isl_pw_qpolynomial_sum: simplify construction of target space

[barvinok.git] / util.c

#include <stdlib.h>
#include <assert.h>
#include <isl_set_polylib.h>
#include <barvinok/util.h>
#include <barvinok/options.h>
#include <polylib/ranking.h>
#include "config.h"
#include "lattice_point.h"

#define ALLOC(type) (type*)malloc(sizeof(type))
#define ALLOCN(type,n) (type*)malloc((n) * sizeof(type))

#ifdef __GNUC__
#define NALLOC(p,n) p = (typeof(p))malloc((n) * sizeof(*p))
#else
#define NALLOC(p,n) p = (void *)malloc((n) * sizeof(*p))
#endif

void manual_count(Polyhedron *P, Value* result)
{
        isl_ctx *ctx = isl_ctx_alloc();
        isl_space *dim;
        isl_set *set;
        isl_int v;
        int nvar = P->Dimension;
        int r;

        dim = isl_space_set_alloc(ctx, 0, nvar);
        set = isl_set_new_from_polylib(P, dim);

        isl_int_init(v);
        r = isl_set_count(set, &v);
        isl_int_get_gmp(v, *result);
        isl_int_clear(v);

        isl_set_free(set);
        isl_ctx_free(ctx);

        assert(r >= 0);
}

#include <barvinok/evalue.h>
#include <barvinok/util.h>
#include <barvinok/barvinok.h>

/* Return random value between 0 and max-1 inclusive
 */
int random_int(int max) {
    return (int) (((double)(max))*rand()/(RAND_MAX+1.0));
}

Polyhedron *Polyhedron_Read(unsigned MaxRays)
{
    int vertices = 0; 
    unsigned NbRows, NbColumns;
    Matrix *M;
    Polyhedron *P;
    char s[128];

    while (fgets(s, sizeof(s), stdin)) {
        if (*s == '#')
            continue;
        if (strncasecmp(s, "vertices", sizeof("vertices")-1) == 0)
            vertices = 1;
        if (sscanf(s, "%u %u", &NbRows, &NbColumns) == 2)
            break;
    }
    if (feof(stdin))
        return NULL;
    M = Matrix_Alloc(NbRows,NbColumns);
    Matrix_Read_Input(M);
    if (vertices)
        P = Rays2Polyhedron(M, MaxRays);
    else
        P = Constraints2Polyhedron(M, MaxRays);
    Matrix_Free(M);
    return P;
}

/* Inplace polarization
 */
void Polyhedron_Polarize(Polyhedron *P)
{
    unsigned NbRows = P->NbConstraints + P->NbRays;
    int i;
    Value **q;

    q = (Value **)malloc(NbRows * sizeof(Value *));
    assert(q);
    for (i = 0; i < P->NbRays; ++i)
        q[i] = P->Ray[i];
    for (; i < NbRows; ++i)
        q[i] = P->Constraint[i-P->NbRays];
    P->NbConstraints = NbRows - P->NbConstraints;
    P->NbRays = NbRows - P->NbRays;
    free(P->Constraint);
    P->Constraint = q;
    P->Ray = q + P->NbConstraints;
}

/*
 * Rather general polar
 * We can optimize it significantly if we assume that
 * P includes zero
 *
 * Also, we calculate the polar as defined in Schrijver
 * The opposite should probably work as well and would
 * eliminate the need for multiplying by -1
 */
Polyhedron* Polyhedron_Polar(Polyhedron *P, unsigned NbMaxRays)
{
    int i;
    Value mone;
    unsigned dim = P->Dimension + 2;
    Matrix *M = Matrix_Alloc(P->NbRays, dim);

    assert(M);
    value_init(mone);
    value_set_si(mone, -1);
    for (i = 0; i < P->NbRays; ++i) {
        Vector_Scale(P->Ray[i], M->p[i], mone, dim);
        value_multiply(M->p[i][0], M->p[i][0], mone);
        value_multiply(M->p[i][dim-1], M->p[i][dim-1], mone);
    }
    P = Constraints2Polyhedron(M, NbMaxRays);
    assert(P);
    Matrix_Free(M);
    value_clear(mone);
    return P;
}

/*
 * Returns the supporting cone of P at the vertex with index v
 */
Polyhedron* supporting_cone(Polyhedron *P, int v)
{
    Matrix *M;
    Value tmp;
    int i, n, j;
    unsigned char *supporting = (unsigned char *)malloc(P->NbConstraints);
    unsigned dim = P->Dimension + 2;

    assert(v >=0 && v < P->NbRays);
    assert(value_pos_p(P->Ray[v][dim-1]));
    assert(supporting);

    value_init(tmp);
    for (i = 0, n = 0; i < P->NbConstraints; ++i) {
        Inner_Product(P->Constraint[i] + 1, P->Ray[v] + 1, dim - 1, &tmp);
        if ((supporting[i] = value_zero_p(tmp)))
            ++n;
    }
    assert(n >= dim - 2);
    value_clear(tmp);
    M = Matrix_Alloc(n, dim);
    assert(M);
    for (i = 0, j = 0; i < P->NbConstraints; ++i)
        if (supporting[i]) {
            value_set_si(M->p[j][dim-1], 0);
            Vector_Copy(P->Constraint[i], M->p[j++], dim-1);
        }
    free(supporting);
    P = Constraints2Polyhedron(M, P->NbRays+1);
    assert(P);
    Matrix_Free(M);
    return P;
}

#define INT_BITS (sizeof(unsigned) * 8)

unsigned *supporting_constraints(Matrix *Constraints, Param_Vertices *v, int *n)
{
    Value lcm, tmp, tmp2;
    unsigned dim = Constraints->NbColumns;
    unsigned nparam = v->Vertex->NbColumns - 2;
    unsigned nvar = dim - nparam - 2;
    int len = (Constraints->NbRows+INT_BITS-1)/INT_BITS;
    unsigned *supporting = (unsigned *)calloc(len, sizeof(unsigned));
    int i, j;
    Vector *row;
    int ix;
    unsigned bx;

    assert(supporting);
    row = Vector_Alloc(nparam+1);
    assert(row);
    value_init(lcm);
    value_init(tmp);
    value_init(tmp2);
    value_set_si(lcm, 1);
    for (i = 0, *n = 0, ix = 0, bx = MSB; i < Constraints->NbRows; ++i) {
        Vector_Set(row->p, 0, nparam+1);
        for (j = 0 ; j < nvar; ++j) {
            value_set_si(tmp, 1);
            value_assign(tmp2,  Constraints->p[i][j+1]);
            if (value_ne(lcm, v->Vertex->p[j][nparam+1])) {
                value_assign(tmp, lcm);
                value_lcm(lcm, lcm, v->Vertex->p[j][nparam+1]);
                value_division(tmp, lcm, tmp);
                value_multiply(tmp2, tmp2, lcm);
                value_division(tmp2, tmp2, v->Vertex->p[j][nparam+1]);
            }
            Vector_Combine(row->p, v->Vertex->p[j], row->p, 
                           tmp, tmp2, nparam+1);
        }
        value_set_si(tmp, 1);
        Vector_Combine(row->p, Constraints->p[i]+1+nvar, row->p, tmp, lcm, nparam+1);
        for (j = 0; j < nparam+1; ++j)
            if (value_notzero_p(row->p[j]))
                break;
        if (j == nparam + 1) {
            supporting[ix] |= bx;
            ++*n;
        }
        NEXT(ix, bx);
    }
    assert(*n >= nvar);
    value_clear(tmp);
    value_clear(tmp2);
    value_clear(lcm);
    Vector_Free(row);

    return supporting;
}

Polyhedron* supporting_cone_p(Polyhedron *P, Param_Vertices *v)
{
    Matrix *M;
    unsigned dim = P->Dimension + 2;
    unsigned nparam = v->Vertex->NbColumns - 2;
    unsigned nvar = dim - nparam - 2;
    int i, n, j;
    int ix;
    unsigned bx;
    unsigned *supporting;
    Matrix View;

