl_int *ineq)
{
        void *saved;

        if (!sol->context)
                goto error;
        saved = sol->context->op->save(sol->context);

        tab = isl_tab_dup(tab);
        if (!tab)
                goto error;

        sol->context->op->add_ineq(sol->context, ineq, 0, 1);

        find_solutions(sol, tab);

        if (!sol->error)
                sol->context->op->restore(sol->context, saved);
        else
                sol->context->op->discard(saved);
        return;
error:
        sol->error = 1;
}

/* Record the absence of solutions for those values of the parameters
 * that do not satisfy the given inequality with equality.
 */
static void no_sol_in_strict(struct isl_sol *sol,
        struct isl_tab *tab, struct isl_vec *ineq)
{
        int empty;
        void *saved;

        if (!sol->context || sol->error)
                goto error;
        saved = sol->context->op->save(sol->context);

        isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);

        sol->context->op->add_ineq(sol->context, ineq->el, 1, 0);
        if (!sol->context)
                goto error;

        empty = tab->empty;
        tab->empty = 1;
        sol_add(sol, tab);
        tab->empty = empty;

        isl_int_add_ui(ineq->el[0], ineq->el[0], 1);

        sol->context->op->restore(sol->context, saved);
        return;
error:
        sol->error = 1;
}

/* Reset all row variables that are marked to have a sign that may
 * be both positive and negative to have an unknown sign.
 */
static void reset_any_to_unknown(struct isl_tab *tab)
{
        int row;

        for (row = tab->n_redundant; row < tab->n_row; ++row) {
                if (!isl_tab_var_from_row(tab, row)->is_nonneg)
                        continue;
                if (tab->row_sign[row] == isl_tab_row_any)
                        tab->row_sign[row] = isl_tab_row_unknown;
        }
}

/* Compute the lexicographic minimum of the set represented by the main
 * tableau "tab" within the context "sol->context_tab".
 * On entry the sample value of the main tableau is lexicographically
 * less than or equal to this lexicographic minimum.
 * Pivots are performed until a feasible point is found, which is then
 * necessarily equal to the minimum, or until the tableau is found to
 * be infeasible.  Some pivots may need to be performed for only some
 * feasible values of the context tableau.  If so, the context tableau
 * is split into a part where the pivot is needed and a part where it is not.
 *
 * Whenever we enter the main loop, the main tableau is such that no
 * "obvious" pivots need to be performed on it, where "obvious" means
 * that the given row can be seen to be negative without looking at
 * the context tableau.  In particular, for non-parametric problems,
 * no pivots need to be performed on the main tableau.
 * The caller of find_solutions is responsible for making this property
 * hold prior to the first iteration of the loop, while restore_lexmin
 * is called before every other iteration.
 *
 * Inside the main loop, we first examine the signs of the rows of
 * the main tableau within the context of the context tableau.
 * If we find a row that is always non-positive for all values of
 * the parameters satisfying the context tableau and negative for at
 * least one value of the parameters, we perform the appropriate pivot
 * and start over.  An exception is the case where no pivot can be
 * performed on the row.  In this case, we require that the sign of
 * the row is negative for all values of the parameters (rather than just
 * non-positive).  This special case is handled inside row_sign, which
 * will say that the row can have any sign if it determines that it can
 * attain both negative and zero values.
 *
 * If we can't find a row that always requires a pivot, but we can find
 * one or more rows that require a pivot for some values of the parameters
 * (i.e., the row can attain both positive and negative signs), then we split
 * the context tableau into two parts, one where we force the sign to be
 * non-negative and one where we force is to be negative.
 * The non-negative part is handled by a recursive call (through find_in_pos).
 * Upon returning from this call, we continue with the negative part and
 * perform the required pivot.
 *
 * If no such rows can be found, all rows are non-negative and we have
 * found a (rational) feasible point.  If we only wanted a rational point
 * then we are done.
 * Otherwise, we check if all values of the sample point of the tableau
 * are integral for the variables.  If so, we have found the minimal
 * integral point and we are done.
 * If the sample point is not integral, then we need to make a distinction
 * based on whether the constant term is non-integral or the coefficients
 * of the parameters.  Furthermore, in order to decide how to handle
 * the non-integrality, we also need to know whether the coefficients
 * of the other columns in the tableau are integral.  This leads
 * to the following table.  The first two rows do not correspond
 * to a non-integral sample point and are only mentioned for completeness.
 *
 *      constant        parameters      other
 *
 *      int             int             int     |
 *      int             int             rat     | -> no problem
 *
 *      rat             int             int       -> fail
 *
 *      rat             int             rat       -> cut
 *
 *      int             rat             rat     |
 *      rat             rat             rat     | -> parametric cut
 *
 *      int             rat             int     |
 *      rat             rat             int     | -> split context
 *
 * If the parametric constant is completely integral, then there is nothing
 * to be done.  If the constant term is non-integral, but all the other
 * coefficient are integral, then there is nothing that can be done
 * and the tableau has no integral solution.
 * If, on the other hand, one or more of the other columns have rational
 * coefficients, but the parameter coefficients are all integral, then
 * we can perform a regular (non-parametric) cut.
 * Finally, if there is any parameter coefficient that is non-integral,
 * then we need to involve the context tableau.  There are two cases here.
 * If at least one other column has a rational coefficient, then we
 * can perform a parametric cut in the main tableau by adding a new
 * integer division in the context tableau.
 * If all other columns have integral coefficients, then we need to
 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
 * is always integral.  We do this by introducing an integer division
 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
 * Since q is expressed in the tableau as
 *      c + \sum a_i y_i - m q >= 0
 *      -c - \sum a_i y_i + m q + m - 1 >= 0
 * it is sufficient to add the inequality
 *      -c - \sum a_i y_i + m q >= 0
 * In the part of the context where this inequality does not hold, the
 * main tableau is marked as being empty.
 */
static void find_solutions(struct isl_sol *sol, struct isl_tab *tab)
{
        struct isl_context *context;
        int r;

        if (!tab || sol->error)
                goto error;

        context = sol->context;

        if (tab->empty)
                goto done;
        if (context->op->is_empty(context))
                goto done;

        for (r = 0; r >= 0 && tab && !tab->empty; r = restore_lexmin(tab)) {
                int flags;
                int row;
                enum isl_tab_row_sign sgn;
                int split = -1;
                int n_split = 0;

                for (row = tab->n_redundant; row < tab->n_row; ++row) {
                        if (!isl_tab_var_from_row(tab, row)->is_nonneg)
                                continue;
                        sgn = row_sign(tab, sol, row);
                        if (!sgn)
                                goto error;
                        tab->row_sign[row] = sgn;
                        if (sgn == isl_tab_row_any)
                                n_split++;
                        if (sgn == isl_tab_row_any && split == -1)
                                split = row;
                        if (sgn == isl_tab_row_neg)
                                break;
                }
                if (row < tab->n_row)
                        continue;
                if (split != -1) {
                        struct isl_vec *ineq;
                        if (n_split != 1)
                                split = context->op->best_split(context, tab);
                        if (split < 0)
                                goto error;
                        ineq = get_row_parameter_ineq(tab, split);
                        if (!ineq)
                                goto error;
                        is_strict(ineq);
                        reset_any_to_unknown(tab);
                        tab->row_sign[split] = isl_tab_row_pos;
                        sol_inc_level(sol);
                        find_in_pos(sol, tab, ineq->el);
                        tab->row_sign[split] = isl_tab_row_neg;
                        isl_seq_neg(ineq->el, ineq->el, ineq->size);
                        isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
                        if (!sol->error)
                                context->op->add_ineq(context, ineq->el, 0, 1);
                        isl_vec_free(ineq);
                        if (sol->error)
                                goto error;
                        continue;
                }
                if (tab->rational)
                        break;
                row = first_non_integer_row(tab, &flags);
                if (row < 0)
                        break;
                if (ISL_FL_ISSET(flags, I_PAR)) {
                        if (ISL_FL_ISSET(flags, I_VAR)) {
                                if (isl_tab_mark_empty(tab) < 0)
                                        goto error;
                                break;
                        }
                        row = add_cut(tab, row);
                } else if (ISL_FL_ISSET(flags, I_VAR)) {
                        struct isl_vec *div;
                        struct isl_vec *ineq;
                        int d;
                        div = get_row_split_div(tab, row);
                        if (!div)
                                goto error;
                        d = context->op->get_div(context, tab, div);
                        isl_vec_free(div);
                        if (d < 0)
                                goto error;
                        ineq = ineq_for_div(context->op->peek_basic_set(context), d);
                        if (!ineq)
                                goto error;
                        sol_inc_level(sol);
                        no_sol_in_strict(sol, tab, ineq);
                        isl_seq_neg(ineq->el, ineq->el, ineq->size);
                        context->op->add_ineq(context, ineq->el, 1, 1);
                        isl_vec_free(ineq);
                        if (sol->error || !context->op->is_ok(context))
                                goto error;
                        tab = set_row_cst_to_div(tab, row, d);
                        if (context->op->is_empty(context))
                                break;
                } else
                        row = add_parametric_cut(tab, row, context);
                if (row < 0)
                        goto error;
        }
        if (r < 0)
                goto error;
done:
        sol_add(sol, tab);
        isl_tab_free(tab);
        return;
error:
        isl_tab_free(tab);
        sol->error = 1;
}

