Polyhedron_public.cc

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/* Polyhedron class implementation (non-inline public functions).
   Copyright (C) 2001-2010 Roberto Bagnara <bagnara@cs.unipr.it>
   Copyright (C) 2010-2012 BUGSENG srl (http://bugseng.com)

This file is part of the Parma Polyhedra Library (PPL).

The PPL is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

The PPL is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software Foundation,
Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02111-1307, USA.

For the most up-to-date information see the Parma Polyhedra Library
site: http://bugseng.com/products/ppl/ . */

#include "ppl-config.h"
#include "Polyhedron.defs.hh"
#include "C_Polyhedron.defs.hh"
#include "NNC_Polyhedron.defs.hh"
#include "Scalar_Products.defs.hh"
#include "MIP_Problem.defs.hh"
#include "wrap_assign.hh"
#include "assert.hh"
#include <cstdlib>
#include <iostream>

#ifndef ENSURE_SORTEDNESS
#define ENSURE_SORTEDNESS 0
#endif

namespace PPL = Parma_Polyhedra_Library;

PPL::dimension_type* PPL::Polyhedron::simplify_num_saturators_p = 0;

size_t PPL::Polyhedron::simplify_num_saturators_size = 0;

void
PPL::Polyhedron::initialize() {
  PPL_ASSERT(simplify_num_saturators_p == 0);
  PPL_ASSERT(simplify_num_saturators_size == 0);
  simplify_num_saturators_p = new dimension_type[simplify_num_saturators_size];
}

void
PPL::Polyhedron::finalize() {
  delete [] simplify_num_saturators_p;
  simplify_num_saturators_p = 0;
  simplify_num_saturators_size = 0;
}

PPL::dimension_type
PPL::Polyhedron::affine_dimension() const {
  if (is_empty())
    return 0;

  const Constraint_System& cs = minimized_constraints();
  dimension_type d = space_dim;
  for (Constraint_System::const_iterator i = cs.begin(),
         cs_end = cs.end(); i != cs_end; ++i)
    if (i->is_equality())
      --d;
  return d;
}

const PPL::Constraint_System&
PPL::Polyhedron::constraints() const {
  if (marked_empty()) {
    // We want `con_sys' to only contain the unsatisfiable constraint
    // of the appropriate dimension.
    if (con_sys.has_no_rows()) {
      // The 0-dim unsatisfiable constraint is extended to
      // the appropriate dimension and then stored in `con_sys'.
      Constraint_System unsat_cs = Constraint_System::zero_dim_empty();
      unsat_cs.adjust_topology_and_space_dimension(topology(), space_dim);
      const_cast<Constraint_System&>(con_sys).m_swap(unsat_cs);
    }
    else {
      // Checking that `con_sys' contains the right thing.
      PPL_ASSERT(con_sys.space_dimension() == space_dim);
      PPL_ASSERT(con_sys.num_rows() == 1);
      PPL_ASSERT(con_sys[0].is_inconsistent());
    }
    return con_sys;
  }

  if (space_dim == 0) {
    // Zero-dimensional universe.
    PPL_ASSERT(con_sys.num_rows() == 0 && con_sys.num_columns() == 0);
    return con_sys;
  }

  // If the polyhedron has pending generators, we process them to obtain
  // the constraints. No processing is needed if the polyhedron has
  // pending constraints.
  if (has_pending_generators())
    process_pending_generators();
  else if (!constraints_are_up_to_date())
    update_constraints();

  // TODO: reconsider whether to really sort constraints at this stage.
#if ENSURE_SORTEDNESS
  // We insist in returning a sorted system of constraints,
  // but sorting is useless if there are pending constraints.
  if (!has_pending_constraints())
    obtain_sorted_constraints();
#endif
  return con_sys;
}

const PPL::Constraint_System&
PPL::Polyhedron::minimized_constraints() const {
  // `minimize()' or `strongly_minimize_constraints()'
  // will process any pending constraints or generators.
  if (is_necessarily_closed())
    minimize();
  else
    strongly_minimize_constraints();
  return constraints();
}

const PPL::Generator_System&
PPL::Polyhedron::generators() const {
  if (marked_empty()) {
    PPL_ASSERT(gen_sys.has_no_rows());
    // We want `gen_sys' to have the appropriate space dimension,
    // even though it is an empty generator system.
    if (gen_sys.space_dimension() != space_dim) {
      Generator_System gs;
      gs.adjust_topology_and_space_dimension(topology(), space_dim);
      const_cast<Generator_System&>(gen_sys).m_swap(gs);
    }
    return gen_sys;
  }

  if (space_dim == 0) {
    PPL_ASSERT(gen_sys.num_rows() == 0 && gen_sys.num_columns() == 0);
    return Generator_System::zero_dim_univ();
  }

  // If the polyhedron has pending constraints, we process them to obtain
  // the generators (we may discover that the polyhedron is empty).
  // No processing is needed if the polyhedron has pending generators.
  if ((has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators())) {
    // We have just discovered that `*this' is empty.
    PPL_ASSERT(gen_sys.has_no_rows());
    // We want `gen_sys' to have the appropriate space dimension,
    // even though it is an empty generator system.
    if (gen_sys.space_dimension() != space_dim) {
      Generator_System gs;
      gs.adjust_topology_and_space_dimension(topology(), space_dim);
      const_cast<Generator_System&>(gen_sys).m_swap(gs);
    }
    return gen_sys;
  }

  // TODO: reconsider whether to really sort generators at this stage.
#if ENSURE_SORTEDNESS
  // We insist in returning a sorted system of generators,
  // but sorting is useless if there are pending generators.
  if (!has_pending_generators())
    obtain_sorted_generators();
#else
  // In the case of an NNC polyhedron, if the generator system is fully
  // minimized (i.e., minimized and with no pending generator), then
  // return a sorted system of generators: this is needed so that the
  // const_iterator could correctly filter out the matched closure points.
  if (!is_necessarily_closed()
      && generators_are_minimized() && !has_pending_generators())
    obtain_sorted_generators();
#endif
  return gen_sys;
}

const PPL::Generator_System&
PPL::Polyhedron::minimized_generators() const {
  // `minimize()' or `strongly_minimize_generators()'
  // will process any pending constraints or generators.
  if (is_necessarily_closed())
    minimize();
  else
    strongly_minimize_generators();
  // Note: calling generators() on a strongly minimized NNC generator
  // system will also ensure sortedness, which is required to correctly
  // filter away the matched closure points.
  return generators();
}

PPL::Poly_Con_Relation
PPL::Polyhedron::relation_with(const Constraint& c) const {
  // Dimension-compatibility check.
  if (space_dim < c.space_dimension())
    throw_dimension_incompatible("relation_with(c)", "c", c);

  if (marked_empty())
    return Poly_Con_Relation::saturates()
      && Poly_Con_Relation::is_included()
      && Poly_Con_Relation::is_disjoint();

  if (space_dim == 0) {
    if (c.is_inconsistent())
      if (c.is_strict_inequality() && c.inhomogeneous_term() == 0)
        // The constraint 0 > 0 implicitly defines the hyperplane 0 = 0;
        // thus, the zero-dimensional point also saturates it.
        return Poly_Con_Relation::saturates()
          && Poly_Con_Relation::is_disjoint();
      else
        return Poly_Con_Relation::is_disjoint();
    else if (c.is_equality() || c.inhomogeneous_term() == 0)
      return Poly_Con_Relation::saturates()
        && Poly_Con_Relation::is_included();
    else
      // The zero-dimensional point saturates
      // neither the positivity constraint 1 >= 0,
      // nor the strict positivity constraint 1 > 0.
      return Poly_Con_Relation::is_included();
  }

  if ((has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators()))
    // The polyhedron is empty.
    return Poly_Con_Relation::saturates()
      && Poly_Con_Relation::is_included()
      && Poly_Con_Relation::is_disjoint();

  return gen_sys.relation_with(c);
}

PPL::Poly_Gen_Relation
PPL::Polyhedron::relation_with(const Generator& g) const {
  // Dimension-compatibility check.
  if (space_dim < g.space_dimension())
    throw_dimension_incompatible("relation_with(g)", "g", g);