    Polyhedron_Matrix_View(P, &View, P->NbConstraints);
    supporting = supporting_constraints(&View, v, &n);
    M = Matrix_Alloc(n, nvar+2);
    assert(M);
    for (i = 0, j = 0, ix = 0, bx = MSB; i < P->NbConstraints; ++i) {
        if (supporting[ix] & bx) {
            value_set_si(M->p[j][nvar+1], 0);
            Vector_Copy(P->Constraint[i], M->p[j++], nvar+1);
        }
        NEXT(ix, bx);
    }
    free(supporting);
    P = Constraints2Polyhedron(M, P->NbRays+1);
    assert(P);
    Matrix_Free(M);
    return P;
}

Polyhedron* triangulate_cone(Polyhedron *P, unsigned NbMaxCons)
{
    struct barvinok_options *options = barvinok_options_new_with_defaults();
    options->MaxRays = NbMaxCons;
    P = triangulate_cone_with_options(P, options);
    barvinok_options_free(options);
    return P;
}

Polyhedron* triangulate_cone_with_options(Polyhedron *P,
                                          struct barvinok_options *options)
{
    const static int MAX_TRY=10;
    int i, j, r, n, t;
    Value tmp;
    unsigned dim = P->Dimension;
    Matrix *M = Matrix_Alloc(P->NbRays+1, dim+3);
    Matrix *M2, *M3;
    Polyhedron *L, *R, *T;
    assert(P->NbEq == 0);

    L = NULL;
    R = NULL;
    value_init(tmp);

    Vector_Set(M->p[0]+1, 0, dim+1);
    value_set_si(M->p[0][0], 1);
    value_set_si(M->p[0][dim+2], 1);
    Vector_Set(M->p[P->NbRays]+1, 0, dim+2);
    value_set_si(M->p[P->NbRays][0], 1);
    value_set_si(M->p[P->NbRays][dim+1], 1);

    for (i = 0, r = 1; i < P->NbRays; ++i) {
        if (value_notzero_p(P->Ray[i][dim+1]))
            continue;
        Vector_Copy(P->Ray[i], M->p[r], dim+1);
        value_set_si(M->p[r][dim+2], 0);
        ++r;
    }

    M2 = Matrix_Alloc(dim+1, dim+2);

    t = 0;
    if (options->try_Delaunay_triangulation) {
        /* Delaunay triangulation */
        for (r = 1; r < P->NbRays; ++r) {
            Inner_Product(M->p[r]+1, M->p[r]+1, dim, &tmp);
            value_assign(M->p[r][dim+1], tmp);
        }
        M3 = Matrix_Copy(M);
        L = Rays2Polyhedron(M3, options->MaxRays);
        Matrix_Free(M3);
        ++t;
    } else {
try_again:
        /* Usually R should still be 0 */
        Domain_Free(R);
        Polyhedron_Free(L);
        for (r = 1; r < P->NbRays; ++r) {
            value_set_si(M->p[r][dim+1], random_int((t+1)*dim*P->NbRays)+1);
        }
        M3 = Matrix_Copy(M);
        L = Rays2Polyhedron(M3, options->MaxRays);
        Matrix_Free(M3);
        ++t;
    }
    assert(t <= MAX_TRY);

    R = NULL;
    n = 0;

    POL_ENSURE_FACETS(L);
    for (i = 0; i < L->NbConstraints; ++i) {
        /* Ignore perpendicular facets, i.e., facets with 0 z-coordinate */
        if (value_negz_p(L->Constraint[i][dim+1]))
            continue;
        if (value_notzero_p(L->Constraint[i][dim+2]))
            continue;
        for (j = 1, r = 1; j < M->NbRows; ++j) {
            Inner_Product(M->p[j]+1, L->Constraint[i]+1, dim+1, &tmp);
            if (value_notzero_p(tmp))
                continue;
            if (r > dim)
                goto try_again;
            Vector_Copy(M->p[j]+1, M2->p[r]+1, dim);
            value_set_si(M2->p[r][0], 1);
            value_set_si(M2->p[r][dim+1], 0);
            ++r;
        }
        assert(r == dim+1);
        Vector_Set(M2->p[0]+1, 0, dim);
        value_set_si(M2->p[0][0], 1);
        value_set_si(M2->p[0][dim+1], 1);
        T = Rays2Polyhedron(M2, P->NbConstraints+1);
        T->next = R;
        R = T;
        ++n;
    }
    Matrix_Free(M2);

    Polyhedron_Free(L);
    value_clear(tmp);
    Matrix_Free(M);

    return R;
}

void check_triangulization(Polyhedron *P, Polyhedron *T)
{
    Polyhedron *C, *D, *E, *F, *G, *U;
    for (C = T; C; C = C->next) {
        if (C == T)
            U = C;
        else 
            U = DomainConvex(DomainUnion(U, C, 100), 100);
        for (D = C->next; D; D = D->next) {
            F = C->next;
            G = D->next;
            C->next = NULL;
            D->next = NULL;
            E = DomainIntersection(C, D, 600);
            assert(E->NbRays == 0 || E->NbEq >= 1);
            Polyhedron_Free(E);
            C->next = F;
            D->next = G;
        }
    }
    assert(PolyhedronIncludes(U, P));
    assert(PolyhedronIncludes(P, U));
}

/* Computes x, y and g such that g = gcd(a,b) and a*x+b*y = g */
void Extended_Euclid(Value a, Value b, Value *x, Value *y, Value *g)
{
    Value c, d, e, f, tmp;

    value_init(c);
    value_init(d);
    value_init(e);
    value_init(f);
    value_init(tmp);
    value_absolute(c, a);
    value_absolute(d, b);
    value_set_si(e, 1);
    value_set_si(f, 0);
    while(value_pos_p(d)) {
        value_division(tmp, c, d);
        value_multiply(tmp, tmp, f);
        value_subtract(e, e, tmp);
        value_division(tmp, c, d);
        value_multiply(tmp, tmp, d);
        value_subtract(c, c, tmp);
        value_swap(c, d);
        value_swap(e, f);
    }
    value_assign(*g, c);
    if (value_zero_p(a))
        value_set_si(*x, 0);
    else if (value_pos_p(a))
        value_assign(*x, e);
    else value_oppose(*x, e);
    if (value_zero_p(b))
        value_set_si(*y, 0);
    else {
        value_multiply(tmp, a, *x);
        value_subtract(tmp, c, tmp);
        value_division(*y, tmp, b);
    }
    value_clear(c);
    value_clear(d);
    value_clear(e);
    value_clear(f);
    value_clear(tmp);
}

static int unimodular_complete_1(Matrix *m) 
{
    Value g, b, c, old, tmp;
    unsigned i, j;
    int ok;