/* Does "sol" contain a pair of partial solutions that could potentially
 * be merged?
 *
 * We currently only check that "sol" is not in an error state
 * and that there are at least two partial solutions of which the final two
 * are defined at the same level.
 */
static int sol_has_mergeable_solutions(struct isl_sol *sol)
{
        if (sol->error)
                return 0;
        if (!sol->partial)
                return 0;
        if (!sol->partial->next)
                return 0;
        return sol->partial->level == sol->partial->next->level;
}

/* Compute the lexicographic minimum of the set represented by the main
 * tableau "tab" within the context "sol->context_tab".
 *
 * As a preprocessing step, we first transfer all the purely parametric
 * equalities from the main tableau to the context tableau, i.e.,
 * parameters that have been pivoted to a row.
 * These equalities are ignored by the main algorithm, because the
 * corresponding rows may not be marked as being non-negative.
 * In parts of the context where the added equality does not hold,
 * the main tableau is marked as being empty.
 *
 * Before we embark on the actual computation, we save a copy
 * of the context.  When we return, we check if there are any
 * partial solutions that can potentially be merged.  If so,
 * we perform a rollback to the initial state of the context.
 * The merging of partial solutions happens inside calls to
 * sol_dec_level that are pushed onto the undo stack of the context.
 * If there are no partial solutions that can potentially be merged
 * then the rollback is skipped as it would just be wasted effort.
 */
static void find_solutions_main(struct isl_sol *sol, struct isl_tab *tab)
{
        int row;
        void *saved;

        if (!tab)
                goto error;

        sol->level = 0;

        for (row = tab->n_redundant; row < tab->n_row; ++row) {
                int p;
                struct isl_vec *eq;

                if (!row_is_parameter_var(tab, row))
                        continue;
                if (tab->row_var[row] < tab->n_param)
                        p = tab->row_var[row];
                else
                        p = tab->row_var[row]
                                + tab->n_param - (tab->n_var - tab->n_div);

                eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
                if (!eq)
                        goto error;
                get_row_parameter_line(tab, row, eq->el);
                isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
                eq = isl_vec_normalize(eq);

                sol_inc_level(sol);
                no_sol_in_strict(sol, tab, eq);

                isl_seq_neg(eq->el, eq->el, eq->size);
                sol_inc_level(sol);
                no_sol_in_strict(sol, tab, eq);
                isl_seq_neg(eq->el, eq->el, eq->size);

                sol->context->op->add_eq(sol->context, eq->el, 1, 1);

                isl_vec_free(eq);

                if (isl_tab_mark_redundant(tab, row) < 0)
                        goto error;

                if (sol->context->op->is_empty(sol->context))
                        break;

                row = tab->n_redundant - 1;
        }

        saved = sol->context->op->save(sol->context);

        find_solutions(sol, tab);

        if (sol_has_mergeable_solutions(sol))
                sol->context->op->restore(sol->context, saved);
        else
                sol->context->op->discard(saved);

        sol->level = 0;
        sol_pop(sol);

        return;
error:
        isl_tab_free(tab);
        sol->error = 1;
}

/* Check if integer division "div" of "dom" also occurs in "bmap".
 * If so, return its position within the divs.
 * If not, return -1.
 */
static int find_context_div(struct isl_basic_map *bmap,
        struct isl_basic_set *dom, unsigned div)
{
        int i;
        unsigned b_dim = isl_space_dim(bmap->dim, isl_dim_all);
        unsigned d_dim = isl_space_dim(dom->dim, isl_dim_all);

        if (isl_int_is_zero(dom->div[div][0]))
                return -1;
        if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
                return -1;

        for (i = 0; i < bmap->n_div; ++i) {
                if (isl_int_is_zero(bmap->div[i][0]))
                        continue;
                if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
                                           (b_dim - d_dim) + bmap->n_div) != -1)
                        continue;
                if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
                        return i;
        }
        return -1;
}

/* The correspondence between the variables in the main tableau,
 * the context tableau, and the input map and domain is as follows.
 * The first n_param and the last n_div variables of the main tableau
 * form the variables of the context tableau.
 * In the basic map, these n_param variables correspond to the
 * parameters and the input dimensions.  In the domain, they correspond
 * to the parameters and the set dimensions.
 * The n_div variables correspond to the integer divisions in the domain.
 * To ensure that everything lines up, we may need to copy some of the
 * integer divisions of the domain to the map.  These have to be placed
 * in the same order as those in the context and they have to be placed
 * after any other integer divisions that the map may have.
 * This function performs the required reordering.
 */
static __isl_give isl_basic_map *align_context_divs(
        __isl_take isl_basic_map *bmap, __isl_keep isl_basic_set *dom)
{
        int i;
        int common = 0;
        int other;

        for (i = 0; i < dom->n_div; ++i)
                if (find_context_div(bmap, dom, i) != -1)
                        common++;
        other = bmap->n_div - common;
        if (dom->n_div - common > 0) {
                bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
                                dom->n_div - common, 0, 0);
                if (!bmap)
                        return NULL;
        }
        for (i = 0; i < dom->n_div; ++i) {
                int pos = find_context_div(bmap, dom, i);
                if (pos < 0) {
                        pos = isl_basic_map_alloc_div(bmap);
                        if (pos < 0)
                                goto error;
                        isl_int_set_si(bmap->div[pos][0], 0);
                }
                if (pos != other + i)
                        isl_basic_map_swap_div(bmap, pos, other + i);
        }
        return bmap;
error:
        isl_basic_map_free(bmap);
        return NULL;
}

/* Base case of isl_tab_basic_map_partial_lexopt, after removing
 * some obvious symmetries.
 *
 * We make sure the divs in the domain are properly ordered,
 * because they will be added one by one in the given order
 * during the construction of the solution map.
 * Furthermore, make sure that the known integer divisions
 * appear before any unknown integer division because the solution
 * may depend on the known integer divisions, while anything that
 * depends on any variable starting from the first unknown integer
 * division is ignored in sol_pma_add.
 */
static struct isl_sol *basic_map_partial_lexopt_base_sol(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max,
        struct isl_sol *(*init)(__isl_keep isl_basic_map *bmap,
                    __isl_take isl_basic_set *dom, int track_empty, int max))
{
        struct isl_tab *tab;
        struct isl_sol *sol = NULL;
        struct isl_context *context;

        if (dom->n_div) {
                dom = isl_basic_set_sort_divs(dom);
                bmap = align_context_divs(bmap, dom);
        }
        sol = init(bmap, dom, !!empty, max);
        if (!sol)
                goto error;

        context = sol->context;
        if (isl_basic_set_plain_is_empty(context->op->peek_basic_set(context)))
                /* nothing */;
        else if (isl_basic_map_plain_is_empty(bmap)) {
                if (sol->add_empty)
                        sol->add_empty(sol,
                    isl_basic_set_copy(context->op->peek_basic_set(context)));
        } else {
                tab = tab_for_lexmin(bmap,
                                    context->op->peek_basic_set(context), 1, max);
                tab = context->op->detect_nonnegative_parameters(context, tab);
                find_solutions_main(sol, tab);
        }
        if (sol->error)
                goto error;

        isl_basic_map_free(bmap);
        return sol;
error:
        sol_free(sol);
        isl_basic_map_free(bmap);
        return NULL;
}

/* Base case of isl_tab_basic_map_partial_lexopt, after removing
 * some obvious symmetries.
 *
 * We call basic_map_partial_lexopt_base_sol and extract the results.
 */
static __isl_give isl_map *basic_map_partial_lexopt_base(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max)
{
        isl_map *result = NULL;
        struct isl_sol *sol;
        struct isl_sol_map *sol_map;

        sol = basic_map_partial_lexopt_base_sol(bmap, dom, empty, max,
                                                &sol_map_init);
        if (!sol)
                return NULL;
        sol_map = (struct isl_sol_map *) sol;

        result = isl_map_copy(sol_map->map);
        if (empty)
                *empty = isl_set_copy(sol_map->empty);
        sol_free(&sol_map->sol);
        return result;
}