  // The empty polyhedron cannot subsume a generator.
  if (marked_empty())
    return Poly_Gen_Relation::nothing();

  // A universe polyhedron in a zero-dimensional space subsumes
  // all the generators of a zero-dimensional space.
  if (space_dim == 0)
    return Poly_Gen_Relation::subsumes();

  if (has_pending_generators())
    process_pending_generators();
  else if (!constraints_are_up_to_date())
    update_constraints();

  return
    con_sys.satisfies_all_constraints(g)
    ? Poly_Gen_Relation::subsumes()
    : Poly_Gen_Relation::nothing();
}

PPL::Poly_Con_Relation
PPL::Polyhedron::relation_with(const Congruence& cg) const {
  dimension_type cg_space_dim = cg.space_dimension();
  // Dimension-compatibility check.
  if (space_dim < cg_space_dim)
    throw_dimension_incompatible("relation_with(cg)", "cg", cg);

  if (cg.is_equality()) {
    const Constraint c(cg);
    return relation_with(c);
  }

  if (marked_empty())
    return Poly_Con_Relation::saturates()
      && Poly_Con_Relation::is_included()
      && Poly_Con_Relation::is_disjoint();

  if (space_dim == 0) {
    if (cg.is_inconsistent())
      return Poly_Con_Relation::is_disjoint();
    else
      return Poly_Con_Relation::saturates()
        && Poly_Con_Relation::is_included();
  }

  if ((has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators()))
    // The polyhedron is empty.
    return Poly_Con_Relation::saturates()
      && Poly_Con_Relation::is_included()
      && Poly_Con_Relation::is_disjoint();

  // Build the equality corresponding to the congruence (ignoring the modulus).
  Linear_Expression expr = Linear_Expression(cg);
  Constraint c(expr == 0);

  // The polyhedron is non-empty so that there exists a point.
  // For an arbitrary generator point, compute the scalar product with
  // the equality.
  PPL_DIRTY_TEMP_COEFFICIENT(sp_point);
  for (Generator_System::const_iterator gs_i = gen_sys.begin(),
         gs_end = gen_sys.end(); gs_i != gs_end; ++gs_i) {
    if (gs_i->is_point()) {
      Scalar_Products::assign(sp_point, c, *gs_i);
      expr -= sp_point;
      break;
    }
  }

  // Find two hyperplanes that satisfy the congruence and are near to
  // the generating point (so that the point lies on or between these
  // two hyperplanes).
  // Then use the relations between the polyhedron and the halfspaces
  // corresponding to the hyperplanes to determine the result.

  // Compute the distance from the point to an hyperplane.
  const Coefficient& modulus = cg.modulus();
  PPL_DIRTY_TEMP_COEFFICIENT(signed_distance);
  signed_distance = sp_point % modulus;
  if (signed_distance == 0)
    // The point is lying on the hyperplane.
    return relation_with(expr == 0);
  else
    // The point is not lying on the hyperplane.
    expr += signed_distance;

  // Build first halfspace constraint.
  const bool positive = (signed_distance > 0);
  Constraint first_halfspace = positive ? (expr >= 0) : (expr <= 0);

  Poly_Con_Relation first_rels = relation_with(first_halfspace);
  PPL_ASSERT(!first_rels.implies(Poly_Con_Relation::saturates())
             && !first_rels.implies(Poly_Con_Relation::is_disjoint()));
  if (first_rels.implies(Poly_Con_Relation::strictly_intersects()))
    return Poly_Con_Relation::strictly_intersects();

  // Build second halfspace.
  if (positive)
    expr -= modulus;
  else
    expr += modulus;
  Constraint second_halfspace = positive ? (expr <= 0) : (expr >= 0);

  PPL_ASSERT(first_rels == Poly_Con_Relation::is_included());
  Poly_Con_Relation second_rels = relation_with(second_halfspace);
  PPL_ASSERT(!second_rels.implies(Poly_Con_Relation::saturates())
             && !second_rels.implies(Poly_Con_Relation::is_disjoint()));
  if (second_rels.implies(Poly_Con_Relation::strictly_intersects()))
    return Poly_Con_Relation::strictly_intersects();

  PPL_ASSERT(second_rels == Poly_Con_Relation::is_included());
  return Poly_Con_Relation::is_disjoint();
}

bool
PPL::Polyhedron::is_universe() const {
  if (marked_empty())
    return false;

  if (space_dim == 0)
    return true;

  if (!has_pending_generators() && constraints_are_up_to_date()) {
    // Search for a constraint that is not a tautology.
    for (dimension_type i = con_sys.num_rows(); i-- > 0; )
      if (!con_sys[i].is_tautological())
        return false;
    // All the constraints are tautologies.
    return true;
  }

  PPL_ASSERT(!has_pending_constraints() && generators_are_up_to_date());

  // Try a fast-fail test.
  dimension_type num_lines = 0;
  dimension_type num_rays = 0;
  const dimension_type first_pending = gen_sys.first_pending_row();
  for (dimension_type i = first_pending; i-- > 0; )
    switch (gen_sys[i].type()) {
    case Generator::RAY:
      ++num_rays;
      break;
    case Generator::LINE:
      ++num_lines;
      break;
    default:
      break;
    }

  if (has_pending_generators()) {
    // The non-pending part of `gen_sys' was minimized:
    // a success-first test is possible in this case.
    PPL_ASSERT(generators_are_minimized());
    if (num_lines == space_dim) {
      PPL_ASSERT(num_rays == 0);
      return true;
    }
    PPL_ASSERT(num_lines < space_dim);
    // Now scan the pending generators.
    dimension_type num_pending_lines = 0;
    dimension_type num_pending_rays = 0;
    const dimension_type gs_num_rows = gen_sys.num_rows();
    for (dimension_type i = first_pending; i < gs_num_rows; ++i)
      switch (gen_sys[i].type()) {
      case Generator::RAY:
        ++num_pending_rays;
        break;
      case Generator::LINE:
        ++num_pending_lines;
        break;
      default:
        break;
      }
    // If no pending rays and lines were found,
    // then it is not the universe polyhedron.
    if (num_pending_rays == 0 && num_pending_lines == 0)
      return false;
    // Factor away the lines already seen (to be on the safe side,
    // we assume they are all linearly independent).
    if (num_lines + num_pending_lines < space_dim) {
      const dimension_type num_dims_missing
        = space_dim - (num_lines + num_pending_lines);
      // In order to span an n dimensional space (where n = num_dims_missing),
      // at least n+1 rays are needed.
      if (num_rays + num_pending_rays <= num_dims_missing)
        return false;
    }
  }
  else {
    // There is nothing pending.
    if (generators_are_minimized()) {
      // The exact test is possible.
      PPL_ASSERT(num_rays == 0 || num_lines < space_dim);
      return num_lines == space_dim;
    }
    else
      // Only the fast-fail test can be computed: in order to span
      // an n dimensional space (where n = space_dim - num_lines),
      // at least n+1 rays are needed.
      if (num_lines < space_dim && num_lines + num_rays <= space_dim)
        return false;
  }