    value_init(b);
    value_init(c);
    value_init(g);
    value_init(old);
    value_init(tmp);
    value_assign(g, m->p[0][0]);
    for (i = 1; value_zero_p(g) && i < m->NbColumns; ++i) {
        for (j = 0; j < m->NbColumns; ++j) {
            if (j == i-1)
                value_set_si(m->p[i][j], 1);
            else
                value_set_si(m->p[i][j], 0);
        }
        value_assign(g, m->p[0][i]);
    }
    for (; i < m->NbColumns; ++i) {
        value_assign(old, g);
        Extended_Euclid(old, m->p[0][i], &c, &b, &g);
        value_oppose(b, b);
        for (j = 0; j < m->NbColumns; ++j) {
            if (j < i) {
                value_multiply(tmp, m->p[0][j], b);
                value_division(m->p[i][j], tmp, old);
            } else if (j == i)
                value_assign(m->p[i][j], c);
            else
                value_set_si(m->p[i][j], 0);
        }
    }
    ok = value_one_p(g);
    value_clear(b);
    value_clear(c);
    value_clear(g);
    value_clear(old);
    value_clear(tmp);
    return ok;
}

int unimodular_complete(Matrix *M, int row)
{
    int r;
    int ok = 1;
    Matrix *H, *Q, *U;

    if (row == 1)
        return unimodular_complete_1(M);

    left_hermite(M, &H, &Q, &U);
    Matrix_Free(U);
    for (r = 0; ok && r < row; ++r)
        if (value_notone_p(H->p[r][r]))
            ok = 0;
    Matrix_Free(H);
    for (r = row; r < M->NbRows; ++r)
        Vector_Copy(Q->p[r], M->p[r], M->NbColumns);
    Matrix_Free(Q);
    return ok;
}

/*
 * left_hermite may leave positive entries below the main diagonal in H.
 * This function postprocesses the output of left_hermite to make
 * the non-zero entries below the main diagonal negative.
 */
void neg_left_hermite(Matrix *A, Matrix **H_p, Matrix **Q_p, Matrix **U_p)
{
    int row, col, i, j;
    Matrix *H, *U, *Q;

    left_hermite(A, &H, &Q, &U);
    *H_p = H;
    *Q_p = Q;
    *U_p = U;

    for (row = 0, col = 0; col < H->NbColumns; ++col, ++row) {
        while (value_zero_p(H->p[row][col]))
            ++row;
        for (i = 0; i < col; ++i) {
            if (value_negz_p(H->p[row][i]))
                continue;

            /* subtract column col from column i in H and U */
            for (j = 0; j < H->NbRows; ++j)
                value_subtract(H->p[j][i], H->p[j][i], H->p[j][col]);
            for (j = 0; j < U->NbRows; ++j)
                value_subtract(U->p[j][i], U->p[j][i], U->p[j][col]);

            /* add row i to row col in Q */
            for (j = 0; j < Q->NbColumns; ++j)
                value_addto(Q->p[col][j], Q->p[col][j], Q->p[i][j]);
        }
    }
}

/*
 * Returns a full-dimensional polyhedron with the same number
 * of integer points as P
 */
Polyhedron *remove_equalities(Polyhedron *P, unsigned MaxRays)
{
    Matrix M;
    Matrix *T;
    Polyhedron *Q = Polyhedron_Copy(P);
    unsigned dim = P->Dimension;

    if (Q->NbEq == 0)
        return Q;

    Q = DomainConstraintSimplify(Q, MaxRays);
    if (emptyQ2(Q))
        return Q;

    Polyhedron_Matrix_View(Q, &M, Q->NbEq);
    T = compress_variables(&M, 0);

    if (!T)
        P = NULL;
    else {
        P = Polyhedron_Preimage(Q, T, MaxRays);
        Matrix_Free(T);
    }

    Polyhedron_Free(Q);

    return P;
}

/*
 * Returns a full-dimensional polyhedron with the same number
 * of integer points as P
 * nvar specifies the number of variables
 * The remaining dimensions are assumed to be parameters
 * Destroys P
 * factor is NbEq x (nparam+2) matrix, containing stride constraints
 * on the parameters; column nparam is the constant;
 * column nparam+1 is the stride
 *
 * if factor is NULL, only remove equalities that don't affect
 * the number of points
 */
Polyhedron *remove_equalities_p(Polyhedron *P, unsigned nvar, Matrix **factor,
                                unsigned MaxRays)
{
    Value g;
    Polyhedron *Q;
    unsigned dim = P->Dimension;
    Matrix *m1, *m2, *f;
    int i, j;

    if (P->NbEq == 0)
        return P;

    m1 = Matrix_Alloc(nvar, nvar);
    P = DomainConstraintSimplify(P, MaxRays);
    if (factor) {
        f = Matrix_Alloc(P->NbEq, dim-nvar+2);
        *factor = f;
    }
    value_init(g);
    for (i = 0, j = 0; i < P->NbEq; ++i) {
        if (First_Non_Zero(P->Constraint[i]+1, nvar) == -1)
            continue;

        Vector_Gcd(P->Constraint[i]+1, nvar, &g);
        if (!factor && value_notone_p(g))
            continue;

        if (factor) {
            Vector_Copy(P->Constraint[i]+1+nvar, f->p[j], dim-nvar+1);
            value_assign(f->p[j][dim-nvar+1], g);
        }

        Vector_Copy(P->Constraint[i]+1, m1->p[j], nvar);

        ++j;
    }
    value_clear(g);

    unimodular_complete(m1, j);

    m2 = Matrix_Alloc(dim+1-j, dim+1);
    for (i = 0; i < nvar-j ; ++i)
        Vector_Copy(m1->p[i+j], m2->p[i], nvar);
    Matrix_Free(m1);
    for (i = nvar-j; i <= dim-j; ++i)
        value_set_si(m2->p[i][i+j], 1);

    Q = Polyhedron_Image(P, m2, MaxRays);
    Matrix_Free(m2);
    Polyhedron_Free(P);

    return Q;
}

void Line_Length(Polyhedron *P, Value *len)
{
    Value tmp, pos, neg;
    int p = 0, n = 0;
    int i;

    assert(P->Dimension == 1);

    if (P->NbEq > 0) {
        if (mpz_divisible_p(P->Constraint[0][2], P->Constraint[0][1]))
            value_set_si(*len, 1);
        else
            value_set_si(*len, 0);
        return;
    }

    value_init(tmp);
    value_init(pos);
    value_init(neg);

    for (i = 0; i < P->NbConstraints; ++i) {
        value_oppose(tmp, P->Constraint[i][2]);
        if (value_pos_p(P->Constraint[i][1])) {
            mpz_cdiv_q(tmp, tmp, P->Constraint[i][1]);
            if (!p || value_gt(tmp, pos))
                value_assign(pos, tmp);
            p = 1;
        } else if (value_neg_p(P->Constraint[i][1])) {
            mpz_fdiv_q(tmp, tmp, P->Constraint[i][1]);
            if (!n || value_lt(tmp, neg))
                value_assign(neg, tmp);
            n = 1;
        }
        if (n && p) {
            value_subtract(tmp, neg, pos);
            value_increment(*len, tmp);
        } else
            value_set_si(*len, -1);
    }

    value_clear(tmp);
    value_clear(pos);
    value_clear(neg);
}

/* Update group[k] to the group column k belongs to.
 * When merging two groups, only the group of the current
 * group leader is changed.  Here we change the group of
 * the other members to also point to the group that the
 * old group leader now points to.
 */
static void update_group(int *group, int *cnt, int k)
{
        int g = group[k];
        while (cnt[g] == 0)
                g = group[g];
        group[k] = g;
}