/* Return a count of the number of occurrences of the "n" first
 * variables in the inequality constraints of "bmap".
 */
static __isl_give int *count_occurrences(__isl_keep isl_basic_map *bmap,
        int n)
{
        int i, j;
        isl_ctx *ctx;
        int *occurrences;

        if (!bmap)
                return NULL;
        ctx = isl_basic_map_get_ctx(bmap);
        occurrences = isl_calloc_array(ctx, int, n);
        if (!occurrences)
                return NULL;

        for (i = 0; i < bmap->n_ineq; ++i) {
                for (j = 0; j < n; ++j) {
                        if (!isl_int_is_zero(bmap->ineq[i][1 + j]))
                                occurrences[j]++;
                }
        }

        return occurrences;
}

/* Do all of the "n" variables with non-zero coefficients in "c"
 * occur in exactly a single constraint.
 * "occurrences" is an array of length "n" containing the number
 * of occurrences of each of the variables in the inequality constraints.
 */
static int single_occurrence(int n, isl_int *c, int *occurrences)
{
        int i;

        for (i = 0; i < n; ++i) {
                if (isl_int_is_zero(c[i]))
                        continue;
                if (occurrences[i] != 1)
                        return 0;
        }

        return 1;
}

/* Do all of the "n" initial variables that occur in inequality constraint
 * "ineq" of "bmap" only occur in that constraint?
 */
static int all_single_occurrence(__isl_keep isl_basic_map *bmap, int ineq,
        int n)
{
        int i, j;

        for (i = 0; i < n; ++i) {
                if (isl_int_is_zero(bmap->ineq[ineq][1 + i]))
                        continue;
                for (j = 0; j < bmap->n_ineq; ++j) {
                        if (j == ineq)
                                continue;
                        if (!isl_int_is_zero(bmap->ineq[j][1 + i]))
                                return 0;
                }
        }

        return 1;
}

/* Structure used during detection of parallel constraints.
 * n_in: number of "input" variables: isl_dim_param + isl_dim_in
 * n_out: number of "output" variables: isl_dim_out + isl_dim_div
 * val: the coefficients of the output variables
 */
struct isl_constraint_equal_info {
        unsigned n_in;
        unsigned n_out;
        isl_int *val;
};

/* Check whether the coefficients of the output variables
 * of the constraint in "entry" are equal to info->val.
 */
static int constraint_equal(const void *entry, const void *val)
{
        isl_int **row = (isl_int **)entry;
        const struct isl_constraint_equal_info *info = val;

        return isl_seq_eq((*row) + 1 + info->n_in, info->val, info->n_out);
}

/* Check whether "bmap" has a pair of constraints that have
 * the same coefficients for the output variables.
 * Note that the coefficients of the existentially quantified
 * variables need to be zero since the existentially quantified
 * of the result are usually not the same as those of the input.
 * Furthermore, check that each of the input variables that occur
 * in those constraints does not occur in any other constraint.
 * If so, return true and return the row indices of the two constraints
 * in *first and *second.
 */
static isl_bool parallel_constraints(__isl_keep isl_basic_map *bmap,
        int *first, int *second)
{
        int i;
        isl_ctx *ctx;
        int *occurrences = NULL;
        struct isl_hash_table *table = NULL;
        struct isl_hash_table_entry *entry;
        struct isl_constraint_equal_info info;
        unsigned n_out;
        unsigned n_div;

        ctx = isl_basic_map_get_ctx(bmap);
        table = isl_hash_table_alloc(ctx, bmap->n_ineq);
        if (!table)
                goto error;

        info.n_in = isl_basic_map_dim(bmap, isl_dim_param) +
                    isl_basic_map_dim(bmap, isl_dim_in);
        occurrences = count_occurrences(bmap, info.n_in);
        if (info.n_in && !occurrences)
                goto error;
        n_out = isl_basic_map_dim(bmap, isl_dim_out);
        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        info.n_out = n_out + n_div;
        for (i = 0; i < bmap->n_ineq; ++i) {
                uint32_t hash;

                info.val = bmap->ineq[i] + 1 + info.n_in;
                if (isl_seq_first_non_zero(info.val, n_out) < 0)
                        continue;
                if (isl_seq_first_non_zero(info.val + n_out, n_div) >= 0)
                        continue;
                if (!single_occurrence(info.n_in, bmap->ineq[i] + 1,
                                        occurrences))
                        continue;
                hash = isl_seq_get_hash(info.val, info.n_out);
                entry = isl_hash_table_find(ctx, table, hash,
                                            constraint_equal, &info, 1);
                if (!entry)
                        goto error;
                if (entry->data)
                        break;
                entry->data = &bmap->ineq[i];
        }

        if (i < bmap->n_ineq) {
                *first = ((isl_int **)entry->data) - bmap->ineq; 
                *second = i;
        }

        isl_hash_table_free(ctx, table);
        free(occurrences);

        return i < bmap->n_ineq;
error:
        isl_hash_table_free(ctx, table);
        free(occurrences);
        return isl_bool_error;
}

/* Given a set of upper bounds in "var", add constraints to "bset"
 * that make the i-th bound smallest.
 *
 * In particular, if there are n bounds b_i, then add the constraints
 *
 *      b_i <= b_j      for j > i
 *      b_i <  b_j      for j < i
 */
static __isl_give isl_basic_set *select_minimum(__isl_take isl_basic_set *bset,
        __isl_keep isl_mat *var, int i)
{
        isl_ctx *ctx;
        int j, k;

        ctx = isl_mat_get_ctx(var);

        for (j = 0; j < var->n_row; ++j) {
                if (j == i)
                        continue;
                k = isl_basic_set_alloc_inequality(bset);
                if (k < 0)
                        goto error;
                isl_seq_combine(bset->ineq[k], ctx->one, var->row[j],
                                ctx->negone, var->row[i], var->n_col);
                isl_int_set_si(bset->ineq[k][var->n_col], 0);
                if (j < i)
                        isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
        }

        bset = isl_basic_set_finalize(bset);

        return bset;
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Given a set of upper bounds on the last "input" variable m,
 * construct a set that assigns the minimal upper bound to m, i.e.,
 * construct a set that divides the space into cells where one
 * of the upper bounds is smaller than all the others and assign
 * this upper bound to m.
 *
 * In particular, if there are n bounds b_i, then the result
 * consists of n basic sets, each one of the form
 *
 *      m = b_i
 *      b_i <= b_j      for j > i
 *      b_i <  b_j      for j < i
 */
static __isl_give isl_set *set_minimum(__isl_take isl_space *dim,
        __isl_take isl_mat *var)
{
        int i, k;
        isl_basic_set *bset = NULL;
        isl_set *set = NULL;

        if (!dim || !var)
                goto error;

        set = isl_set_alloc_space(isl_space_copy(dim),
                                var->n_row, ISL_SET_DISJOINT);

        for (i = 0; i < var->n_row; ++i) {
                bset = isl_basic_set_alloc_space(isl_space_copy(dim), 0,
                                               1, var->n_row - 1);
                k = isl_basic_set_alloc_equality(bset);
                if (k < 0)
                        goto error;
                isl_seq_cpy(bset->eq[k], var->row[i], var->n_col);
                isl_int_set_si(bset->eq[k][var->n_col], -1);
                bset = select_minimum(bset, var, i);
                set = isl_set_add_basic_set(set, bset);
        }

        isl_space_free(dim);
        isl_mat_free(var);
        return set;
error:
        isl_basic_set_free(bset);
        isl_set_free(set);
        isl_space_free(dim);
        isl_mat_free(var);
        return NULL;
}

/* Given that the last input variable of "bmap" represents the minimum
 * of the bounds in "cst", check whether we need to split the domain
 * based on which bound attains the minimum.
 *
 * A split is needed when the minimum appears in an integer division
 * or in an equality.  Otherwise, it is only needed if it appears in
 * an upper bound that is different from the upper bounds on which it
 * is defined.
 */
static isl_bool need_split_basic_map(__isl_keep isl_basic_map *bmap,
        __isl_keep isl_mat *cst)
{
        int i, j;
        unsigned total;
        unsigned pos;

        pos = cst->n_col - 1;
        total = isl_basic_map_dim(bmap, isl_dim_all);

        for (i = 0; i < bmap->n_div; ++i)
                if (!isl_int_is_zero(bmap->div[i][2 + pos]))
                        return isl_bool_true;

        for (i = 0; i < bmap->n_eq; ++i)
                if (!isl_int_is_zero(bmap->eq[i][1 + pos]))
                        return isl_bool_true;

        for (i = 0; i < bmap->n_ineq; ++i) {
                if (isl_int_is_nonneg(bmap->ineq[i][1 + pos]))
                        continue;
                if (!isl_int_is_negone(bmap->ineq[i][1 + pos]))
                        return isl_bool_true;
                if (isl_seq_first_non_zero(bmap->ineq[i] + 1 + pos + 1,
                                           total - pos - 1) >= 0)
                        return isl_bool_true;

                for (j = 0; j < cst->n_row; ++j)
                        if (isl_seq_eq(bmap->ineq[i], cst->row[j], cst->n_col))
                                break;
                if (j >= cst->n_row)
                        return isl_bool_true;
        }

        return isl_bool_false;
}

/* Given that the last set variable of "bset" represents the minimum
 * of the bounds in "cst", check whether we need to split the domain
 * based on which bound attains the minimum.
 *
 * We simply call need_split_basic_map here.  This is safe because
 * the position of the minimum is computed from "cst" and not
 * from "bmap".
 */
static isl_bool need_split_basic_set(__isl_keep isl_basic_set *bset,
        __isl_keep isl_mat *cst)
{
        return need_split_basic_map(bset_to_bmap(bset), cst);
}