  // We need the polyhedron in minimal form.
  if (has_pending_generators())
    process_pending_generators();
  else if (!constraints_are_minimized())
    minimize();
  if (is_necessarily_closed())
    return con_sys.num_rows() == 1
      && con_sys[0].is_inequality()
      && con_sys[0].is_tautological();
  else {
    // NNC polyhedron.
    if (con_sys.num_rows() != 2
        || con_sys[0].is_equality()
        || con_sys[1].is_equality())
      return false;
    else {
      // If the system of constraints contains two rows that
      // are not equalities, we are sure that they are
      // epsilon constraints: in this case we know that
      // the polyhedron is universe.
#ifndef NDEBUG
      obtain_sorted_constraints();
      const Constraint& eps_leq_one = con_sys[0];
      const Constraint& eps_geq_zero = con_sys[1];
      const dimension_type eps_index = con_sys.num_columns() - 1;
      PPL_ASSERT(eps_leq_one[0] > 0 && eps_leq_one[eps_index] < 0
             && eps_geq_zero[0] == 0 && eps_geq_zero[eps_index] > 0);
      for (dimension_type i = 1; i < eps_index; ++i)
        PPL_ASSERT(eps_leq_one[i] == 0 && eps_geq_zero[i] == 0);
#endif
      return true;
    }
  }
}

bool
PPL::Polyhedron::is_bounded() const {
  // A zero-dimensional or empty polyhedron is bounded.
  if (space_dim == 0
      || marked_empty()
      || (has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators()))
    return true;

  // If the system of generators contains any line or a ray,
  // then the polyhedron is unbounded.
  for (dimension_type i = gen_sys.num_rows(); i-- > 0; )
    if (gen_sys[i].is_line_or_ray())
      return false;

  // The system of generators is composed only by
  // points and closure points: the polyhedron is bounded.
  return true;
}

bool
PPL::Polyhedron::is_topologically_closed() const {
  // Necessarily closed polyhedra are trivially closed.
  if (is_necessarily_closed())
    return true;
  // Any empty or zero-dimensional polyhedron is closed.
  if (marked_empty()
      || space_dim == 0
      || (has_something_pending() && !process_pending()))
    return true;

  // At this point there are no pending constraints or generators.
  PPL_ASSERT(!has_something_pending());

  if (generators_are_minimized()) {
    // A polyhedron is closed if and only if all of its (non-redundant)
    // closure points are matched by a corresponding point.
    const dimension_type n_rows = gen_sys.num_rows();
    const dimension_type n_lines = gen_sys.num_lines();
    for (dimension_type i = n_rows; i-- > n_lines; ) {
      const Generator& gen_sys_i = gen_sys[i];
      if (gen_sys_i.is_closure_point()) {
        bool gen_sys_i_has_no_matching_point = true;
        for (dimension_type j = n_rows; j-- > n_lines; ) {
          const Generator& gen_sys_j = gen_sys[j];
          if (i != j
              && gen_sys_j.is_point()
              && gen_sys_i.is_matching_closure_point(gen_sys_j)) {
            gen_sys_i_has_no_matching_point = false;
            break;
          }
        }
        if (gen_sys_i_has_no_matching_point)
          return false;
      }
    }
    // All closure points are matched.
    return true;
  }

  // A polyhedron is closed if, after strong minimization
  // of its constraint system, it has no strict inequalities.
  strongly_minimize_constraints();
  return marked_empty() || !con_sys.has_strict_inequalities();
}

bool
PPL::Polyhedron::contains_integer_point() const {
  // Any empty polyhedron does not contain integer points.
  if (marked_empty())
    return false;

  // A zero-dimensional, universe polyhedron has, by convention, an
  // integer point.
  if (space_dim == 0)
    return true;

  // CHECKME: do we really want to call conversion to check for emptiness?
  if (has_pending_constraints() && !process_pending())
    // Empty again.
    return false;

  // FIXME: do also exploit info regarding rays and lines, if possible.
  // Is any integer point already available?
  PPL_ASSERT(!has_pending_constraints());
  if (generators_are_up_to_date())
    for (dimension_type i = gen_sys.num_rows(); i-- > 0; )
      if (gen_sys[i].is_point() && gen_sys[i].divisor() == 1)
        return true;

  const Constraint_System& cs = constraints();
#if 0 // TEMPORARILY DISABLED.
  MIP_Problem mip(space_dim,
                  cs.begin(), cs.end(),
                  Variables_Set(Variable(0), Variable(space_dim-1)));
#else
  // FIXME: temporary workaround, to be removed as soon as the MIP
  // problem class will correctly and precisely handle
  // ((strict) in-) equality constraints having all integer variables.
  MIP_Problem mip(space_dim);
  mip.add_to_integer_space_dimensions(Variables_Set(Variable(0),
                                                    Variable(space_dim-1)));
  PPL_DIRTY_TEMP_COEFFICIENT(homogeneous_gcd);
  PPL_DIRTY_TEMP_COEFFICIENT(gcd);
  PPL_DIRTY_TEMP(mpq_class, rational_inhomogeneous);
  PPL_DIRTY_TEMP_COEFFICIENT(tightened_inhomogeneous);
  for (Constraint_System::const_iterator cs_i = cs.begin(),
         cs_end = cs.end(); cs_i != cs_end; ++cs_i) {
    const Constraint& c = *cs_i;
    const Constraint::Type c_type = c.type();
    const Coefficient& inhomogeneous = c.inhomogeneous_term();
    if (c_type == Constraint::STRICT_INEQUALITY) {
      // CHECKME: should we change the behavior of Linear_Expression(c) ?
      // Compute the GCD of the coefficients of c
      // (disregarding the inhomogeneous term and the epsilon dimension).
      homogeneous_gcd = 0;
      for (dimension_type i = space_dim; i-- > 0; )
        gcd_assign(homogeneous_gcd,
                   homogeneous_gcd, c.coefficient(Variable(i)));
      if (homogeneous_gcd == 0) {
        // NOTE: since tautological constraints are already filtered away
        // by iterators, here we must have an inconsistent constraint.
        PPL_ASSERT(c.is_inconsistent());
        return false;
      }
      Linear_Expression le;
      for (dimension_type i = space_dim; i-- > 0; )
        le += (c.coefficient(Variable(i)) / homogeneous_gcd) * Variable(i);
      // Add the integer part of `inhomogeneous'.
      le += (inhomogeneous / homogeneous_gcd);
      // Further tighten the constraint if the inhomogeneous term
      // was integer, i.e., if `homogeneous_gcd' divides `inhomogeneous'.
      gcd_assign(gcd, homogeneous_gcd, inhomogeneous);
      if (gcd == homogeneous_gcd)
        le -= 1;
      mip.add_constraint(le >= 0);
    }
    else {
      // Equality or non-strict inequality.
      // If possible, avoid useless gcd computations.
      if (inhomogeneous == 0)
        // The inhomogeneous term cannot be tightened.
        mip.add_constraint(c);
      else {
        // Compute the GCD of the coefficients of c
        // (disregarding the inhomogeneous term)
        // to see whether or not the inhomogeneous term can be tightened.
        homogeneous_gcd = 0;
        for (dimension_type i = space_dim; i-- > 0; )
          gcd_assign(homogeneous_gcd,
                     homogeneous_gcd, c.coefficient(Variable(i)));
        if (homogeneous_gcd == 0) {
          // NOTE: since tautological constraints are already filtered away
          // by iterators, here we must have an inconsistent constraint.
          PPL_ASSERT(c.is_inconsistent());
          return false;
        }
        else if (homogeneous_gcd == 1)
          // The normalized inhomogeneous term is integer:
          // add the constraint as-is.
          mip.add_constraint(c);
        else {
          PPL_ASSERT(homogeneous_gcd > 1);
          // Here the normalized inhomogeneous term is rational:
          // the constraint has to be tightened.
#ifndef NDEBUG
          // `homogeneous_gcd' does not divide `inhomogeneous'.
          // FIXME: add a divisibility test for Coefficient.
          gcd_assign(gcd, homogeneous_gcd, inhomogeneous);
          PPL_ASSERT(gcd == 1);
#endif
          if (c.type() == Constraint::EQUALITY)
            return false;
          // Extract the homogeneous part of the constraint.
          Linear_Expression le = Linear_Expression(c);
          le -= inhomogeneous;
          // Tighten the inhomogeneous term.
          assign_r(rational_inhomogeneous.get_num(),
                   inhomogeneous, ROUND_NOT_NEEDED);
          assign_r(rational_inhomogeneous.get_den(),
                   homogeneous_gcd, ROUND_NOT_NEEDED);
          // Note: canonicalization is not needed (as gcd == 1).
          PPL_ASSERT(is_canonical(rational_inhomogeneous));
          assign_r(tightened_inhomogeneous,
                   rational_inhomogeneous, ROUND_DOWN);
          tightened_inhomogeneous *= homogeneous_gcd;
          le += tightened_inhomogeneous;
          mip.add_constraint(le >= 0);
        }
      }
    }
  }
#endif // TEMPORARY WORKAROUND.
  return mip.is_satisfiable();
}

bool
PPL::Polyhedron::constrains(const Variable var) const {
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("constrains(v)", "v", var);