/*
 * Factors the polyhedron P into polyhedra Q_i such that
 * the number of integer points in P is equal to the product
 * of the number of integer points in the individual Q_i
 *
 * If no factors can be found, NULL is returned.
 * Otherwise, a linked list of the factors is returned.
 *
 * If there are factors and if T is not NULL, then a matrix will be
 * returned through T expressing the old variables in terms of the
 * new variables as they appear in the sequence of factors.
 *
 * The algorithm works by first computing the Hermite normal form
 * and then grouping columns linked by one or more constraints together,
 * where a constraints "links" two or more columns if the constraint
 * has nonzero coefficients in the columns.
 */
Polyhedron* Polyhedron_Factor(Polyhedron *P, unsigned nparam, Matrix **T,
                              unsigned NbMaxRays)
{
    int i, j, k;
    Matrix *M, *H, *Q, *U;
    int *pos;           /* for each column: row position of pivot */
    int *group;         /* group to which a column belongs */
    int *cnt;           /* number of columns in the group */
    int *rowgroup;      /* group to which a constraint belongs */
    int nvar = P->Dimension - nparam;
    Polyhedron *F = NULL;

    if (nvar <= 1)
        return NULL;

    NALLOC(pos, nvar);
    NALLOC(group, nvar);
    NALLOC(cnt, nvar);
    NALLOC(rowgroup, P->NbConstraints);

    M = Matrix_Alloc(P->NbConstraints, nvar);
    for (i = 0; i < P->NbConstraints; ++i)
        Vector_Copy(P->Constraint[i]+1, M->p[i], nvar);
    left_hermite(M, &H, &Q, &U);
    Matrix_Free(M);
    Matrix_Free(Q);

    for (i = 0; i < P->NbConstraints; ++i)
        rowgroup[i] = -1;
    for (i = 0, j = 0; i < H->NbColumns; ++i) {
        for ( ; j < H->NbRows; ++j)
            if (value_notzero_p(H->p[j][i]))
                break;
        pos[i] = j;
    }
    for (i = 0; i < nvar; ++i) {
        group[i] = i;
        cnt[i] = 1;
    }
    for (i = 0; i < H->NbColumns && cnt[0] < nvar; ++i) {
        if (pos[i] == H->NbRows)
            continue; /* A line direction */
        if (rowgroup[pos[i]] == -1)
            rowgroup[pos[i]] = i;
        for (j = pos[i]+1; j <  H->NbRows; ++j) {
            if (value_zero_p(H->p[j][i]))
                continue;
            if (rowgroup[j] != -1)
                continue;
            rowgroup[j] = group[i];
            for (k = i+1; k < H->NbColumns && j >= pos[k]; ++k) {
                update_group(group, cnt, k);
                update_group(group, cnt, i);
                if (group[k] != group[i] && value_notzero_p(H->p[j][k])) {
                    assert(cnt[group[k]] != 0);
                    assert(cnt[group[i]] != 0);
                    if (group[i] < group[k]) {
                        cnt[group[i]] += cnt[group[k]];
                        cnt[group[k]] = 0;
                        group[group[k]] = group[i];
                    } else {
                        cnt[group[k]] += cnt[group[i]];
                        cnt[group[i]] = 0;
                        group[group[i]] = group[k];
                    }
                }
            }
        }
    }

    if (cnt[0] != nvar) {
        /* Extract out pure context constraints separately */
        Polyhedron **next = &F;
        int tot_d = 0;
        if (T)
            *T = Matrix_Alloc(nvar, nvar);
        for (i = nparam ? -1 : 0; i < nvar; ++i) {
            int d;

            if (i == -1) {
                for (j = 0, k = 0; j < P->NbConstraints; ++j)
                    if (rowgroup[j] == -1) {
                        if (First_Non_Zero(P->Constraint[j]+1+nvar, 
                                           nparam) == -1)
                            rowgroup[j] = -2;
                        else 
                            ++k;
                    }
                if (k == 0)
                    continue;
                d = 0;
            } else {
                if (cnt[i] == 0)
                    continue;
                d = cnt[i];
                for (j = 0, k = 0; j < P->NbConstraints; ++j)
                    if (rowgroup[j] >= 0 && group[rowgroup[j]] == i) {
                        rowgroup[j] = i;
                        ++k;
                    }
            }

            if (T)
                for (j = 0; j < nvar; ++j) {
                    int l, m;
                    for (l = 0, m = 0; m < d; ++l) {
                        if (group[l] != i)
                            continue;
                        value_assign((*T)->p[j][tot_d+m++], U->p[j][l]);
                    }
                }

            M = Matrix_Alloc(k, d+nparam+2);
            for (j = 0, k = 0; j < P->NbConstraints; ++j) {
                int l, m;
                if (rowgroup[j] != i)
                    continue;
                value_assign(M->p[k][0], P->Constraint[j][0]);
                for (l = 0, m = 0; m < d; ++l) {
                    if (group[l] != i)
                        continue;
                    value_assign(M->p[k][1+m++], H->p[j][l]);
                }
                Vector_Copy(P->Constraint[j]+1+nvar, M->p[k]+1+m, nparam+1);
                ++k;
            }
            *next = Constraints2Polyhedron(M, NbMaxRays);
            next = &(*next)->next;
            Matrix_Free(M);
            tot_d += d;
        }
    }
    Matrix_Free(U);
    Matrix_Free(H);
    free(pos);
    free(group);
    free(cnt);
    free(rowgroup);
    return F;
}

/* Computes the intersection of the contexts of a list of factors */
Polyhedron *Factor_Context(Polyhedron *F, unsigned nparam, unsigned MaxRays)
{
    Polyhedron *Q;
    Polyhedron *C = NULL;

    for (Q = F; Q; Q = Q->next) {
        Polyhedron *QC = Q;
        Polyhedron *next = Q->next;
        Q->next = NULL;

        if (Q->Dimension != nparam)
            QC = Polyhedron_Project(Q, nparam);

        if (!C)
            C = Q == QC ? Polyhedron_Copy(QC) : QC;
        else {
            Polyhedron *C2 = C;
            C = DomainIntersection(C, QC, MaxRays);
            Polyhedron_Free(C2);
            if (QC != Q)
                Polyhedron_Free(QC);
        }
        Q->next = next;
    }
    return C;
}

/*
 * Project on final dim dimensions
 */
Polyhedron* Polyhedron_Project(Polyhedron *P, int dim)
{
    int i;
    int remove = P->Dimension - dim;
    Matrix *T;
    Polyhedron *I;

    if (P->Dimension == dim)
        return Polyhedron_Copy(P);

    T = Matrix_Alloc(dim+1, P->Dimension+1);
    for (i = 0; i < dim+1; ++i)
        value_set_si(T->p[i][i+remove], 1);
    I = Polyhedron_Image(P, T, P->NbConstraints);
    Matrix_Free(T);
    return I;
}

/* Constructs a new constraint that ensures that
 * the first constraint is (strictly) smaller than
 * the second.
 */
static void smaller_constraint(Value *a, Value *b, Value *c, int pos, int shift,
                        int len, int strict, Value *tmp)
{
    value_oppose(*tmp, b[pos+1]);
    value_set_si(c[0], 1);
    Vector_Combine(a+1+shift, b+1+shift, c+1, *tmp, a[pos+1], len-shift-1);
    if (strict)
        value_decrement(c[len-shift-1], c[len-shift-1]);
    ConstraintSimplify(c, c, len-shift, tmp);
}


