isl_basic_set_opt: reuse isl_basic_set_get_divs

[isl.git] / isl_polynomial.c

/*
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
 * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
 * 91893 Orsay, France 
 */

#include <stdlib.h>
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl_factorization.h>
#include <isl_lp_private.h>
#include <isl_seq.h>
#include <isl_union_map_private.h>
#include <isl_constraint_private.h>
#include <isl_polynomial_private.h>
#include <isl_point_private.h>
#include <isl_space_private.h>
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl_range.h>
#include <isl_local.h>
#include <isl_local_space_private.h>
#include <isl_aff_private.h>
#include <isl_val_private.h>
#include <isl_config.h>

#undef BASE
#define BASE pw_qpolynomial

#include <isl_list_templ.c>

static unsigned pos(__isl_keep isl_space *dim, enum isl_dim_type type)
{
        switch (type) {
        case isl_dim_param:     return 0;
        case isl_dim_in:        return dim->nparam;
        case isl_dim_out:       return dim->nparam + dim->n_in;
        default:                return 0;
        }
}

int isl_upoly_is_cst(__isl_keep struct isl_upoly *up)
{
        if (!up)
                return -1;

        return up->var < 0;
}

__isl_keep struct isl_upoly_cst *isl_upoly_as_cst(__isl_keep struct isl_upoly *up)
{
        if (!up)
                return NULL;

        isl_assert(up->ctx, up->var < 0, return NULL);

        return (struct isl_upoly_cst *)up;
}

__isl_keep struct isl_upoly_rec *isl_upoly_as_rec(__isl_keep struct isl_upoly *up)
{
        if (!up)
                return NULL;

        isl_assert(up->ctx, up->var >= 0, return NULL);

        return (struct isl_upoly_rec *)up;
}

/* Compare two polynomials.
 *
 * Return -1 if "up1" is "smaller" than "up2", 1 if "up1" is "greater"
 * than "up2" and 0 if they are equal.
 */
static int isl_upoly_plain_cmp(__isl_keep struct isl_upoly *up1,
        __isl_keep struct isl_upoly *up2)
{
        int i;
        struct isl_upoly_rec *rec1, *rec2;

        if (up1 == up2)
                return 0;
        if (!up1)
                return -1;
        if (!up2)
                return 1;
        if (up1->var != up2->var)
                return up1->var - up2->var;

        if (isl_upoly_is_cst(up1)) {
                struct isl_upoly_cst *cst1, *cst2;
                int cmp;

                cst1 = isl_upoly_as_cst(up1);
                cst2 = isl_upoly_as_cst(up2);
                if (!cst1 || !cst2)
                        return 0;
                cmp = isl_int_cmp(cst1->n, cst2->n);
                if (cmp != 0)
                        return cmp;
                return isl_int_cmp(cst1->d, cst2->d);
        }

        rec1 = isl_upoly_as_rec(up1);
        rec2 = isl_upoly_as_rec(up2);
        if (!rec1 || !rec2)
                return 0;

        if (rec1->n != rec2->n)
                return rec1->n - rec2->n;

        for (i = 0; i < rec1->n; ++i) {
                int cmp = isl_upoly_plain_cmp(rec1->p[i], rec2->p[i]);
                if (cmp != 0)
                        return cmp;
        }

        return 0;
}

isl_bool isl_upoly_is_equal(__isl_keep struct isl_upoly *up1,
        __isl_keep struct isl_upoly *up2)
{
        int i;
        struct isl_upoly_rec *rec1, *rec2;

        if (!up1 || !up2)
                return isl_bool_error;
        if (up1 == up2)
                return isl_bool_true;
        if (up1->var != up2->var)
                return isl_bool_false;
        if (isl_upoly_is_cst(up1)) {
                struct isl_upoly_cst *cst1, *cst2;
                cst1 = isl_upoly_as_cst(up1);
                cst2 = isl_upoly_as_cst(up2);
                if (!cst1 || !cst2)
                        return isl_bool_error;
                return isl_int_eq(cst1->n, cst2->n) &&
                       isl_int_eq(cst1->d, cst2->d);
        }

        rec1 = isl_upoly_as_rec(up1);
        rec2 = isl_upoly_as_rec(up2);
        if (!rec1 || !rec2)
                return isl_bool_error;

        if (rec1->n != rec2->n)
                return isl_bool_false;

        for (i = 0; i < rec1->n; ++i) {
                isl_bool eq = isl_upoly_is_equal(rec1->p[i], rec2->p[i]);
                if (eq < 0 || !eq)
                        return eq;
        }

        return isl_bool_true;
}

int isl_upoly_is_zero(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return -1;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return -1;

        return isl_int_is_zero(cst->n) && isl_int_is_pos(cst->d);
}

int isl_upoly_sgn(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return 0;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return 0;

        return isl_int_sgn(cst->n);
}

int isl_upoly_is_nan(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return -1;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return -1;

        return isl_int_is_zero(cst->n) && isl_int_is_zero(cst->d);
}

int isl_upoly_is_infty(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return -1;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return -1;

        return isl_int_is_pos(cst->n) && isl_int_is_zero(cst->d);
}

int isl_upoly_is_neginfty(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return -1;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return -1;

        return isl_int_is_neg(cst->n) && isl_int_is_zero(cst->d);
}

int isl_upoly_is_one(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return -1;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return -1;

        return isl_int_eq(cst->n, cst->d) && isl_int_is_pos(cst->d);
}

int isl_upoly_is_negone(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return -1;
        if (!isl_upoly_is_cst(up))
                return 0;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return -1;

        return isl_int_is_negone(cst->n) && isl_int_is_one(cst->d);
}

__isl_give struct isl_upoly_cst *isl_upoly_cst_alloc(struct isl_ctx *ctx)
{
        struct isl_upoly_cst *cst;

        cst = isl_alloc_type(ctx, struct isl_upoly_cst);
        if (!cst)
                return NULL;

        cst->up.ref = 1;
        cst->up.ctx = ctx;
        isl_ctx_ref(ctx);
        cst->up.var = -1;

        isl_int_init(cst->n);
        isl_int_init(cst->d);

        return cst;
}

__isl_give struct isl_upoly *isl_upoly_zero(struct isl_ctx *ctx)
{
        struct isl_upoly_cst *cst;

        cst = isl_upoly_cst_alloc(ctx);
        if (!cst)
                return NULL;

        isl_int_set_si(cst->n, 0);
        isl_int_set_si(cst->d, 1);

        return &cst->up;
}

__isl_give struct isl_upoly *isl_upoly_one(struct isl_ctx *ctx)
{
        struct isl_upoly_cst *cst;

        cst = isl_upoly_cst_alloc(ctx);
        if (!cst)
                return NULL;

        isl_int_set_si(cst->n, 1);
        isl_int_set_si(cst->d, 1);

        return &cst->up;
}

__isl_give struct isl_upoly *isl_upoly_infty(struct isl_ctx *ctx)
{
        struct isl_upoly_cst *cst;

        cst = isl_upoly_cst_alloc(ctx);
        if (!cst)
                return NULL;

        isl_int_set_si(cst->n, 1);
        isl_int_set_si(cst->d, 0);

        return &cst->up;
}

__isl_give struct isl_upoly *isl_upoly_neginfty(struct isl_ctx *ctx)
{
        struct isl_upoly_cst *cst;

        cst = isl_upoly_cst_alloc(ctx);
        if (!cst)
                return NULL;

        isl_int_set_si(cst->n, -1);
        isl_int_set_si(cst->d, 0);

        return &cst->up;
}

__isl_give struct isl_upoly *isl_upoly_nan(struct isl_ctx *ctx)
{
        struct isl_upoly_cst *cst;

        cst = isl_upoly_cst_alloc(ctx);
        if (!cst)
                return NULL;

        isl_int_set_si(cst->n, 0);
        isl_int_set_si(cst->d, 0);

        return &cst->up;
}

__isl_give struct isl_upoly *isl_upoly_rat_cst(struct isl_ctx *ctx,
        isl_int n, isl_int d)
{
        struct isl_upoly_cst *cst;

        cst = isl_upoly_cst_alloc(ctx);
        if (!cst)
                return NULL;

        isl_int_set(cst->n, n);
        isl_int_set(cst->d, d);

        return &cst->up;
}

__isl_give struct isl_upoly_rec *isl_upoly_alloc_rec(struct isl_ctx *ctx,
        int var, int size)
{
        struct isl_upoly_rec *rec;

        isl_assert(ctx, var >= 0, return NULL);
        isl_assert(ctx, size >= 0, return NULL);
        rec = isl_calloc(ctx, struct isl_upoly_rec,
                        sizeof(struct isl_upoly_rec) +
                        size * sizeof(struct isl_upoly *));
        if (!rec)
                return NULL;

        rec->up.ref = 1;
        rec->up.ctx = ctx;
        isl_ctx_ref(ctx);
        rec->up.var = var;

        rec->n = 0;
        rec->size = size;

        return rec;
}

__isl_give isl_qpolynomial *isl_qpolynomial_reset_domain_space(
        __isl_take isl_qpolynomial *qp, __isl_take isl_space *dim)
{
        qp = isl_qpolynomial_cow(qp);
        if (!qp || !dim)
                goto error;

        isl_space_free(qp->dim);
        qp->dim = dim;

        return qp;
error:
        isl_qpolynomial_free(qp);
        isl_space_free(dim);
        return NULL;
}

/* Reset the space of "qp".  This function is called from isl_pw_templ.c
 * and doesn't know if the space of an element object is represented
 * directly or through its domain.  It therefore passes along both.
 */
__isl_give isl_qpolynomial *isl_qpolynomial_reset_space_and_domain(
        __isl_take isl_qpolynomial *qp, __isl_take isl_space *space,
        __isl_take isl_space *domain)
{
        isl_space_free(space);
        return isl_qpolynomial_reset_domain_space(qp, domain);
}

isl_ctx *isl_qpolynomial_get_ctx(__isl_keep isl_qpolynomial *qp)
{
        return qp ? qp->dim->ctx : NULL;
}

__isl_give isl_space *isl_qpolynomial_get_domain_space(
        __isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_space_copy(qp->dim) : NULL;
}

/* Return a copy of the local space on which "qp" is defined.
 */
static __isl_give isl_local_space *isl_qpolynomial_get_domain_local_space(
        __isl_keep isl_qpolynomial *qp)
{
        isl_space *space;

        if (!qp)
                return NULL;

        space = isl_qpolynomial_get_domain_space(qp);
        return isl_local_space_alloc_div(space, isl_mat_copy(qp->div));
}

__isl_give isl_space *isl_qpolynomial_get_space(__isl_keep isl_qpolynomial *qp)
{
        isl_space *space;
        if (!qp)
                return NULL;
        space = isl_space_copy(qp->dim);
        space = isl_space_from_domain(space);
        space = isl_space_add_dims(space, isl_dim_out, 1);
        return space;
}

/* Return the number of variables of the given type in the domain of "qp".
 */
unsigned isl_qpolynomial_domain_dim(__isl_keep isl_qpolynomial *qp,
        enum isl_dim_type type)
{
        if (!qp)
                return 0;
        if (type == isl_dim_div)
                return qp->div->n_row;
        if (type == isl_dim_all)
                return isl_space_dim(qp->dim, isl_dim_all) +
                                    isl_qpolynomial_domain_dim(qp, isl_dim_div);
        return isl_space_dim(qp->dim, type);
}

/* Externally, an isl_qpolynomial has a map space, but internally, the
 * ls field corresponds to the domain of that space.
 */
unsigned isl_qpolynomial_dim(__isl_keep isl_qpolynomial *qp,
        enum isl_dim_type type)
{
        if (!qp)
                return 0;
        if (type == isl_dim_out)
                return 1;
        if (type == isl_dim_in)
                type = isl_dim_set;
        return isl_qpolynomial_domain_dim(qp, type);
}

/* Return the offset of the first coefficient of type "type" in
 * the domain of "qp".
 */
unsigned isl_qpolynomial_domain_offset(__isl_keep isl_qpolynomial *qp,
        enum isl_dim_type type)
{
        if (!qp)
                return 0;
        switch (type) {
        case isl_dim_cst:
                return 0;
        case isl_dim_param:
        case isl_dim_set:
                return 1 + isl_space_offset(qp->dim, type);
        case isl_dim_div:
                return 1 + isl_space_dim(qp->dim, isl_dim_all);
        default:
                return 0;
        }
}

isl_bool isl_qpolynomial_is_zero(__isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_upoly_is_zero(qp->upoly) : isl_bool_error;
}

isl_bool isl_qpolynomial_is_one(__isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_upoly_is_one(qp->upoly) : isl_bool_error;
}

isl_bool isl_qpolynomial_is_nan(__isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_upoly_is_nan(qp->upoly) : isl_bool_error;
}

isl_bool isl_qpolynomial_is_infty(__isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_upoly_is_infty(qp->upoly) : isl_bool_error;
}

isl_bool isl_qpolynomial_is_neginfty(__isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_upoly_is_neginfty(qp->upoly) : isl_bool_error;
}

int isl_qpolynomial_sgn(__isl_keep isl_qpolynomial *qp)
{
        return qp ? isl_upoly_sgn(qp->upoly) : 0;
}

static void upoly_free_cst(__isl_take struct isl_upoly_cst *cst)
{
        isl_int_clear(cst->n);
        isl_int_clear(cst->d);
}

static void upoly_free_rec(__isl_take struct isl_upoly_rec *rec)
{
        int i;

        for (i = 0; i < rec->n; ++i)
                isl_upoly_free(rec->p[i]);
}

__isl_give struct isl_upoly *isl_upoly_copy(__isl_keep struct isl_upoly *up)
{
        if (!up)
                return NULL;

        up->ref++;
        return up;
}

__isl_give struct isl_upoly *isl_upoly_dup_cst(__isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;
        struct isl_upoly_cst *dup;

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return NULL;

        dup = isl_upoly_as_cst(isl_upoly_zero(up->ctx));
        if (!dup)
                return NULL;
        isl_int_set(dup->n, cst->n);
        isl_int_set(dup->d, cst->d);

        return &dup->up;
}

__isl_give struct isl_upoly *isl_upoly_dup_rec(__isl_keep struct isl_upoly *up)
{
        int i;
        struct isl_upoly_rec *rec;
        struct isl_upoly_rec *dup;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                return NULL;

        dup = isl_upoly_alloc_rec(up->ctx, up->var, rec->n);
        if (!dup)
                return NULL;

        for (i = 0; i < rec->n; ++i) {
                dup->p[i] = isl_upoly_copy(rec->p[i]);
                if (!dup->p[i])
                        goto error;
                dup->n++;
        }

        return &dup->up;
error:
        isl_upoly_free(&dup->up);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_dup(__isl_keep struct isl_upoly *up)
{
        if (!up)
                return NULL;

        if (isl_upoly_is_cst(up))
                return isl_upoly_dup_cst(up);
        else
                return isl_upoly_dup_rec(up);
}

__isl_give struct isl_upoly *isl_upoly_cow(__isl_take struct isl_upoly *up)
{
        if (!up)
                return NULL;

        if (up->ref == 1)
                return up;
        up->ref--;
        return isl_upoly_dup(up);
}

__isl_null struct isl_upoly *isl_upoly_free(__isl_take struct isl_upoly *up)
{
        if (!up)
                return NULL;

        if (--up->ref > 0)
                return NULL;

        if (up->var < 0)
                upoly_free_cst((struct isl_upoly_cst *)up);
        else
                upoly_free_rec((struct isl_upoly_rec *)up);

        isl_ctx_deref(up->ctx);
        free(up);
        return NULL;
}

static void isl_upoly_cst_reduce(__isl_keep struct isl_upoly_cst *cst)
{
        isl_int gcd;

        isl_int_init(gcd);
        isl_int_gcd(gcd, cst->n, cst->d);
        if (!isl_int_is_zero(gcd) && !isl_int_is_one(gcd)) {
                isl_int_divexact(cst->n, cst->n, gcd);
                isl_int_divexact(cst->d, cst->d, gcd);
        }
        isl_int_clear(gcd);
}