  // An empty polyhedron constrains all variables.
  if (marked_empty())
    return true;

  if (generators_are_up_to_date() && !has_pending_constraints()) {
    // Since generators are up-to-date and there are no pending
    // constraints, the generator system (since it is well formed)
    // contains a point.  Hence the polyhedron is not empty.
    if (constraints_are_up_to_date() && !has_pending_generators())
      // Here a variable is constrained if and only if it is
      // syntactically constrained.
      goto syntactic_check;

    if (generators_are_minimized()) {
      // Try a quick, incomplete check for the universe polyhedron:
      // a universe polyhedron constrains no variable.
      // Count the number of non-pending
      // (hence, linearly independent) lines.
      dimension_type num_lines = 0;
      const dimension_type first_pending = gen_sys.first_pending_row();
      for (dimension_type i = first_pending; i-- > 0; )
        if (gen_sys[i].is_line())
          ++num_lines;

      if (num_lines == space_dim)
        return false;
    }

    // Scan generators: perhaps we will find a generator equivalent to
    // line(var) or a pair of generators equivalent to ray(-var) and
    // ray(var).
    bool have_positive_ray = false;
    bool have_negative_ray = false;
    const dimension_type var_id = var.id();
    for (dimension_type i = gen_sys.num_rows(); i-- > 0; ) {
      const Generator& gen_sys_i = gen_sys[i];
      if (gen_sys_i.is_line_or_ray()) {
        const Linear_Row& row = gen_sys_i;
        const int sign = sgn(row.coefficient(var_id));
        if (sign != 0) {
          for (dimension_type j = space_dim+1; --j > 0; )
            if (j != var_id && row[j] != 0)
              goto next;
          if (gen_sys_i.is_line())
            return true;
          if (sign > 0)
            if (have_negative_ray)
              return true;
            else
              have_positive_ray = true;
          else if (have_positive_ray)
            return true;
          else
            have_negative_ray = true;
        }
      }
    next:
      ;
    }

    // We are still here: at least we know that, since generators are
    // up-to-date and there are no pending constraints, then the
    // generator system (since it is well formed) contains a point.
    // Hence the polyhedron is not empty.
    if (has_pending_generators())
      process_pending_generators();
    else if (!constraints_are_up_to_date())
      update_constraints();
    goto syntactic_check;
  }

  // We must minimize to detect emptiness and obtain constraints.
  if (!minimize())
    return true;

 syntactic_check:
  for (dimension_type i = con_sys.num_rows(); i-- > 0; )
    if (con_sys[i].coefficient(var) != 0)
      return true;
  return false;
}

bool
PPL::Polyhedron::OK(bool check_not_empty) const {
#ifndef NDEBUG
  using std::endl;
  using std::cerr;
#endif

  // The expected number of columns in the constraint and generator
  // systems, if they are not empty.
  const dimension_type poly_num_columns
    = space_dim + (is_necessarily_closed() ? 1U : 2U);

  // Check whether the topologies of `con_sys' and `gen_sys' agree.
  if (con_sys.topology() != gen_sys.topology()) {
#ifndef NDEBUG
    cerr << "Constraints and generators have different topologies!"
         << endl;
#endif
    goto bomb;
  }

  // Check whether the saturation matrices are well-formed.
  if (!sat_c.OK())
    goto bomb;
  if (!sat_g.OK())
    goto bomb;

  // Check whether the status information is legal.
  if (!status.OK())
    goto bomb;

  if (marked_empty()) {
    if (check_not_empty) {
      // The caller does not want the polyhedron to be empty.
#ifndef NDEBUG
      cerr << "Empty polyhedron!" << endl;
#endif
      goto bomb;
    }

    // An empty polyhedron is allowed if the system of constraints
    // either has no rows or only contains an unsatisfiable constraint
    // and if it has no pending constraints or generators.
    if (has_something_pending()) {
#ifndef NDEBUG
      cerr << "The polyhedron is empty, "
           << "but it has something pending" << endl;
#endif
      goto bomb;
    }
    if (con_sys.has_no_rows())
      return true;
    else {
      if (con_sys.space_dimension() != space_dim) {
#ifndef NDEBUG
        cerr << "The polyhedron is in a space of dimension "
             << space_dim
             << " while the system of constraints is in a space of dimension "
             << con_sys.space_dimension()
             << endl;
#endif
        goto bomb;
      }
      if (con_sys.num_rows() != 1) {
#ifndef NDEBUG
        cerr << "The system of constraints for an empty polyhedron "
             << "has more then one row"
             << endl;
#endif
        goto bomb;
      }
      if (!con_sys[0].is_inconsistent()) {
#ifndef NDEBUG
        cerr << "Empty polyhedron with a satisfiable system of constraints"
             << endl;
#endif
        goto bomb;
      }
      // Here we have only one, inconsistent constraint.
      return true;
    }
  }

  // A zero-dimensional, non-empty polyhedron is legal only if the
  // system of constraint `con_sys' and the system of generators
  // `gen_sys' have no rows.
  if (space_dim == 0) {
    if (has_something_pending()) {
#ifndef NDEBUG
      cerr << "Zero-dimensional polyhedron with something pending"
           << endl;
#endif
      goto bomb;
    }
    if (!con_sys.has_no_rows() || !gen_sys.has_no_rows()) {
#ifndef NDEBUG
      cerr << "Zero-dimensional polyhedron with a non-empty"
           << endl
           << "system of constraints or generators."
           << endl;
#endif
      goto bomb;
    }
    return true;
  }

  // A polyhedron is defined by a system of constraints
  // or a system of generators: at least one of them must be up to date.
  if (!constraints_are_up_to_date() && !generators_are_up_to_date()) {
#ifndef NDEBUG
    cerr << "Polyhedron not empty, not zero-dimensional"
         << endl
         << "and with neither constraints nor generators up-to-date!"
         << endl;
#endif
    goto bomb;
  }

