isl_equalities.c

Go to the documentation of this file.

/*
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege, K.U.Leuven, Departement
 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
 */
 
#include <isl_mat_private.h>
#include <isl_vec_private.h>
#include <isl_seq.h>
#include "isl_map_private.h"
#include "isl_equalities.h"
#include <isl_val_private.h>
 
/* Given a set of modulo constraints
 *
 *    c + A y = 0 mod d
 *
 * this function computes a particular solution y_0
 *
 * The input is given as a matrix B = [ c A ] and a vector d.
 *
 * The output is matrix containing the solution y_0 or
 * a zero-column matrix if the constraints admit no integer solution.
 *
 * The given set of constrains is equivalent to
 *
 *    c + A y = -D x
 *
 * with D = diag d and x a fresh set of variables.
 * Reducing both c and A modulo d does not change the
 * value of y in the solution and may lead to smaller coefficients.
 * Let M = [ D A ] and [ H 0 ] = M U, the Hermite normal form of M.
 * Then
 *      [ x ]
 *    M [ y ] = - c
 * and so
 *                   [ x ]
 *    [ H 0 ] U^{-1} [ y ] = - c
 * Let
 *    [ A ]          [ x ]
 *    [ B ] = U^{-1} [ y ]
 * then
 *    H A + 0 B = -c
 *
 * so B may be chosen arbitrarily, e.g., B = 0, and then
 *
 *           [ x ] = [ -c ]
 *    U^{-1} [ y ] = [  0 ]
 * or
 *    [ x ]     [ -c ]
 *    [ y ] = U [  0 ]
 * specifically,
 *
 *    y = U_{2,1} (-c)
 *
 * If any of the coordinates of this y are non-integer
 * then the constraints admit no integer solution and
 * a zero-column matrix is returned.
 */
static __isl_give isl_mat *particular_solution(__isl_keep isl_mat *B,
  __isl_keep isl_vec *d)
{
  int i, j;
  struct isl_mat *M = NULL;
  struct isl_mat *C = NULL;
  struct isl_mat *U = NULL;
  struct isl_mat *H = NULL;
  struct isl_mat *cst = NULL;
  struct isl_mat *T = NULL;
 
  M = isl_mat_alloc(B->ctx, B->n_row, B->n_row + B->n_col - 1);
  C = isl_mat_alloc(B->ctx, 1 + B->n_row, 1);
  if (!M || !C)
    goto error;
  isl_int_set_si(C->row[0][0], 1);
  for (i = 0; i < B->n_row; ++i) {
    isl_seq_clr(M->row[i], B->n_row);
    isl_int_set(M->row[i][i], d->block.data[i]);
    isl_int_neg(C->row[1 + i][0], B->row[i][0]);
    isl_int_fdiv_r(C->row[1+i][0], C->row[1+i][0], M->row[i][i]);
    for (j = 0; j < B->n_col - 1; ++j)
      isl_int_fdiv_r(M->row[i][B->n_row + j],
          B->row[i][1 + j], M->row[i][i]);
  }
  M = isl_mat_left_hermite(M, 0, &U, NULL);
  if (!M || !U)
    goto error;
  H = isl_mat_sub_alloc(M, 0, B->n_row, 0, B->n_row);
  H = isl_mat_lin_to_aff(H);
  C = isl_mat_inverse_product(H, C);
  if (!C)
    goto error;
  for (i = 0; i < B->n_row; ++i) {
    if (!isl_int_is_divisible_by(C->row[1+i][0], C->row[0][0]))
      break;
    isl_int_divexact(C->row[1+i][0], C->row[1+i][0], C->row[0][0]);
  }
  if (i < B->n_row)
    cst = isl_mat_alloc(B->ctx, B->n_row, 0);
  else
    cst = isl_mat_sub_alloc(C, 1, B->n_row, 0, 1);
  T = isl_mat_sub_alloc(U, B->n_row, B->n_col - 1, 0, B->n_row);
  cst = isl_mat_product(T, cst);
  isl_mat_free(M);
  isl_mat_free(C);
  isl_mat_free(U);
  return cst;
error:
  isl_mat_free(M);
  isl_mat_free(C);
  isl_mat_free(U);
  return NULL;
}
 