__isl_give struct isl_upoly *isl_upoly_sum_cst(__isl_take struct isl_upoly *up1,
        __isl_take struct isl_upoly *up2)
{
        struct isl_upoly_cst *cst1;
        struct isl_upoly_cst *cst2;

        up1 = isl_upoly_cow(up1);
        if (!up1 || !up2)
                goto error;

        cst1 = isl_upoly_as_cst(up1);
        cst2 = isl_upoly_as_cst(up2);

        if (isl_int_eq(cst1->d, cst2->d))
                isl_int_add(cst1->n, cst1->n, cst2->n);
        else {
                isl_int_mul(cst1->n, cst1->n, cst2->d);
                isl_int_addmul(cst1->n, cst2->n, cst1->d);
                isl_int_mul(cst1->d, cst1->d, cst2->d);
        }

        isl_upoly_cst_reduce(cst1);

        isl_upoly_free(up2);
        return up1;
error:
        isl_upoly_free(up1);
        isl_upoly_free(up2);
        return NULL;
}

static __isl_give struct isl_upoly *replace_by_zero(
        __isl_take struct isl_upoly *up)
{
        struct isl_ctx *ctx;

        if (!up)
                return NULL;
        ctx = up->ctx;
        isl_upoly_free(up);
        return isl_upoly_zero(ctx);
}

static __isl_give struct isl_upoly *replace_by_constant_term(
        __isl_take struct isl_upoly *up)
{
        struct isl_upoly_rec *rec;
        struct isl_upoly *cst;

        if (!up)
                return NULL;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;
        cst = isl_upoly_copy(rec->p[0]);
        isl_upoly_free(up);
        return cst;
error:
        isl_upoly_free(up);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_sum(__isl_take struct isl_upoly *up1,
        __isl_take struct isl_upoly *up2)
{
        int i;
        struct isl_upoly_rec *rec1, *rec2;

        if (!up1 || !up2)
                goto error;

        if (isl_upoly_is_nan(up1)) {
                isl_upoly_free(up2);
                return up1;
        }

        if (isl_upoly_is_nan(up2)) {
                isl_upoly_free(up1);
                return up2;
        }

        if (isl_upoly_is_zero(up1)) {
                isl_upoly_free(up1);
                return up2;
        }

        if (isl_upoly_is_zero(up2)) {
                isl_upoly_free(up2);
                return up1;
        }

        if (up1->var < up2->var)
                return isl_upoly_sum(up2, up1);

        if (up2->var < up1->var) {
                struct isl_upoly_rec *rec;
                if (isl_upoly_is_infty(up2) || isl_upoly_is_neginfty(up2)) {
                        isl_upoly_free(up1);
                        return up2;
                }
                up1 = isl_upoly_cow(up1);
                rec = isl_upoly_as_rec(up1);
                if (!rec)
                        goto error;
                rec->p[0] = isl_upoly_sum(rec->p[0], up2);
                if (rec->n == 1)
                        up1 = replace_by_constant_term(up1);
                return up1;
        }

        if (isl_upoly_is_cst(up1))
                return isl_upoly_sum_cst(up1, up2);

        rec1 = isl_upoly_as_rec(up1);
        rec2 = isl_upoly_as_rec(up2);
        if (!rec1 || !rec2)
                goto error;

        if (rec1->n < rec2->n)
                return isl_upoly_sum(up2, up1);

        up1 = isl_upoly_cow(up1);
        rec1 = isl_upoly_as_rec(up1);
        if (!rec1)
                goto error;

        for (i = rec2->n - 1; i >= 0; --i) {
                rec1->p[i] = isl_upoly_sum(rec1->p[i],
                                            isl_upoly_copy(rec2->p[i]));
                if (!rec1->p[i])
                        goto error;
                if (i == rec1->n - 1 && isl_upoly_is_zero(rec1->p[i])) {
                        isl_upoly_free(rec1->p[i]);
                        rec1->n--;
                }
        }

        if (rec1->n == 0)
                up1 = replace_by_zero(up1);
        else if (rec1->n == 1)
                up1 = replace_by_constant_term(up1);

        isl_upoly_free(up2);

        return up1;
error:
        isl_upoly_free(up1);
        isl_upoly_free(up2);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_cst_add_isl_int(
        __isl_take struct isl_upoly *up, isl_int v)
{
        struct isl_upoly_cst *cst;

        up = isl_upoly_cow(up);
        if (!up)
                return NULL;

        cst = isl_upoly_as_cst(up);

        isl_int_addmul(cst->n, cst->d, v);

        return up;
}

__isl_give struct isl_upoly *isl_upoly_add_isl_int(
        __isl_take struct isl_upoly *up, isl_int v)
{
        struct isl_upoly_rec *rec;

        if (!up)
                return NULL;

        if (isl_upoly_is_cst(up))
                return isl_upoly_cst_add_isl_int(up, v);

        up = isl_upoly_cow(up);
        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        rec->p[0] = isl_upoly_add_isl_int(rec->p[0], v);
        if (!rec->p[0])
                goto error;

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_cst_mul_isl_int(
        __isl_take struct isl_upoly *up, isl_int v)
{
        struct isl_upoly_cst *cst;

        if (isl_upoly_is_zero(up))
                return up;

        up = isl_upoly_cow(up);
        if (!up)
                return NULL;

        cst = isl_upoly_as_cst(up);

        isl_int_mul(cst->n, cst->n, v);

        return up;
}

__isl_give struct isl_upoly *isl_upoly_mul_isl_int(
        __isl_take struct isl_upoly *up, isl_int v)
{
        int i;
        struct isl_upoly_rec *rec;

        if (!up)
                return NULL;

        if (isl_upoly_is_cst(up))
                return isl_upoly_cst_mul_isl_int(up, v);

        up = isl_upoly_cow(up);
        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                rec->p[i] = isl_upoly_mul_isl_int(rec->p[i], v);
                if (!rec->p[i])
                        goto error;
        }

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

/* Multiply the constant polynomial "up" by "v".
 */
static __isl_give struct isl_upoly *isl_upoly_cst_scale_val(
        __isl_take struct isl_upoly *up, __isl_keep isl_val *v)
{
        struct isl_upoly_cst *cst;

        if (isl_upoly_is_zero(up))
                return up;

        up = isl_upoly_cow(up);
        if (!up)
                return NULL;

        cst = isl_upoly_as_cst(up);

        isl_int_mul(cst->n, cst->n, v->n);
        isl_int_mul(cst->d, cst->d, v->d);
        isl_upoly_cst_reduce(cst);

        return up;
}

/* Multiply the polynomial "up" by "v".
 */
static __isl_give struct isl_upoly *isl_upoly_scale_val(
        __isl_take struct isl_upoly *up, __isl_keep isl_val *v)
{
        int i;
        struct isl_upoly_rec *rec;

        if (!up)
                return NULL;

        if (isl_upoly_is_cst(up))
                return isl_upoly_cst_scale_val(up, v);

        up = isl_upoly_cow(up);
        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                rec->p[i] = isl_upoly_scale_val(rec->p[i], v);
                if (!rec->p[i])
                        goto error;
        }

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_mul_cst(__isl_take struct isl_upoly *up1,
        __isl_take struct isl_upoly *up2)
{
        struct isl_upoly_cst *cst1;
        struct isl_upoly_cst *cst2;

        up1 = isl_upoly_cow(up1);
        if (!up1 || !up2)
                goto error;

        cst1 = isl_upoly_as_cst(up1);
        cst2 = isl_upoly_as_cst(up2);

        isl_int_mul(cst1->n, cst1->n, cst2->n);
        isl_int_mul(cst1->d, cst1->d, cst2->d);

        isl_upoly_cst_reduce(cst1);

        isl_upoly_free(up2);
        return up1;
error:
        isl_upoly_free(up1);
        isl_upoly_free(up2);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_mul_rec(__isl_take struct isl_upoly *up1,
        __isl_take struct isl_upoly *up2)
{
        struct isl_upoly_rec *rec1;
        struct isl_upoly_rec *rec2;
        struct isl_upoly_rec *res = NULL;
        int i, j;
        int size;

        rec1 = isl_upoly_as_rec(up1);
        rec2 = isl_upoly_as_rec(up2);
        if (!rec1 || !rec2)
                goto error;
        size = rec1->n + rec2->n - 1;
        res = isl_upoly_alloc_rec(up1->ctx, up1->var, size);
        if (!res)
                goto error;

        for (i = 0; i < rec1->n; ++i) {
                res->p[i] = isl_upoly_mul(isl_upoly_copy(rec2->p[0]),
                                            isl_upoly_copy(rec1->p[i]));
                if (!res->p[i])
                        goto error;
                res->n++;
        }
        for (; i < size; ++i) {
                res->p[i] = isl_upoly_zero(up1->ctx);
                if (!res->p[i])
                        goto error;
                res->n++;
        }
        for (i = 0; i < rec1->n; ++i) {
                for (j = 1; j < rec2->n; ++j) {
                        struct isl_upoly *up;
                        up = isl_upoly_mul(isl_upoly_copy(rec2->p[j]),
                                            isl_upoly_copy(rec1->p[i]));
                        res->p[i + j] = isl_upoly_sum(res->p[i + j], up);
                        if (!res->p[i + j])
                                goto error;
                }
        }

        isl_upoly_free(up1);
        isl_upoly_free(up2);

        return &res->up;
error:
        isl_upoly_free(up1);
        isl_upoly_free(up2);
        isl_upoly_free(&res->up);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_mul(__isl_take struct isl_upoly *up1,
        __isl_take struct isl_upoly *up2)
{
        if (!up1 || !up2)
                goto error;

        if (isl_upoly_is_nan(up1)) {
                isl_upoly_free(up2);
                return up1;
        }

        if (isl_upoly_is_nan(up2)) {
                isl_upoly_free(up1);
                return up2;
        }

        if (isl_upoly_is_zero(up1)) {
                isl_upoly_free(up2);
                return up1;
        }

        if (isl_upoly_is_zero(up2)) {
                isl_upoly_free(up1);
                return up2;
        }

        if (isl_upoly_is_one(up1)) {
                isl_upoly_free(up1);
                return up2;
        }

        if (isl_upoly_is_one(up2)) {
                isl_upoly_free(up2);
                return up1;
        }

        if (up1->var < up2->var)
                return isl_upoly_mul(up2, up1);

        if (up2->var < up1->var) {
                int i;
                struct isl_upoly_rec *rec;
                if (isl_upoly_is_infty(up2) || isl_upoly_is_neginfty(up2)) {
                        isl_ctx *ctx = up1->ctx;
                        isl_upoly_free(up1);
                        isl_upoly_free(up2);
                        return isl_upoly_nan(ctx);
                }
                up1 = isl_upoly_cow(up1);
                rec = isl_upoly_as_rec(up1);
                if (!rec)
                        goto error;

                for (i = 0; i < rec->n; ++i) {
                        rec->p[i] = isl_upoly_mul(rec->p[i],
                                                    isl_upoly_copy(up2));
                        if (!rec->p[i])
                                goto error;
                }
                isl_upoly_free(up2);
                return up1;
        }

        if (isl_upoly_is_cst(up1))
                return isl_upoly_mul_cst(up1, up2);

        return isl_upoly_mul_rec(up1, up2);
error:
        isl_upoly_free(up1);
        isl_upoly_free(up2);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_pow(__isl_take struct isl_upoly *up,
        unsigned power)
{
        struct isl_upoly *res;

        if (!up)
                return NULL;
        if (power == 1)
                return up;

        if (power % 2)
                res = isl_upoly_copy(up);
        else
                res = isl_upoly_one(up->ctx);

        while (power >>= 1) {
                up = isl_upoly_mul(up, isl_upoly_copy(up));
                if (power % 2)
                        res = isl_upoly_mul(res, isl_upoly_copy(up));
        }

        isl_upoly_free(up);
        return res;
}

__isl_give isl_qpolynomial *isl_qpolynomial_alloc(__isl_take isl_space *space,
        unsigned n_div, __isl_take struct isl_upoly *up)
{
        struct isl_qpolynomial *qp = NULL;
        unsigned total;

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

        if (!isl_space_is_set(space))
                isl_die(isl_space_get_ctx(space), isl_error_invalid,
                        "domain of polynomial should be a set", goto error);

        total = isl_space_dim(space, isl_dim_all);

        qp = isl_calloc_type(space->ctx, struct isl_qpolynomial);
        if (!qp)
                goto error;

        qp->ref = 1;
        qp->div = isl_mat_alloc(space->ctx, n_div, 1 + 1 + total + n_div);
        if (!qp->div)
                goto error;

        qp->dim = space;
        qp->upoly = up;

        return qp;
error:
        isl_space_free(space);
        isl_upoly_free(up);
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_copy(__isl_keep isl_qpolynomial *qp)
{
        if (!qp)
                return NULL;

        qp->ref++;
        return qp;
}

__isl_give isl_qpolynomial *isl_qpolynomial_dup(__isl_keep isl_qpolynomial *qp)
{
        struct isl_qpolynomial *dup;

        if (!qp)
                return NULL;

        dup = isl_qpolynomial_alloc(isl_space_copy(qp->dim), qp->div->n_row,
                                    isl_upoly_copy(qp->upoly));
        if (!dup)
                return NULL;
        isl_mat_free(dup->div);
        dup->div = isl_mat_copy(qp->div);
        if (!dup->div)
                goto error;

        return dup;
error:
        isl_qpolynomial_free(dup);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_cow(__isl_take isl_qpolynomial *qp)
{
        if (!qp)
                return NULL;

        if (qp->ref == 1)
                return qp;
        qp->ref--;
        return isl_qpolynomial_dup(qp);
}

__isl_null isl_qpolynomial *isl_qpolynomial_free(
        __isl_take isl_qpolynomial *qp)
{
        if (!qp)
                return NULL;

        if (--qp->ref > 0)
                return NULL;

        isl_space_free(qp->dim);
        isl_mat_free(qp->div);
        isl_upoly_free(qp->upoly);

        free(qp);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_var_pow(isl_ctx *ctx, int pos, int power)
{
        int i;
        struct isl_upoly_rec *rec;
        struct isl_upoly_cst *cst;

        rec = isl_upoly_alloc_rec(ctx, pos, 1 + power);
        if (!rec)
                return NULL;
        for (i = 0; i < 1 + power; ++i) {
                rec->p[i] = isl_upoly_zero(ctx);
                if (!rec->p[i])
                        goto error;
                rec->n++;
        }
        cst = isl_upoly_as_cst(rec->p[power]);
        isl_int_set_si(cst->n, 1);

        return &rec->up;
error:
        isl_upoly_free(&rec->up);
        return NULL;
}

/* r array maps original positions to new positions.
 */
static __isl_give struct isl_upoly *reorder(__isl_take struct isl_upoly *up,
        int *r)
{
        int i;
        struct isl_upoly_rec *rec;
        struct isl_upoly *base;
        struct isl_upoly *res;

        if (isl_upoly_is_cst(up))
                return up;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        isl_assert(up->ctx, rec->n >= 1, goto error);

        base = isl_upoly_var_pow(up->ctx, r[up->var], 1);
        res = reorder(isl_upoly_copy(rec->p[rec->n - 1]), r);

        for (i = rec->n - 2; i >= 0; --i) {
                res = isl_upoly_mul(res, isl_upoly_copy(base));
                res = isl_upoly_sum(res, reorder(isl_upoly_copy(rec->p[i]), r));
        }

        isl_upoly_free(base);
        isl_upoly_free(up);

        return res;
error:
        isl_upoly_free(up);
        return NULL;
}

static isl_bool compatible_divs(__isl_keep isl_mat *div1,
        __isl_keep isl_mat *div2)
{
        int n_row, n_col;
        isl_bool equal;

        isl_assert(div1->ctx, div1->n_row >= div2->n_row &&
                                div1->n_col >= div2->n_col,
                    return isl_bool_error);

        if (div1->n_row == div2->n_row)
                return isl_mat_is_equal(div1, div2);

        n_row = div1->n_row;
        n_col = div1->n_col;
        div1->n_row = div2->n_row;
        div1->n_col = div2->n_col;

        equal = isl_mat_is_equal(div1, div2);

        div1->n_row = n_row;
        div1->n_col = n_col;

        return equal;
}

static int cmp_row(__isl_keep isl_mat *div, int i, int j)
{
        int li, lj;

        li = isl_seq_last_non_zero(div->row[i], div->n_col);
        lj = isl_seq_last_non_zero(div->row[j], div->n_col);

        if (li != lj)
                return li - lj;

        return isl_seq_cmp(div->row[i], div->row[j], div->n_col);
}

struct isl_div_sort_info {
        isl_mat *div;
        int      row;
};

static int div_sort_cmp(const void *p1, const void *p2)
{
        const struct isl_div_sort_info *i1, *i2;
        i1 = (const struct isl_div_sort_info *) p1;
        i2 = (const struct isl_div_sort_info *) p2;

        return cmp_row(i1->div, i1->row, i2->row);
}

/* Sort divs and remove duplicates.
 */
static __isl_give isl_qpolynomial *sort_divs(__isl_take isl_qpolynomial *qp)
{
        int i;
        int skip;
        int len;
        struct isl_div_sort_info *array = NULL;
        int *pos = NULL, *at = NULL;
        int *reordering = NULL;
        unsigned div_pos;

        if (!qp)
                return NULL;
        if (qp->div->n_row <= 1)
                return qp;

        div_pos = isl_space_dim(qp->dim, isl_dim_all);

        array = isl_alloc_array(qp->div->ctx, struct isl_div_sort_info,
                                qp->div->n_row);
        pos = isl_alloc_array(qp->div->ctx, int, qp->div->n_row);
        at = isl_alloc_array(qp->div->ctx, int, qp->div->n_row);
        len = qp->div->n_col - 2;
        reordering = isl_alloc_array(qp->div->ctx, int, len);
        if (!array || !pos || !at || !reordering)
                goto error;

        for (i = 0; i < qp->div->n_row; ++i) {
                array[i].div = qp->div;
                array[i].row = i;
                pos[i] = i;
                at[i] = i;
        }

        qsort(array, qp->div->n_row, sizeof(struct isl_div_sort_info),
                div_sort_cmp);

        for (i = 0; i < div_pos; ++i)
                reordering[i] = i;

        for (i = 0; i < qp->div->n_row; ++i) {
                if (pos[array[i].row] == i)
                        continue;
                qp->div = isl_mat_swap_rows(qp->div, i, pos[array[i].row]);
                pos[at[i]] = pos[array[i].row];
                at[pos[array[i].row]] = at[i];
                at[i] = array[i].row;
                pos[array[i].row] = i;
        }

        skip = 0;
        for (i = 0; i < len - div_pos; ++i) {
                if (i > 0 &&
                    isl_seq_eq(qp->div->row[i - skip - 1],
                               qp->div->row[i - skip], qp->div->n_col)) {
                        qp->div = isl_mat_drop_rows(qp->div, i - skip, 1);
                        isl_mat_col_add(qp->div, 2 + div_pos + i - skip - 1,
                                                 2 + div_pos + i - skip);
                        qp->div = isl_mat_drop_cols(qp->div,
                                                    2 + div_pos + i - skip, 1);
                        skip++;
                }
                reordering[div_pos + array[i].row] = div_pos + i - skip;
        }

        qp->upoly = reorder(qp->upoly, reordering);

        if (!qp->upoly || !qp->div)
                goto error;

        free(at);
        free(pos);
        free(array);
        free(reordering);

        return qp;
error:
        free(at);
        free(pos);
        free(array);
        free(reordering);
        isl_qpolynomial_free(qp);
        return NULL;
}

static __isl_give struct isl_upoly *expand(__isl_take struct isl_upoly *up,
        int *exp, int first)
{
        int i;
        struct isl_upoly_rec *rec;

        if (isl_upoly_is_cst(up))
                return up;

        if (up->var < first)
                return up;

        if (exp[up->var - first] == up->var - first)
                return up;

        up = isl_upoly_cow(up);
        if (!up)
                goto error;

        up->var = exp[up->var - first] + first;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                rec->p[i] = expand(rec->p[i], exp, first);
                if (!rec->p[i])
                        goto error;
        }