  // Here we check if the size of the matrices is consistent.
  // Let us suppose that all the matrices are up-to-date; this means:
  // `con_sys' : number of constraints x poly_num_columns
  // `gen_sys' : number of generators  x poly_num_columns
  // `sat_c'   : number of generators  x number of constraints
  // `sat_g'   : number of constraints x number of generators.
  if (constraints_are_up_to_date()) {
    if (con_sys.num_columns() != poly_num_columns) {
#ifndef NDEBUG
      cerr << "Incompatible size! (con_sys and space_dim)"
           << endl;
#endif
      goto bomb;
    }
    if (sat_c_is_up_to_date())
      if (con_sys.first_pending_row() != sat_c.num_columns()) {
#ifndef NDEBUG
        cerr << "Incompatible size! (con_sys and sat_c)"
             << endl;
#endif
        goto bomb;
      }
    if (sat_g_is_up_to_date())
      if (con_sys.first_pending_row() != sat_g.num_rows()) {
#ifndef NDEBUG
        cerr << "Incompatible size! (con_sys and sat_g)"
             << endl;
#endif
        goto bomb;
      }
    if (generators_are_up_to_date())
      if (con_sys.num_columns() != gen_sys.num_columns()) {
#ifndef NDEBUG
        cerr << "Incompatible size! (con_sys and gen_sys)"
             << endl;
#endif
        goto bomb;
      }
  }

  if (generators_are_up_to_date()) {
    if (gen_sys.num_columns() != poly_num_columns) {
#ifndef NDEBUG
      cerr << "Incompatible size! (gen_sys and space_dim)"
           << endl;
#endif
      goto bomb;
    }
    if (sat_c_is_up_to_date())
      if (gen_sys.first_pending_row() != sat_c.num_rows()) {
#ifndef NDEBUG
        cerr << "Incompatible size! (gen_sys and sat_c)"
             << endl;
#endif
        goto bomb;
      }
    if (sat_g_is_up_to_date())
      if (gen_sys.first_pending_row() != sat_g.num_columns()) {
#ifndef NDEBUG
        cerr << "Incompatible size! (gen_sys and sat_g)"
             << endl;
#endif
        goto bomb;
      }

    // Check if the system of generators is well-formed.
    if (!gen_sys.OK())
      goto bomb;

    if (gen_sys.first_pending_row() == 0) {
#ifndef NDEBUG
      cerr << "Up-to-date generator system with all rows pending!"
           << endl;
#endif
      goto bomb;
    }

    // A non-empty system of generators describing a polyhedron
    // is valid if and only if it contains a point.
    if (!gen_sys.has_no_rows() && !gen_sys.has_points()) {
#ifndef NDEBUG
      cerr << "Non-empty generator system declared up-to-date "
           << "has no points!"
           << endl;
#endif
      goto bomb;
    }

#if 0 // To be activated when Status keeps strong minimization flags.
    //=================================================
    // TODO: this test is wrong in the general case.
    // However, such an invariant does hold for a
    // strongly-minimized Generator_System.
    // We will activate this test as soon as the Status
    // flags will be able to remember if a system is
    // strongly minimized.

    // Checking that the number of closure points is always
    // greater than the number of points.
    if (!is_necessarily_closed()) {
      dimension_type num_points = 0;
      dimension_type num_closure_points = 0;
      dimension_type eps_index = gen_sys.num_columns() - 1;
      for (dimension_type i = gen_sys.num_rows(); i-- > 0; )
        if (!gen_sys[i].is_line_or_ray())
          if (gen_sys[i][eps_index] > 0)
            ++num_points;
          else
            ++num_closure_points;
      if (num_points > num_closure_points) {
#ifndef NDEBUG
        cerr << "# POINTS > # CLOSURE_POINTS" << endl;
#endif
        goto bomb;
      }
    }
    //=================================================
#endif

    if (generators_are_minimized()) {
      // If the system of generators is minimized, the number of
      // lines, rays and points of the polyhedron must be the same as
      // of a temporary, minimized one. If this does not happen then
      // the polyhedron is not OK.
      Constraint_System new_con_sys(topology());
      Generator_System gs_without_pending = gen_sys;
      gs_without_pending.remove_trailing_rows(gs_without_pending.num_rows()
                                              - gen_sys.first_pending_row());
      gs_without_pending.unset_pending_rows();
      Generator_System copy_of_gen_sys = gs_without_pending;
      Bit_Matrix new_sat_c;
      minimize(false, copy_of_gen_sys, new_con_sys, new_sat_c);
      const dimension_type copy_num_lines = copy_of_gen_sys.num_lines();
      if (gs_without_pending.num_rows() != copy_of_gen_sys.num_rows()
          || gs_without_pending.num_lines() != copy_num_lines
          || gs_without_pending.num_rays() != copy_of_gen_sys.num_rays()) {
#ifndef NDEBUG
        cerr << "Generators are declared minimized, but they are not!\n"
             << "Here is the minimized form of the generators:\n";
        copy_of_gen_sys.ascii_dump(cerr);
        cerr << endl;
#endif
        goto bomb;
      }

      // CHECKME : the following observation is not formally true
      //           for a NNC_Polyhedron. But it may be true for its
      //           representation ...

      // If the corresponding polyhedral cone is _pointed_, then
      // a minimal system of generators is unique up to positive scaling.
      // We thus verify if the cone is pointed (i.e., there are no lines)
      // and, after normalizing and sorting a copy of the system `gen_sys'
      // of the polyhedron (we use a copy not to modify the polyhedron's
      // system) and the system `copy_of_gen_sys' that has been just
      // minimized, we check if the two matrices are identical.  If
      // they are different it means that the generators of the
      // polyhedron are declared minimized, but they are not.
      if (copy_num_lines == 0) {
        copy_of_gen_sys.strong_normalize();
        copy_of_gen_sys.sort_rows();
        gs_without_pending.strong_normalize();
        gs_without_pending.sort_rows();
        if (copy_of_gen_sys != gs_without_pending) {
#ifndef NDEBUG
          cerr << "Generators are declared minimized, but they are not!\n"
               << "(we are in the case:\n"
               << "dimension of lineality space equal to 0)\n"
               << "Here is the minimized form of the generators:\n";
          copy_of_gen_sys.ascii_dump(cerr);
          cerr << endl;
#endif
            goto bomb;
        }
      }
    }
  }

  if (constraints_are_up_to_date()) {
    // Check if the system of constraints is well-formed.
    if (!con_sys.OK())
      goto bomb;

    if (con_sys.first_pending_row() == 0) {
#ifndef NDEBUG
      cerr << "Up-to-date constraint system with all rows pending!"
           << endl;
#endif
      goto bomb;
    }

    // A non-empty system of constraints describing a polyhedron
    // must contain a constraint with a non-zero inhomogeneous term;
    // such a constraint corresponds to (a combination of other
    // constraints with):
    // -* the positivity constraint, for necessarily closed polyhedra;
    // -* the epsilon <= 1 constraint, for NNC polyhedra.
    bool no_positivity_constraint = true;
    for (dimension_type i = con_sys.num_rows(); i-- > 0; )
      if (con_sys[i].inhomogeneous_term() != 0) {
        no_positivity_constraint = false;
        break;
      }
    if (no_positivity_constraint) {
#ifndef NDEBUG
      cerr << "Non-empty constraint system has no positivity constraint"
           << endl;
#endif
      goto bomb;
    }

    if (!is_necessarily_closed()) {
      // A non-empty system of constraints describing a NNC polyhedron
      // must also contain a (combination of) the constraint epsilon >= 0,
      // i.e., a constraint with a positive epsilon coefficient.
      bool no_epsilon_geq_zero = true;
      const dimension_type eps_index = con_sys.num_columns() - 1;
      for (dimension_type i = con_sys.num_rows(); i-- > 0; )
        if (con_sys[i][eps_index] > 0) {
          no_epsilon_geq_zero = false;
          break;
        }
      if (no_epsilon_geq_zero) {
#ifndef NDEBUG
        cerr << "Non-empty constraint system for NNC polyhedron "
             << "has no epsilon >= 0 constraint"
             << endl;
#endif
        goto bomb;
      }
    }