/* Given that the last set variable of "set" represents the minimum
 * of the bounds in "cst", check whether we need to split the domain
 * based on which bound attains the minimum.
 */
static isl_bool need_split_set(__isl_keep isl_set *set, __isl_keep isl_mat *cst)
{
        int i;

        for (i = 0; i < set->n; ++i) {
                isl_bool split;

                split = need_split_basic_set(set->p[i], cst);
                if (split < 0 || split)
                        return split;
        }

        return isl_bool_false;
}

/* Given a set of which the last set variable is the minimum
 * of the bounds in "cst", split each basic set in the set
 * in pieces where one of the bounds is (strictly) smaller than the others.
 * This subdivision is given in "min_expr".
 * The variable is subsequently projected out.
 *
 * We only do the split when it is needed.
 * For example if the last input variable m = min(a,b) and the only
 * constraints in the given basic set are lower bounds on m,
 * i.e., l <= m = min(a,b), then we can simply project out m
 * to obtain l <= a and l <= b, without having to split on whether
 * m is equal to a or b.
 */
static __isl_give isl_set *split(__isl_take isl_set *empty,
        __isl_take isl_set *min_expr, __isl_take isl_mat *cst)
{
        int n_in;
        int i;
        isl_space *dim;
        isl_set *res;

        if (!empty || !min_expr || !cst)
                goto error;

        n_in = isl_set_dim(empty, isl_dim_set);
        dim = isl_set_get_space(empty);
        dim = isl_space_drop_dims(dim, isl_dim_set, n_in - 1, 1);
        res = isl_set_empty(dim);

        for (i = 0; i < empty->n; ++i) {
                isl_bool split;
                isl_set *set;

                set = isl_set_from_basic_set(isl_basic_set_copy(empty->p[i]));
                split = need_split_basic_set(empty->p[i], cst);
                if (split < 0)
                        set = isl_set_free(set);
                else if (split)
                        set = isl_set_intersect(set, isl_set_copy(min_expr));
                set = isl_set_remove_dims(set, isl_dim_set, n_in - 1, 1);

                res = isl_set_union_disjoint(res, set);
        }

        isl_set_free(empty);
        isl_set_free(min_expr);
        isl_mat_free(cst);
        return res;
error:
        isl_set_free(empty);
        isl_set_free(min_expr);
        isl_mat_free(cst);
        return NULL;
}

/* Given a map of which the last input variable is the minimum
 * of the bounds in "cst", split each basic set in the set
 * in pieces where one of the bounds is (strictly) smaller than the others.
 * This subdivision is given in "min_expr".
 * The variable is subsequently projected out.
 *
 * The implementation is essentially the same as that of "split".
 */
static __isl_give isl_map *split_domain(__isl_take isl_map *opt,
        __isl_take isl_set *min_expr, __isl_take isl_mat *cst)
{
        int n_in;
        int i;
        isl_space *dim;
        isl_map *res;

        if (!opt || !min_expr || !cst)
                goto error;

        n_in = isl_map_dim(opt, isl_dim_in);
        dim = isl_map_get_space(opt);
        dim = isl_space_drop_dims(dim, isl_dim_in, n_in - 1, 1);
        res = isl_map_empty(dim);

        for (i = 0; i < opt->n; ++i) {
                isl_map *map;
                isl_bool split;

                map = isl_map_from_basic_map(isl_basic_map_copy(opt->p[i]));
                split = need_split_basic_map(opt->p[i], cst);
                if (split < 0)
                        map = isl_map_free(map);
                else if (split)
                        map = isl_map_intersect_domain(map,
                                                       isl_set_copy(min_expr));
                map = isl_map_remove_dims(map, isl_dim_in, n_in - 1, 1);

                res = isl_map_union_disjoint(res, map);
        }

        isl_map_free(opt);
        isl_set_free(min_expr);
        isl_mat_free(cst);
        return res;
error:
        isl_map_free(opt);
        isl_set_free(min_expr);
        isl_mat_free(cst);
        return NULL;
}

static __isl_give isl_map *basic_map_partial_lexopt(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max);

/* This function is called from basic_map_partial_lexopt_symm.
 * The last variable of "bmap" and "dom" corresponds to the minimum
 * of the bounds in "cst".  "map_space" is the space of the original
 * input relation (of basic_map_partial_lexopt_symm) and "set_space"
 * is the space of the original domain.
 *
 * We recursively call basic_map_partial_lexopt and then plug in
 * the definition of the minimum in the result.
 */
static __isl_give isl_map *basic_map_partial_lexopt_symm_core(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max, __isl_take isl_mat *cst,
        __isl_take isl_space *map_space, __isl_take isl_space *set_space)
{
        isl_map *opt;
        isl_set *min_expr;

        min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst));

        opt = basic_map_partial_lexopt(bmap, dom, empty, max);

        if (empty) {
                *empty = split(*empty,
                               isl_set_copy(min_expr), isl_mat_copy(cst));
                *empty = isl_set_reset_space(*empty, set_space);
        }

        opt = split_domain(opt, min_expr, cst);
        opt = isl_map_reset_space(opt, map_space);

        return opt;
}

/* Extract a domain from "bmap" for the purpose of computing
 * a lexicographic optimum.
 *
 * This function is only called when the caller wants to compute a full
 * lexicographic optimum, i.e., without specifying a domain.  In this case,
 * the caller is not interested in the part of the domain space where
 * there is no solution and the domain can be initialized to those constraints
 * of "bmap" that only involve the parameters and the input dimensions.
 * This relieves the parametric programming engine from detecting those
 * inequalities and transferring them to the context.  More importantly,
 * it ensures that those inequalities are transferred first and not
 * intermixed with inequalities that actually split the domain.
 *
 * If the caller does not require the absence of existentially quantified
 * variables in the result (i.e., if ISL_OPT_QE is not set in "flags"),
 * then the actual domain of "bmap" can be used.  This ensures that
 * the domain does not need to be split at all just to separate out
 * pieces of the domain that do not have a solution from piece that do.
 * This domain cannot be used in general because it may involve
 * (unknown) existentially quantified variables which will then also
 * appear in the solution.
 */
static __isl_give isl_basic_set *extract_domain(__isl_keep isl_basic_map *bmap,
        unsigned flags)
{
        int n_div;
        int n_out;

        n_div = isl_basic_map_dim(bmap, isl_dim_div);
        n_out = isl_basic_map_dim(bmap, isl_dim_out);
        bmap = isl_basic_map_copy(bmap);
        if (ISL_FL_ISSET(flags, ISL_OPT_QE)) {
                bmap = isl_basic_map_drop_constraints_involving_dims(bmap,
                                                        isl_dim_div, 0, n_div);
                bmap = isl_basic_map_drop_constraints_involving_dims(bmap,
                                                        isl_dim_out, 0, n_out);
        }
        return isl_basic_map_domain(bmap);
}