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

static __isl_give isl_qpolynomial *with_merged_divs(
        __isl_give isl_qpolynomial *(*fn)(__isl_take isl_qpolynomial *qp1,
                                          __isl_take isl_qpolynomial *qp2),
        __isl_take isl_qpolynomial *qp1, __isl_take isl_qpolynomial *qp2)
{
        int *exp1 = NULL;
        int *exp2 = NULL;
        isl_mat *div = NULL;
        int n_div1, n_div2;

        qp1 = isl_qpolynomial_cow(qp1);
        qp2 = isl_qpolynomial_cow(qp2);

        if (!qp1 || !qp2)
                goto error;

        isl_assert(qp1->div->ctx, qp1->div->n_row >= qp2->div->n_row &&
                                qp1->div->n_col >= qp2->div->n_col, goto error);

        n_div1 = qp1->div->n_row;
        n_div2 = qp2->div->n_row;
        exp1 = isl_alloc_array(qp1->div->ctx, int, n_div1);
        exp2 = isl_alloc_array(qp2->div->ctx, int, n_div2);
        if ((n_div1 && !exp1) || (n_div2 && !exp2))
                goto error;

        div = isl_merge_divs(qp1->div, qp2->div, exp1, exp2);
        if (!div)
                goto error;

        isl_mat_free(qp1->div);
        qp1->div = isl_mat_copy(div);
        isl_mat_free(qp2->div);
        qp2->div = isl_mat_copy(div);

        qp1->upoly = expand(qp1->upoly, exp1, div->n_col - div->n_row - 2);
        qp2->upoly = expand(qp2->upoly, exp2, div->n_col - div->n_row - 2);

        if (!qp1->upoly || !qp2->upoly)
                goto error;

        isl_mat_free(div);
        free(exp1);
        free(exp2);

        return fn(qp1, qp2);
error:
        isl_mat_free(div);
        free(exp1);
        free(exp2);
        isl_qpolynomial_free(qp1);
        isl_qpolynomial_free(qp2);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_add(__isl_take isl_qpolynomial *qp1,
        __isl_take isl_qpolynomial *qp2)
{
        isl_bool compatible;

        qp1 = isl_qpolynomial_cow(qp1);

        if (!qp1 || !qp2)
                goto error;

        if (qp1->div->n_row < qp2->div->n_row)
                return isl_qpolynomial_add(qp2, qp1);

        isl_assert(qp1->dim->ctx, isl_space_is_equal(qp1->dim, qp2->dim), goto error);
        compatible = compatible_divs(qp1->div, qp2->div);
        if (compatible < 0)
                goto error;
        if (!compatible)
                return with_merged_divs(isl_qpolynomial_add, qp1, qp2);

        qp1->upoly = isl_upoly_sum(qp1->upoly, isl_upoly_copy(qp2->upoly));
        if (!qp1->upoly)
                goto error;

        isl_qpolynomial_free(qp2);

        return qp1;
error:
        isl_qpolynomial_free(qp1);
        isl_qpolynomial_free(qp2);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_add_on_domain(
        __isl_keep isl_set *dom,
        __isl_take isl_qpolynomial *qp1,
        __isl_take isl_qpolynomial *qp2)
{
        qp1 = isl_qpolynomial_add(qp1, qp2);
        qp1 = isl_qpolynomial_gist(qp1, isl_set_copy(dom));
        return qp1;
}

__isl_give isl_qpolynomial *isl_qpolynomial_sub(__isl_take isl_qpolynomial *qp1,
        __isl_take isl_qpolynomial *qp2)
{
        return isl_qpolynomial_add(qp1, isl_qpolynomial_neg(qp2));
}

__isl_give isl_qpolynomial *isl_qpolynomial_add_isl_int(
        __isl_take isl_qpolynomial *qp, isl_int v)
{
        if (isl_int_is_zero(v))
                return qp;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        qp->upoly = isl_upoly_add_isl_int(qp->upoly, v);
        if (!qp->upoly)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;

}

__isl_give isl_qpolynomial *isl_qpolynomial_neg(__isl_take isl_qpolynomial *qp)
{
        if (!qp)
                return NULL;

        return isl_qpolynomial_mul_isl_int(qp, qp->dim->ctx->negone);
}

__isl_give isl_qpolynomial *isl_qpolynomial_mul_isl_int(
        __isl_take isl_qpolynomial *qp, isl_int v)
{
        if (isl_int_is_one(v))
                return qp;

        if (qp && isl_int_is_zero(v)) {
                isl_qpolynomial *zero;
                zero = isl_qpolynomial_zero_on_domain(isl_space_copy(qp->dim));
                isl_qpolynomial_free(qp);
                return zero;
        }
        
        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        qp->upoly = isl_upoly_mul_isl_int(qp->upoly, v);
        if (!qp->upoly)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_scale(
        __isl_take isl_qpolynomial *qp, isl_int v)
{
        return isl_qpolynomial_mul_isl_int(qp, v);
}

/* Multiply "qp" by "v".
 */
__isl_give isl_qpolynomial *isl_qpolynomial_scale_val(
        __isl_take isl_qpolynomial *qp, __isl_take isl_val *v)
{
        if (!qp || !v)
                goto error;

        if (!isl_val_is_rat(v))
                isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
                        "expecting rational factor", goto error);

        if (isl_val_is_one(v)) {
                isl_val_free(v);
                return qp;
        }

        if (isl_val_is_zero(v)) {
                isl_space *space;

                space = isl_qpolynomial_get_domain_space(qp);
                isl_qpolynomial_free(qp);
                isl_val_free(v);
                return isl_qpolynomial_zero_on_domain(space);
        }

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                goto error;

        qp->upoly = isl_upoly_scale_val(qp->upoly, v);
        if (!qp->upoly)
                qp = isl_qpolynomial_free(qp);

        isl_val_free(v);
        return qp;
error:
        isl_val_free(v);
        isl_qpolynomial_free(qp);
        return NULL;
}

/* Divide "qp" by "v".
 */
__isl_give isl_qpolynomial *isl_qpolynomial_scale_down_val(
        __isl_take isl_qpolynomial *qp, __isl_take isl_val *v)
{
        if (!qp || !v)
                goto error;

        if (!isl_val_is_rat(v))
                isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
                        "expecting rational factor", goto error);
        if (isl_val_is_zero(v))
                isl_die(isl_val_get_ctx(v), isl_error_invalid,
                        "cannot scale down by zero", goto error);

        return isl_qpolynomial_scale_val(qp, isl_val_inv(v));
error:
        isl_val_free(v);
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_mul(__isl_take isl_qpolynomial *qp1,
        __isl_take isl_qpolynomial *qp2)
{
        isl_bool compatible;

        qp1 = isl_qpolynomial_cow(qp1);

        if (!qp1 || !qp2)
                goto error;

        if (qp1->div->n_row < qp2->div->n_row)
                return isl_qpolynomial_mul(qp2, qp1);

        isl_assert(qp1->dim->ctx, isl_space_is_equal(qp1->dim, qp2->dim), goto error);
        compatible = compatible_divs(qp1->div, qp2->div);
        if (compatible < 0)
                goto error;
        if (!compatible)
                return with_merged_divs(isl_qpolynomial_mul, qp1, qp2);

        qp1->upoly = isl_upoly_mul(qp1->upoly, isl_upoly_copy(qp2->upoly));
        if (!qp1->upoly)
                goto error;

        isl_qpolynomial_free(qp2);

        return qp1;
error:
        isl_qpolynomial_free(qp1);
        isl_qpolynomial_free(qp2);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_pow(__isl_take isl_qpolynomial *qp,
        unsigned power)
{
        qp = isl_qpolynomial_cow(qp);

        if (!qp)
                return NULL;

        qp->upoly = isl_upoly_pow(qp->upoly, power);
        if (!qp->upoly)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_pow(
        __isl_take isl_pw_qpolynomial *pwqp, unsigned power)
{
        int i;

        if (power == 1)
                return pwqp;

        pwqp = isl_pw_qpolynomial_cow(pwqp);
        if (!pwqp)
                return NULL;

        for (i = 0; i < pwqp->n; ++i) {
                pwqp->p[i].qp = isl_qpolynomial_pow(pwqp->p[i].qp, power);
                if (!pwqp->p[i].qp)
                        return isl_pw_qpolynomial_free(pwqp);
        }

        return pwqp;
}

__isl_give isl_qpolynomial *isl_qpolynomial_zero_on_domain(
        __isl_take isl_space *domain)
{
        if (!domain)
                return NULL;
        return isl_qpolynomial_alloc(domain, 0, isl_upoly_zero(domain->ctx));
}

__isl_give isl_qpolynomial *isl_qpolynomial_one_on_domain(
        __isl_take isl_space *domain)
{
        if (!domain)
                return NULL;
        return isl_qpolynomial_alloc(domain, 0, isl_upoly_one(domain->ctx));
}

__isl_give isl_qpolynomial *isl_qpolynomial_infty_on_domain(
        __isl_take isl_space *domain)
{
        if (!domain)
                return NULL;
        return isl_qpolynomial_alloc(domain, 0, isl_upoly_infty(domain->ctx));
}

__isl_give isl_qpolynomial *isl_qpolynomial_neginfty_on_domain(
        __isl_take isl_space *domain)
{
        if (!domain)
                return NULL;
        return isl_qpolynomial_alloc(domain, 0,
                                        isl_upoly_neginfty(domain->ctx));
}

__isl_give isl_qpolynomial *isl_qpolynomial_nan_on_domain(
        __isl_take isl_space *domain)
{
        if (!domain)
                return NULL;
        return isl_qpolynomial_alloc(domain, 0, isl_upoly_nan(domain->ctx));
}

__isl_give isl_qpolynomial *isl_qpolynomial_cst_on_domain(
        __isl_take isl_space *domain,
        isl_int v)
{
        struct isl_qpolynomial *qp;
        struct isl_upoly_cst *cst;

        if (!domain)
                return NULL;

        qp = isl_qpolynomial_alloc(domain, 0, isl_upoly_zero(domain->ctx));
        if (!qp)
                return NULL;

        cst = isl_upoly_as_cst(qp->upoly);
        isl_int_set(cst->n, v);

        return qp;
}

int isl_qpolynomial_is_cst(__isl_keep isl_qpolynomial *qp,
        isl_int *n, isl_int *d)
{
        struct isl_upoly_cst *cst;

        if (!qp)
                return -1;

        if (!isl_upoly_is_cst(qp->upoly))
                return 0;

        cst = isl_upoly_as_cst(qp->upoly);
        if (!cst)
                return -1;

        if (n)
                isl_int_set(*n, cst->n);
        if (d)
                isl_int_set(*d, cst->d);

        return 1;
}

/* Return the constant term of "up".
 */
static __isl_give isl_val *isl_upoly_get_constant_val(
        __isl_keep struct isl_upoly *up)
{
        struct isl_upoly_cst *cst;

        if (!up)
                return NULL;

        while (!isl_upoly_is_cst(up)) {
                struct isl_upoly_rec *rec;

                rec = isl_upoly_as_rec(up);
                if (!rec)
                        return NULL;
                up = rec->p[0];
        }

        cst = isl_upoly_as_cst(up);
        if (!cst)
                return NULL;
        return isl_val_rat_from_isl_int(cst->up.ctx, cst->n, cst->d);
}

/* Return the constant term of "qp".
 */
__isl_give isl_val *isl_qpolynomial_get_constant_val(
        __isl_keep isl_qpolynomial *qp)
{
        if (!qp)
                return NULL;

        return isl_upoly_get_constant_val(qp->upoly);
}

int isl_upoly_is_affine(__isl_keep struct isl_upoly *up)
{
        int is_cst;
        struct isl_upoly_rec *rec;

        if (!up)
                return -1;

        if (up->var < 0)
                return 1;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                return -1;

        if (rec->n > 2)
                return 0;

        isl_assert(up->ctx, rec->n > 1, return -1);

        is_cst = isl_upoly_is_cst(rec->p[1]);
        if (is_cst < 0)
                return -1;
        if (!is_cst)
                return 0;

        return isl_upoly_is_affine(rec->p[0]);
}

int isl_qpolynomial_is_affine(__isl_keep isl_qpolynomial *qp)
{
        if (!qp)
                return -1;

        if (qp->div->n_row > 0)
                return 0;

        return isl_upoly_is_affine(qp->upoly);
}

static void update_coeff(__isl_keep isl_vec *aff,
        __isl_keep struct isl_upoly_cst *cst, int pos)
{
        isl_int gcd;
        isl_int f;

        if (isl_int_is_zero(cst->n))
                return;

        isl_int_init(gcd);
        isl_int_init(f);
        isl_int_gcd(gcd, cst->d, aff->el[0]);
        isl_int_divexact(f, cst->d, gcd);
        isl_int_divexact(gcd, aff->el[0], gcd);
        isl_seq_scale(aff->el, aff->el, f, aff->size);
        isl_int_mul(aff->el[1 + pos], gcd, cst->n);
        isl_int_clear(gcd);
        isl_int_clear(f);
}

int isl_upoly_update_affine(__isl_keep struct isl_upoly *up,
        __isl_keep isl_vec *aff)
{
        struct isl_upoly_cst *cst;
        struct isl_upoly_rec *rec;

        if (!up || !aff)
                return -1;

        if (up->var < 0) {
                struct isl_upoly_cst *cst;

                cst = isl_upoly_as_cst(up);
                if (!cst)
                        return -1;
                update_coeff(aff, cst, 0);
                return 0;
        }

        rec = isl_upoly_as_rec(up);
        if (!rec)
                return -1;
        isl_assert(up->ctx, rec->n == 2, return -1);

        cst = isl_upoly_as_cst(rec->p[1]);
        if (!cst)
                return -1;
        update_coeff(aff, cst, 1 + up->var);

        return isl_upoly_update_affine(rec->p[0], aff);
}

__isl_give isl_vec *isl_qpolynomial_extract_affine(
        __isl_keep isl_qpolynomial *qp)
{
        isl_vec *aff;
        unsigned d;

        if (!qp)
                return NULL;

        d = isl_space_dim(qp->dim, isl_dim_all);
        aff = isl_vec_alloc(qp->div->ctx, 2 + d + qp->div->n_row);
        if (!aff)
                return NULL;

        isl_seq_clr(aff->el + 1, 1 + d + qp->div->n_row);
        isl_int_set_si(aff->el[0], 1);

        if (isl_upoly_update_affine(qp->upoly, aff) < 0)
                goto error;

        return aff;
error:
        isl_vec_free(aff);
        return NULL;
}

/* Compare two quasi-polynomials.
 *
 * Return -1 if "qp1" is "smaller" than "qp2", 1 if "qp1" is "greater"
 * than "qp2" and 0 if they are equal.
 */
int isl_qpolynomial_plain_cmp(__isl_keep isl_qpolynomial *qp1,
        __isl_keep isl_qpolynomial *qp2)
{
        int cmp;

        if (qp1 == qp2)
                return 0;
        if (!qp1)
                return -1;
        if (!qp2)
                return 1;

        cmp = isl_space_cmp(qp1->dim, qp2->dim);
        if (cmp != 0)
                return cmp;

        cmp = isl_local_cmp(qp1->div, qp2->div);
        if (cmp != 0)
                return cmp;

        return isl_upoly_plain_cmp(qp1->upoly, qp2->upoly);
}

/* Is "qp1" obviously equal to "qp2"?
 *
 * NaN is not equal to anything, not even to another NaN.
 */
isl_bool isl_qpolynomial_plain_is_equal(__isl_keep isl_qpolynomial *qp1,
        __isl_keep isl_qpolynomial *qp2)
{
        isl_bool equal;

        if (!qp1 || !qp2)
                return isl_bool_error;

        if (isl_qpolynomial_is_nan(qp1) || isl_qpolynomial_is_nan(qp2))
                return isl_bool_false;

        equal = isl_space_is_equal(qp1->dim, qp2->dim);
        if (equal < 0 || !equal)
                return equal;

        equal = isl_mat_is_equal(qp1->div, qp2->div);
        if (equal < 0 || !equal)
                return equal;

        return isl_upoly_is_equal(qp1->upoly, qp2->upoly);
}

static void upoly_update_den(__isl_keep struct isl_upoly *up, isl_int *d)
{
        int i;
        struct isl_upoly_rec *rec;

        if (isl_upoly_is_cst(up)) {
                struct isl_upoly_cst *cst;
                cst = isl_upoly_as_cst(up);
                if (!cst)
                        return;
                isl_int_lcm(*d, *d, cst->d);
                return;
        }

        rec = isl_upoly_as_rec(up);
        if (!rec)
                return;

        for (i = 0; i < rec->n; ++i)
                upoly_update_den(rec->p[i], d);
}

void isl_qpolynomial_get_den(__isl_keep isl_qpolynomial *qp, isl_int *d)
{
        isl_int_set_si(*d, 1);
        if (!qp)
                return;
        upoly_update_den(qp->upoly, d);
}