    Constraint_System cs_without_pending = con_sys;
    cs_without_pending.remove_trailing_rows(cs_without_pending.num_rows()
                                            - con_sys.first_pending_row());
    cs_without_pending.unset_pending_rows();
    Constraint_System copy_of_con_sys = cs_without_pending;
    bool empty = false;
    if (check_not_empty || constraints_are_minimized()) {
      Generator_System new_gen_sys(topology());
      Bit_Matrix new_sat_g;
      empty = minimize(true, copy_of_con_sys, new_gen_sys, new_sat_g);
    }

    if (empty && check_not_empty) {
#ifndef NDEBUG
      cerr << "Unsatisfiable system of constraints!"
           << endl;
#endif
      goto bomb;
    }

    if (constraints_are_minimized()) {
      // If the constraints are minimized, the number of equalities
      // and of inequalities of the system of the polyhedron must be
      // the same of the temporary minimized one.
      // If it does not happen, the polyhedron is not OK.
      if (cs_without_pending.num_rows() != copy_of_con_sys.num_rows()
          || cs_without_pending.num_equalities()
          != copy_of_con_sys.num_equalities()) {
#ifndef NDEBUG
        cerr << "Constraints are declared minimized, but they are not!\n"
             << "Here is the minimized form of the constraints:\n";
        copy_of_con_sys.ascii_dump(cerr);
        cerr << endl;
#endif
        goto bomb;
      }
      // The system `copy_of_con_sys' has the form that is obtained
      // after applying methods gauss() and back_substitute().
      // A system of constraints can be minimal even if it does not
      // have this form. So, to verify if the polyhedron is correct,
      // we copy the system `con_sys' in a temporary one and then
      // modify it using method simplify() (which calls both gauss()
      // and back_substitute()).
      // If the temporary system and `copy_of_con_sys' are different,
      // the polyhedron is not OK.
      copy_of_con_sys.strong_normalize();
      copy_of_con_sys.sort_rows();
      cs_without_pending.simplify();
      cs_without_pending.strong_normalize();
      cs_without_pending.sort_rows();
      if (cs_without_pending != copy_of_con_sys) {
#ifndef NDEBUG
        cerr << "Constraints are declared minimized, but they are not!\n"
             << "Here is the minimized form of the constraints:\n";
        copy_of_con_sys.ascii_dump(cerr);
        cerr << endl;
#endif
        goto bomb;
      }
    }
  }

  if (sat_c_is_up_to_date())
    for (dimension_type i = sat_c.num_rows(); i-- > 0; ) {
      const Generator tmp_gen = gen_sys[i];
      const Bit_Row tmp_sat = sat_c[i];
      for (dimension_type j = sat_c.num_columns(); j-- > 0; ) {
        const bool sat_j = (Scalar_Products::sign(con_sys[j], tmp_gen) == 0);
        if (sat_j == tmp_sat[j]) {
#ifndef NDEBUG
          cerr << "sat_c is declared up-to-date, but it is not!"
               << endl;
#endif
          goto bomb;
        }
      }
    }

  if (sat_g_is_up_to_date())
    for (dimension_type i = sat_g.num_rows(); i-- > 0; ) {
      const Constraint tmp_con = con_sys[i];
      const Bit_Row tmp_sat = sat_g[i];
      for (dimension_type j = sat_g.num_columns(); j-- > 0; ) {
        const bool sat_j = (Scalar_Products::sign(tmp_con, gen_sys[j]) == 0);
        if (sat_j == tmp_sat[j]) {
#ifndef NDEBUG
          cerr << "sat_g is declared up-to-date, but it is not!"
               << endl;
#endif
          goto bomb;
        }
      }
    }

  if (has_pending_constraints()) {
    if (con_sys.num_pending_rows() == 0) {
#ifndef NDEBUG
      cerr << "The polyhedron is declared to have pending constraints, "
           << "but con_sys has no pending rows!"
           << endl;
#endif
      goto bomb;
    }
  }

  if (has_pending_generators()) {
    if (gen_sys.num_pending_rows() == 0) {
#ifndef NDEBUG
      cerr << "The polyhedron is declared to have pending generators, "
           << "but gen_sys has no pending rows!"
           << endl;
#endif
      goto bomb;
    }
  }

  return true;

 bomb:
#ifndef NDEBUG
  cerr << "Here is the guilty polyhedron:"
       << endl;
  ascii_dump(cerr);
#endif
  return false;
}

void
PPL::Polyhedron::add_constraint(const Constraint& c) {
  // Topology-compatibility check.
  if (c.is_strict_inequality() && is_necessarily_closed()) {
    // Trivially true/false strict inequalities are legal.
    if (c.is_tautological())
      return;
    if (c.is_inconsistent()) {
      set_empty();
      return;
    }
    // Here c is a non-trivial strict inequality.
    throw_topology_incompatible("add_constraint(c)", "c", c);
  }

  // Dimension-compatibility check:
  // the dimension of `c' can not be greater than space_dim.
  if (space_dim < c.space_dimension())
    throw_dimension_incompatible("add_constraint(c)", "c", c);

  if (!marked_empty())
    refine_no_check(c);
}

void
PPL::Polyhedron::add_congruence(const Congruence& cg) {
  // Dimension-compatibility check:
  // the dimension of `cg' can not be greater than space_dim.
  if (space_dim < cg.space_dimension())
    throw_dimension_incompatible("add_congruence(cg)", "cg", cg);

  // Handle the case of proper congruences first.
  if (cg.is_proper_congruence()) {
    if (cg.is_tautological())
      return;
    if (cg.is_inconsistent()) {
      set_empty();
      return;
    }
    // Non-trivial and proper congruences are not allowed.
    throw_invalid_argument("add_congruence(cg)",
                           "cg is a non-trivial, proper congruence");
  }

  PPL_ASSERT(cg.is_equality());
  // Handle empty and 0-dim cases first.
  if (marked_empty())
    return;
  if (space_dim == 0) {
    if (cg.is_inconsistent())
      set_empty();
    return;
  }

  // Add the equality.
  Linear_Expression le(cg);
  Constraint c(le, Constraint::EQUALITY, NECESSARILY_CLOSED);
  // Enforce normalization.
  c.strong_normalize();
  refine_no_check(c);
}

void
PPL::Polyhedron::add_generator(const Generator& g) {
  // Topology-compatibility check.
  if (g.is_closure_point() && is_necessarily_closed())
    throw_topology_incompatible("add_generator(g)", "g", g);
  // Dimension-compatibility check:
  // the dimension of `g' can not be greater than space_dim.
  const dimension_type g_space_dim = g.space_dimension();
  if (space_dim < g_space_dim)
    throw_dimension_incompatible("add_generator(g)", "g", g);

  // Dealing with a zero-dimensional space polyhedron first.
  if (space_dim == 0) {
    // It is not possible to create 0-dim rays or lines.
    PPL_ASSERT(g.is_point() || g.is_closure_point());
    // Closure points can only be inserted in non-empty polyhedra.
    if (marked_empty()) {
      if (g.type() != Generator::POINT)
        throw_invalid_generator("add_generator(g)", "g");
      else
        set_zero_dim_univ();
    }
    PPL_ASSERT_HEAVY(OK());
    return;
  }