#undef TYPE
#define TYPE    isl_map
#undef SUFFIX
#define SUFFIX
#include "isl_tab_lexopt_templ.c"

struct isl_sol_for {
        struct isl_sol  sol;
        isl_stat        (*fn)(__isl_take isl_basic_set *dom,
                                __isl_take isl_aff_list *list, void *user);
        void            *user;
};

static void sol_for_free(struct isl_sol *sol)
{
}

/* Add the solution identified by the tableau and the context tableau.
 * In particular, "dom" represents the context and "ma" expresses
 * the solution on that context.
 *
 * See documentation of sol_add for more details.
 *
 * Instead of constructing a basic map, this function calls a user
 * defined function with the current context as a basic set and
 * a list of affine expressions representing the relation between
 * the input and output.  The space over which the affine expressions
 * are defined is the same as that of the domain.  The number of
 * affine expressions in the list is equal to the number of output variables.
 */
static void sol_for_add(struct isl_sol_for *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma)
{
        int i, n;
        isl_ctx *ctx;
        isl_aff *aff;
        isl_aff_list *list;

        if (sol->sol.error || !dom || !ma)
                goto error;

        ctx = isl_basic_set_get_ctx(dom);
        n = isl_multi_aff_dim(ma, isl_dim_out);
        list = isl_aff_list_alloc(ctx, n);
        for (i = 0; i < n; ++i) {
                aff = isl_multi_aff_get_aff(ma, i);
                list = isl_aff_list_add(list, aff);
        }

        dom = isl_basic_set_finalize(dom);

        if (sol->fn(isl_basic_set_copy(dom), list, sol->user) < 0)
                goto error;

        isl_basic_set_free(dom);
        isl_multi_aff_free(ma);
        return;
error:
        isl_basic_set_free(dom);
        isl_multi_aff_free(ma);
        sol->sol.error = 1;
}

static void sol_for_add_wrap(struct isl_sol *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma)
{
        sol_for_add((struct isl_sol_for *)sol, dom, ma);
}

static struct isl_sol_for *sol_for_init(__isl_keep isl_basic_map *bmap, int max,
        isl_stat (*fn)(__isl_take isl_basic_set *dom,
                __isl_take isl_aff_list *list, void *user),
        void *user)
{
        struct isl_sol_for *sol_for = NULL;
        isl_space *dom_dim;
        struct isl_basic_set *dom = NULL;

        sol_for = isl_calloc_type(bmap->ctx, struct isl_sol_for);
        if (!sol_for)
                goto error;

        dom_dim = isl_space_domain(isl_space_copy(bmap->dim));
        dom = isl_basic_set_universe(dom_dim);

        sol_for->sol.free = &sol_for_free;
        if (sol_init(&sol_for->sol, bmap, dom, max) < 0)
                goto error;
        sol_for->fn = fn;
        sol_for->user = user;
        sol_for->sol.add = &sol_for_add_wrap;
        sol_for->sol.add_empty = NULL;

        isl_basic_set_free(dom);
        return sol_for;
error:
        isl_basic_set_free(dom);
        sol_free(&sol_for->sol);
        return NULL;
}

static void sol_for_find_solutions(struct isl_sol_for *sol_for,
        struct isl_tab *tab)
{
        find_solutions_main(&sol_for->sol, tab);
}

isl_stat isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
        isl_stat (*fn)(__isl_take isl_basic_set *dom,
                __isl_take isl_aff_list *list, void *user),
        void *user)
{
        struct isl_sol_for *sol_for = NULL;

        bmap = isl_basic_map_copy(bmap);
        bmap = isl_basic_map_detect_equalities(bmap);
        if (!bmap)
                return isl_stat_error;

        sol_for = sol_for_init(bmap, max, fn, user);
        if (!sol_for)
                goto error;

        if (isl_basic_map_plain_is_empty(bmap))
                /* nothing */;
        else {
                struct isl_tab *tab;
                struct isl_context *context = sol_for->sol.context;
                tab = tab_for_lexmin(bmap,
                                context->op->peek_basic_set(context), 1, max);
                tab = context->op->detect_nonnegative_parameters(context, tab);
                sol_for_find_solutions(sol_for, tab);
                if (sol_for->sol.error)
                        goto error;
        }

        sol_free(&sol_for->sol);
        isl_basic_map_free(bmap);
        return isl_stat_ok;
error:
        sol_free(&sol_for->sol);
        isl_basic_map_free(bmap);
        return isl_stat_error;
}

/* Extract the subsequence of the sample value of "tab"
 * starting at "pos" and of length "len".
 */
static __isl_give isl_vec *extract_sample_sequence(struct isl_tab *tab,
        int pos, int len)
{
        int i;
        isl_ctx *ctx;
        isl_vec *v;

        ctx = isl_tab_get_ctx(tab);
        v = isl_vec_alloc(ctx, len);
        if (!v)
                return NULL;
        for (i = 0; i < len; ++i) {
                if (!tab->var[pos + i].is_row) {
                        isl_int_set_si(v->el[i], 0);
                } else {
                        int row;

                        row = tab->var[pos + i].index;
                        isl_int_divexact(v->el[i], tab->mat->row[row][1],
                                        tab->mat->row[row][0]);
                }
        }

        return v;
}

/* Check if the sequence of variables starting at "pos"
 * represents a trivial solution according to "trivial".
 * That is, is the result of applying "trivial" to this sequence
 * equal to the zero vector?
 */
static isl_bool region_is_trivial(struct isl_tab *tab, int pos,
        __isl_keep isl_mat *trivial)
{
        int n, len;
        isl_vec *v;
        isl_bool is_trivial;

        if (!trivial)
                return isl_bool_error;

        n = isl_mat_rows(trivial);
        if (n == 0)
                return isl_bool_false;

        len = isl_mat_cols(trivial);
        v = extract_sample_sequence(tab, pos, len);
        v = isl_mat_vec_product(isl_mat_copy(trivial), v);
        is_trivial = isl_vec_is_zero(v);
        isl_vec_free(v);

        return is_trivial;
}

/* Global internal data for isl_tab_basic_set_non_trivial_lexmin.
 *
 * "n_op" is the number of initial coordinates to optimize,
 * as passed to isl_tab_basic_set_non_trivial_lexmin.
 * "region" is the "n_region"-sized array of regions passed
 * to isl_tab_basic_set_non_trivial_lexmin.
 *
 * "tab" is the tableau that corresponds to the ILP problem.
 * "local" is an array of local data structure, one for each
 * (potential) level of the backtracking procedure of
 * isl_tab_basic_set_non_trivial_lexmin.
 * "v" is a pre-allocated vector that can be used for adding
 * constraints to the tableau.
 *
 * "sol" contains the best solution found so far.
 * It is initialized to a vector of size zero.
 */
struct isl_lexmin_data {
        int n_op;
        int n_region;
        struct isl_trivial_region *region;

        struct isl_tab *tab;
        struct isl_local_region *local;
        isl_vec *v;

        isl_vec *sol;
};

/* Return the index of the first trivial region, "n_region" if all regions
 * are non-trivial or -1 in case of error.
 */
static int first_trivial_region(struct isl_lexmin_data *data)
{
        int i;

        for (i = 0; i < data->n_region; ++i) {
                isl_bool trivial;
                trivial = region_is_trivial(data->tab, data->region[i].pos,
                                        data->region[i].trivial);
                if (trivial < 0)
                        return -1;
                if (trivial)
                        return i;
        }

        return data->n_region;
}

/* Check if the solution is optimal, i.e., whether the first
 * n_op entries are zero.
 */
static int is_optimal(__isl_keep isl_vec *sol, int n_op)
{
        int i;

        for (i = 0; i < n_op; ++i)
                if (!isl_int_is_zero(sol->el[1 + i]))
                        return 0;
        return 1;
}

/* Add constraints to "tab" that ensure that any solution is significantly
 * better than that represented by "sol".  That is, find the first
 * relevant (within first n_op) non-zero coefficient and force it (along
 * with all previous coefficients) to be zero.
 * If the solution is already optimal (all relevant coefficients are zero),
 * then just mark the table as empty.
 * "n_zero" is the number of coefficients that have been forced zero
 * by previous calls to this function at the same level.
 * Return the updated number of forced zero coefficients or -1 on error.
 *
 * This function assumes that at least 2 * (n_op - n_zero) more rows and
 * at least 2 * (n_op - n_zero) more elements in the constraint array
 * are available in the tableau.
 */
static int force_better_solution(struct isl_tab *tab,
        __isl_keep isl_vec *sol, int n_op, int n_zero)
{
        int i, n;
        isl_ctx *ctx;
        isl_vec *v = NULL;

        if (!sol)
                return -1;

        for (i = n_zero; i < n_op; ++i)
                if (!isl_int_is_zero(sol->el[1 + i]))
                        break;

        if (i == n_op) {
                if (isl_tab_mark_empty(tab) < 0)
                        return -1;
                return n_op;
        }

        ctx = isl_vec_get_ctx(sol);
        v = isl_vec_alloc(ctx, 1 + tab->n_var);
        if (!v)
                return -1;

        n = i + 1;
        for (; i >= n_zero; --i) {
                v = isl_vec_clr(v);
                isl_int_set_si(v->el[1 + i], -1);
                if (add_lexmin_eq(tab, v->el) < 0)
                        goto error;
        }

        isl_vec_free(v);
        return n;
error:
        isl_vec_free(v);
        return -1;
}