/* For each pair of lower and upper bounds on the first variable,
 * calls fn with the set of constraints on the remaining variables
 * where these bounds are active, i.e., (stricly) larger/smaller than
 * the other lower/upper bounds, the lower and upper bound and the
 * call back data.
 *
 * If the first variable is equal to an affine combination of the
 * other variables then fn is called with both lower and upper
 * pointing to the corresponding equality.
 *
 * If there is no lower (or upper) bound, then NULL is passed
 * as the corresponding bound.
 */
void for_each_lower_upper_bound(Polyhedron *P,
                                for_each_lower_upper_bound_init init,
                                for_each_lower_upper_bound_fn fn,
                                void *cb_data)
{
    unsigned dim = P->Dimension;
    Matrix *M;
    int *pos;
    int i, j, p, n, z;
    int k, l, k2, l2, q;
    Value g;

    if (value_zero_p(P->Constraint[0][0]) &&
            value_notzero_p(P->Constraint[0][1])) {
        M = Matrix_Alloc(P->NbConstraints-1, dim-1+2);
        for (i = 1; i < P->NbConstraints; ++i) {
            value_assign(M->p[i-1][0], P->Constraint[i][0]);
            Vector_Copy(P->Constraint[i]+2, M->p[i-1]+1, dim);
        }
        if (init)
            init(1, cb_data);
        fn(M, P->Constraint[0], P->Constraint[0], cb_data);
        Matrix_Free(M);
        return;
    }

    value_init(g);
    pos = ALLOCN(int, P->NbConstraints);

    for (i = 0, z = 0; i < P->NbConstraints; ++i)
        if (value_zero_p(P->Constraint[i][1]))
            pos[P->NbConstraints-1 - z++] = i;
    /* put those with positive coefficients first; number: p */
    for (i = 0, p = 0, n = P->NbConstraints-z-1; i < P->NbConstraints; ++i)
        if (value_pos_p(P->Constraint[i][1]))
            pos[p++] = i;
        else if (value_neg_p(P->Constraint[i][1]))
            pos[n--] = i;
    n = P->NbConstraints-z-p;

    if (init)
        init(p*n, cb_data);

    M = Matrix_Alloc((p ? p-1 : 0) + (n ? n-1 : 0) + z + 1, dim-1+2);
    for (i = 0; i < z; ++i) {
        value_assign(M->p[i][0], P->Constraint[pos[P->NbConstraints-1 - i]][0]);
        Vector_Copy(P->Constraint[pos[P->NbConstraints-1 - i]]+2,
                    M->p[i]+1, dim);
    }
    for (k = p ? 0 : -1; k < p; ++k) {
        for (k2 = 0; k2 < p; ++k2) {
            if (k2 == k)
                continue;
            q = 1 + z + k2 - (k2 > k);
            smaller_constraint(
                P->Constraint[pos[k]],
                P->Constraint[pos[k2]],
                M->p[q], 0, 1, dim+2, k2 > k, &g);
        }
        for (l = n ? p : p-1; l < p+n; ++l) {
            Value *lower;
            Value *upper;
            for (l2 = p; l2 < p+n; ++l2) {
                if (l2 == l)
                    continue;
                q = 1 + z + l2-1 - (l2 > l);
                smaller_constraint(
                    P->Constraint[pos[l2]],
                    P->Constraint[pos[l]],
                    M->p[q], 0, 1, dim+2, l2 > l, &g);
            }
            if (p && n)
                smaller_constraint(P->Constraint[pos[k]],
                                    P->Constraint[pos[l]],
                                    M->p[z], 0, 1, dim+2, 0, &g);
            lower = p ? P->Constraint[pos[k]] : NULL;
            upper = n ? P->Constraint[pos[l]] : NULL;
            fn(M, lower, upper, cb_data);
        }
    }
    Matrix_Free(M);

    free(pos);
    value_clear(g);
}

struct section { Polyhedron * D; evalue E; };

struct PLL_data {
    int nd;
    unsigned MaxRays;
    Polyhedron *C;
    evalue mone;
    struct section *s;
};

static void PLL_init(unsigned n, void *cb_data)
{
    struct PLL_data *data = (struct PLL_data *)cb_data;

    data->s = ALLOCN(struct section, n);
}

/* Computes ceil(-coef/abs(d)) */
static evalue* bv_ceil3(Value *coef, int len, Value d, Polyhedron *P)
{
    Value t;
    evalue *EP, *f;
    Vector *val = Vector_Alloc(len);

    value_init(t);
    Vector_Oppose(coef, val->p, len);
    value_absolute(t, d);

    EP = ceiling(val->p, t, len-1, P);

    value_clear(t);
    Vector_Free(val);

    return EP;
}

static void PLL_cb(Matrix *M, Value *lower, Value *upper, void *cb_data)
{
    struct PLL_data *data = (struct PLL_data *)cb_data;
    unsigned dim = M->NbColumns-1;
    Matrix *M2;
    Polyhedron *T;
    evalue *L, *U;

    assert(lower);
    assert(upper);

    M2 = Matrix_Copy(M);
    T = Constraints2Polyhedron(M2, data->MaxRays);
    Matrix_Free(M2);
    data->s[data->nd].D = DomainIntersection(T, data->C, data->MaxRays);
    Domain_Free(T);

    POL_ENSURE_VERTICES(data->s[data->nd].D);
    if (emptyQ(data->s[data->nd].D)) {
        Polyhedron_Free(data->s[data->nd].D);
        return;
    }
    L = bv_ceil3(lower+1+1, dim-1+1, lower[0+1], data->s[data->nd].D);
    U = bv_ceil3(upper+1+1, dim-1+1, upper[0+1], data->s[data->nd].D);
    eadd(L, U);
    eadd(&data->mone, U);
    emul(&data->mone, U);
    data->s[data->nd].E = *U;
    evalue_free(L);
    free(U);
    ++data->nd;
}

static evalue *ParamLine_Length_mod(Polyhedron *P, Polyhedron *C, unsigned MaxRays)
{
    unsigned dim = P->Dimension;
    unsigned nvar = dim - C->Dimension;
    struct PLL_data data;
    evalue *F;
    int k;

    assert(nvar == 1);

    value_init(data.mone.d);
    evalue_set_si(&data.mone, -1, 1);

    data.nd = 0;
    data.MaxRays = MaxRays;
    data.C = C;
    for_each_lower_upper_bound(P, PLL_init, PLL_cb, &data);

    F = ALLOC(evalue);
    value_init(F->d);
    value_set_si(F->d, 0);
    F->x.p = new_enode(partition, 2*data.nd, dim-nvar);
    for (k = 0; k < data.nd; ++k) {
        EVALUE_SET_DOMAIN(F->x.p->arr[2*k], data.s[k].D);
        value_clear(F->x.p->arr[2*k+1].d);
        F->x.p->arr[2*k+1] = data.s[k].E;
    }
    free(data.s);

    free_evalue_refs(&data.mone); 

    return F;
}

evalue* ParamLine_Length(Polyhedron *P, Polyhedron *C,
                         struct barvinok_options *options)
{
    evalue* tmp;
    tmp = ParamLine_Length_mod(P, C, options->MaxRays);
    if (options->lookup_table) {
        evalue_mod2table(tmp, C->Dimension);
        reduce_evalue(tmp);
    }
    return tmp;
}

Bool isIdentity(Matrix *M)
{
    unsigned i, j;
    if (M->NbRows != M->NbColumns)
        return False;

    for (i = 0;i < M->NbRows; i ++)
        for (j = 0; j < M->NbColumns; j ++)
            if (i == j) {
                if(value_notone_p(M->p[i][j]))
                    return False;
            } else {
                if(value_notzero_p(M->p[i][j]))
                    return False;
            }
    return True;
}

void Param_Polyhedron_Print(FILE* DST, Param_Polyhedron *PP,
                                const char **param_names)
{
  Param_Domain *P;
  Param_Vertices *V;

  for(P=PP->D;P;P=P->next) {
    
    /* prints current val. dom. */
    fprintf(DST, "---------------------------------------\n");
    fprintf(DST, "Domain :\n");
    Print_Domain(DST, P->Domain, param_names);
    