/* Compute and return the matrix
 *
 *    U_1^{-1} diag(d_1, 1, ..., 1)
 *
 * with U_1 the unimodular completion of the first (and only) row of B.
 * The columns of this matrix generate the lattice that satisfies
 * the single (linear) modulo constraint.
 */
static __isl_take isl_mat *parameter_compression_1(__isl_keep isl_mat *B,
  __isl_keep isl_vec *d)
{
  struct isl_mat *U;
 
  U = isl_mat_alloc(B->ctx, B->n_col - 1, B->n_col - 1);
  if (!U)
    return NULL;
  isl_seq_cpy(U->row[0], B->row[0] + 1, B->n_col - 1);
  U = isl_mat_unimodular_complete(U, 1);
  U = isl_mat_right_inverse(U);
  if (!U)
    return NULL;
  isl_mat_col_mul(U, 0, d->block.data[0], 0);
  U = isl_mat_lin_to_aff(U);
  return U;
}
 
/* Compute a common lattice of solutions to the linear modulo
 * constraints specified by B and d.
 * See also the documentation of isl_mat_parameter_compression.
 * We put the matrix
 * 
 *    A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 *
 * on a common denominator.  This denominator D is the lcm of modulos d.
 * Since L_i = U_i^{-1} diag(d_i, 1, ... 1), we have
 * L_i^{-T} = U_i^T diag(d_i, 1, ... 1)^{-T} = U_i^T diag(1/d_i, 1, ..., 1).
 * Putting this on the common denominator, we have
 * D * L_i^{-T} = U_i^T diag(D/d_i, D, ..., D).
 */
static __isl_give isl_mat *parameter_compression_multi(__isl_keep isl_mat *B,
  __isl_keep isl_vec *d)
{
  int i, j, k;
  isl_int D;
  struct isl_mat *A = NULL, *U = NULL;
  struct isl_mat *T;
  unsigned size;
 
  isl_int_init(D);
 
  isl_vec_lcm(d, &D);
 
  size = B->n_col - 1;
  A = isl_mat_alloc(B->ctx, size, B->n_row * size);
  U = isl_mat_alloc(B->ctx, size, size);
  if (!U || !A)
    goto error;
  for (i = 0; i < B->n_row; ++i) {
    isl_seq_cpy(U->row[0], B->row[i] + 1, size);
    U = isl_mat_unimodular_complete(U, 1);
    if (!U)
      goto error;
    isl_int_divexact(D, D, d->block.data[i]);
    for (k = 0; k < U->n_col; ++k)
      isl_int_mul(A->row[k][i*size+0], D, U->row[0][k]);
    isl_int_mul(D, D, d->block.data[i]);
    for (j = 1; j < U->n_row; ++j)
      for (k = 0; k < U->n_col; ++k)
        isl_int_mul(A->row[k][i*size+j],
            D, U->row[j][k]);
  }
  A = isl_mat_left_hermite(A, 0, NULL, NULL);
  T = isl_mat_sub_alloc(A, 0, A->n_row, 0, A->n_row);
  T = isl_mat_lin_to_aff(T);
  if (!T)
    goto error;
  isl_int_set(T->row[0][0], D);
  T = isl_mat_right_inverse(T);
  if (!T)
    goto error;
  isl_assert(T->ctx, isl_int_is_one(T->row[0][0]), goto error);
  T = isl_mat_transpose(T);
  isl_mat_free(A);
  isl_mat_free(U);
 
  isl_int_clear(D);
  return T;
error:
  isl_mat_free(A);
  isl_mat_free(U);
  isl_int_clear(D);
  return NULL;
}
 