/* Fix triviality direction "dir" of the given region to zero.
 *
 * This function assumes that at least two more rows and at least
 * two more elements in the constraint array are available in the tableau.
 */
static isl_stat fix_zero(struct isl_tab *tab, struct isl_trivial_region *region,
        int dir, struct isl_lexmin_data *data)
{
        int len;

        data->v = isl_vec_clr(data->v);
        if (!data->v)
                return isl_stat_error;
        len = isl_mat_cols(region->trivial);
        isl_seq_cpy(data->v->el + 1 + region->pos, region->trivial->row[dir],
                    len);
        if (add_lexmin_eq(tab, data->v->el) < 0)
                return isl_stat_error;

        return isl_stat_ok;
}

/* This function selects case "side" for non-triviality region "region",
 * assuming all the equality constraints have been imposed already.
 * In particular, the triviality direction side/2 is made positive
 * if side is even and made negative if side is odd.
 *
 * This function assumes that at least one more row and at least
 * one more element in the constraint array are available in the tableau.
 */
static struct isl_tab *pos_neg(struct isl_tab *tab,
        struct isl_trivial_region *region,
        int side, struct isl_lexmin_data *data)
{
        int len;

        data->v = isl_vec_clr(data->v);
        if (!data->v)
                goto error;
        isl_int_set_si(data->v->el[0], -1);
        len = isl_mat_cols(region->trivial);
        if (side % 2 == 0)
                isl_seq_cpy(data->v->el + 1 + region->pos,
                            region->trivial->row[side / 2], len);
        else
                isl_seq_neg(data->v->el + 1 + region->pos,
                            region->trivial->row[side / 2], len);
        return add_lexmin_ineq(tab, data->v->el);
error:
        isl_tab_free(tab);
        return NULL;
}

/* Local data at each level of the backtracking procedure of
 * isl_tab_basic_set_non_trivial_lexmin.
 *
 * "update" is set if a solution has been found in the current case
 * of this level, such that a better solution needs to be enforced
 * in the next case.
 * "n_zero" is the number of initial coordinates that have already
 * been forced to be zero at this level.
 * "region" is the non-triviality region considered at this level.
 * "side" is the index of the current case at this level.
 * "n" is the number of triviality directions.
 * "snap" is a snapshot of the tableau holding a state that needs
 * to be satisfied by all subsequent cases.
 */
struct isl_local_region {
        int update;
        int n_zero;
        int region;
        int side;
        int n;
        struct isl_tab_undo *snap;
};

/* Initialize the global data structure "data" used while solving
 * the ILP problem "bset".
 */
static isl_stat init_lexmin_data(struct isl_lexmin_data *data,
        __isl_keep isl_basic_set *bset)
{
        isl_ctx *ctx;

        ctx = isl_basic_set_get_ctx(bset);

        data->tab = tab_for_lexmin(bset, NULL, 0, 0);
        if (!data->tab)
                return isl_stat_error;

        data->v = isl_vec_alloc(ctx, 1 + data->tab->n_var);
        if (!data->v)
                return isl_stat_error;
        data->local = isl_calloc_array(ctx, struct isl_local_region,
                                        data->n_region);
        if (data->n_region && !data->local)
                return isl_stat_error;

        data->sol = isl_vec_alloc(ctx, 0);

        return isl_stat_ok;
}

/* Mark all outer levels as requiring a better solution
 * in the next cases.
 */
static void update_outer_levels(struct isl_lexmin_data *data, int level)
{
        int i;

        for (i = 0; i < level; ++i)
                data->local[i].update = 1;
}

/* Initialize "local" to refer to region "region" and
 * to initiate processing at this level.
 */
static void init_local_region(struct isl_local_region *local, int region,
        struct isl_lexmin_data *data)
{
        local->n = isl_mat_rows(data->region[region].trivial);
        local->region = region;
        local->side = 0;
        local->update = 0;
        local->n_zero = 0;
}

/* What to do next after entering a level of the backtracking procedure.
 *
 * error: some error has occurred; abort
 * done: an optimal solution has been found; stop search
 * backtrack: backtrack to the previous level
 * handle: add the constraints for the current level and
 *      move to the next level
 */
enum isl_next {
        isl_next_error = -1,
        isl_next_done,
        isl_next_backtrack,
        isl_next_handle,
};

/* Have all cases of the current region been considered?
 * If there are n directions, then there are 2n cases.
 *
 * The constraints in the current tableau are imposed
 * in all subsequent cases.  This means that if the current
 * tableau is empty, then none of those cases should be considered
 * anymore and all cases have effectively been considered.
 */
static int finished_all_cases(struct isl_local_region *local,
        struct isl_lexmin_data *data)
{
        if (data->tab->empty)
                return 1;
        return local->side >= 2 * local->n;
}

/* Enter level "level" of the backtracking search and figure out
 * what to do next.  "init" is set if the level was entered
 * from a higher level and needs to be initialized.
 * Otherwise, the level is entered as a result of backtracking and
 * the tableau needs to be restored to a position that can
 * be used for the next case at this level.
 * The snapshot is assumed to have been saved in the previous case,
 * before the constraints specific to that case were added.
 *
 * In the initialization case, the local region is initialized
 * to point to the first violated region.
 * If the constraints of all regions are satisfied by the current
 * sample of the tableau, then tell the caller to continue looking
 * for a better solution or to stop searching if an optimal solution
 * has been found.
 *
 * If the tableau is empty or if all cases at the current level
 * have been considered, then the caller needs to backtrack as well.
 */
static enum isl_next enter_level(int level, int init,
        struct isl_lexmin_data *data)
{
        struct isl_local_region *local = &data->local[level];

        if (init) {
                int r;

                data->tab = cut_to_integer_lexmin(data->tab, CUT_ONE);
                if (!data->tab)
                        return isl_next_error;
                if (data->tab->empty)
                        return isl_next_backtrack;
                r = first_trivial_region(data);
                if (r < 0)
                        return isl_next_error;
                if (r == data->n_region) {
                        update_outer_levels(data, level);
                        isl_vec_free(data->sol);
                        data->sol = isl_tab_get_sample_value(data->tab);
                        if (!data->sol)
                                return isl_next_error;
                        if (is_optimal(data->sol, data->n_op))
                                return isl_next_done;
                        return isl_next_backtrack;
                }
                if (level >= data->n_region)
                        isl_die(isl_vec_get_ctx(data->v), isl_error_internal,
                                "nesting level too deep",
                                return isl_next_error);
                init_local_region(local, r, data);
                if (isl_tab_extend_cons(data->tab,
                                    2 * local->n + 2 * data->n_op) < 0)
                        return isl_next_error;
        } else {
                if (isl_tab_rollback(data->tab, local->snap) < 0)
                        return isl_next_error;
        }

        if (finished_all_cases(local, data))
                return isl_next_backtrack;
        return isl_next_handle;
}

/* If a solution has been found in the previous case at this level
 * (marked by local->update being set), then add constraints
 * that enforce a better solution in the present and all following cases.
 * The constraints only need to be imposed once because they are
 * included in the snapshot (taken in pick_side) that will be used in
 * subsequent cases.
 */
static isl_stat better_next_side(struct isl_local_region *local,
        struct isl_lexmin_data *data)
{
        if (!local->update)
                return isl_stat_ok;

        local->n_zero = force_better_solution(data->tab,
                                data->sol, data->n_op, local->n_zero);
        if (local->n_zero < 0)
                return isl_stat_error;

        local->update = 0;

        return isl_stat_ok;
}

/* Add constraints to data->tab that select the current case (local->side)
 * at the current level.
 *
 * If the linear combinations v should not be zero, then the cases are
 *      v_0 >= 1
 *      v_0 <= -1
 *      v_0 = 0 and v_1 >= 1
 *      v_0 = 0 and v_1 <= -1
 *      v_0 = 0 and v_1 = 0 and v_2 >= 1
 *      v_0 = 0 and v_1 = 0 and v_2 <= -1
 *      ...
 * in this order.
 *
 * A snapshot is taken after the equality constraint (if any) has been added
 * such that the next case can start off from this position.
 * The rollback to this position is performed in enter_level.
 */
static isl_stat pick_side(struct isl_local_region *local,
        struct isl_lexmin_data *data)
{
        struct isl_trivial_region *region;
        int side, base;

        region = &data->region[local->region];
        side = local->side;
        base = 2 * (side/2);

        if (side == base && base >= 2 &&
            fix_zero(data->tab, region, base / 2 - 1, data) < 0)
                return isl_stat_error;

        local->snap = isl_tab_snap(data->tab);
        if (isl_tab_push_basis(data->tab) < 0)
                return isl_stat_error;

        data->tab = pos_neg(data->tab, region, side, data);
        if (!data->tab)
                return isl_stat_error;
        return isl_stat_ok;
}

/* Free the memory associated to "data".
 */
static void clear_lexmin_data(struct isl_lexmin_data *data)
{
        free(data->local);
        isl_vec_free(data->v);
        isl_tab_free(data->tab);
}