__isl_give isl_qpolynomial *isl_qpolynomial_var_pow_on_domain(
        __isl_take isl_space *domain, int pos, int power)
{
        struct isl_ctx *ctx;

        if (!domain)
                return NULL;

        ctx = domain->ctx;

        return isl_qpolynomial_alloc(domain, 0,
                                        isl_upoly_var_pow(ctx, pos, power));
}

__isl_give isl_qpolynomial *isl_qpolynomial_var_on_domain(
        __isl_take isl_space *domain, enum isl_dim_type type, unsigned pos)
{
        if (isl_space_check_is_set(domain ) < 0)
                goto error;
        isl_assert(domain->ctx, pos < isl_space_dim(domain, type), goto error);

        if (type == isl_dim_set)
                pos += isl_space_dim(domain, isl_dim_param);

        return isl_qpolynomial_var_pow_on_domain(domain, pos, 1);
error:
        isl_space_free(domain);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_subs(__isl_take struct isl_upoly *up,
        unsigned first, unsigned n, __isl_keep struct isl_upoly **subs)
{
        int i;
        struct isl_upoly_rec *rec;
        struct isl_upoly *base, *res;

        if (!up)
                return NULL;

        if (isl_upoly_is_cst(up))
                return up;

        if (up->var < first)
                return up;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        isl_assert(up->ctx, rec->n >= 1, goto error);

        if (up->var >= first + n)
                base = isl_upoly_var_pow(up->ctx, up->var, 1);
        else
                base = isl_upoly_copy(subs[up->var - first]);

        res = isl_upoly_subs(isl_upoly_copy(rec->p[rec->n - 1]), first, n, subs);
        for (i = rec->n - 2; i >= 0; --i) {
                struct isl_upoly *t;
                t = isl_upoly_subs(isl_upoly_copy(rec->p[i]), first, n, subs);
                res = isl_upoly_mul(res, isl_upoly_copy(base));
                res = isl_upoly_sum(res, t);
        }

        isl_upoly_free(base);
        isl_upoly_free(up);
                                
        return res;
error:
        isl_upoly_free(up);
        return NULL;
}       

__isl_give struct isl_upoly *isl_upoly_from_affine(isl_ctx *ctx, isl_int *f,
        isl_int denom, unsigned len)
{
        int i;
        struct isl_upoly *up;

        isl_assert(ctx, len >= 1, return NULL);

        up = isl_upoly_rat_cst(ctx, f[0], denom);
        for (i = 0; i < len - 1; ++i) {
                struct isl_upoly *t;
                struct isl_upoly *c;

                if (isl_int_is_zero(f[1 + i]))
                        continue;

                c = isl_upoly_rat_cst(ctx, f[1 + i], denom);
                t = isl_upoly_var_pow(ctx, i, 1);
                t = isl_upoly_mul(c, t);
                up = isl_upoly_sum(up, t);
        }

        return up;
}

/* Remove common factor of non-constant terms and denominator.
 */
static void normalize_div(__isl_keep isl_qpolynomial *qp, int div)
{
        isl_ctx *ctx = qp->div->ctx;
        unsigned total = qp->div->n_col - 2;

        isl_seq_gcd(qp->div->row[div] + 2, total, &ctx->normalize_gcd);
        isl_int_gcd(ctx->normalize_gcd,
                    ctx->normalize_gcd, qp->div->row[div][0]);
        if (isl_int_is_one(ctx->normalize_gcd))
                return;

        isl_seq_scale_down(qp->div->row[div] + 2, qp->div->row[div] + 2,
                            ctx->normalize_gcd, total);
        isl_int_divexact(qp->div->row[div][0], qp->div->row[div][0],
                            ctx->normalize_gcd);
        isl_int_fdiv_q(qp->div->row[div][1], qp->div->row[div][1],
                            ctx->normalize_gcd);
}

/* Replace the integer division identified by "div" by the polynomial "s".
 * The integer division is assumed not to appear in the definition
 * of any other integer divisions.
 */
static __isl_give isl_qpolynomial *substitute_div(
        __isl_take isl_qpolynomial *qp,
        int div, __isl_take struct isl_upoly *s)
{
        int i;
        int total;
        int *reordering;

        if (!qp || !s)
                goto error;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                goto error;

        total = isl_space_dim(qp->dim, isl_dim_all);
        qp->upoly = isl_upoly_subs(qp->upoly, total + div, 1, &s);
        if (!qp->upoly)
                goto error;

        reordering = isl_alloc_array(qp->dim->ctx, int, total + qp->div->n_row);
        if (!reordering)
                goto error;
        for (i = 0; i < total + div; ++i)
                reordering[i] = i;
        for (i = total + div + 1; i < total + qp->div->n_row; ++i)
                reordering[i] = i - 1;
        qp->div = isl_mat_drop_rows(qp->div, div, 1);
        qp->div = isl_mat_drop_cols(qp->div, 2 + total + div, 1);
        qp->upoly = reorder(qp->upoly, reordering);
        free(reordering);

        if (!qp->upoly || !qp->div)
                goto error;

        isl_upoly_free(s);
        return qp;
error:
        isl_qpolynomial_free(qp);
        isl_upoly_free(s);
        return NULL;
}

/* Replace all integer divisions [e/d] that turn out to not actually be integer
 * divisions because d is equal to 1 by their definition, i.e., e.
 */
static __isl_give isl_qpolynomial *substitute_non_divs(
        __isl_take isl_qpolynomial *qp)
{
        int i, j;
        int total;
        struct isl_upoly *s;

        if (!qp)
                return NULL;

        total = isl_space_dim(qp->dim, isl_dim_all);
        for (i = 0; qp && i < qp->div->n_row; ++i) {
                if (!isl_int_is_one(qp->div->row[i][0]))
                        continue;
                for (j = i + 1; j < qp->div->n_row; ++j) {
                        if (isl_int_is_zero(qp->div->row[j][2 + total + i]))
                                continue;
                        isl_seq_combine(qp->div->row[j] + 1,
                                qp->div->ctx->one, qp->div->row[j] + 1,
                                qp->div->row[j][2 + total + i],
                                qp->div->row[i] + 1, 1 + total + i);
                        isl_int_set_si(qp->div->row[j][2 + total + i], 0);
                        normalize_div(qp, j);
                }
                s = isl_upoly_from_affine(qp->dim->ctx, qp->div->row[i] + 1,
                                        qp->div->row[i][0], qp->div->n_col - 1);
                qp = substitute_div(qp, i, s);
                --i;
        }

        return qp;
}

/* Reduce the coefficients of div "div" to lie in the interval [0, d-1],
 * with d the denominator.  When replacing the coefficient e of x by
 * d * frac(e/d) = e - d * floor(e/d), we are subtracting d * floor(e/d) * x
 * inside the division, so we need to add floor(e/d) * x outside.
 * That is, we replace q by q' + floor(e/d) * x and we therefore need
 * to adjust the coefficient of x in each later div that depends on the
 * current div "div" and also in the affine expressions in the rows of "mat"
 * (if they too depend on "div").
 */
static void reduce_div(__isl_keep isl_qpolynomial *qp, int div,
        __isl_keep isl_mat **mat)
{
        int i, j;
        isl_int v;
        unsigned total = qp->div->n_col - qp->div->n_row - 2;

        isl_int_init(v);
        for (i = 0; i < 1 + total + div; ++i) {
                if (isl_int_is_nonneg(qp->div->row[div][1 + i]) &&
                    isl_int_lt(qp->div->row[div][1 + i], qp->div->row[div][0]))
                        continue;
                isl_int_fdiv_q(v, qp->div->row[div][1 + i], qp->div->row[div][0]);
                isl_int_fdiv_r(qp->div->row[div][1 + i],
                                qp->div->row[div][1 + i], qp->div->row[div][0]);
                *mat = isl_mat_col_addmul(*mat, i, v, 1 + total + div);
                for (j = div + 1; j < qp->div->n_row; ++j) {
                        if (isl_int_is_zero(qp->div->row[j][2 + total + div]))
                                continue;
                        isl_int_addmul(qp->div->row[j][1 + i],
                                        v, qp->div->row[j][2 + total + div]);
                }
        }
        isl_int_clear(v);
}

/* Check if the last non-zero coefficient is bigger that half of the
 * denominator.  If so, we will invert the div to further reduce the number
 * of distinct divs that may appear.
 * If the last non-zero coefficient is exactly half the denominator,
 * then we continue looking for earlier coefficients that are bigger
 * than half the denominator.
 */
static int needs_invert(__isl_keep isl_mat *div, int row)
{
        int i;
        int cmp;

        for (i = div->n_col - 1; i >= 1; --i) {
                if (isl_int_is_zero(div->row[row][i]))
                        continue;
                isl_int_mul_ui(div->row[row][i], div->row[row][i], 2);
                cmp = isl_int_cmp(div->row[row][i], div->row[row][0]);
                isl_int_divexact_ui(div->row[row][i], div->row[row][i], 2);
                if (cmp)
                        return cmp > 0;
                if (i == 1)
                        return 1;
        }

        return 0;
}

/* Replace div "div" q = [e/d] by -[(-e+(d-1))/d].
 * We only invert the coefficients of e (and the coefficient of q in
 * later divs and in the rows of "mat").  After calling this function, the
 * coefficients of e should be reduced again.
 */
static void invert_div(__isl_keep isl_qpolynomial *qp, int div,
        __isl_keep isl_mat **mat)
{
        unsigned total = qp->div->n_col - qp->div->n_row - 2;

        isl_seq_neg(qp->div->row[div] + 1,
                    qp->div->row[div] + 1, qp->div->n_col - 1);
        isl_int_sub_ui(qp->div->row[div][1], qp->div->row[div][1], 1);
        isl_int_add(qp->div->row[div][1],
                    qp->div->row[div][1], qp->div->row[div][0]);
        *mat = isl_mat_col_neg(*mat, 1 + total + div);
        isl_mat_col_mul(qp->div, 2 + total + div,
                        qp->div->ctx->negone, 2 + total + div);
}

/* Reduce all divs of "qp" to have coefficients
 * in the interval [0, d-1], with d the denominator and such that the
 * last non-zero coefficient that is not equal to d/2 is smaller than d/2.
 * The modifications to the integer divisions need to be reflected
 * in the factors of the polynomial that refer to the original
 * integer divisions.  To this end, the modifications are collected
 * as a set of affine expressions and then plugged into the polynomial.
 *
 * After the reduction, some divs may have become redundant or identical,
 * so we call substitute_non_divs and sort_divs.  If these functions
 * eliminate divs or merge two or more divs into one, the coefficients
 * of the enclosing divs may have to be reduced again, so we call
 * ourselves recursively if the number of divs decreases.
 */
static __isl_give isl_qpolynomial *reduce_divs(__isl_take isl_qpolynomial *qp)
{
        int i;
        isl_ctx *ctx;
        isl_mat *mat;
        struct isl_upoly **s;
        unsigned o_div, n_div, total;

        if (!qp)
                return NULL;

        total = isl_qpolynomial_domain_dim(qp, isl_dim_all);
        n_div = isl_qpolynomial_domain_dim(qp, isl_dim_div);
        o_div = isl_qpolynomial_domain_offset(qp, isl_dim_div);
        ctx = isl_qpolynomial_get_ctx(qp);
        mat = isl_mat_zero(ctx, n_div, 1 + total);

        for (i = 0; i < n_div; ++i)
                mat = isl_mat_set_element_si(mat, i, o_div + i, 1);

        for (i = 0; i < qp->div->n_row; ++i) {
                normalize_div(qp, i);
                reduce_div(qp, i, &mat);
                if (needs_invert(qp->div, i)) {
                        invert_div(qp, i, &mat);
                        reduce_div(qp, i, &mat);
                }
        }
        if (!mat)
                goto error;

        s = isl_alloc_array(ctx, struct isl_upoly *, n_div);
        if (n_div && !s)
                goto error;
        for (i = 0; i < n_div; ++i)
                s[i] = isl_upoly_from_affine(ctx, mat->row[i], ctx->one,
                                            1 + total);
        qp->upoly = isl_upoly_subs(qp->upoly, o_div - 1, n_div, s);
        for (i = 0; i < n_div; ++i)
                isl_upoly_free(s[i]);
        free(s);
        if (!qp->upoly)
                goto error;

        isl_mat_free(mat);

        qp = substitute_non_divs(qp);
        qp = sort_divs(qp);
        if (qp && isl_qpolynomial_domain_dim(qp, isl_dim_div) < n_div)
                return reduce_divs(qp);

        return qp;
error:
        isl_qpolynomial_free(qp);
        isl_mat_free(mat);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_rat_cst_on_domain(
        __isl_take isl_space *domain, const isl_int n, const isl_int d)
{
        struct isl_qpolynomial *qp;
        struct isl_upoly_cst *cst;

        if (!domain)
                return NULL;

        qp = isl_qpolynomial_alloc(domain, 0, isl_upoly_zero(domain->ctx));
        if (!qp)
                return NULL;

        cst = isl_upoly_as_cst(qp->upoly);
        isl_int_set(cst->n, n);
        isl_int_set(cst->d, d);

        return qp;
}

/* Return an isl_qpolynomial that is equal to "val" on domain space "domain".
 */
__isl_give isl_qpolynomial *isl_qpolynomial_val_on_domain(
        __isl_take isl_space *domain, __isl_take isl_val *val)
{
        isl_qpolynomial *qp;
        struct isl_upoly_cst *cst;

        if (!domain || !val)
                goto error;

        qp = isl_qpolynomial_alloc(isl_space_copy(domain), 0,
                                        isl_upoly_zero(domain->ctx));
        if (!qp)
                goto error;

        cst = isl_upoly_as_cst(qp->upoly);
        isl_int_set(cst->n, val->n);
        isl_int_set(cst->d, val->d);

        isl_space_free(domain);
        isl_val_free(val);
        return qp;
error:
        isl_space_free(domain);
        isl_val_free(val);
        return NULL;
}

static int up_set_active(__isl_keep struct isl_upoly *up, int *active, int d)
{
        struct isl_upoly_rec *rec;
        int i;

        if (!up)
                return -1;

        if (isl_upoly_is_cst(up))
                return 0;

        if (up->var < d)
                active[up->var] = 1;

        rec = isl_upoly_as_rec(up);
        for (i = 0; i < rec->n; ++i)
                if (up_set_active(rec->p[i], active, d) < 0)
                        return -1;

        return 0;
}

static int set_active(__isl_keep isl_qpolynomial *qp, int *active)
{
        int i, j;
        int d = isl_space_dim(qp->dim, isl_dim_all);

        if (!qp || !active)
                return -1;

        for (i = 0; i < d; ++i)
                for (j = 0; j < qp->div->n_row; ++j) {
                        if (isl_int_is_zero(qp->div->row[j][2 + i]))
                                continue;
                        active[i] = 1;
                        break;
                }

        return up_set_active(qp->upoly, active, d);
}

isl_bool isl_qpolynomial_involves_dims(__isl_keep isl_qpolynomial *qp,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        int i;
        int *active = NULL;
        isl_bool involves = isl_bool_false;

        if (!qp)
                return isl_bool_error;
        if (n == 0)
                return isl_bool_false;

        isl_assert(qp->dim->ctx,
                    first + n <= isl_qpolynomial_dim(qp, type),
                    return isl_bool_error);
        isl_assert(qp->dim->ctx, type == isl_dim_param ||
                                 type == isl_dim_in, return isl_bool_error);

        active = isl_calloc_array(qp->dim->ctx, int,
                                        isl_space_dim(qp->dim, isl_dim_all));
        if (set_active(qp, active) < 0)
                goto error;

        if (type == isl_dim_in)
                first += isl_space_dim(qp->dim, isl_dim_param);
        for (i = 0; i < n; ++i)
                if (active[first + i]) {
                        involves = isl_bool_true;
                        break;
                }

        free(active);

        return involves;
error:
        free(active);
        return isl_bool_error;
}

/* Remove divs that do not appear in the quasi-polynomial, nor in any
 * of the divs that do appear in the quasi-polynomial.
 */
static __isl_give isl_qpolynomial *remove_redundant_divs(
        __isl_take isl_qpolynomial *qp)
{
        int i, j;
        int d;
        int len;
        int skip;
        int *active = NULL;
        int *reordering = NULL;
        int redundant = 0;
        int n_div;
        isl_ctx *ctx;

        if (!qp)
                return NULL;
        if (qp->div->n_row == 0)
                return qp;

        d = isl_space_dim(qp->dim, isl_dim_all);
        len = qp->div->n_col - 2;
        ctx = isl_qpolynomial_get_ctx(qp);
        active = isl_calloc_array(ctx, int, len);
        if (!active)
                goto error;

        if (up_set_active(qp->upoly, active, len) < 0)
                goto error;

        for (i = qp->div->n_row - 1; i >= 0; --i) {
                if (!active[d + i]) {
                        redundant = 1;
                        continue;
                }
                for (j = 0; j < i; ++j) {
                        if (isl_int_is_zero(qp->div->row[i][2 + d + j]))
                                continue;
                        active[d + j] = 1;
                        break;
                }
        }

        if (!redundant) {
                free(active);
                return qp;
        }

        reordering = isl_alloc_array(qp->div->ctx, int, len);
        if (!reordering)
                goto error;

        for (i = 0; i < d; ++i)
                reordering[i] = i;

        skip = 0;
        n_div = qp->div->n_row;
        for (i = 0; i < n_div; ++i) {
                if (!active[d + i]) {
                        qp->div = isl_mat_drop_rows(qp->div, i - skip, 1);
                        qp->div = isl_mat_drop_cols(qp->div,
                                                    2 + d + i - skip, 1);
                        skip++;
                }
                reordering[d + i] = d + i - skip;
        }

        qp->upoly = reorder(qp->upoly, reordering);

        if (!qp->upoly || !qp->div)
                goto error;

        free(active);
        free(reordering);

        return qp;
error:
        free(active);
        free(reordering);
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give struct isl_upoly *isl_upoly_drop(__isl_take struct isl_upoly *up,
        unsigned first, unsigned n)
{
        int i;
        struct isl_upoly_rec *rec;

        if (!up)
                return NULL;
        if (n == 0 || up->var < 0 || up->var < first)
                return up;
        if (up->var < first + n) {
                up = replace_by_constant_term(up);
                return isl_upoly_drop(up, first, n);
        }
        up = isl_upoly_cow(up);
        if (!up)
                return NULL;
        up->var -= n;
        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                rec->p[i] = isl_upoly_drop(rec->p[i], first, n);
                if (!rec->p[i])
                        goto error;
        }