/* Given a set of modulo constraints
 *
 *    c + A y = 0 mod d
 *
 * this function returns an affine transformation T,
 *
 *    y = T y'
 *
 * that bijectively maps the integer vectors y' to integer
 * vectors y that satisfy the modulo constraints.
 *
 * This function is inspired by Section 2.5.3
 * of B. Meister, "Stating and Manipulating Periodicity in the Polytope
 * Model.  Applications to Program Analysis and Optimization".
 * However, the implementation only follows the algorithm of that
 * section for computing a particular solution and not for computing
 * a general homogeneous solution.  The latter is incomplete and
 * may remove some valid solutions.
 * Instead, we use an adaptation of the algorithm in Section 7 of
 * B. Meister, S. Verdoolaege, "Polynomial Approximations in the Polytope
 * Model: Bringing the Power of Quasi-Polynomials to the Masses".
 *
 * The input is given as a matrix B = [ c A ] and a vector d.
 * Each element of the vector d corresponds to a row in B.
 * The output is a lower triangular matrix.
 * If no integer vector y satisfies the given constraints then
 * a matrix with zero columns is returned.
 *
 * We first compute a particular solution y_0 to the given set of
 * modulo constraints in particular_solution.  If no such solution
 * exists, then we return a zero-columned transformation matrix.
 * Otherwise, we compute the generic solution to
 *
 *    A y = 0 mod d
 *
 * That is we want to compute G such that
 *
 *    y = G y''
 *
 * with y'' integer, describes the set of solutions.
 *
 * We first remove the common factors of each row.
 * In particular if gcd(A_i,d_i) != 1, then we divide the whole
 * row i (including d_i) by this common factor.  If afterwards gcd(A_i) != 1,
 * then we divide this row of A by the common factor, unless gcd(A_i) = 0.
 * In the later case, we simply drop the row (in both A and d).
 *
 * If there are no rows left in A, then G is the identity matrix. Otherwise,
 * for each row i, we now determine the lattice of integer vectors
 * that satisfies this row.  Let U_i be the unimodular extension of the
 * row A_i.  This unimodular extension exists because gcd(A_i) = 1.
 * The first component of
 *
 *    y' = U_i y
 *
 * needs to be a multiple of d_i.  Let y' = diag(d_i, 1, ..., 1) y''.
 * Then,
 *
 *    y = U_i^{-1} diag(d_i, 1, ..., 1) y''
 *
 * for arbitrary integer vectors y''.  That is, y belongs to the lattice
 * generated by the columns of L_i = U_i^{-1} diag(d_i, 1, ..., 1).
 * If there is only one row, then G = L_1.
 *
 * If there is more than one row left, we need to compute the intersection
 * of the lattices.  That is, we need to compute an L such that
 *
 *    L = L_i L_i'  for all i
 *
 * with L_i' some integer matrices.  Let A be constructed as follows
 *
 *    A = [ L_1^{-T} L_2^{-T} ... L_k^{-T} ]
 *
 * and computed the Hermite Normal Form of A = [ H 0 ] U
 * Then,
 *
 *    L_i^{-T} = H U_{1,i}
 *
 * or
 *
 *    H^{-T} = L_i U_{1,i}^T
 *
 * In other words G = L = H^{-T}.
 * To ensure that G is lower triangular, we compute and use its Hermite
 * normal form.
 *
 * The affine transformation matrix returned is then
 *
 *    [  1   0  ]
 *    [ y_0  G  ]
 *
 * as any y = y_0 + G y' with y' integer is a solution to the original
 * modulo constraints.
 */
__isl_give isl_mat *isl_mat_parameter_compression(__isl_take isl_mat *B,
  __isl_take isl_vec *d)
{
  int i;
  struct isl_mat *cst = NULL;
  struct isl_mat *T = NULL;
  isl_int D;
 