  if (marked_empty()
      || (has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators())) {
    // Here the polyhedron is empty:
    // the specification says we can only insert a point.
    if (!g.is_point())
      throw_invalid_generator("add_generator(g)", "g");
    if (g.is_necessarily_closed() || !is_necessarily_closed()) {
      gen_sys.insert(g);
      // Since `gen_sys' was empty, after inserting `g' we have to resize
      // the system of generators to have the right dimension.
      gen_sys.adjust_topology_and_space_dimension(topology(), space_dim);
      if (!is_necessarily_closed()) {
        // In the NNC topology, each point has to be matched by
        // a corresponding closure point:
        // turn the just inserted point into the corresponding
        // (normalized) closure point.
        Generator& cp = gen_sys[gen_sys.num_rows() - 1];
        cp[space_dim + 1] = 0;
        cp.normalize();
        // Re-insert the point (which is already normalized).
        gen_sys.insert(g);
      }
    }
    else {
      // Note: here we have a _legal_ topology mismatch,
      // because `g' is NOT a closure point (it is a point!)
      // However, by barely invoking `gen_sys.insert(g)' we would
      // cause a change in the topology of `gen_sys', which is wrong.
      // Thus, we insert a "topology corrected" copy of `g'.
      const Linear_Expression nc_expr = Linear_Expression(g);
      gen_sys.insert(Generator::point(nc_expr, g.divisor()));
      // Since `gen_sys' was empty, after inserting `g' we have to resize
      // the system of generators to have the right dimension.
      gen_sys.adjust_topology_and_space_dimension(topology(), space_dim);
    }
    // No longer empty, generators up-to-date and minimized.
    clear_empty();
    set_generators_minimized();
  }
  else {
    PPL_ASSERT(generators_are_up_to_date());
    const bool has_pending = can_have_something_pending();
    if (g.is_necessarily_closed() || !is_necessarily_closed()) {
      // Since `gen_sys' is not empty, the topology and space dimension
      // of the inserted generator are automatically adjusted.
      if (has_pending)
        gen_sys.insert_pending(g);
      else
        gen_sys.insert(g);
      if (!is_necessarily_closed() && g.is_point()) {
        // In the NNC topology, each point has to be matched by
        // a corresponding closure point:
        // turn the just inserted point into the corresponding
        // (normalized) closure point.
        Generator& cp = gen_sys[gen_sys.num_rows() - 1];
        cp[space_dim + 1] = 0;
        cp.normalize();
        // Re-insert the point (which is already normalized).
        if (has_pending)
          gen_sys.insert_pending(g);
        else
          gen_sys.insert(g);
      }
    }
    else {
      PPL_ASSERT(!g.is_closure_point());
      // Note: here we have a _legal_ topology mismatch, because
      // `g' is NOT a closure point.
      // However, by barely invoking `gen_sys.insert(g)' we would
      // cause a change in the topology of `gen_sys', which is wrong.
      // Thus, we insert a "topology corrected" copy of `g'.
      const Linear_Expression nc_expr = Linear_Expression(g);
      switch (g.type()) {
      case Generator::LINE:
        if (has_pending)
          gen_sys.insert_pending(Generator::line(nc_expr));
        else
          gen_sys.insert(Generator::line(nc_expr));
        break;
      case Generator::RAY:
        if (has_pending)
          gen_sys.insert_pending(Generator::ray(nc_expr));
        else
          gen_sys.insert(Generator::ray(nc_expr));
        break;
      case Generator::POINT:
        if (has_pending)
          gen_sys.insert_pending(Generator::point(nc_expr, g.divisor()));
        else
          gen_sys.insert(Generator::point(nc_expr, g.divisor()));
        break;
      case Generator::CLOSURE_POINT:
        PPL_UNREACHABLE;
        break;
      }
    }

    if (has_pending)
      set_generators_pending();
    else {
      // After adding the new generator,
      // constraints are no longer up-to-date.
      clear_generators_minimized();
      clear_constraints_up_to_date();
    }
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::add_recycled_constraints(Constraint_System& cs) {
  // Topology compatibility check.
  if (is_necessarily_closed() && cs.has_strict_inequalities()) {
    // We check if _all_ strict inequalities in cs are trivially false.
    // (The iterators already filter away trivially true constraints.)
    for (Constraint_System::const_iterator i = cs.begin(),
           i_end = cs.end(); i != i_end; ++i) {
      if (i->is_strict_inequality()
          && !i->is_inconsistent())
        throw_topology_incompatible("add_recycled_constraints(cs)",
                                    "cs", cs);
    }
    // If we reach this point, all strict inequalities were inconsistent.
    set_empty();
    return;
  }

  // Dimension-compatibility check:
  // the dimension of `cs' can not be greater than space_dim.
  const dimension_type cs_space_dim = cs.space_dimension();
  if (space_dim < cs_space_dim)
    throw_dimension_incompatible("add_recycled_constraints(cs)", "cs", cs);

  // Adding no constraints is a no-op.
  if (cs.has_no_rows())
    return;

  if (space_dim == 0) {
    // In a 0-dimensional space the constraints are
    // tautologies (e.g., 0 == 0 or 1 >= 0 or 1 > 0) or
    // inconsistent (e.g., 1 == 0 or -1 >= 0 or 0 > 0).
    // In a system of constraints `begin()' and `end()' are equal
    // if and only if the system only contains tautologies.
    if (cs.begin() != cs.end())
      // There is a constraint, it must be inconsistent,
      // the polyhedron is empty.
      status.set_empty();
    return;
  }

  if (marked_empty())
    return;

  // The constraints (possibly with pending rows) are required.
  if (has_pending_generators())
    process_pending_generators();
  else if (!constraints_are_up_to_date())
    update_constraints();

  // Adjust `cs' to the right topology and space dimension.
  // NOTE: we already checked for topology compatibility.
  cs.adjust_topology_and_space_dimension(topology(), space_dim);

  const bool adding_pending = can_have_something_pending();

  // Here we do not require `con_sys' to be sorted.
  // also, we _swap_ (instead of copying) the coefficients of `cs'
  // (which is not a const).
  const dimension_type old_num_rows = con_sys.num_rows();
  const dimension_type cs_num_rows = cs.num_rows();
  const dimension_type cs_num_columns = cs.num_columns();
  con_sys.add_zero_rows(cs_num_rows,
                        Linear_Row::Flags(topology(),
                                          Linear_Row::RAY_OR_POINT_OR_INEQUALITY));
  for (dimension_type i = cs_num_rows; i-- > 0; ) {
    // NOTE: we cannot directly swap the rows, since they might have
    // different capacities (besides possibly having different sizes):
    // thus, we steal one coefficient at a time.
    Constraint& new_c = con_sys[old_num_rows + i];
    Constraint& old_c = cs[i];
    if (old_c.is_equality())
      new_c.set_is_equality();
    using std::swap;
    for (dimension_type j = cs_num_columns; j-- > 0; )
      swap(new_c[j], old_c[j]);
  }

  if (adding_pending)
    set_constraints_pending();
  else {
    // The newly added ones are not pending constraints.
    con_sys.unset_pending_rows();
    // They have been simply appended.
    con_sys.set_sorted(false);
    // Constraints are not minimized and generators are not up-to-date.
    clear_constraints_minimized();
    clear_generators_up_to_date();
  }
  // Note: the constraint system may have become unsatisfiable, thus
  // we do not check for satisfiability.
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::add_constraints(const Constraint_System& cs) {
  // TODO: this is just an executable specification.
  Constraint_System cs_copy = cs;
  add_recycled_constraints(cs_copy);
}

void
PPL::Polyhedron::add_recycled_generators(Generator_System& gs) {
  // Topology compatibility check.
  if (is_necessarily_closed() && gs.has_closure_points())
    throw_topology_incompatible("add_recycled_generators(gs)", "gs", gs);
  // Dimension-compatibility check:
  // the dimension of `gs' can not be greater than space_dim.
  const dimension_type gs_space_dim = gs.space_dimension();
  if (space_dim < gs_space_dim)
    throw_dimension_incompatible("add_recycled_generators(gs)", "gs", gs);

  // Adding no generators is a no-op.
  if (gs.has_no_rows())
    return;

  // Adding valid generators to a zero-dimensional polyhedron
  // transform it in the zero-dimensional universe polyhedron.
  if (space_dim == 0) {
    if (marked_empty() && !gs.has_points())
      throw_invalid_generators("add_recycled_generators(gs)", "gs");
    set_zero_dim_univ();
    PPL_ASSERT_HEAVY(OK(true));
    return;
  }

  // Adjust `gs' to the right topology and dimensions.
  // NOTE: we already checked for topology compatibility.
  gs.adjust_topology_and_space_dimension(topology(), space_dim);
  // For NNC polyhedra, each point must be matched by
  // the corresponding closure point.
  if (!is_necessarily_closed())
    gs.add_corresponding_closure_points();