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
        __isl_take isl_qpolynomial *qp,
        enum isl_dim_type type, unsigned pos, const char *s)
{
        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;
        if (type == isl_dim_out)
                isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
                        "cannot set name of output/set dimension",
                        return isl_qpolynomial_free(qp));
        if (type == isl_dim_in)
                type = isl_dim_set;
        qp->dim = isl_space_set_dim_name(qp->dim, type, pos, s);
        if (!qp->dim)
                goto error;
        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_drop_dims(
        __isl_take isl_qpolynomial *qp,
        enum isl_dim_type type, unsigned first, unsigned n)
{
        if (!qp)
                return NULL;
        if (type == isl_dim_out)
                isl_die(qp->dim->ctx, isl_error_invalid,
                        "cannot drop output/set dimension",
                        goto error);
        if (type == isl_dim_in)
                type = isl_dim_set;
        if (n == 0 && !isl_space_is_named_or_nested(qp->dim, type))
                return qp;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        isl_assert(qp->dim->ctx, first + n <= isl_space_dim(qp->dim, type),
                        goto error);
        isl_assert(qp->dim->ctx, type == isl_dim_param ||
                                 type == isl_dim_set, goto error);

        qp->dim = isl_space_drop_dims(qp->dim, type, first, n);
        if (!qp->dim)
                goto error;

        if (type == isl_dim_set)
                first += isl_space_dim(qp->dim, isl_dim_param);

        qp->div = isl_mat_drop_cols(qp->div, 2 + first, n);
        if (!qp->div)
                goto error;

        qp->upoly = isl_upoly_drop(qp->upoly, first, n);
        if (!qp->upoly)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

/* Project the domain of the quasi-polynomial onto its parameter space.
 * The quasi-polynomial may not involve any of the domain dimensions.
 */
__isl_give isl_qpolynomial *isl_qpolynomial_project_domain_on_params(
        __isl_take isl_qpolynomial *qp)
{
        isl_space *space;
        unsigned n;
        isl_bool involves;

        n = isl_qpolynomial_dim(qp, isl_dim_in);
        involves = isl_qpolynomial_involves_dims(qp, isl_dim_in, 0, n);
        if (involves < 0)
                return isl_qpolynomial_free(qp);
        if (involves)
                isl_die(isl_qpolynomial_get_ctx(qp), isl_error_invalid,
                        "polynomial involves some of the domain dimensions",
                        return isl_qpolynomial_free(qp));
        qp = isl_qpolynomial_drop_dims(qp, isl_dim_in, 0, n);
        space = isl_qpolynomial_get_domain_space(qp);
        space = isl_space_params(space);
        qp = isl_qpolynomial_reset_domain_space(qp, space);
        return qp;
}

static __isl_give isl_qpolynomial *isl_qpolynomial_substitute_equalities_lifted(
        __isl_take isl_qpolynomial *qp, __isl_take isl_basic_set *eq)
{
        int i, j, k;
        isl_int denom;
        unsigned total;
        unsigned n_div;
        struct isl_upoly *up;

        if (!eq)
                goto error;
        if (eq->n_eq == 0) {
                isl_basic_set_free(eq);
                return qp;
        }

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                goto error;
        qp->div = isl_mat_cow(qp->div);
        if (!qp->div)
                goto error;

        total = 1 + isl_space_dim(eq->dim, isl_dim_all);
        n_div = eq->n_div;
        isl_int_init(denom);
        for (i = 0; i < eq->n_eq; ++i) {
                j = isl_seq_last_non_zero(eq->eq[i], total + n_div);
                if (j < 0 || j == 0 || j >= total)
                        continue;

                for (k = 0; k < qp->div->n_row; ++k) {
                        if (isl_int_is_zero(qp->div->row[k][1 + j]))
                                continue;
                        isl_seq_elim(qp->div->row[k] + 1, eq->eq[i], j, total,
                                        &qp->div->row[k][0]);
                        normalize_div(qp, k);
                }

                if (isl_int_is_pos(eq->eq[i][j]))
                        isl_seq_neg(eq->eq[i], eq->eq[i], total);
                isl_int_abs(denom, eq->eq[i][j]);
                isl_int_set_si(eq->eq[i][j], 0);

                up = isl_upoly_from_affine(qp->dim->ctx,
                                                   eq->eq[i], denom, total);
                qp->upoly = isl_upoly_subs(qp->upoly, j - 1, 1, &up);
                isl_upoly_free(up);
        }
        isl_int_clear(denom);

        if (!qp->upoly)
                goto error;

        isl_basic_set_free(eq);

        qp = substitute_non_divs(qp);
        qp = sort_divs(qp);

        return qp;
error:
        isl_basic_set_free(eq);
        isl_qpolynomial_free(qp);
        return NULL;
}

/* Exploit the equalities in "eq" to simplify the quasi-polynomial.
 */
__isl_give isl_qpolynomial *isl_qpolynomial_substitute_equalities(
        __isl_take isl_qpolynomial *qp, __isl_take isl_basic_set *eq)
{
        if (!qp || !eq)
                goto error;
        if (qp->div->n_row > 0)
                eq = isl_basic_set_add_dims(eq, isl_dim_set, qp->div->n_row);
        return isl_qpolynomial_substitute_equalities_lifted(qp, eq);
error:
        isl_basic_set_free(eq);
        isl_qpolynomial_free(qp);
        return NULL;
}

/* Look for equalities among the variables shared by context and qp
 * and the integer divisions of qp, if any.
 * The equalities are then used to eliminate variables and/or integer
 * divisions from qp.
 */
__isl_give isl_qpolynomial *isl_qpolynomial_gist(
        __isl_take isl_qpolynomial *qp, __isl_take isl_set *context)
{
        isl_local_space *ls;
        isl_basic_set *aff;

        ls = isl_qpolynomial_get_domain_local_space(qp);
        context = isl_local_space_lift_set(ls, context);

        aff = isl_set_affine_hull(context);
        return isl_qpolynomial_substitute_equalities_lifted(qp, aff);
}

__isl_give isl_qpolynomial *isl_qpolynomial_gist_params(
        __isl_take isl_qpolynomial *qp, __isl_take isl_set *context)
{
        isl_space *space = isl_qpolynomial_get_domain_space(qp);
        isl_set *dom_context = isl_set_universe(space);
        dom_context = isl_set_intersect_params(dom_context, context);
        return isl_qpolynomial_gist(qp, dom_context);
}

__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_qpolynomial(
        __isl_take isl_qpolynomial *qp)
{
        isl_set *dom;

        if (!qp)
                return NULL;
        if (isl_qpolynomial_is_zero(qp)) {
                isl_space *dim = isl_qpolynomial_get_space(qp);
                isl_qpolynomial_free(qp);
                return isl_pw_qpolynomial_zero(dim);
        }

        dom = isl_set_universe(isl_qpolynomial_get_domain_space(qp));
        return isl_pw_qpolynomial_alloc(dom, qp);
}

#define isl_qpolynomial_involves_nan isl_qpolynomial_is_nan

#undef PW
#define PW isl_pw_qpolynomial
#undef EL
#define EL isl_qpolynomial
#undef EL_IS_ZERO
#define EL_IS_ZERO is_zero
#undef ZERO
#define ZERO zero
#undef IS_ZERO
#define IS_ZERO is_zero
#undef FIELD
#define FIELD qp
#undef DEFAULT_IS_ZERO
#define DEFAULT_IS_ZERO 1

#define NO_PULLBACK

#include <isl_pw_templ.c>
#include <isl_pw_eval.c>

#undef BASE
#define BASE pw_qpolynomial

#include <isl_union_single.c>
#include <isl_union_eval.c>
#include <isl_union_neg.c>

int isl_pw_qpolynomial_is_one(__isl_keep isl_pw_qpolynomial *pwqp)
{
        if (!pwqp)
                return -1;

        if (pwqp->n != -1)
                return 0;

        if (!isl_set_plain_is_universe(pwqp->p[0].set))
                return 0;

        return isl_qpolynomial_is_one(pwqp->p[0].qp);
}

__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
        __isl_take isl_pw_qpolynomial *pwqp1,
        __isl_take isl_pw_qpolynomial *pwqp2)
{
        return isl_pw_qpolynomial_union_add_(pwqp1, pwqp2);
}

__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
        __isl_take isl_pw_qpolynomial *pwqp1,
        __isl_take isl_pw_qpolynomial *pwqp2)
{
        int i, j, n;
        struct isl_pw_qpolynomial *res;

        if (!pwqp1 || !pwqp2)
                goto error;

        isl_assert(pwqp1->dim->ctx, isl_space_is_equal(pwqp1->dim, pwqp2->dim),
                        goto error);

        if (isl_pw_qpolynomial_is_zero(pwqp1)) {
                isl_pw_qpolynomial_free(pwqp2);
                return pwqp1;
        }

        if (isl_pw_qpolynomial_is_zero(pwqp2)) {
                isl_pw_qpolynomial_free(pwqp1);
                return pwqp2;
        }

        if (isl_pw_qpolynomial_is_one(pwqp1)) {
                isl_pw_qpolynomial_free(pwqp1);
                return pwqp2;
        }

        if (isl_pw_qpolynomial_is_one(pwqp2)) {
                isl_pw_qpolynomial_free(pwqp2);
                return pwqp1;
        }

        n = pwqp1->n * pwqp2->n;
        res = isl_pw_qpolynomial_alloc_size(isl_space_copy(pwqp1->dim), n);

        for (i = 0; i < pwqp1->n; ++i) {
                for (j = 0; j < pwqp2->n; ++j) {
                        struct isl_set *common;
                        struct isl_qpolynomial *prod;
                        common = isl_set_intersect(isl_set_copy(pwqp1->p[i].set),
                                                isl_set_copy(pwqp2->p[j].set));
                        if (isl_set_plain_is_empty(common)) {
                                isl_set_free(common);
                                continue;
                        }

                        prod = isl_qpolynomial_mul(
                                isl_qpolynomial_copy(pwqp1->p[i].qp),
                                isl_qpolynomial_copy(pwqp2->p[j].qp));

                        res = isl_pw_qpolynomial_add_piece(res, common, prod);
                }
        }

        isl_pw_qpolynomial_free(pwqp1);
        isl_pw_qpolynomial_free(pwqp2);

        return res;
error:
        isl_pw_qpolynomial_free(pwqp1);
        isl_pw_qpolynomial_free(pwqp2);
        return NULL;
}

__isl_give isl_val *isl_upoly_eval(__isl_take struct isl_upoly *up,
        __isl_take isl_vec *vec)
{
        int i;
        struct isl_upoly_rec *rec;
        isl_val *res;
        isl_val *base;

        if (isl_upoly_is_cst(up)) {
                isl_vec_free(vec);
                res = isl_upoly_get_constant_val(up);
                isl_upoly_free(up);
                return res;
        }

        rec = isl_upoly_as_rec(up);
        if (!rec || !vec)
                goto error;

        isl_assert(up->ctx, rec->n >= 1, goto error);

        base = isl_val_rat_from_isl_int(up->ctx,
                                        vec->el[1 + up->var], vec->el[0]);

        res = isl_upoly_eval(isl_upoly_copy(rec->p[rec->n - 1]),
                                isl_vec_copy(vec));

        for (i = rec->n - 2; i >= 0; --i) {
                res = isl_val_mul(res, isl_val_copy(base));
                res = isl_val_add(res,
                            isl_upoly_eval(isl_upoly_copy(rec->p[i]),
                                                            isl_vec_copy(vec)));
        }

        isl_val_free(base);
        isl_upoly_free(up);
        isl_vec_free(vec);
        return res;
error:
        isl_upoly_free(up);
        isl_vec_free(vec);
        return NULL;
}

/* Evaluate "qp" in the void point "pnt".
 * In particular, return the value NaN.
 */
static __isl_give isl_val *eval_void(__isl_take isl_qpolynomial *qp,
        __isl_take isl_point *pnt)
{
        isl_ctx *ctx;

        ctx = isl_point_get_ctx(pnt);
        isl_qpolynomial_free(qp);
        isl_point_free(pnt);
        return isl_val_nan(ctx);
}

__isl_give isl_val *isl_qpolynomial_eval(__isl_take isl_qpolynomial *qp,
        __isl_take isl_point *pnt)
{
        isl_bool is_void;
        isl_vec *ext;
        isl_val *v;

        if (!qp || !pnt)
                goto error;
        isl_assert(pnt->dim->ctx, isl_space_is_equal(pnt->dim, qp->dim), goto error);
        is_void = isl_point_is_void(pnt);
        if (is_void < 0)
                goto error;
        if (is_void)
                return eval_void(qp, pnt);

        ext = isl_local_extend_point_vec(qp->div, isl_vec_copy(pnt->vec));

        v = isl_upoly_eval(isl_upoly_copy(qp->upoly), ext);

        isl_qpolynomial_free(qp);
        isl_point_free(pnt);

        return v;
error:
        isl_qpolynomial_free(qp);
        isl_point_free(pnt);
        return NULL;
}

int isl_upoly_cmp(__isl_keep struct isl_upoly_cst *cst1,
        __isl_keep struct isl_upoly_cst *cst2)
{
        int cmp;
        isl_int t;
        isl_int_init(t);
        isl_int_mul(t, cst1->n, cst2->d);
        isl_int_submul(t, cst2->n, cst1->d);
        cmp = isl_int_sgn(t);
        isl_int_clear(t);
        return cmp;
}