  if (!B || !d)
    goto error;
  isl_assert(B->ctx, B->n_row == d->size, goto error);
  cst = particular_solution(B, d);
  if (!cst)
    goto error;
  if (cst->n_col == 0) {
    T = isl_mat_alloc(B->ctx, B->n_col, 0);
    isl_mat_free(cst);
    isl_mat_free(B);
    isl_vec_free(d);
    return T;
  }
  isl_int_init(D);
  /* Replace a*g*row = 0 mod g*m by row = 0 mod m */
  for (i = 0; i < B->n_row; ++i) {
    isl_seq_gcd(B->row[i] + 1, B->n_col - 1, &D);
    if (isl_int_is_one(D))
      continue;
    if (isl_int_is_zero(D)) {
      B = isl_mat_drop_rows(B, i, 1);
      d = isl_vec_cow(d);
      if (!B || !d)
        goto error2;
      isl_seq_cpy(d->block.data+i, d->block.data+i+1,
              d->size - (i+1));
      d->size--;
      i--;
      continue;
    }
    B = isl_mat_cow(B);
    if (!B)
      goto error2;
    isl_seq_scale_down(B->row[i] + 1, B->row[i] + 1, D, B->n_col-1);
    isl_int_gcd(D, D, d->block.data[i]);
    d = isl_vec_cow(d);
    if (!d)
      goto error2;
    isl_int_divexact(d->block.data[i], d->block.data[i], D);
  }
  isl_int_clear(D);
  if (B->n_row == 0)
    T = isl_mat_identity(B->ctx, B->n_col);
  else if (B->n_row == 1)
    T = parameter_compression_1(B, d);
  else
    T = parameter_compression_multi(B, d);
  T = isl_mat_left_hermite(T, 0, NULL, NULL);
  if (!T)
    goto error;
  isl_mat_sub_copy(T->ctx, T->row + 1, cst->row, cst->n_row, 0, 0, 1);
  isl_mat_free(cst);
  isl_mat_free(B);
  isl_vec_free(d);
  return T;
error2:
  isl_int_clear(D);
error:
  isl_mat_free(cst);
  isl_mat_free(B);
  isl_vec_free(d);
  return NULL;
}
 
/* Given a set of equalities
 *
 *    B(y) + A x = 0            (*)
 *
 * compute and return an affine transformation T,
 *
 *    y = T y'
 *
 * that bijectively maps the integer vectors y' to integer
 * vectors y that satisfy the modulo constraints for some value of x.
 *
 * Let [H 0] be the Hermite Normal Form of A, i.e.,
 *
 *    A = [H 0] Q
 *
 * Then y is a solution of (*) iff
 *
 *    H^-1 B(y) (= - [I 0] Q x)
 *
 * is an integer vector.  Let d be the common denominator of H^-1.
 * We impose
 *
 *    d H^-1 B(y) = 0 mod d
 *
 * and compute the solution using isl_mat_parameter_compression.
 */
__isl_give isl_mat *isl_mat_parameter_compression_ext(__isl_take isl_mat *B,
  __isl_take isl_mat *A)
{
  isl_ctx *ctx;
  isl_vec *d;
  int n_row, n_col;
 
  if (!A)
    return isl_mat_free(B);
 
  ctx = isl_mat_get_ctx(A);
  n_row = A->n_row;
  n_col = A->n_col;
  A = isl_mat_left_hermite(A, 0, NULL, NULL);
  A = isl_mat_drop_cols(A, n_row, n_col - n_row);
  A = isl_mat_lin_to_aff(A);
  A = isl_mat_right_inverse(A);
  d = isl_vec_alloc(ctx, n_row);
  if (A)
    d = isl_vec_set(d, A->row[0][0]);
  A = isl_mat_drop_rows(A, 0, 1);
  A = isl_mat_drop_cols(A, 0, 1);
  B = isl_mat_product(A, B);
 
  return isl_mat_parameter_compression(B, d);
}
 
/* Return a compression matrix that indicates that there are no solutions
 * to the original constraints.  In particular, return a zero-column
 * matrix with 1 + dim rows.  If "T2" is not NULL, then assign *T2
 * the inverse of this matrix.  *T2 may already have been assigned
 * matrix, so free it first.
 * "free1", "free2" and "free3" are temporary matrices that are
 * not useful when an empty compression is returned.  They are
 * simply freed.
 */
static __isl_give isl_mat *empty_compression(isl_ctx *ctx, unsigned dim,
  __isl_give isl_mat **T2, __isl_take isl_mat *free1,
  __isl_take isl_mat *free2, __isl_take isl_mat *free3)
{
  isl_mat_free(free1);
  isl_mat_free(free2);
  isl_mat_free(free3);
  if (T2) {
    isl_mat_free(*T2);
    *T2 = isl_mat_alloc(ctx, 0, 1 + dim);
  }
  return isl_mat_alloc(ctx, 1 + dim, 0);
}
 