    /* scan the vertices */
    fprintf(DST, "Vertices :\n");
    FORALL_PVertex_in_ParamPolyhedron(V,P,PP) {
        
      /* prints each vertex */
      Print_Vertex(DST, V->Vertex, param_names);
      fprintf(DST, "\n");
    }
    END_FORALL_PVertex_in_ParamPolyhedron;
  }
}

void Enumeration_Print(FILE *Dst, Enumeration *en, const char **params)
{
    for (; en; en = en->next) {
        Print_Domain(Dst, en->ValidityDomain, params);
        print_evalue(Dst, &en->EP, params);
    }
}

void Enumeration_Free(Enumeration *en)
{
  Enumeration *ee;

  while( en )
  {
          free_evalue_refs( &(en->EP) );
          Domain_Free( en->ValidityDomain );
          ee = en ->next;
          free( en );
          en = ee;
  }
}

void Enumeration_mod2table(Enumeration *en, unsigned nparam)
{
    for (; en; en = en->next) {
        evalue_mod2table(&en->EP, nparam);
        reduce_evalue(&en->EP);
    }
}

size_t Enumeration_size(Enumeration *en)
{
    size_t s = 0;

    for (; en; en = en->next) {
        s += domain_size(en->ValidityDomain);
        s += evalue_size(&en->EP);
    }
    return s;
}

/* Check whether every set in D2 is included in some set of D1 */
int DomainIncludes(Polyhedron *D1, Polyhedron *D2)
{
    for ( ; D2; D2 = D2->next) {
        Polyhedron *P1;
        for (P1 = D1; P1; P1 = P1->next)
            if (PolyhedronIncludes(P1, D2))
                break;
        if (!P1)
            return 0;
    }
    return 1;
}

int line_minmax(Polyhedron *I, Value *min, Value *max)
{
    int i;

    if (I->NbEq >= 1) {
        value_oppose(I->Constraint[0][2], I->Constraint[0][2]);
        /* There should never be a remainder here */
        if (value_pos_p(I->Constraint[0][1]))
            mpz_fdiv_q(*min, I->Constraint[0][2], I->Constraint[0][1]);
        else
            mpz_fdiv_q(*min, I->Constraint[0][2], I->Constraint[0][1]);
        value_assign(*max, *min);
    } else for (i = 0; i < I->NbConstraints; ++i) {
        if (value_zero_p(I->Constraint[i][1])) {
            Polyhedron_Free(I);
            return 0;
        }

        value_oppose(I->Constraint[i][2], I->Constraint[i][2]);
        if (value_pos_p(I->Constraint[i][1]))
            mpz_cdiv_q(*min, I->Constraint[i][2], I->Constraint[i][1]);
        else
            mpz_fdiv_q(*max, I->Constraint[i][2], I->Constraint[i][1]);
    }
    Polyhedron_Free(I);
    return 1;
}

int DomainContains(Polyhedron *P, Value *list_args, int len, 
                   unsigned MaxRays, int set)
{
    int i;
    Value m;

    if (P->Dimension == len)
        return in_domain(P, list_args);

    assert(set); // assume list_args is large enough
    assert((P->Dimension - len) % 2 == 0);
    value_init(m);
    for (i = 0; i < P->Dimension - len; i += 2) {
        int j, k;
        for (j = 0 ; j < P->NbEq; ++j)
            if (value_notzero_p(P->Constraint[j][1+len+i]))
                break;
        assert(j < P->NbEq);
        value_absolute(m, P->Constraint[j][1+len+i]);
        k = First_Non_Zero(P->Constraint[j]+1, len);
        assert(k != -1);
        assert(First_Non_Zero(P->Constraint[j]+1+k+1, len - k - 1) == -1);
        mpz_fdiv_q(list_args[len+i], list_args[k], m);
        mpz_fdiv_r(list_args[len+i+1], list_args[k], m);
    }
    value_clear(m);

    return in_domain(P, list_args);
}

Polyhedron *DomainConcat(Polyhedron *head, Polyhedron *tail)
{
    Polyhedron *S;
    if (!head)
        return tail;
    for (S = head; S->next; S = S->next)
        ;
    S->next = tail;
    return head;
}

evalue *barvinok_lexsmaller_ev(Polyhedron *P, Polyhedron *D, unsigned dim,
                               Polyhedron *C, unsigned MaxRays)
{
    evalue *ranking;
    Polyhedron *RC, *RD, *Q;
    unsigned nparam = dim + C->Dimension;
    unsigned exist;
    Polyhedron *CA;

    RC = LexSmaller(P, D, dim, C, MaxRays);
    RD = RC->next;
    RC->next = NULL;

    exist = RD->Dimension - nparam - dim;
    CA = align_context(RC, RD->Dimension, MaxRays);
    Q = DomainIntersection(RD, CA, MaxRays);
    Polyhedron_Free(CA);
    Domain_Free(RD);
    Polyhedron_Free(RC);
    RD = Q;

    for (Q = RD; Q; Q = Q->next) {
        evalue *t;
        Polyhedron *next = Q->next;
        Q->next = 0;

        t = barvinok_enumerate_e(Q, exist, nparam, MaxRays);

        if (Q == RD)
            ranking = t;
        else {
            eadd(t, ranking);
            evalue_free(t);
        }

        Q->next = next;
    }

    Domain_Free(RD);

    return ranking;
}

Enumeration *barvinok_lexsmaller(Polyhedron *P, Polyhedron *D, unsigned dim,
                                 Polyhedron *C, unsigned MaxRays)
{
    evalue *EP = barvinok_lexsmaller_ev(P, D, dim, C, MaxRays);

    return partition2enumeration(EP);
}

/* "align" matrix to have nrows by inserting
 * the necessary number of rows and an equal number of columns in front
 */
Matrix *align_matrix(Matrix *M, int nrows)
{
    int i;
    int newrows = nrows - M->NbRows;
    Matrix *M2 = Matrix_Alloc(nrows, newrows + M->NbColumns);
    for (i = 0; i < newrows; ++i)
        value_set_si(M2->p[i][i], 1);
    for (i = 0; i < M->NbRows; ++i)
        Vector_Copy(M->p[i], M2->p[newrows+i]+newrows, M->NbColumns);
    return M2;
}

static void print_varlist(FILE *out, int n, char **names)
{
    int i;
    fprintf(out, "[");
    for (i = 0; i < n; ++i) {
        if (i)
            fprintf(out, ",");
        fprintf(out, "%s", names[i]);
    }
    fprintf(out, "]");
}

static void print_term(FILE *out, Value v, int pos, int dim, int nparam,
                       char **iter_names, char **param_names, int *first)
{
    if (value_zero_p(v)) {
        if (first && *first && pos >= dim + nparam)
            fprintf(out, "0");
        return;
    }

    if (first) {
        if (!*first && value_pos_p(v))
            fprintf(out, "+");
        *first = 0;
    }
    if (pos < dim + nparam) {
        if (value_mone_p(v))
            fprintf(out, "-");
        else if (!value_one_p(v))
            value_print(out, VALUE_FMT, v);
        if (pos < dim)
            fprintf(out, "%s", iter_names[pos]);
        else
            fprintf(out, "%s", param_names[pos-dim]);
    } else
        value_print(out, VALUE_FMT, v);
}