__isl_give isl_qpolynomial *isl_qpolynomial_insert_dims(
        __isl_take isl_qpolynomial *qp, enum isl_dim_type type,
        unsigned first, unsigned n)
{
        unsigned total;
        unsigned g_pos;
        int *exp;

        if (!qp)
                return NULL;
        if (type == isl_dim_out)
                isl_die(qp->div->ctx, isl_error_invalid,
                        "cannot insert output/set dimensions",
                        goto error);
        if (type == isl_dim_in)
                type = isl_dim_set;
        if (n == 0 && !isl_space_is_named_or_nested(qp->dim, type))
                return qp;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        isl_assert(qp->div->ctx, first <= isl_space_dim(qp->dim, type),
                    goto error);

        g_pos = pos(qp->dim, type) + first;

        qp->div = isl_mat_insert_zero_cols(qp->div, 2 + g_pos, n);
        if (!qp->div)
                goto error;

        total = qp->div->n_col - 2;
        if (total > g_pos) {
                int i;
                exp = isl_alloc_array(qp->div->ctx, int, total - g_pos);
                if (!exp)
                        goto error;
                for (i = 0; i < total - g_pos; ++i)
                        exp[i] = i + n;
                qp->upoly = expand(qp->upoly, exp, g_pos);
                free(exp);
                if (!qp->upoly)
                        goto error;
        }

        qp->dim = isl_space_insert_dims(qp->dim, type, first, n);
        if (!qp->dim)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_add_dims(
        __isl_take isl_qpolynomial *qp, enum isl_dim_type type, unsigned n)
{
        unsigned pos;

        pos = isl_qpolynomial_dim(qp, type);

        return isl_qpolynomial_insert_dims(qp, type, pos, n);
}

__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_dims(
        __isl_take isl_pw_qpolynomial *pwqp,
        enum isl_dim_type type, unsigned n)
{
        unsigned pos;

        pos = isl_pw_qpolynomial_dim(pwqp, type);

        return isl_pw_qpolynomial_insert_dims(pwqp, type, pos, n);
}

static int *reordering_move(isl_ctx *ctx,
        unsigned len, unsigned dst, unsigned src, unsigned n)
{
        int i;
        int *reordering;

        reordering = isl_alloc_array(ctx, int, len);
        if (!reordering)
                return NULL;

        if (dst <= src) {
                for (i = 0; i < dst; ++i)
                        reordering[i] = i;
                for (i = 0; i < n; ++i)
                        reordering[src + i] = dst + i;
                for (i = 0; i < src - dst; ++i)
                        reordering[dst + i] = dst + n + i;
                for (i = 0; i < len - src - n; ++i)
                        reordering[src + n + i] = src + n + i;
        } else {
                for (i = 0; i < src; ++i)
                        reordering[i] = i;
                for (i = 0; i < n; ++i)
                        reordering[src + i] = dst + i;
                for (i = 0; i < dst - src; ++i)
                        reordering[src + n + i] = src + i;
                for (i = 0; i < len - dst - n; ++i)
                        reordering[dst + n + i] = dst + n + i;
        }

        return reordering;
}

__isl_give isl_qpolynomial *isl_qpolynomial_move_dims(
        __isl_take isl_qpolynomial *qp,
        enum isl_dim_type dst_type, unsigned dst_pos,
        enum isl_dim_type src_type, unsigned src_pos, unsigned n)
{
        unsigned g_dst_pos;
        unsigned g_src_pos;
        int *reordering;

        if (!qp)
                return NULL;

        if (dst_type == isl_dim_out || src_type == isl_dim_out)
                isl_die(qp->dim->ctx, isl_error_invalid,
                        "cannot move output/set dimension",
                        goto error);
        if (dst_type == isl_dim_in)
                dst_type = isl_dim_set;
        if (src_type == isl_dim_in)
                src_type = isl_dim_set;

        if (n == 0 &&
            !isl_space_is_named_or_nested(qp->dim, src_type) &&
            !isl_space_is_named_or_nested(qp->dim, dst_type))
                return qp;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        isl_assert(qp->dim->ctx, src_pos + n <= isl_space_dim(qp->dim, src_type),
                goto error);

        g_dst_pos = pos(qp->dim, dst_type) + dst_pos;
        g_src_pos = pos(qp->dim, src_type) + src_pos;
        if (dst_type > src_type)
                g_dst_pos -= n;

        qp->div = isl_mat_move_cols(qp->div, 2 + g_dst_pos, 2 + g_src_pos, n);
        if (!qp->div)
                goto error;
        qp = sort_divs(qp);
        if (!qp)
                goto error;

        reordering = reordering_move(qp->dim->ctx,
                                qp->div->n_col - 2, g_dst_pos, g_src_pos, n);
        if (!reordering)
                goto error;

        qp->upoly = reorder(qp->upoly, reordering);
        free(reordering);
        if (!qp->upoly)
                goto error;

        qp->dim = isl_space_move_dims(qp->dim, dst_type, dst_pos, src_type, src_pos, n);
        if (!qp->dim)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_from_affine(
        __isl_take isl_space *space, isl_int *f, isl_int denom)
{
        struct isl_upoly *up;

        space = isl_space_domain(space);
        if (!space)
                return NULL;

        up = isl_upoly_from_affine(space->ctx, f, denom,
                                        1 + isl_space_dim(space, isl_dim_all));

        return isl_qpolynomial_alloc(space, 0, up);
}

__isl_give isl_qpolynomial *isl_qpolynomial_from_aff(__isl_take isl_aff *aff)
{
        isl_ctx *ctx;
        struct isl_upoly *up;
        isl_qpolynomial *qp;

        if (!aff)
                return NULL;

        ctx = isl_aff_get_ctx(aff);
        up = isl_upoly_from_affine(ctx, aff->v->el + 1, aff->v->el[0],
                                    aff->v->size - 1);

        qp = isl_qpolynomial_alloc(isl_aff_get_domain_space(aff),
                                    aff->ls->div->n_row, up);
        if (!qp)
                goto error;

        isl_mat_free(qp->div);
        qp->div = isl_mat_copy(aff->ls->div);
        qp->div = isl_mat_cow(qp->div);
        if (!qp->div)
                goto error;

        isl_aff_free(aff);
        qp = reduce_divs(qp);
        qp = remove_redundant_divs(qp);
        return qp;
error:
        isl_aff_free(aff);
        return isl_qpolynomial_free(qp);
}

__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_pw_aff(
        __isl_take isl_pw_aff *pwaff)
{
        int i;
        isl_pw_qpolynomial *pwqp;

        if (!pwaff)
                return NULL;

        pwqp = isl_pw_qpolynomial_alloc_size(isl_pw_aff_get_space(pwaff),
                                                pwaff->n);

        for (i = 0; i < pwaff->n; ++i) {
                isl_set *dom;
                isl_qpolynomial *qp;

                dom = isl_set_copy(pwaff->p[i].set);
                qp = isl_qpolynomial_from_aff(isl_aff_copy(pwaff->p[i].aff));
                pwqp = isl_pw_qpolynomial_add_piece(pwqp,  dom, qp);
        }

        isl_pw_aff_free(pwaff);
        return pwqp;
}

__isl_give isl_qpolynomial *isl_qpolynomial_from_constraint(
        __isl_take isl_constraint *c, enum isl_dim_type type, unsigned pos)
{
        isl_aff *aff;

        aff = isl_constraint_get_bound(c, type, pos);
        isl_constraint_free(c);
        return isl_qpolynomial_from_aff(aff);
}

/* For each 0 <= i < "n", replace variable "first" + i of type "type"
 * in "qp" by subs[i].
 */
__isl_give isl_qpolynomial *isl_qpolynomial_substitute(
        __isl_take isl_qpolynomial *qp,
        enum isl_dim_type type, unsigned first, unsigned n,
        __isl_keep isl_qpolynomial **subs)
{
        int i;
        struct isl_upoly **ups;

        if (n == 0)
                return qp;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        if (type == isl_dim_out)
                isl_die(qp->dim->ctx, isl_error_invalid,
                        "cannot substitute output/set dimension",
                        goto error);
        if (type == isl_dim_in)
                type = isl_dim_set;

        for (i = 0; i < n; ++i)
                if (!subs[i])
                        goto error;

        isl_assert(qp->dim->ctx, first + n <= isl_space_dim(qp->dim, type),
                        goto error);

        for (i = 0; i < n; ++i)
                isl_assert(qp->dim->ctx, isl_space_is_equal(qp->dim, subs[i]->dim),
                                goto error);

        isl_assert(qp->dim->ctx, qp->div->n_row == 0, goto error);
        for (i = 0; i < n; ++i)
                isl_assert(qp->dim->ctx, subs[i]->div->n_row == 0, goto error);

        first += pos(qp->dim, type);

        ups = isl_alloc_array(qp->dim->ctx, struct isl_upoly *, n);
        if (!ups)
                goto error;
        for (i = 0; i < n; ++i)
                ups[i] = subs[i]->upoly;

        qp->upoly = isl_upoly_subs(qp->upoly, first, n, ups);

        free(ups);

        if (!qp->upoly)
                goto error;

        return qp;
error:
        isl_qpolynomial_free(qp);
        return NULL;
}

/* Extend "bset" with extra set dimensions for each integer division
 * in "qp" and then call "fn" with the extended bset and the polynomial
 * that results from replacing each of the integer divisions by the
 * corresponding extra set dimension.
 */
isl_stat isl_qpolynomial_as_polynomial_on_domain(__isl_keep isl_qpolynomial *qp,
        __isl_keep isl_basic_set *bset,
        isl_stat (*fn)(__isl_take isl_basic_set *bset,
                  __isl_take isl_qpolynomial *poly, void *user), void *user)
{
        isl_space *space;
        isl_local_space *ls;
        isl_qpolynomial *poly;

        if (!qp || !bset)
                return isl_stat_error;
        if (qp->div->n_row == 0)
                return fn(isl_basic_set_copy(bset), isl_qpolynomial_copy(qp),
                          user);

        space = isl_space_copy(qp->dim);
        space = isl_space_add_dims(space, isl_dim_set, qp->div->n_row);
        poly = isl_qpolynomial_alloc(space, 0, isl_upoly_copy(qp->upoly));
        bset = isl_basic_set_copy(bset);
        ls = isl_qpolynomial_get_domain_local_space(qp);
        bset = isl_local_space_lift_basic_set(ls, bset);

        return fn(bset, poly, user);
}

/* Return total degree in variables first (inclusive) up to last (exclusive).
 */
int isl_upoly_degree(__isl_keep struct isl_upoly *up, int first, int last)
{
        int deg = -1;
        int i;
        struct isl_upoly_rec *rec;

        if (!up)
                return -2;
        if (isl_upoly_is_zero(up))
                return -1;
        if (isl_upoly_is_cst(up) || up->var < first)
                return 0;

        rec = isl_upoly_as_rec(up);
        if (!rec)
                return -2;

        for (i = 0; i < rec->n; ++i) {
                int d;

                if (isl_upoly_is_zero(rec->p[i]))
                        continue;
                d = isl_upoly_degree(rec->p[i], first, last);
                if (up->var < last)
                        d += i;
                if (d > deg)
                        deg = d;
        }

        return deg;
}

/* Return total degree in set variables.
 */
int isl_qpolynomial_degree(__isl_keep isl_qpolynomial *poly)
{
        unsigned ovar;
        unsigned nvar;

        if (!poly)
                return -2;

        ovar = isl_space_offset(poly->dim, isl_dim_set);
        nvar = isl_space_dim(poly->dim, isl_dim_set);
        return isl_upoly_degree(poly->upoly, ovar, ovar + nvar);
}

__isl_give struct isl_upoly *isl_upoly_coeff(__isl_keep struct isl_upoly *up,
        unsigned pos, int deg)
{
        int i;
        struct isl_upoly_rec *rec;

        if (!up)
                return NULL;

        if (isl_upoly_is_cst(up) || up->var < pos) {
                if (deg == 0)
                        return isl_upoly_copy(up);
                else
                        return isl_upoly_zero(up->ctx);
        }

        rec = isl_upoly_as_rec(up);
        if (!rec)
                return NULL;

        if (up->var == pos) {
                if (deg < rec->n)
                        return isl_upoly_copy(rec->p[deg]);
                else
                        return isl_upoly_zero(up->ctx);
        }

        up = isl_upoly_copy(up);
        up = isl_upoly_cow(up);
        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                struct isl_upoly *t;
                t = isl_upoly_coeff(rec->p[i], pos, deg);
                if (!t)
                        goto error;
                isl_upoly_free(rec->p[i]);
                rec->p[i] = t;
        }

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

/* Return coefficient of power "deg" of variable "t_pos" of type "type".
 */
__isl_give isl_qpolynomial *isl_qpolynomial_coeff(
        __isl_keep isl_qpolynomial *qp,
        enum isl_dim_type type, unsigned t_pos, int deg)
{
        unsigned g_pos;
        struct isl_upoly *up;
        isl_qpolynomial *c;

        if (!qp)
                return NULL;

        if (type == isl_dim_out)
                isl_die(qp->div->ctx, isl_error_invalid,
                        "output/set dimension does not have a coefficient",
                        return NULL);
        if (type == isl_dim_in)
                type = isl_dim_set;

        isl_assert(qp->div->ctx, t_pos < isl_space_dim(qp->dim, type),
                        return NULL);

        g_pos = pos(qp->dim, type) + t_pos;
        up = isl_upoly_coeff(qp->upoly, g_pos, deg);

        c = isl_qpolynomial_alloc(isl_space_copy(qp->dim), qp->div->n_row, up);
        if (!c)
                return NULL;
        isl_mat_free(c->div);
        c->div = isl_mat_copy(qp->div);
        if (!c->div)
                goto error;
        return c;
error:
        isl_qpolynomial_free(c);
        return NULL;
}

/* Homogenize the polynomial in the variables first (inclusive) up to
 * last (exclusive) by inserting powers of variable first.
 * Variable first is assumed not to appear in the input.
 */
__isl_give struct isl_upoly *isl_upoly_homogenize(
        __isl_take struct isl_upoly *up, int deg, int target,
        int first, int last)
{
        int i;
        struct isl_upoly_rec *rec;

        if (!up)
                return NULL;
        if (isl_upoly_is_zero(up))
                return up;
        if (deg == target)
                return up;
        if (isl_upoly_is_cst(up) || up->var < first) {
                struct isl_upoly *hom;

                hom = isl_upoly_var_pow(up->ctx, first, target - deg);
                if (!hom)
                        goto error;
                rec = isl_upoly_as_rec(hom);
                rec->p[target - deg] = isl_upoly_mul(rec->p[target - deg], up);

                return hom;
        }

        up = isl_upoly_cow(up);
        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                if (isl_upoly_is_zero(rec->p[i]))
                        continue;
                rec->p[i] = isl_upoly_homogenize(rec->p[i],
                                up->var < last ? deg + i : i, target,
                                first, last);
                if (!rec->p[i])
                        goto error;
        }

        return up;
error:
        isl_upoly_free(up);
        return NULL;
}

/* Homogenize the polynomial in the set variables by introducing
 * powers of an extra set variable at position 0.
 */
__isl_give isl_qpolynomial *isl_qpolynomial_homogenize(
        __isl_take isl_qpolynomial *poly)
{
        unsigned ovar;
        unsigned nvar;
        int deg = isl_qpolynomial_degree(poly);

        if (deg < -1)
                goto error;

        poly = isl_qpolynomial_insert_dims(poly, isl_dim_in, 0, 1);
        poly = isl_qpolynomial_cow(poly);
        if (!poly)
                goto error;

        ovar = isl_space_offset(poly->dim, isl_dim_set);
        nvar = isl_space_dim(poly->dim, isl_dim_set);
        poly->upoly = isl_upoly_homogenize(poly->upoly, 0, deg,
                                                ovar, ovar + nvar);
        if (!poly->upoly)
                goto error;

        return poly;
error:
        isl_qpolynomial_free(poly);
        return NULL;
}

__isl_give isl_term *isl_term_alloc(__isl_take isl_space *space,
        __isl_take isl_mat *div)
{
        isl_term *term;
        int n;

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

        n = isl_space_dim(space, isl_dim_all) + div->n_row;

        term = isl_calloc(space->ctx, struct isl_term,
                        sizeof(struct isl_term) + (n - 1) * sizeof(int));
        if (!term)
                goto error;

        term->ref = 1;
        term->dim = space;
        term->div = div;
        isl_int_init(term->n);
        isl_int_init(term->d);
        
        return term;
error:
        isl_space_free(space);
        isl_mat_free(div);
        return NULL;
}

__isl_give isl_term *isl_term_copy(__isl_keep isl_term *term)
{
        if (!term)
                return NULL;

        term->ref++;
        return term;
}

__isl_give isl_term *isl_term_dup(__isl_keep isl_term *term)
{
        int i;
        isl_term *dup;
        unsigned total;

        if (!term)
                return NULL;

        total = isl_space_dim(term->dim, isl_dim_all) + term->div->n_row;

        dup = isl_term_alloc(isl_space_copy(term->dim), isl_mat_copy(term->div));
        if (!dup)
                return NULL;

        isl_int_set(dup->n, term->n);
        isl_int_set(dup->d, term->d);

        for (i = 0; i < total; ++i)
                dup->pow[i] = term->pow[i];

        return dup;
}

__isl_give isl_term *isl_term_cow(__isl_take isl_term *term)
{
        if (!term)
                return NULL;

        if (term->ref == 1)
                return term;
        term->ref--;
        return isl_term_dup(term);
}

__isl_null isl_term *isl_term_free(__isl_take isl_term *term)
{
        if (!term)
                return NULL;

        if (--term->ref > 0)
                return NULL;

        isl_space_free(term->dim);
        isl_mat_free(term->div);
        isl_int_clear(term->n);
        isl_int_clear(term->d);
        free(term);

        return NULL;
}

unsigned isl_term_dim(__isl_keep isl_term *term, enum isl_dim_type type)
{
        if (!term)
                return 0;

        switch (type) {
        case isl_dim_param:
        case isl_dim_in:
        case isl_dim_out:       return isl_space_dim(term->dim, type);
        case isl_dim_div:       return term->div->n_row;
        case isl_dim_all:       return isl_space_dim(term->dim, isl_dim_all) +
                                                                term->div->n_row;
        default:                return 0;
        }
}

isl_ctx *isl_term_get_ctx(__isl_keep isl_term *term)
{
        return term ? term->dim->ctx : NULL;
}

void isl_term_get_num(__isl_keep isl_term *term, isl_int *n)
{
        if (!term)
                return;
        isl_int_set(*n, term->n);
}