/* Given a matrix that maps a (possibly) parametric domain to
 * a parametric domain, add in rows that map the "nparam" parameters onto
 * themselves.
 */
static __isl_give isl_mat *insert_parameter_rows(__isl_take isl_mat *mat,
  unsigned nparam)
{
  int i;
 
  if (nparam == 0)
    return mat;
  if (!mat)
    return NULL;
 
  mat = isl_mat_insert_rows(mat, 1, nparam);
  if (!mat)
    return NULL;
 
  for (i = 0; i < nparam; ++i) {
    isl_seq_clr(mat->row[1 + i], mat->n_col);
    isl_int_set(mat->row[1 + i][1 + i], mat->row[0][0]);
  }
 
  return mat;
}
 
/* Given a set of equalities
 *
 *    -C(y) + M x = 0
 *
 * this function computes a unimodular transformation from a lower-dimensional
 * space to the original space that bijectively maps the integer points x'
 * in the lower-dimensional space to the integer points x in the original
 * space that satisfy the equalities.
 *
 * The input is given as a matrix B = [ -C M ] and the output is a
 * matrix that maps [1 x'] to [1 x].
 * The number of equality constraints in B is assumed to be smaller than
 * or equal to the number of variables x.
 * "first" is the position of the first x variable.
 * The preceding variables are considered to be y-variables.
 * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 *
 * First compute the (left) Hermite normal form of M,
 *
 *    M [U1 U2] = M U = H = [H1 0]
 * or
 *                  M = H Q = [H1 0] [Q1]
 *                                             [Q2]
 *
 * with U, Q unimodular, Q = U^{-1} (and H lower triangular).
 * Define the transformed variables as
 *
 *    x = [U1 U2] [ x1' ] = [U1 U2] [Q1] x
 *                [ x2' ]           [Q2]
 *
 * The equalities then become
 *
 *    -C(y) + H1 x1' = 0   or   x1' = H1^{-1} C(y) = C'(y)
 *
 * If the denominator of the constant term does not divide the
 * the common denominator of the coefficients of y, then every
 * integer point is mapped to a non-integer point and then the original set
 * has no integer solutions (since the x' are a unimodular transformation
 * of the x).  In this case, a zero-column matrix is returned.
 * Otherwise, the transformation is given by
 *
 *    x = U1 H1^{-1} C(y) + U2 x2'
 *
 * The inverse transformation is simply
 *
 *    x2' = Q2 x
 */
__isl_give isl_mat *isl_mat_final_variable_compression(__isl_take isl_mat *B,
  int first, __isl_give isl_mat **T2)
{
  int i, n;
  isl_ctx *ctx;
  isl_mat *H = NULL, *C, *H1, *U = NULL, *U1, *U2;
  unsigned dim;
 
  if (T2)
    *T2 = NULL;
  if (!B)
    goto error;
 
  ctx = isl_mat_get_ctx(B);
  dim = B->n_col - 1;
  n = dim - first;
  if (n < B->n_row)
    isl_die(ctx, isl_error_invalid, "too many equality constraints",
      goto error);
  H = isl_mat_sub_alloc(B, 0, B->n_row, 1 + first, n);
  H = isl_mat_left_hermite(H, 0, &U, T2);
  if (!H || !U || (T2 && !*T2))
    goto error;
  if (T2) {
    *T2 = isl_mat_drop_rows(*T2, 0, B->n_row);
    *T2 = isl_mat_diagonal(isl_mat_identity(ctx, 1 + first), *T2);
    if (!*T2)
      goto error;
  }
  C = isl_mat_alloc(ctx, 1 + B->n_row, 1 + first);
  if (!C)
    goto error;
  isl_int_set_si(C->row[0][0], 1);
  isl_seq_clr(C->row[0] + 1, first);
  isl_mat_sub_neg(ctx, C->row + 1, B->row, B->n_row, 0, 0, 1 + first);
  H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
  H1 = isl_mat_lin_to_aff(H1);
  C = isl_mat_inverse_product(H1, C);
  if (!C)
    goto error;
  isl_mat_free(H);
  if (!isl_int_is_one(C->row[0][0])) {
    isl_int g;
 