char **util_generate_names(int n, const char *prefix)
{
    int i;
    int len = (prefix ? strlen(prefix) : 0) + 10;
    char **names = ALLOCN(char*, n);
    if (!names) {
        fprintf(stderr, "ERROR: memory overflow.\n");
        exit(1);
    }
    for (i = 0; i < n; ++i) {
        names[i] = ALLOCN(char, len);
        if (!names[i]) {
            fprintf(stderr, "ERROR: memory overflow.\n");
            exit(1);
        }
        if (!prefix)
            snprintf(names[i], len, "%d", i);
        else
            snprintf(names[i], len, "%s%d", prefix, i);
    }

    return names;
}

void util_free_names(int n, char **names)
{
    int i;
    for (i = 0; i < n; ++i)
        free(names[i]);
    free(names);
}

void Polyhedron_pprint(FILE *out, Polyhedron *P, int dim, int nparam,
                       char **iter_names, char **param_names)
{
    int i, j;
    Value tmp;

    assert(dim + nparam == P->Dimension);

    value_init(tmp);

    fprintf(out, "{ ");
    if (nparam) {
        print_varlist(out, nparam, param_names);
        fprintf(out, " -> ");
    }
    print_varlist(out, dim, iter_names);
    fprintf(out, " : ");

    if (emptyQ2(P))
        fprintf(out, "FALSE");
    else for (i = 0; i < P->NbConstraints; ++i) {
        int first = 1;
        int v = First_Non_Zero(P->Constraint[i]+1, P->Dimension);
        if (v == -1 && value_pos_p(P->Constraint[i][0]))
            continue;
        if (i)
            fprintf(out, " && ");
        if (v == -1 && value_notzero_p(P->Constraint[i][1+P->Dimension]))
            fprintf(out, "FALSE");
        else if (value_pos_p(P->Constraint[i][v+1])) {
            print_term(out, P->Constraint[i][v+1], v, dim, nparam, 
                       iter_names, param_names, NULL);
            if (value_zero_p(P->Constraint[i][0]))
                fprintf(out, " = ");
            else
                fprintf(out, " >= ");
            for (j = v+1; j <= dim+nparam; ++j) {
                value_oppose(tmp, P->Constraint[i][1+j]);
                print_term(out, tmp, j, dim, nparam, 
                           iter_names, param_names, &first);
            }
        } else {
            value_oppose(tmp, P->Constraint[i][1+v]);
            print_term(out, tmp, v, dim, nparam, 
                       iter_names, param_names, NULL);
            fprintf(out, " <= ");
            for (j = v+1; j <= dim+nparam; ++j)
                print_term(out, P->Constraint[i][1+j], j, dim, nparam, 
                           iter_names, param_names, &first);
        }
    }

    fprintf(out, " }\n");

    value_clear(tmp);
}

/* Construct a cone over P with P placed at x_d = 1, with
 * x_d the coordinate of an extra dimension
 *
 * It's probably a mistake to depend so much on the internal
 * representation.  We should probably simply compute the
 * vertices/facets first.
 */
Polyhedron *Cone_over_Polyhedron(Polyhedron *P)
{
    unsigned NbConstraints = 0;
    unsigned NbRays = 0;
    Polyhedron *C;
    int i;

    if (POL_HAS(P, POL_INEQUALITIES))
        NbConstraints = P->NbConstraints + 1;
    if (POL_HAS(P, POL_POINTS))
        NbRays = P->NbRays + 1;

    C = Polyhedron_Alloc(P->Dimension+1, NbConstraints, NbRays);
    if (POL_HAS(P, POL_INEQUALITIES)) {
        C->NbEq = P->NbEq;
        for (i = 0; i < P->NbConstraints; ++i)
            Vector_Copy(P->Constraint[i], C->Constraint[i], P->Dimension+2);
        /* n >= 0 */
        value_set_si(C->Constraint[P->NbConstraints][0], 1);
        value_set_si(C->Constraint[P->NbConstraints][1+P->Dimension], 1);
    }
    if (POL_HAS(P, POL_POINTS)) {
        C->NbBid = P->NbBid;
        for (i = 0; i < P->NbRays; ++i)
            Vector_Copy(P->Ray[i], C->Ray[i], P->Dimension+2);
        /* vertex 0 */
        value_set_si(C->Ray[P->NbRays][0], 1);
        value_set_si(C->Ray[P->NbRays][1+C->Dimension], 1);
    }
    POL_SET(C, POL_VALID);
    if (POL_HAS(P, POL_INEQUALITIES))
        POL_SET(C, POL_INEQUALITIES);
    if (POL_HAS(P, POL_POINTS))
        POL_SET(C, POL_POINTS);
    if (POL_HAS(P, POL_VERTICES))
        POL_SET(C, POL_VERTICES);
    return C;
}

/* Returns a (dim+nparam+1)x((dim-n)+nparam+1) matrix
 * mapping the transformed subspace back to the original space.
 * n is the number of equalities involving the variables
 * (i.e., not purely the parameters).
 * The remaining n coordinates in the transformed space would
 * have constant (parametric) values and are therefore not
 * included in the variables of the new space.
 */
Matrix *compress_variables(Matrix *Equalities, unsigned nparam)
{
    unsigned dim = (Equalities->NbColumns-2) - nparam;
    Matrix *M, *H, *Q, *U, *C, *ratH, *invH, *Ul, *T1, *T2, *T;
    Value mone;
    int n, i, j;
    int ok;

    for (n = 0; n < Equalities->NbRows; ++n)
        if (First_Non_Zero(Equalities->p[n]+1, dim) == -1)
            break;
    if (n == 0)
        return Identity(dim+nparam+1);
    value_init(mone);
    value_set_si(mone, -1);
    M = Matrix_Alloc(n, dim);
    C = Matrix_Alloc(n+1, nparam+1);
    for (i = 0; i < n; ++i) {
        Vector_Copy(Equalities->p[i]+1, M->p[i], dim);
        Vector_Scale(Equalities->p[i]+1+dim, C->p[i], mone, nparam+1);
    }
    value_set_si(C->p[n][nparam], 1);
    left_hermite(M, &H, &Q, &U);
    Matrix_Free(M);
    Matrix_Free(Q);
    value_clear(mone);

    ratH = Matrix_Alloc(n+1, n+1);
    invH = Matrix_Alloc(n+1, n+1);
    for (i = 0; i < n; ++i)
        Vector_Copy(H->p[i], ratH->p[i], n);
    value_set_si(ratH->p[n][n], 1);
    ok = Matrix_Inverse(ratH, invH);
    assert(ok);
    Matrix_Free(H);
    Matrix_Free(ratH);
    T1 = Matrix_Alloc(n+1, nparam+1);
    Matrix_Product(invH, C, T1);
    Matrix_Free(C);
    Matrix_Free(invH);
    if (value_notone_p(T1->p[n][nparam])) {
        for (i = 0; i < n; ++i) {
            if (!mpz_divisible_p(T1->p[i][nparam], T1->p[n][nparam])) {
                Matrix_Free(T1);
                Matrix_Free(U);
                return NULL;
            }
            /* compress_params should have taken care of this */
            for (j = 0; j < nparam; ++j)
                assert(mpz_divisible_p(T1->p[i][j], T1->p[n][nparam]));
            Vector_AntiScale(T1->p[i], T1->p[i], T1->p[n][nparam], nparam+1);
        }
        value_set_si(T1->p[n][nparam], 1);
    }
    Ul = Matrix_Alloc(dim+1, n+1);
    for (i = 0; i < dim; ++i)
        Vector_Copy(U->p[i], Ul->p[i], n);
    value_set_si(Ul->p[dim][n], 1);
    T2 = Matrix_Alloc(dim+1, nparam+1);
    Matrix_Product(Ul, T1, T2);
    Matrix_Free(Ul);
    Matrix_Free(T1);