/* Return the lexicographically smallest non-trivial solution of the
 * given ILP problem.
 *
 * All variables are assumed to be non-negative.
 *
 * n_op is the number of initial coordinates to optimize.
 * That is, once a solution has been found, we will only continue looking
 * for solutions that result in significantly better values for those
 * initial coordinates.  That is, we only continue looking for solutions
 * that increase the number of initial zeros in this sequence.
 *
 * A solution is non-trivial, if it is non-trivial on each of the
 * specified regions.  Each region represents a sequence of
 * triviality directions on a sequence of variables that starts
 * at a given position.  A solution is non-trivial on such a region if
 * at least one of the triviality directions is non-zero
 * on that sequence of variables.
 *
 * Whenever a conflict is encountered, all constraints involved are
 * reported to the caller through a call to "conflict".
 *
 * We perform a simple branch-and-bound backtracking search.
 * Each level in the search represents an initially trivial region
 * that is forced to be non-trivial.
 * At each level we consider 2 * n cases, where n
 * is the number of triviality directions.
 * In terms of those n directions v_i, we consider the cases
 *      v_0 >= 1
 *      v_0 <= -1
 *      v_0 = 0 and v_1 >= 1
 *      v_0 = 0 and v_1 <= -1
 *      v_0 = 0 and v_1 = 0 and v_2 >= 1
 *      v_0 = 0 and v_1 = 0 and v_2 <= -1
 *      ...
 * in this order.
 */
__isl_give isl_vec *isl_tab_basic_set_non_trivial_lexmin(
        __isl_take isl_basic_set *bset, int n_op, int n_region,
        struct isl_trivial_region *region,
        int (*conflict)(int con, void *user), void *user)
{
        struct isl_lexmin_data data = { n_op, n_region, region };
        int level, init;

        if (!bset)
                return NULL;

        if (init_lexmin_data(&data, bset) < 0)
                goto error;
        data.tab->conflict = conflict;
        data.tab->conflict_user = user;

        level = 0;
        init = 1;

        while (level >= 0) {
                enum isl_next next;
                struct isl_local_region *local = &data.local[level];

                next = enter_level(level, init, &data);
                if (next < 0)
                        goto error;
                if (next == isl_next_done)
                        break;
                if (next == isl_next_backtrack) {
                        level--;
                        init = 0;
                        continue;
                }

                if (better_next_side(local, &data) < 0)
                        goto error;
                if (pick_side(local, &data) < 0)
                        goto error;

                local->side++;
                level++;
                init = 1;
        }

        clear_lexmin_data(&data);
        isl_basic_set_free(bset);

        return data.sol;
error:
        clear_lexmin_data(&data);
        isl_basic_set_free(bset);
        isl_vec_free(data.sol);
        return NULL;
}

/* Wrapper for a tableau that is used for computing
 * the lexicographically smallest rational point of a non-negative set.
 * This point is represented by the sample value of "tab",
 * unless "tab" is empty.
 */
struct isl_tab_lexmin {
        isl_ctx *ctx;
        struct isl_tab *tab;
};

/* Free "tl" and return NULL.
 */
__isl_null isl_tab_lexmin *isl_tab_lexmin_free(__isl_take isl_tab_lexmin *tl)
{
        if (!tl)
                return NULL;
        isl_ctx_deref(tl->ctx);
        isl_tab_free(tl->tab);
        free(tl);

        return NULL;
}

/* Construct an isl_tab_lexmin for computing
 * the lexicographically smallest rational point in "bset",
 * assuming that all variables are non-negative.
 */
__isl_give isl_tab_lexmin *isl_tab_lexmin_from_basic_set(
        __isl_take isl_basic_set *bset)
{
        isl_ctx *ctx;
        isl_tab_lexmin *tl;

        if (!bset)
                return NULL;

        ctx = isl_basic_set_get_ctx(bset);
        tl = isl_calloc_type(ctx, struct isl_tab_lexmin);
        if (!tl)
                goto error;
        tl->ctx = ctx;
        isl_ctx_ref(ctx);
        tl->tab = tab_for_lexmin(bset, NULL, 0, 0);
        isl_basic_set_free(bset);
        if (!tl->tab)
                return isl_tab_lexmin_free(tl);
        return tl;
error:
        isl_basic_set_free(bset);
        isl_tab_lexmin_free(tl);
        return NULL;
}

/* Return the dimension of the set represented by "tl".
 */
int isl_tab_lexmin_dim(__isl_keep isl_tab_lexmin *tl)
{
        return tl ? tl->tab->n_var : -1;
}

/* Add the equality with coefficients "eq" to "tl", updating the optimal
 * solution if needed.
 * The equality is added as two opposite inequality constraints.
 */
__isl_give isl_tab_lexmin *isl_tab_lexmin_add_eq(__isl_take isl_tab_lexmin *tl,
        isl_int *eq)
{
        unsigned n_var;

        if (!tl || !eq)
                return isl_tab_lexmin_free(tl);

        if (isl_tab_extend_cons(tl->tab, 2) < 0)
                return isl_tab_lexmin_free(tl);
        n_var = tl->tab->n_var;
        isl_seq_neg(eq, eq, 1 + n_var);
        tl->tab = add_lexmin_ineq(tl->tab, eq);
        isl_seq_neg(eq, eq, 1 + n_var);
        tl->tab = add_lexmin_ineq(tl->tab, eq);

        if (!tl->tab)
                return isl_tab_lexmin_free(tl);

        return tl;
}

/* Add cuts to "tl" until the sample value reaches an integer value or
 * until the result becomes empty.
 */
__isl_give isl_tab_lexmin *isl_tab_lexmin_cut_to_integer(
        __isl_take isl_tab_lexmin *tl)
{
        if (!tl)
                return NULL;
        tl->tab = cut_to_integer_lexmin(tl->tab, CUT_ONE);
        if (!tl->tab)
                return isl_tab_lexmin_free(tl);
        return tl;
}

/* Return the lexicographically smallest rational point in the basic set
 * from which "tl" was constructed.
 * If the original input was empty, then return a zero-length vector.
 */
__isl_give isl_vec *isl_tab_lexmin_get_solution(__isl_keep isl_tab_lexmin *tl)
{
        if (!tl)
                return NULL;
        if (tl->tab->empty)
                return isl_vec_alloc(tl->ctx, 0);
        else
                return isl_tab_get_sample_value(tl->tab);
}

struct isl_sol_pma {
        struct isl_sol  sol;
        isl_pw_multi_aff *pma;
        isl_set *empty;
};

static void sol_pma_free(struct isl_sol *sol)
{
        struct isl_sol_pma *sol_pma = (struct isl_sol_pma *) sol;
        isl_pw_multi_aff_free(sol_pma->pma);
        isl_set_free(sol_pma->empty);
}

/* This function is called for parts of the context where there is
 * no solution, with "bset" corresponding to the context tableau.
 * Simply add the basic set to the set "empty".
 */
static void sol_pma_add_empty(struct isl_sol_pma *sol,
        __isl_take isl_basic_set *bset)
{
        if (!bset || !sol->empty)
                goto error;

        sol->empty = isl_set_grow(sol->empty, 1);
        bset = isl_basic_set_simplify(bset);
        bset = isl_basic_set_finalize(bset);
        sol->empty = isl_set_add_basic_set(sol->empty, bset);
        if (!sol->empty)
                sol->sol.error = 1;
        return;
error:
        isl_basic_set_free(bset);
        sol->sol.error = 1;
}

/* Given a basic set "dom" that represents the context and a tuple of
 * affine expressions "maff" defined over this domain, construct
 * an isl_pw_multi_aff with a single cell corresponding to "dom" and
 * the affine expressions in "maff".
 */
static void sol_pma_add(struct isl_sol_pma *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *maff)
{
        isl_pw_multi_aff *pma;

        dom = isl_basic_set_simplify(dom);
        dom = isl_basic_set_finalize(dom);
        pma = isl_pw_multi_aff_alloc(isl_set_from_basic_set(dom), maff);
        sol->pma = isl_pw_multi_aff_add_disjoint(sol->pma, pma);
        if (!sol->pma)
                sol->sol.error = 1;
}

static void sol_pma_add_empty_wrap(struct isl_sol *sol,
        __isl_take isl_basic_set *bset)
{
        sol_pma_add_empty((struct isl_sol_pma *)sol, bset);
}

static void sol_pma_add_wrap(struct isl_sol *sol,
        __isl_take isl_basic_set *dom, __isl_take isl_multi_aff *ma)
{
        sol_pma_add((struct isl_sol_pma *)sol, dom, ma);
}