    isl_int_init(g);
    for (i = 0; i < B->n_row; ++i) {
      isl_seq_gcd(C->row[1 + i] + 1, first, &g);
      isl_int_gcd(g, g, C->row[0][0]);
      if (!isl_int_is_divisible_by(C->row[1 + i][0], g))
        break;
    }
    isl_int_clear(g);
 
    if (i < B->n_row)
      return empty_compression(ctx, dim, T2, B, C, U);
    C = isl_mat_normalize(C);
  }
  U1 = isl_mat_sub_alloc(U, 0, U->n_row, 0, B->n_row);
  U1 = isl_mat_lin_to_aff(U1);
  U2 = isl_mat_sub_alloc(U, 0, U->n_row, B->n_row, U->n_row - B->n_row);
  U2 = isl_mat_lin_to_aff(U2);
  isl_mat_free(U);
  C = isl_mat_product(U1, C);
  C = isl_mat_aff_direct_sum(C, U2);
  C = insert_parameter_rows(C, first);
 
  isl_mat_free(B);
 
  return C;
error:
  isl_mat_free(B);
  isl_mat_free(H);
  isl_mat_free(U);
  if (T2) {
    isl_mat_free(*T2);
    *T2 = NULL;
  }
  return NULL;
}
 
/* Given a set of equalities
 *
 *    M x - c = 0
 *
 * this function computes a unimodular transformation from a lower-dimensional
 * space to the original space that bijectively maps the integer points x'
 * in the lower-dimensional space to the integer points x in the original
 * space that satisfy the equalities.
 *
 * The input is given as a matrix B = [ -c M ] and the output is a
 * matrix that maps [1 x'] to [1 x].
 * The number of equality constraints in B is assumed to be smaller than
 * or equal to the number of variables x.
 * If T2 is not NULL, then *T2 is set to a matrix mapping [1 x] to [1 x'].
 */
__isl_give isl_mat *isl_mat_variable_compression(__isl_take isl_mat *B,
  __isl_give isl_mat **T2)
{
  return isl_mat_final_variable_compression(B, 0, T2);
}
 
/* Return "bset" and set *T and *T2 to the identity transformation
 * on "bset" (provided T and T2 are not NULL).
 */
static __isl_give isl_basic_set *return_with_identity(
  __isl_take isl_basic_set *bset, __isl_give isl_mat **T,
  __isl_give isl_mat **T2)
{
  isl_size dim;
  isl_mat *id;
 
  dim = isl_basic_set_dim(bset, isl_dim_set);
  if (dim < 0)
    return isl_basic_set_free(bset);
  if (!T && !T2)
    return bset;
 
  id = isl_mat_identity(isl_basic_map_get_ctx(bset), 1 + dim);
  if (T)
    *T = isl_mat_copy(id);
  if (T2)
    *T2 = isl_mat_copy(id);
  isl_mat_free(id);
 
  return bset;
}
 
/* Use the n equalities of bset to unimodularly transform the
 * variables x such that n transformed variables x1' have a constant value
 * and rewrite the constraints of bset in terms of the remaining
 * transformed variables x2'.  The matrix pointed to by T maps
 * the new variables x2' back to the original variables x, while T2
 * maps the original variables to the new variables.
 */
static __isl_give isl_basic_set *compress_variables(
  __isl_take isl_basic_set *bset,
  __isl_give isl_mat **T, __isl_give isl_mat **T2)
{
  struct isl_mat *B, *TC;
  isl_size dim;
 
  if (T)
    *T = NULL;
  if (T2)
    *T2 = NULL;
  if (isl_basic_set_check_no_params(bset) < 0 ||
      isl_basic_set_check_no_locals(bset) < 0)
    return isl_basic_set_free(bset);
  dim = isl_basic_set_dim(bset, isl_dim_set);
  if (dim < 0)
    return isl_basic_set_free(bset);
  isl_assert(bset->ctx, bset->n_eq <= dim, goto error);
  if (bset->n_eq == 0)
    return return_with_identity(bset, T, T2);
 