    T = Matrix_Alloc(dim+nparam+1, (dim-n)+nparam+1);
    for (i = 0; i < dim; ++i) {
        Vector_Copy(U->p[i]+n, T->p[i], dim-n);
        Vector_Copy(T2->p[i], T->p[i]+dim-n, nparam+1);
    }
    for (i = 0; i < nparam+1; ++i)
        value_set_si(T->p[dim+i][(dim-n)+i], 1);
    assert(value_one_p(T2->p[dim][nparam]));
    Matrix_Free(U);
    Matrix_Free(T2);

    return T;
}

/* Computes the left inverse of an affine embedding M and, if Eq is not NULL,
 * the equalities that define the affine subspace onto which M maps
 * its argument.
 */
Matrix *left_inverse(Matrix *M, Matrix **Eq)
{
    int i, ok;
    Matrix *L, *H, *Q, *U, *ratH, *invH, *Ut, *inv;
    Vector *t;

    if (M->NbColumns == 1) {
        inv = Matrix_Alloc(1, M->NbRows);
        value_set_si(inv->p[0][M->NbRows-1], 1);
        if (Eq) {
            *Eq = Matrix_Alloc(M->NbRows-1, 1+(M->NbRows-1)+1);
            for (i = 0; i < M->NbRows-1; ++i) {
                value_oppose((*Eq)->p[i][1+i], M->p[M->NbRows-1][0]);
                value_assign((*Eq)->p[i][1+(M->NbRows-1)], M->p[i][0]);
            }
        }
        return inv;
    }
    if (Eq)
        *Eq = NULL;
    L = Matrix_Alloc(M->NbRows-1, M->NbColumns-1);
    for (i = 0; i < L->NbRows; ++i)
        Vector_Copy(M->p[i], L->p[i], L->NbColumns);
    right_hermite(L, &H, &U, &Q);
    Matrix_Free(L);
    Matrix_Free(Q);
    t = Vector_Alloc(U->NbColumns);
    for (i = 0; i < U->NbColumns; ++i)
        value_oppose(t->p[i], M->p[i][M->NbColumns-1]);
    if (Eq) {
        *Eq = Matrix_Alloc(H->NbRows - H->NbColumns, 2 + U->NbColumns);
        for (i = 0; i < H->NbRows - H->NbColumns; ++i) {
            Vector_Copy(U->p[H->NbColumns+i], (*Eq)->p[i]+1, U->NbColumns);
            Inner_Product(U->p[H->NbColumns+i], t->p, U->NbColumns,
                          (*Eq)->p[i]+1+U->NbColumns);
        }
    }
    ratH = Matrix_Alloc(H->NbColumns+1, H->NbColumns+1);
    invH = Matrix_Alloc(H->NbColumns+1, H->NbColumns+1);
    for (i = 0; i < H->NbColumns; ++i)
        Vector_Copy(H->p[i], ratH->p[i], H->NbColumns);
    value_set_si(ratH->p[ratH->NbRows-1][ratH->NbColumns-1], 1);
    Matrix_Free(H);
    ok = Matrix_Inverse(ratH, invH);
    assert(ok);
    Matrix_Free(ratH);
    Ut = Matrix_Alloc(invH->NbRows, U->NbColumns+1);
    for (i = 0; i < Ut->NbRows-1; ++i) {
        Vector_Copy(U->p[i], Ut->p[i], U->NbColumns);
        Inner_Product(U->p[i], t->p, U->NbColumns, &Ut->p[i][Ut->NbColumns-1]);
    }
    Matrix_Free(U);
    Vector_Free(t);
    value_set_si(Ut->p[Ut->NbRows-1][Ut->NbColumns-1], 1);
    inv = Matrix_Alloc(invH->NbRows, Ut->NbColumns);
    Matrix_Product(invH, Ut, inv);
    Matrix_Free(Ut);
    Matrix_Free(invH);
    return inv;
}

/* Check whether all rays are revlex positive in the parameters
 */
int Polyhedron_has_revlex_positive_rays(Polyhedron *P, unsigned nparam)
{
    int r;
    for (r = 0; r < P->NbRays; ++r) {
        int i;
        if (value_notzero_p(P->Ray[r][P->Dimension+1]))
            continue;
        for (i = P->Dimension-1; i >= P->Dimension-nparam; --i) {
            if (value_neg_p(P->Ray[r][i+1]))
                return 0;
            if (value_pos_p(P->Ray[r][i+1]))
                break;
        }
        /* A ray independent of the parameters */
        if (i < P->Dimension-nparam)
            return 0;
    }
    return 1;
}

static Polyhedron *Recession_Cone(Polyhedron *P, unsigned nparam, unsigned MaxRays)
{
    int i;
    unsigned nvar = P->Dimension - nparam;
    Matrix *M = Matrix_Alloc(P->NbConstraints, 1 + nvar + 1);
    Polyhedron *R;
    for (i = 0; i < P->NbConstraints; ++i)
        Vector_Copy(P->Constraint[i], M->p[i], 1+nvar);
    R = Constraints2Polyhedron(M, MaxRays);
    Matrix_Free(M);
    return R;
}

int Polyhedron_is_unbounded(Polyhedron *P, unsigned nparam, unsigned MaxRays)
{
    int i;
    int is_unbounded;
    Polyhedron *R = Recession_Cone(P, nparam, MaxRays);
    POL_ENSURE_VERTICES(R);
    if (R->NbBid == 0)
        for (i = 0; i < R->NbRays; ++i)
            if (value_zero_p(R->Ray[i][1+R->Dimension]))
                break;
    is_unbounded = R->NbBid > 0 || i < R->NbRays;
    Polyhedron_Free(R);
    return is_unbounded;
}

static void SwapColumns(Value **V, int n, int i, int j)
{
    int r;

    for (r = 0; r < n; ++r)
        value_swap(V[r][i], V[r][j]);
}

void Polyhedron_ExchangeColumns(Polyhedron *P, int Column1, int Column2)
{
    SwapColumns(P->Constraint, P->NbConstraints, Column1, Column2);
    SwapColumns(P->Ray, P->NbRays, Column1, Column2);
    if (P->NbEq) {
        Matrix M;
        Polyhedron_Matrix_View(P, &M, P->NbConstraints);
        Gauss(&M, P->NbEq, P->Dimension+1);
    }
}

/* perform transposition inline; assumes M is a square matrix */
void Matrix_Transposition(Matrix *M)
{
    int i, j;

    assert(M->NbRows == M->NbColumns);
    for (i = 0; i < M->NbRows; ++i)
        for (j = i+1; j < M->NbColumns; ++j)
            value_swap(M->p[i][j], M->p[j][i]);
}

/* Matrix "view" of first rows rows */
void Polyhedron_Matrix_View(Polyhedron *P, Matrix *M, unsigned rows)
{
    M->NbRows = rows;
    M->NbColumns = P->Dimension+2;
    M->p_Init = P->p_Init;
    M->p = P->Constraint;
}

int Last_Non_Zero(Value *p, unsigned len)
{
        int i;

        for (i = len - 1; i >= 0; --i)
                if (value_notzero_p(p[i]))
                        return i;

        return -1;
}