/* Construct an isl_sol_pma structure for accumulating the solution.
 * If track_empty is set, then we also keep track of the parts
 * of the context where there is no solution.
 * If max is set, then we are solving a maximization, rather than
 * a minimization problem, which means that the variables in the
 * tableau have value "M - x" rather than "M + x".
 */
static struct isl_sol *sol_pma_init(__isl_keep isl_basic_map *bmap,
        __isl_take isl_basic_set *dom, int track_empty, int max)
{
        struct isl_sol_pma *sol_pma = NULL;
        isl_space *space;

        if (!bmap)
                goto error;

        sol_pma = isl_calloc_type(bmap->ctx, struct isl_sol_pma);
        if (!sol_pma)
                goto error;

        sol_pma->sol.free = &sol_pma_free;
        if (sol_init(&sol_pma->sol, bmap, dom, max) < 0)
                goto error;
        sol_pma->sol.add = &sol_pma_add_wrap;
        sol_pma->sol.add_empty = track_empty ? &sol_pma_add_empty_wrap : NULL;
        space = isl_space_copy(sol_pma->sol.space);
        sol_pma->pma = isl_pw_multi_aff_empty(space);
        if (!sol_pma->pma)
                goto error;

        if (track_empty) {
                sol_pma->empty = isl_set_alloc_space(isl_basic_set_get_space(dom),
                                                        1, ISL_SET_DISJOINT);
                if (!sol_pma->empty)
                        goto error;
        }

        isl_basic_set_free(dom);
        return &sol_pma->sol;
error:
        isl_basic_set_free(dom);
        sol_free(&sol_pma->sol);
        return NULL;
}

/* Base case of isl_tab_basic_map_partial_lexopt, after removing
 * some obvious symmetries.
 *
 * We call basic_map_partial_lexopt_base_sol and extract the results.
 */
static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_base_pw_multi_aff(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max)
{
        isl_pw_multi_aff *result = NULL;
        struct isl_sol *sol;
        struct isl_sol_pma *sol_pma;

        sol = basic_map_partial_lexopt_base_sol(bmap, dom, empty, max,
                                                &sol_pma_init);
        if (!sol)
                return NULL;
        sol_pma = (struct isl_sol_pma *) sol;

        result = isl_pw_multi_aff_copy(sol_pma->pma);
        if (empty)
                *empty = isl_set_copy(sol_pma->empty);
        sol_free(&sol_pma->sol);
        return result;
}

/* Given that the last input variable of "maff" represents the minimum
 * of some bounds, check whether we need to plug in the expression
 * of the minimum.
 *
 * In particular, check if the last input variable appears in any
 * of the expressions in "maff".
 */
static int need_substitution(__isl_keep isl_multi_aff *maff)
{
        int i;
        unsigned pos;

        pos = isl_multi_aff_dim(maff, isl_dim_in) - 1;

        for (i = 0; i < maff->n; ++i)
                if (isl_aff_involves_dims(maff->p[i], isl_dim_in, pos, 1))
                        return 1;

        return 0;
}

/* Given a set of upper bounds on the last "input" variable m,
 * construct a piecewise affine expression that selects
 * the minimal upper bound to m, i.e.,
 * divide the space into cells where one
 * of the upper bounds is smaller than all the others and select
 * this upper bound on that cell.
 *
 * In particular, if there are n bounds b_i, then the result
 * consists of n cell, each one of the form
 *
 *      b_i <= b_j      for j > i
 *      b_i <  b_j      for j < i
 *
 * The affine expression on this cell is
 *
 *      b_i
 */
static __isl_give isl_pw_aff *set_minimum_pa(__isl_take isl_space *space,
        __isl_take isl_mat *var)
{
        int i;
        isl_aff *aff = NULL;
        isl_basic_set *bset = NULL;
        isl_pw_aff *paff = NULL;
        isl_space *pw_space;
        isl_local_space *ls = NULL;

        if (!space || !var)
                goto error;

        ls = isl_local_space_from_space(isl_space_copy(space));
        pw_space = isl_space_copy(space);
        pw_space = isl_space_from_domain(pw_space);
        pw_space = isl_space_add_dims(pw_space, isl_dim_out, 1);
        paff = isl_pw_aff_alloc_size(pw_space, var->n_row);

        for (i = 0; i < var->n_row; ++i) {
                isl_pw_aff *paff_i;

                aff = isl_aff_alloc(isl_local_space_copy(ls));
                bset = isl_basic_set_alloc_space(isl_space_copy(space), 0,
                                               0, var->n_row - 1);
                if (!aff || !bset)
                        goto error;
                isl_int_set_si(aff->v->el[0], 1);
                isl_seq_cpy(aff->v->el + 1, var->row[i], var->n_col);
                isl_int_set_si(aff->v->el[1 + var->n_col], 0);
                bset = select_minimum(bset, var, i);
                paff_i = isl_pw_aff_alloc(isl_set_from_basic_set(bset), aff);
                paff = isl_pw_aff_add_disjoint(paff, paff_i);
        }

        isl_local_space_free(ls);
        isl_space_free(space);
        isl_mat_free(var);
        return paff;
error:
        isl_aff_free(aff);
        isl_basic_set_free(bset);
        isl_pw_aff_free(paff);
        isl_local_space_free(ls);
        isl_space_free(space);
        isl_mat_free(var);
        return NULL;
}

/* Given a piecewise multi-affine expression of which the last input variable
 * is the minimum of the bounds in "cst", plug in the value of the minimum.
 * This minimum expression is given in "min_expr_pa".
 * The set "min_expr" contains the same information, but in the form of a set.
 * The variable is subsequently projected out.
 *
 * The implementation is similar to those of "split" and "split_domain".
 * If the variable appears in a given expression, then minimum expression
 * is plugged in.  Otherwise, if the variable appears in the constraints
 * and a split is required, then the domain is split.  Otherwise, no split
 * is performed.
 */
static __isl_give isl_pw_multi_aff *split_domain_pma(
        __isl_take isl_pw_multi_aff *opt, __isl_take isl_pw_aff *min_expr_pa,
        __isl_take isl_set *min_expr, __isl_take isl_mat *cst)
{
        int n_in;
        int i;
        isl_space *space;
        isl_pw_multi_aff *res;

        if (!opt || !min_expr || !cst)
                goto error;

        n_in = isl_pw_multi_aff_dim(opt, isl_dim_in);
        space = isl_pw_multi_aff_get_space(opt);
        space = isl_space_drop_dims(space, isl_dim_in, n_in - 1, 1);
        res = isl_pw_multi_aff_empty(space);

        for (i = 0; i < opt->n; ++i) {
                isl_pw_multi_aff *pma;

                pma = isl_pw_multi_aff_alloc(isl_set_copy(opt->p[i].set),
                                         isl_multi_aff_copy(opt->p[i].maff));
                if (need_substitution(opt->p[i].maff))
                        pma = isl_pw_multi_aff_substitute(pma,
                                        isl_dim_in, n_in - 1, min_expr_pa);
                else {
                        isl_bool split;
                        split = need_split_set(opt->p[i].set, cst);
                        if (split < 0)
                                pma = isl_pw_multi_aff_free(pma);
                        else if (split)
                                pma = isl_pw_multi_aff_intersect_domain(pma,
                                                       isl_set_copy(min_expr));
                }
                pma = isl_pw_multi_aff_project_out(pma,
                                                    isl_dim_in, n_in - 1, 1);

                res = isl_pw_multi_aff_add_disjoint(res, pma);
        }

        isl_pw_multi_aff_free(opt);
        isl_pw_aff_free(min_expr_pa);
        isl_set_free(min_expr);
        isl_mat_free(cst);
        return res;
error:
        isl_pw_multi_aff_free(opt);
        isl_pw_aff_free(min_expr_pa);
        isl_set_free(min_expr);
        isl_mat_free(cst);
        return NULL;
}

static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pw_multi_aff(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max);

/* This function is called from basic_map_partial_lexopt_symm.
 * The last variable of "bmap" and "dom" corresponds to the minimum
 * of the bounds in "cst".  "map_space" is the space of the original
 * input relation (of basic_map_partial_lexopt_symm) and "set_space"
 * is the space of the original domain.
 *
 * We recursively call basic_map_partial_lexopt and then plug in
 * the definition of the minimum in the result.
 */
static __isl_give isl_pw_multi_aff *
basic_map_partial_lexopt_symm_core_pw_multi_aff(
        __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
        __isl_give isl_set **empty, int max, __isl_take isl_mat *cst,
        __isl_take isl_space *map_space, __isl_take isl_space *set_space)
{
        isl_pw_multi_aff *opt;
        isl_pw_aff *min_expr_pa;
        isl_set *min_expr;

        min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst));
        min_expr_pa = set_minimum_pa(isl_basic_set_get_space(dom),
                                        isl_mat_copy(cst));

        opt = basic_map_partial_lexopt_pw_multi_aff(bmap, dom, empty, max);

        if (empty) {
                *empty = split(*empty,
                               isl_set_copy(min_expr), isl_mat_copy(cst));
                *empty = isl_set_reset_space(*empty, set_space);
        }

        opt = split_domain_pma(opt, min_expr_pa, min_expr, cst);
        opt = isl_pw_multi_aff_reset_space(opt, map_space);

        return opt;
}

#undef TYPE
#define TYPE    isl_pw_multi_aff
#undef SUFFIX
#define SUFFIX  _pw_multi_aff
#include "isl_tab_lexopt_templ.c"