  B = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq, 0, 1 + dim);
  TC = isl_mat_variable_compression(B, T2);
  if (!TC)
    goto error;
  if (TC->n_col == 0) {
    isl_mat_free(TC);
    if (T2) {
      isl_mat_free(*T2);
      *T2 = NULL;
    }
    bset = isl_basic_set_set_to_empty(bset);
    return return_with_identity(bset, T, T2);
  }
 
  bset = isl_basic_set_preimage(bset, T ? isl_mat_copy(TC) : TC);
  if (T)
    *T = TC;
  return bset;
error:
  isl_basic_set_free(bset);
  return NULL;
}
 
__isl_give isl_basic_set *isl_basic_set_remove_equalities(
  __isl_take isl_basic_set *bset, __isl_give isl_mat **T,
  __isl_give isl_mat **T2)
{
  if (T)
    *T = NULL;
  if (T2)
    *T2 = NULL;
  if (isl_basic_set_check_no_params(bset) < 0)
    return isl_basic_set_free(bset);
  bset = isl_basic_set_gauss(bset, NULL);
  if (ISL_F_ISSET(bset, ISL_BASIC_SET_EMPTY))
    return return_with_identity(bset, T, T2);
  bset = compress_variables(bset, T, T2);
  return bset;
}
 
/* Check if dimension dim belongs to a residue class
 *    i_dim \equiv r mod m
 * with m != 1 and if so return m in *modulo and r in *residue.
 * As a special case, when i_dim has a fixed value v, then
 * *modulo is set to 0 and *residue to v.
 *
 * If i_dim does not belong to such a residue class, then *modulo
 * is set to 1 and *residue is set to 0.
 */
isl_stat isl_basic_set_dim_residue_class(__isl_keep isl_basic_set *bset,
  int pos, isl_int *modulo, isl_int *residue)
{
  isl_bool fixed;
  struct isl_ctx *ctx;
  struct isl_mat *H = NULL, *U = NULL, *C, *H1, *U1;
  isl_size total;
  isl_size nparam;
 
  if (!bset || !modulo || !residue)
    return isl_stat_error;
 
  fixed = isl_basic_set_plain_dim_is_fixed(bset, pos, residue);
  if (fixed < 0)
    return isl_stat_error;
  if (fixed) {
    isl_int_set_si(*modulo, 0);
    return isl_stat_ok;
  }
 
  ctx = isl_basic_set_get_ctx(bset);
  total = isl_basic_set_dim(bset, isl_dim_all);
  nparam = isl_basic_set_dim(bset, isl_dim_param);
  if (total < 0 || nparam < 0)
    return isl_stat_error;
  H = isl_mat_sub_alloc6(ctx, bset->eq, 0, bset->n_eq, 1, total);
  H = isl_mat_left_hermite(H, 0, &U, NULL);
  if (!H)
    return isl_stat_error;
 
  isl_seq_gcd(U->row[nparam + pos]+bset->n_eq,
      total-bset->n_eq, modulo);
  if (isl_int_is_zero(*modulo))
    isl_int_set_si(*modulo, 1);
  if (isl_int_is_one(*modulo)) {
    isl_int_set_si(*residue, 0);
    isl_mat_free(H);
    isl_mat_free(U);
    return isl_stat_ok;
  }
 