/* Return the coefficient of the term "term".
 */
__isl_give isl_val *isl_term_get_coefficient_val(__isl_keep isl_term *term)
{
        if (!term)
                return NULL;

        return isl_val_rat_from_isl_int(isl_term_get_ctx(term),
                                        term->n, term->d);
}

int isl_term_get_exp(__isl_keep isl_term *term,
        enum isl_dim_type type, unsigned pos)
{
        if (!term)
                return -1;

        isl_assert(term->dim->ctx, pos < isl_term_dim(term, type), return -1);

        if (type >= isl_dim_set)
                pos += isl_space_dim(term->dim, isl_dim_param);
        if (type >= isl_dim_div)
                pos += isl_space_dim(term->dim, isl_dim_set);

        return term->pow[pos];
}

__isl_give isl_aff *isl_term_get_div(__isl_keep isl_term *term, unsigned pos)
{
        isl_local_space *ls;
        isl_aff *aff;

        if (!term)
                return NULL;

        isl_assert(term->dim->ctx, pos < isl_term_dim(term, isl_dim_div),
                        return NULL);

        ls = isl_local_space_alloc_div(isl_space_copy(term->dim),
                                        isl_mat_copy(term->div));
        aff = isl_aff_alloc(ls);
        if (!aff)
                return NULL;

        isl_seq_cpy(aff->v->el, term->div->row[pos], aff->v->size);

        aff = isl_aff_normalize(aff);

        return aff;
}

__isl_give isl_term *isl_upoly_foreach_term(__isl_keep struct isl_upoly *up,
        isl_stat (*fn)(__isl_take isl_term *term, void *user),
        __isl_take isl_term *term, void *user)
{
        int i;
        struct isl_upoly_rec *rec;

        if (!up || !term)
                goto error;

        if (isl_upoly_is_zero(up))
                return term;

        isl_assert(up->ctx, !isl_upoly_is_nan(up), goto error);
        isl_assert(up->ctx, !isl_upoly_is_infty(up), goto error);
        isl_assert(up->ctx, !isl_upoly_is_neginfty(up), goto error);

        if (isl_upoly_is_cst(up)) {
                struct isl_upoly_cst *cst;
                cst = isl_upoly_as_cst(up);
                if (!cst)
                        goto error;
                term = isl_term_cow(term);
                if (!term)
                        goto error;
                isl_int_set(term->n, cst->n);
                isl_int_set(term->d, cst->d);
                if (fn(isl_term_copy(term), user) < 0)
                        goto error;
                return term;
        }

        rec = isl_upoly_as_rec(up);
        if (!rec)
                goto error;

        for (i = 0; i < rec->n; ++i) {
                term = isl_term_cow(term);
                if (!term)
                        goto error;
                term->pow[up->var] = i;
                term = isl_upoly_foreach_term(rec->p[i], fn, term, user);
                if (!term)
                        goto error;
        }
        term->pow[up->var] = 0;

        return term;
error:
        isl_term_free(term);
        return NULL;
}

isl_stat isl_qpolynomial_foreach_term(__isl_keep isl_qpolynomial *qp,
        isl_stat (*fn)(__isl_take isl_term *term, void *user), void *user)
{
        isl_term *term;

        if (!qp)
                return isl_stat_error;

        term = isl_term_alloc(isl_space_copy(qp->dim), isl_mat_copy(qp->div));
        if (!term)
                return isl_stat_error;

        term = isl_upoly_foreach_term(qp->upoly, fn, term, user);

        isl_term_free(term);

        return term ? isl_stat_ok : isl_stat_error;
}

__isl_give isl_qpolynomial *isl_qpolynomial_from_term(__isl_take isl_term *term)
{
        struct isl_upoly *up;
        isl_qpolynomial *qp;
        int i, n;

        if (!term)
                return NULL;

        n = isl_space_dim(term->dim, isl_dim_all) + term->div->n_row;

        up = isl_upoly_rat_cst(term->dim->ctx, term->n, term->d);
        for (i = 0; i < n; ++i) {
                if (!term->pow[i])
                        continue;
                up = isl_upoly_mul(up,
                        isl_upoly_var_pow(term->dim->ctx, i, term->pow[i]));
        }

        qp = isl_qpolynomial_alloc(isl_space_copy(term->dim), term->div->n_row, up);
        if (!qp)
                goto error;
        isl_mat_free(qp->div);
        qp->div = isl_mat_copy(term->div);
        if (!qp->div)
                goto error;

        isl_term_free(term);
        return qp;
error:
        isl_qpolynomial_free(qp);
        isl_term_free(term);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_lift(__isl_take isl_qpolynomial *qp,
        __isl_take isl_space *space)
{
        int i;
        int extra;
        unsigned total;

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

        if (isl_space_is_equal(qp->dim, space)) {
                isl_space_free(space);
                return qp;
        }

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                goto error;

        extra = isl_space_dim(space, isl_dim_set) -
                        isl_space_dim(qp->dim, isl_dim_set);
        total = isl_space_dim(qp->dim, isl_dim_all);
        if (qp->div->n_row) {
                int *exp;

                exp = isl_alloc_array(qp->div->ctx, int, qp->div->n_row);
                if (!exp)
                        goto error;
                for (i = 0; i < qp->div->n_row; ++i)
                        exp[i] = extra + i;
                qp->upoly = expand(qp->upoly, exp, total);
                free(exp);
                if (!qp->upoly)
                        goto error;
        }
        qp->div = isl_mat_insert_cols(qp->div, 2 + total, extra);
        if (!qp->div)
                goto error;
        for (i = 0; i < qp->div->n_row; ++i)
                isl_seq_clr(qp->div->row[i] + 2 + total, extra);

        isl_space_free(qp->dim);
        qp->dim = space;

        return qp;
error:
        isl_space_free(space);
        isl_qpolynomial_free(qp);
        return NULL;
}

/* For each parameter or variable that does not appear in qp,
 * first eliminate the variable from all constraints and then set it to zero.
 */
static __isl_give isl_set *fix_inactive(__isl_take isl_set *set,
        __isl_keep isl_qpolynomial *qp)
{
        int *active = NULL;
        int i;
        int d;
        unsigned nparam;
        unsigned nvar;

        if (!set || !qp)
                goto error;

        d = isl_space_dim(set->dim, isl_dim_all);
        active = isl_calloc_array(set->ctx, int, d);
        if (set_active(qp, active) < 0)
                goto error;

        for (i = 0; i < d; ++i)
                if (!active[i])
                        break;

        if (i == d) {
                free(active);
                return set;
        }

        nparam = isl_space_dim(set->dim, isl_dim_param);
        nvar = isl_space_dim(set->dim, isl_dim_set);
        for (i = 0; i < nparam; ++i) {
                if (active[i])
                        continue;
                set = isl_set_eliminate(set, isl_dim_param, i, 1);
                set = isl_set_fix_si(set, isl_dim_param, i, 0);
        }
        for (i = 0; i < nvar; ++i) {
                if (active[nparam + i])
                        continue;
                set = isl_set_eliminate(set, isl_dim_set, i, 1);
                set = isl_set_fix_si(set, isl_dim_set, i, 0);
        }

        free(active);

        return set;
error:
        free(active);
        isl_set_free(set);
        return NULL;
}

struct isl_opt_data {
        isl_qpolynomial *qp;
        int first;
        isl_val *opt;
        int max;
};

static isl_stat opt_fn(__isl_take isl_point *pnt, void *user)
{
        struct isl_opt_data *data = (struct isl_opt_data *)user;
        isl_val *val;

        val = isl_qpolynomial_eval(isl_qpolynomial_copy(data->qp), pnt);
        if (data->first) {
                data->first = 0;
                data->opt = val;
        } else if (data->max) {
                data->opt = isl_val_max(data->opt, val);
        } else {
                data->opt = isl_val_min(data->opt, val);
        }

        return isl_stat_ok;
}

__isl_give isl_val *isl_qpolynomial_opt_on_domain(
        __isl_take isl_qpolynomial *qp, __isl_take isl_set *set, int max)
{
        struct isl_opt_data data = { NULL, 1, NULL, max };

        if (!set || !qp)
                goto error;

        if (isl_upoly_is_cst(qp->upoly)) {
                isl_set_free(set);
                data.opt = isl_qpolynomial_get_constant_val(qp);
                isl_qpolynomial_free(qp);
                return data.opt;
        }

        set = fix_inactive(set, qp);

        data.qp = qp;
        if (isl_set_foreach_point(set, opt_fn, &data) < 0)
                goto error;

        if (data.first)
                data.opt = isl_val_zero(isl_set_get_ctx(set));

        isl_set_free(set);
        isl_qpolynomial_free(qp);
        return data.opt;
error:
        isl_set_free(set);
        isl_qpolynomial_free(qp);
        isl_val_free(data.opt);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_morph_domain(
        __isl_take isl_qpolynomial *qp, __isl_take isl_morph *morph)
{
        int i;
        int n_sub;
        isl_ctx *ctx;
        struct isl_upoly **subs;
        isl_mat *mat, *diag;

        qp = isl_qpolynomial_cow(qp);
        if (!qp || !morph)
                goto error;

        ctx = qp->dim->ctx;
        isl_assert(ctx, isl_space_is_equal(qp->dim, morph->dom->dim), goto error);

        n_sub = morph->inv->n_row - 1;
        if (morph->inv->n_row != morph->inv->n_col)
                n_sub += qp->div->n_row;
        subs = isl_calloc_array(ctx, struct isl_upoly *, n_sub);
        if (n_sub && !subs)
                goto error;

        for (i = 0; 1 + i < morph->inv->n_row; ++i)
                subs[i] = isl_upoly_from_affine(ctx, morph->inv->row[1 + i],
                                        morph->inv->row[0][0], morph->inv->n_col);
        if (morph->inv->n_row != morph->inv->n_col)
                for (i = 0; i < qp->div->n_row; ++i)
                        subs[morph->inv->n_row - 1 + i] =
                            isl_upoly_var_pow(ctx, morph->inv->n_col - 1 + i, 1);

        qp->upoly = isl_upoly_subs(qp->upoly, 0, n_sub, subs);

        for (i = 0; i < n_sub; ++i)
                isl_upoly_free(subs[i]);
        free(subs);

        diag = isl_mat_diag(ctx, 1, morph->inv->row[0][0]);
        mat = isl_mat_diagonal(diag, isl_mat_copy(morph->inv));
        diag = isl_mat_diag(ctx, qp->div->n_row, morph->inv->row[0][0]);
        mat = isl_mat_diagonal(mat, diag);
        qp->div = isl_mat_product(qp->div, mat);
        isl_space_free(qp->dim);
        qp->dim = isl_space_copy(morph->ran->dim);

        if (!qp->upoly || !qp->div || !qp->dim)
                goto error;

        isl_morph_free(morph);

        return qp;
error:
        isl_qpolynomial_free(qp);
        isl_morph_free(morph);
        return NULL;
}

__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
        __isl_take isl_union_pw_qpolynomial *upwqp1,
        __isl_take isl_union_pw_qpolynomial *upwqp2)
{
        return isl_union_pw_qpolynomial_match_bin_op(upwqp1, upwqp2,
                                                &isl_pw_qpolynomial_mul);
}

/* Reorder the dimension of "qp" according to the given reordering.
 */
__isl_give isl_qpolynomial *isl_qpolynomial_realign_domain(
        __isl_take isl_qpolynomial *qp, __isl_take isl_reordering *r)
{
        isl_space *space;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                goto error;

        r = isl_reordering_extend(r, qp->div->n_row);
        if (!r)
                goto error;

        qp->div = isl_local_reorder(qp->div, isl_reordering_copy(r));
        if (!qp->div)
                goto error;

        qp->upoly = reorder(qp->upoly, r->pos);
        if (!qp->upoly)
                goto error;

        space = isl_reordering_get_space(r);
        qp = isl_qpolynomial_reset_domain_space(qp, space);

        isl_reordering_free(r);
        return qp;
error:
        isl_qpolynomial_free(qp);
        isl_reordering_free(r);
        return NULL;
}

__isl_give isl_qpolynomial *isl_qpolynomial_align_params(
        __isl_take isl_qpolynomial *qp, __isl_take isl_space *model)
{
        isl_bool equal_params;

        if (!qp || !model)
                goto error;

        equal_params = isl_space_has_equal_params(qp->dim, model);
        if (equal_params < 0)
                goto error;
        if (!equal_params) {
                isl_reordering *exp;

                exp = isl_parameter_alignment_reordering(qp->dim, model);
                exp = isl_reordering_extend_space(exp,
                                        isl_qpolynomial_get_domain_space(qp));
                qp = isl_qpolynomial_realign_domain(qp, exp);
        }

        isl_space_free(model);
        return qp;
error:
        isl_space_free(model);
        isl_qpolynomial_free(qp);
        return NULL;
}

struct isl_split_periods_data {
        int max_periods;
        isl_pw_qpolynomial *res;
};

/* Create a slice where the integer division "div" has the fixed value "v".
 * In particular, if "div" refers to floor(f/m), then create a slice
 *
 *      m v <= f <= m v + (m - 1)
 *
 * or
 *
 *      f - m v >= 0
 *      -f + m v + (m - 1) >= 0
 */
static __isl_give isl_set *set_div_slice(__isl_take isl_space *space,
        __isl_keep isl_qpolynomial *qp, int div, isl_int v)
{
        int total;
        isl_basic_set *bset = NULL;
        int k;

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

        total = isl_space_dim(space, isl_dim_all);
        bset = isl_basic_set_alloc_space(isl_space_copy(space), 0, 0, 2);

        k = isl_basic_set_alloc_inequality(bset);
        if (k < 0)
                goto error;
        isl_seq_cpy(bset->ineq[k], qp->div->row[div] + 1, 1 + total);
        isl_int_submul(bset->ineq[k][0], v, qp->div->row[div][0]);

        k = isl_basic_set_alloc_inequality(bset);
        if (k < 0)
                goto error;
        isl_seq_neg(bset->ineq[k], qp->div->row[div] + 1, 1 + total);
        isl_int_addmul(bset->ineq[k][0], v, qp->div->row[div][0]);
        isl_int_add(bset->ineq[k][0], bset->ineq[k][0], qp->div->row[div][0]);
        isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);

        isl_space_free(space);
        return isl_set_from_basic_set(bset);
error:
        isl_basic_set_free(bset);
        isl_space_free(space);
        return NULL;
}

static isl_stat split_periods(__isl_take isl_set *set,
        __isl_take isl_qpolynomial *qp, void *user);

/* Create a slice of the domain "set" such that integer division "div"
 * has the fixed value "v" and add the results to data->res,
 * replacing the integer division by "v" in "qp".
 */
static isl_stat set_div(__isl_take isl_set *set,
        __isl_take isl_qpolynomial *qp, int div, isl_int v,
        struct isl_split_periods_data *data)
{
        int i;
        int total;
        isl_set *slice;
        struct isl_upoly *cst;

        slice = set_div_slice(isl_set_get_space(set), qp, div, v);
        set = isl_set_intersect(set, slice);

        if (!qp)
                goto error;

        total = isl_space_dim(qp->dim, isl_dim_all);

        for (i = div + 1; i < qp->div->n_row; ++i) {
                if (isl_int_is_zero(qp->div->row[i][2 + total + div]))
                        continue;
                isl_int_addmul(qp->div->row[i][1],
                                qp->div->row[i][2 + total + div], v);
                isl_int_set_si(qp->div->row[i][2 + total + div], 0);
        }

        cst = isl_upoly_rat_cst(qp->dim->ctx, v, qp->dim->ctx->one);
        qp = substitute_div(qp, div, cst);

        return split_periods(set, qp, data);
error:
        isl_set_free(set);
        isl_qpolynomial_free(qp);
        return isl_stat_error;
}

/* Split the domain "set" such that integer division "div"
 * has a fixed value (ranging from "min" to "max") on each slice
 * and add the results to data->res.
 */
static isl_stat split_div(__isl_take isl_set *set,
        __isl_take isl_qpolynomial *qp, int div, isl_int min, isl_int max,
        struct isl_split_periods_data *data)
{
        for (; isl_int_le(min, max); isl_int_add_ui(min, min, 1)) {
                isl_set *set_i = isl_set_copy(set);
                isl_qpolynomial *qp_i = isl_qpolynomial_copy(qp);

                if (set_div(set_i, qp_i, div, min, data) < 0)
                        goto error;
        }
        isl_set_free(set);
        isl_qpolynomial_free(qp);
        return isl_stat_ok;
error:
        isl_set_free(set);
        isl_qpolynomial_free(qp);
        return isl_stat_error;
}

/* If "qp" refers to any integer division
 * that can only attain "max_periods" distinct values on "set"
 * then split the domain along those distinct values.
 * Add the results (or the original if no splitting occurs)
 * to data->res.
 */
static isl_stat split_periods(__isl_take isl_set *set,
        __isl_take isl_qpolynomial *qp, void *user)
{
        int i;
        isl_pw_qpolynomial *pwqp;
        struct isl_split_periods_data *data;
        isl_int min, max;
        int total;
        isl_stat r = isl_stat_ok;

        data = (struct isl_split_periods_data *)user;

        if (!set || !qp)
                goto error;

        if (qp->div->n_row == 0) {
                pwqp = isl_pw_qpolynomial_alloc(set, qp);
                data->res = isl_pw_qpolynomial_add_disjoint(data->res, pwqp);
                return isl_stat_ok;
        }

        isl_int_init(min);
        isl_int_init(max);
        total = isl_space_dim(qp->dim, isl_dim_all);
        for (i = 0; i < qp->div->n_row; ++i) {
                enum isl_lp_result lp_res;

                if (isl_seq_first_non_zero(qp->div->row[i] + 2 + total,
                                                qp->div->n_row) != -1)
                        continue;

                lp_res = isl_set_solve_lp(set, 0, qp->div->row[i] + 1,
                                          set->ctx->one, &min, NULL, NULL);
                if (lp_res == isl_lp_error)
                        goto error2;
                if (lp_res == isl_lp_unbounded || lp_res == isl_lp_empty)
                        continue;
                isl_int_fdiv_q(min, min, qp->div->row[i][0]);

                lp_res = isl_set_solve_lp(set, 1, qp->div->row[i] + 1,
                                          set->ctx->one, &max, NULL, NULL);
                if (lp_res == isl_lp_error)
                        goto error2;
                if (lp_res == isl_lp_unbounded || lp_res == isl_lp_empty)
                        continue;
                isl_int_fdiv_q(max, max, qp->div->row[i][0]);

                isl_int_sub(max, max, min);
                if (isl_int_cmp_si(max, data->max_periods) < 0) {
                        isl_int_add(max, max, min);
                        break;
                }
        }

        if (i < qp->div->n_row) {
                r = split_div(set, qp, i, min, max, data);
        } else {
                pwqp = isl_pw_qpolynomial_alloc(set, qp);
                data->res = isl_pw_qpolynomial_add_disjoint(data->res, pwqp);
        }

        isl_int_clear(max);
        isl_int_clear(min);

        return r;
error2:
        isl_int_clear(max);
        isl_int_clear(min);
error:
        isl_set_free(set);
        isl_qpolynomial_free(qp);
        return isl_stat_error;
}

/* If any quasi-polynomial in pwqp refers to any integer division
 * that can only attain "max_periods" distinct values on its domain
 * then split the domain along those distinct values.
 */
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_split_periods(
        __isl_take isl_pw_qpolynomial *pwqp, int max_periods)
{
        struct isl_split_periods_data data;

        data.max_periods = max_periods;
        data.res = isl_pw_qpolynomial_zero(isl_pw_qpolynomial_get_space(pwqp));

        if (isl_pw_qpolynomial_foreach_piece(pwqp, &split_periods, &data) < 0)
                goto error;

        isl_pw_qpolynomial_free(pwqp);

        return data.res;
error:
        isl_pw_qpolynomial_free(data.res);
        isl_pw_qpolynomial_free(pwqp);
        return NULL;
}

/* Construct a piecewise quasipolynomial that is constant on the given
 * domain.  In particular, it is
 *      0       if cst == 0
 *      1       if cst == 1
 *  infinity    if cst == -1
 *
 * If cst == -1, then explicitly check whether the domain is empty and,
 * if so, return 0 instead.
 */
static __isl_give isl_pw_qpolynomial *constant_on_domain(
        __isl_take isl_basic_set *bset, int cst)
{
        isl_space *dim;
        isl_qpolynomial *qp;

        if (cst < 0 && isl_basic_set_is_empty(bset) == isl_bool_true)
                cst = 0;
        if (!bset)
                return NULL;

        bset = isl_basic_set_params(bset);
        dim = isl_basic_set_get_space(bset);
        if (cst < 0)
                qp = isl_qpolynomial_infty_on_domain(dim);
        else if (cst == 0)
                qp = isl_qpolynomial_zero_on_domain(dim);
        else
                qp = isl_qpolynomial_one_on_domain(dim);
        return isl_pw_qpolynomial_alloc(isl_set_from_basic_set(bset), qp);
}

/* Factor bset, call fn on each of the factors and return the product.
 *
 * If no factors can be found, simply call fn on the input.
 * Otherwise, construct the factors based on the factorizer,
 * call fn on each factor and compute the product.
 */
static __isl_give isl_pw_qpolynomial *compressed_multiplicative_call(
        __isl_take isl_basic_set *bset,
        __isl_give isl_pw_qpolynomial *(*fn)(__isl_take isl_basic_set *bset))
{
        int i, n;
        isl_space *space;
        isl_set *set;
        isl_factorizer *f;
        isl_qpolynomial *qp;
        isl_pw_qpolynomial *pwqp;
        unsigned nparam;
        unsigned nvar;

        f = isl_basic_set_factorizer(bset);
        if (!f)
                goto error;
        if (f->n_group == 0) {
                isl_factorizer_free(f);
                return fn(bset);
        }

        nparam = isl_basic_set_dim(bset, isl_dim_param);
        nvar = isl_basic_set_dim(bset, isl_dim_set);

        space = isl_basic_set_get_space(bset);
        space = isl_space_params(space);
        set = isl_set_universe(isl_space_copy(space));
        qp = isl_qpolynomial_one_on_domain(space);
        pwqp = isl_pw_qpolynomial_alloc(set, qp);

        bset = isl_morph_basic_set(isl_morph_copy(f->morph), bset);

        for (i = 0, n = 0; i < f->n_group; ++i) {
                isl_basic_set *bset_i;
                isl_pw_qpolynomial *pwqp_i;

                bset_i = isl_basic_set_copy(bset);
                bset_i = isl_basic_set_drop_constraints_involving(bset_i,
                            nparam + n + f->len[i], nvar - n - f->len[i]);
                bset_i = isl_basic_set_drop_constraints_involving(bset_i,
                            nparam, n);
                bset_i = isl_basic_set_drop(bset_i, isl_dim_set,
                            n + f->len[i], nvar - n - f->len[i]);
                bset_i = isl_basic_set_drop(bset_i, isl_dim_set, 0, n);

                pwqp_i = fn(bset_i);
                pwqp = isl_pw_qpolynomial_mul(pwqp, pwqp_i);

                n += f->len[i];
        }

        isl_basic_set_free(bset);
        isl_factorizer_free(f);

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

/* Factor bset, call fn on each of the factors and return the product.
 * The function is assumed to evaluate to zero on empty domains,
 * to one on zero-dimensional domains and to infinity on unbounded domains
 * and will not be called explicitly on zero-dimensional or unbounded domains.
 *
 * We first check for some special cases and remove all equalities.
 * Then we hand over control to compressed_multiplicative_call.
 */
__isl_give isl_pw_qpolynomial *isl_basic_set_multiplicative_call(
        __isl_take isl_basic_set *bset,
        __isl_give isl_pw_qpolynomial *(*fn)(__isl_take isl_basic_set *bset))
{
        isl_bool bounded;
        isl_morph *morph;
        isl_pw_qpolynomial *pwqp;

        if (!bset)
                return NULL;

        if (isl_basic_set_plain_is_empty(bset))
                return constant_on_domain(bset, 0);

        if (isl_basic_set_dim(bset, isl_dim_set) == 0)
                return constant_on_domain(bset, 1);

        bounded = isl_basic_set_is_bounded(bset);
        if (bounded < 0)
                goto error;
        if (!bounded)
                return constant_on_domain(bset, -1);

        if (bset->n_eq == 0)
                return compressed_multiplicative_call(bset, fn);

        morph = isl_basic_set_full_compression(bset);
        bset = isl_morph_basic_set(isl_morph_copy(morph), bset);

        pwqp = compressed_multiplicative_call(bset, fn);

        morph = isl_morph_dom_params(morph);
        morph = isl_morph_ran_params(morph);
        morph = isl_morph_inverse(morph);

        pwqp = isl_pw_qpolynomial_morph_domain(pwqp, morph);

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

/* Drop all floors in "qp", turning each integer division [a/m] into
 * a rational division a/m.  If "down" is set, then the integer division
 * is replaced by (a-(m-1))/m instead.
 */
static __isl_give isl_qpolynomial *qp_drop_floors(
        __isl_take isl_qpolynomial *qp, int down)
{
        int i;
        struct isl_upoly *s;

        if (!qp)
                return NULL;
        if (qp->div->n_row == 0)
                return qp;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;

        for (i = qp->div->n_row - 1; i >= 0; --i) {
                if (down) {
                        isl_int_sub(qp->div->row[i][1],
                                    qp->div->row[i][1], qp->div->row[i][0]);
                        isl_int_add_ui(qp->div->row[i][1],
                                       qp->div->row[i][1], 1);
                }
                s = isl_upoly_from_affine(qp->dim->ctx, qp->div->row[i] + 1,
                                        qp->div->row[i][0], qp->div->n_col - 1);
                qp = substitute_div(qp, i, s);
                if (!qp)
                        return NULL;
        }

        return qp;
}

/* Drop all floors in "pwqp", turning each integer division [a/m] into
 * a rational division a/m.
 */
static __isl_give isl_pw_qpolynomial *pwqp_drop_floors(
        __isl_take isl_pw_qpolynomial *pwqp)
{
        int i;

        if (!pwqp)
                return NULL;

        if (isl_pw_qpolynomial_is_zero(pwqp))
                return pwqp;

        pwqp = isl_pw_qpolynomial_cow(pwqp);
        if (!pwqp)
                return NULL;

        for (i = 0; i < pwqp->n; ++i) {
                pwqp->p[i].qp = qp_drop_floors(pwqp->p[i].qp, 0);
                if (!pwqp->p[i].qp)
                        goto error;
        }

        return pwqp;
error:
        isl_pw_qpolynomial_free(pwqp);
        return NULL;
}

/* Adjust all the integer divisions in "qp" such that they are at least
 * one over the given orthant (identified by "signs").  This ensures
 * that they will still be non-negative even after subtracting (m-1)/m.
 *
 * In particular, f is replaced by f' + v, changing f = [a/m]
 * to f' = [(a - m v)/m].
 * If the constant term k in a is smaller than m,
 * the constant term of v is set to floor(k/m) - 1.
 * For any other term, if the coefficient c and the variable x have
 * the same sign, then no changes are needed.
 * Otherwise, if the variable is positive (and c is negative),
 * then the coefficient of x in v is set to floor(c/m).
 * If the variable is negative (and c is positive),
 * then the coefficient of x in v is set to ceil(c/m).
 */
static __isl_give isl_qpolynomial *make_divs_pos(__isl_take isl_qpolynomial *qp,
        int *signs)
{
        int i, j;
        int total;
        isl_vec *v = NULL;
        struct isl_upoly *s;

        qp = isl_qpolynomial_cow(qp);
        if (!qp)
                return NULL;
        qp->div = isl_mat_cow(qp->div);
        if (!qp->div)
                goto error;

        total = isl_space_dim(qp->dim, isl_dim_all);
        v = isl_vec_alloc(qp->div->ctx, qp->div->n_col - 1);

        for (i = 0; i < qp->div->n_row; ++i) {
                isl_int *row = qp->div->row[i];
                v = isl_vec_clr(v);
                if (!v)
                        goto error;
                if (isl_int_lt(row[1], row[0])) {
                        isl_int_fdiv_q(v->el[0], row[1], row[0]);
                        isl_int_sub_ui(v->el[0], v->el[0], 1);
                        isl_int_submul(row[1], row[0], v->el[0]);
                }
                for (j = 0; j < total; ++j) {
                        if (isl_int_sgn(row[2 + j]) * signs[j] >= 0)
                                continue;
                        if (signs[j] < 0)
                                isl_int_cdiv_q(v->el[1 + j], row[2 + j], row[0]);
                        else
                                isl_int_fdiv_q(v->el[1 + j], row[2 + j], row[0]);
                        isl_int_submul(row[2 + j], row[0], v->el[1 + j]);
                }
                for (j = 0; j < i; ++j) {
                        if (isl_int_sgn(row[2 + total + j]) >= 0)
                                continue;
                        isl_int_fdiv_q(v->el[1 + total + j],
                                        row[2 + total + j], row[0]);
                        isl_int_submul(row[2 + total + j],
                                        row[0], v->el[1 + total + j]);
                }
                for (j = i + 1; j < qp->div->n_row; ++j) {
                        if (isl_int_is_zero(qp->div->row[j][2 + total + i]))
                                continue;
                        isl_seq_combine(qp->div->row[j] + 1,
                                qp->div->ctx->one, qp->div->row[j] + 1,
                                qp->div->row[j][2 + total + i], v->el, v->size);
                }
                isl_int_set_si(v->el[1 + total + i], 1);
                s = isl_upoly_from_affine(qp->dim->ctx, v->el,
                                        qp->div->ctx->one, v->size);
                qp->upoly = isl_upoly_subs(qp->upoly, total + i, 1, &s);
                isl_upoly_free(s);
                if (!qp->upoly)
                        goto error;
        }

        isl_vec_free(v);
        return qp;
error:
        isl_vec_free(v);
        isl_qpolynomial_free(qp);
        return NULL;
}

struct isl_to_poly_data {
        int sign;
        isl_pw_qpolynomial *res;
        isl_qpolynomial *qp;
};

/* Appoximate data->qp by a polynomial on the orthant identified by "signs".
 * We first make all integer divisions positive and then split the
 * quasipolynomials into terms with sign data->sign (the direction
 * of the requested approximation) and terms with the opposite sign.
 * In the first set of terms, each integer division [a/m] is
 * overapproximated by a/m, while in the second it is underapproximated
 * by (a-(m-1))/m.
 */
static isl_stat to_polynomial_on_orthant(__isl_take isl_set *orthant,
        int *signs, void *user)
{
        struct isl_to_poly_data *data = user;
        isl_pw_qpolynomial *t;
        isl_qpolynomial *qp, *up, *down;

        qp = isl_qpolynomial_copy(data->qp);
        qp = make_divs_pos(qp, signs);

        up = isl_qpolynomial_terms_of_sign(qp, signs, data->sign);
        up = qp_drop_floors(up, 0);
        down = isl_qpolynomial_terms_of_sign(qp, signs, -data->sign);
        down = qp_drop_floors(down, 1);

        isl_qpolynomial_free(qp);
        qp = isl_qpolynomial_add(up, down);

        t = isl_pw_qpolynomial_alloc(orthant, qp);
        data->res = isl_pw_qpolynomial_add_disjoint(data->res, t);

        return isl_stat_ok;
}

/* Approximate each quasipolynomial by a polynomial.  If "sign" is positive,
 * the polynomial will be an overapproximation.  If "sign" is negative,
 * it will be an underapproximation.  If "sign" is zero, the approximation
 * will lie somewhere in between.
 *
 * In particular, is sign == 0, we simply drop the floors, turning
 * the integer divisions into rational divisions.
 * Otherwise, we split the domains into orthants, make all integer divisions
 * positive and then approximate each [a/m] by either a/m or (a-(m-1))/m,
 * depending on the requested sign and the sign of the term in which
 * the integer division appears.
 */
__isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
        __isl_take isl_pw_qpolynomial *pwqp, int sign)
{
        int i;
        struct isl_to_poly_data data;

        if (sign == 0)
                return pwqp_drop_floors(pwqp);

        if (!pwqp)
                return NULL;

        data.sign = sign;
        data.res = isl_pw_qpolynomial_zero(isl_pw_qpolynomial_get_space(pwqp));

        for (i = 0; i < pwqp->n; ++i) {
                if (pwqp->p[i].qp->div->n_row == 0) {
                        isl_pw_qpolynomial *t;
                        t = isl_pw_qpolynomial_alloc(
                                        isl_set_copy(pwqp->p[i].set),
                                        isl_qpolynomial_copy(pwqp->p[i].qp));
                        data.res = isl_pw_qpolynomial_add_disjoint(data.res, t);
                        continue;
                }
                data.qp = pwqp->p[i].qp;
                if (isl_set_foreach_orthant(pwqp->p[i].set,
                                        &to_polynomial_on_orthant, &data) < 0)
                        goto error;
        }

        isl_pw_qpolynomial_free(pwqp);

        return data.res;
error:
        isl_pw_qpolynomial_free(pwqp);
        isl_pw_qpolynomial_free(data.res);
        return NULL;
}

static __isl_give isl_pw_qpolynomial *poly_entry(
        __isl_take isl_pw_qpolynomial *pwqp, void *user)
{
        int *sign = user;

        return isl_pw_qpolynomial_to_polynomial(pwqp, *sign);
}

__isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_to_polynomial(
        __isl_take isl_union_pw_qpolynomial *upwqp, int sign)
{
        return isl_union_pw_qpolynomial_transform_inplace(upwqp,
                                   &poly_entry, &sign);
}

__isl_give isl_basic_map *isl_basic_map_from_qpolynomial(
        __isl_take isl_qpolynomial *qp)
{
        int i, k;
        isl_space *dim;
        isl_vec *aff = NULL;
        isl_basic_map *bmap = NULL;
        unsigned pos;
        unsigned n_div;

        if (!qp)
                return NULL;
        if (!isl_upoly_is_affine(qp->upoly))
                isl_die(qp->dim->ctx, isl_error_invalid,
                        "input quasi-polynomial not affine", goto error);
        aff = isl_qpolynomial_extract_affine(qp);
        if (!aff)
                goto error;
        dim = isl_qpolynomial_get_space(qp);
        pos = 1 + isl_space_offset(dim, isl_dim_out);
        n_div = qp->div->n_row;
        bmap = isl_basic_map_alloc_space(dim, n_div, 1, 2 * n_div);

        for (i = 0; i < n_div; ++i) {
                k = isl_basic_map_alloc_div(bmap);
                if (k < 0)
                        goto error;
                isl_seq_cpy(bmap->div[k], qp->div->row[i], qp->div->n_col);
                isl_int_set_si(bmap->div[k][qp->div->n_col], 0);
                if (isl_basic_map_add_div_constraints(bmap, k) < 0)
                        goto error;
        }
        k = isl_basic_map_alloc_equality(bmap);
        if (k < 0)
                goto error;
        isl_int_neg(bmap->eq[k][pos], aff->el[0]);
        isl_seq_cpy(bmap->eq[k], aff->el + 1, pos);
        isl_seq_cpy(bmap->eq[k] + pos + 1, aff->el + 1 + pos, n_div);

        isl_vec_free(aff);
        isl_qpolynomial_free(qp);
        bmap = isl_basic_map_finalize(bmap);
        return bmap;
error:
        isl_vec_free(aff);
        isl_qpolynomial_free(qp);
        isl_basic_map_free(bmap);
        return NULL;
}