  C = isl_mat_alloc(ctx, 1 + bset->n_eq, 1);
  if (!C)
    goto error;
  isl_int_set_si(C->row[0][0], 1);
  isl_mat_sub_neg(ctx, C->row + 1, bset->eq, bset->n_eq, 0, 0, 1);
  H1 = isl_mat_sub_alloc(H, 0, H->n_row, 0, H->n_row);
  H1 = isl_mat_lin_to_aff(H1);
  C = isl_mat_inverse_product(H1, C);
  isl_mat_free(H);
  U1 = isl_mat_sub_alloc(U, nparam+pos, 1, 0, bset->n_eq);
  U1 = isl_mat_lin_to_aff(U1);
  isl_mat_free(U);
  C = isl_mat_product(U1, C);
  if (!C)
    return isl_stat_error;
  if (!isl_int_is_divisible_by(C->row[1][0], C->row[0][0])) {
    bset = isl_basic_set_copy(bset);
    bset = isl_basic_set_set_to_empty(bset);
    isl_basic_set_free(bset);
    isl_int_set_si(*modulo, 1);
    isl_int_set_si(*residue, 0);
    return isl_stat_ok;
  }
  isl_int_divexact(*residue, C->row[1][0], C->row[0][0]);
  isl_int_fdiv_r(*residue, *residue, *modulo);
  isl_mat_free(C);
  return isl_stat_ok;
error:
  isl_mat_free(H);
  isl_mat_free(U);
  return isl_stat_error;
}
 
/* Check if dimension dim belongs to a residue class
 *    i_dim \equiv r mod m
 * with m != 1 and if so return m in *modulo and r in *residue.
 * As a special case, when i_dim has a fixed value v, then
 * *modulo is set to 0 and *residue to v.
 *
 * If i_dim does not belong to such a residue class, then *modulo
 * is set to 1 and *residue is set to 0.
 */
isl_stat isl_set_dim_residue_class(__isl_keep isl_set *set,
  int pos, isl_int *modulo, isl_int *residue)
{
  isl_int m;
  isl_int r;
  int i;
 
  if (!set || !modulo || !residue)
    return isl_stat_error;
 
  if (set->n == 0) {
    isl_int_set_si(*modulo, 0);
    isl_int_set_si(*residue, 0);
    return isl_stat_ok;
  }
 
  if (isl_basic_set_dim_residue_class(set->p[0], pos, modulo, residue)<0)
    return isl_stat_error;
 
  if (set->n == 1)
    return isl_stat_ok;
 
  if (isl_int_is_one(*modulo))
    return isl_stat_ok;
 
  isl_int_init(m);
  isl_int_init(r);
 
  for (i = 1; i < set->n; ++i) {
    if (isl_basic_set_dim_residue_class(set->p[i], pos, &m, &r) < 0)
      goto error;
    isl_int_gcd(*modulo, *modulo, m);
    isl_int_sub(m, *residue, r);
    isl_int_gcd(*modulo, *modulo, m);
    if (!isl_int_is_zero(*modulo))
      isl_int_fdiv_r(*residue, *residue, *modulo);
    if (isl_int_is_one(*modulo))
      break;
  }
 
  isl_int_clear(m);
  isl_int_clear(r);
 
  return isl_stat_ok;
error:
  isl_int_clear(m);
  isl_int_clear(r);
  return isl_stat_error;
}
 
/* Check if dimension "dim" belongs to a residue class
 *    i_dim \equiv r mod m
 * with m != 1 and if so return m in *modulo and r in *residue.
 * As a special case, when i_dim has a fixed value v, then
 * *modulo is set to 0 and *residue to v.
 *
 * If i_dim does not belong to such a residue class, then *modulo
 * is set to 1 and *residue is set to 0.
 */
isl_stat isl_set_dim_residue_class_val(__isl_keep isl_set *set,
  int pos, __isl_give isl_val **modulo, __isl_give isl_val **residue)
{
  *modulo = NULL;
  *residue = NULL;
  if (!set)
    return isl_stat_error;
  *modulo = isl_val_alloc(isl_set_get_ctx(set));
  *residue = isl_val_alloc(isl_set_get_ctx(set));
  if (!*modulo || !*residue)
    goto error;
  if (isl_set_dim_residue_class(set, pos,
          &(*modulo)->n, &(*residue)->n) < 0)
    goto error;
  isl_int_set_si((*modulo)->d, 1);
  isl_int_set_si((*residue)->d, 1);
  return isl_stat_ok;
error:
  isl_val_free(*modulo);
  isl_val_free(*residue);
  return isl_stat_error;
}