  // The generators (possibly with pending rows) are required.
  if ((has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !minimize())) {
    // We have just discovered that `*this' is empty.
    // So `gs' must contain at least one point.
    if (!gs.has_points())
      throw_invalid_generators("add_recycled_generators(gs)", "gs");
    // The polyhedron is no longer empty and generators are up-to-date.
    using std::swap;
    swap(gen_sys, gs);
    if (gen_sys.num_pending_rows() > 0) {
      // Even though `gs' has pending generators, since the constraints
      // of the polyhedron are not up-to-date, the polyhedron cannot
      // have pending generators. By integrating the pending part
      // of `gen_sys' we may loose sortedness.
      gen_sys.unset_pending_rows();
      gen_sys.set_sorted(false);
    }
    set_generators_up_to_date();
    clear_empty();
    PPL_ASSERT_HEAVY(OK());
    return;
  }

  const bool adding_pending = can_have_something_pending();

  // Here we do not require `gen_sys' to be sorted.
  // also, we _swap_ (instead of copying) the coefficients of `gs'
  // (which is not a const).
  const dimension_type old_num_rows = gen_sys.num_rows();
  const dimension_type gs_num_rows = gs.num_rows();
  const dimension_type gs_num_columns = gs.num_columns();
  gen_sys.add_zero_rows(gs_num_rows,
                        Linear_Row::Flags(topology(),
                                          Linear_Row::RAY_OR_POINT_OR_INEQUALITY));
  for (dimension_type i = gs_num_rows; i-- > 0; ) {
    // NOTE: we cannot directly swap the rows, since they might have
    // different capacities (besides possibly having different sizes):
    // thus, we steal one coefficient at a time.
    Generator& new_g = gen_sys[old_num_rows + i];
    Generator& old_g = gs[i];
    if (old_g.is_line())
      new_g.set_is_line();
    using std::swap;
    for (dimension_type j = gs_num_columns; j-- > 0; )
      swap(new_g[j], old_g[j]);
  }

  if (adding_pending)
    set_generators_pending();
  else {
    // The newly added ones are not pending generators.
    gen_sys.unset_pending_rows();
    // They have been simply appended.
    gen_sys.set_sorted(false);
    // Constraints are not up-to-date and generators are not minimized.
    clear_constraints_up_to_date();
    clear_generators_minimized();
  }
  PPL_ASSERT_HEAVY(OK(true));
}

void
PPL::Polyhedron::add_generators(const Generator_System& gs) {
  // TODO: this is just an executable specification.
  Generator_System gs_copy = gs;
  add_recycled_generators(gs_copy);
}

void
PPL::Polyhedron::add_congruences(const Congruence_System& cgs) {
  // Dimension-compatibility check.
  if (space_dim < cgs.space_dimension())
    throw_dimension_incompatible("add_congruences(cgs)", "cgs", cgs);

  Constraint_System cs;
  bool inserted = false;
  for (Congruence_System::const_iterator i = cgs.begin(),
         cgs_end = cgs.end(); i != cgs_end; ++i) {
    const Congruence& cg = *i;
    if (cg.is_equality()) {
      Linear_Expression le(cg);
      Constraint c(le, Constraint::EQUALITY, NECESSARILY_CLOSED);
      // Enforce normalization.
      c.strong_normalize();
      // TODO: Consider stealing the row in c when adding it to cs.
      cs.insert(c);
      inserted = true;
    }
    else {
      PPL_ASSERT(cg.is_proper_congruence());
      if (cg.is_inconsistent()) {
        set_empty();
        return;
      }
      if (!cg.is_tautological())
        throw_invalid_argument("add_congruences(cgs)",
                               "cgs has a non-trivial, proper congruence");
    }
  }
  // Only add cs if it contains something.
  if (inserted)
    add_recycled_constraints(cs);
}

void
PPL::Polyhedron::refine_with_constraint(const Constraint& c) {
  // Dimension-compatibility check.
  if (space_dim < c.space_dimension())
    throw_dimension_incompatible("refine_with_constraint(c)", "c", c);
  // If the polyhedron is known to be empty, do nothing.
  if (!marked_empty())
    refine_no_check(c);
}

void
PPL::Polyhedron::refine_with_congruence(const Congruence& cg) {
  // Dimension-compatibility check.
  if (space_dim < cg.space_dimension())
    throw_dimension_incompatible("refine_with_congruence(cg)", "cg", cg);

  // If the polyhedron is known to be empty, do nothing.
  if (marked_empty())
    return;

  // Dealing with a zero-dimensional space polyhedron first.
  if (space_dim == 0) {
    if (!cg.is_tautological())
      set_empty();
    return;
  }

  if (cg.is_equality()) {
    Linear_Expression le(cg);
    Constraint c(le, Constraint::EQUALITY, NECESSARILY_CLOSED);
    // Enforce normalization.
    c.strong_normalize();
    refine_no_check(c);
  }
}

void
PPL::Polyhedron::refine_with_constraints(const Constraint_System& cs) {
  // TODO: this is just an executable specification.

  // Dimension-compatibility check.
  const dimension_type cs_space_dim = cs.space_dimension();
  if (space_dim < cs_space_dim)
    throw_dimension_incompatible("refine_with_constraints(cs)a",
                                 "cs", cs);

  // Adding no constraints is a no-op.
  if (cs.has_no_rows())
    return;

  if (space_dim == 0) {
    // In a 0-dimensional space the constraints are
    // tautologies (e.g., 0 == 0 or 1 >= 0 or 1 > 0) or
    // inconsistent (e.g., 1 == 0 or -1 >= 0 or 0 > 0).
    // In a system of constraints `begin()' and `end()' are equal
    // if and only if the system only contains tautologies.
    if (cs.begin() != cs.end())
      // There is a constraint, it must be inconsistent,
      // the polyhedron is empty.
      status.set_empty();
    return;
  }

  if (marked_empty())
    return;

  // The constraints (possibly with pending rows) are required.
  if (has_pending_generators())
    process_pending_generators();
  else if (!constraints_are_up_to_date())
    update_constraints();

  const bool adding_pending = can_have_something_pending();
  for (dimension_type i = cs.num_rows(); i-- > 0; ) {
    const Constraint& c = cs[i];

    if (c.is_necessarily_closed() || !is_necessarily_closed())
      // Since `con_sys' is not empty, the topology and space dimension
      // of the inserted constraint are automatically adjusted.
      if (adding_pending)
        con_sys.insert_pending(c);
      else
        con_sys.insert(c);
    else {
      // Here we know that *this is necessarily closed so even if c is
      // topologically closed, by barely invoking `con_sys.insert(c)' we
      // would cause a change in the topology of `con_sys', which is
      // wrong.  Thus, we insert a topology closed and "topology
      // corrected" version of `c'.
      Linear_Expression nc_expr = Linear_Expression(c);
      if (c.is_equality())
        if (adding_pending)
          con_sys.insert_pending(nc_expr == 0);
        else
          con_sys.insert(nc_expr == 0);
      else
        if (adding_pending)
          con_sys.insert_pending(nc_expr >= 0);
        else
          con_sys.insert(nc_expr >= 0);
    }
  }

  if (adding_pending)
    set_constraints_pending();
  else {
    // Constraints are not minimized and generators are not up-to-date.
    clear_constraints_minimized();
    clear_generators_up_to_date();
  }

  // Note: the constraint system may have become unsatisfiable, thus
  // we do not check for satisfiability.
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::refine_with_congruences(const Congruence_System& cgs) {
  // Dimension-compatibility check.
  if (space_dim < cgs.space_dimension())
    throw_dimension_incompatible("refine_with_congruences(cgs)", "cgs", cgs);

  Constraint_System cs;
  bool inserted = false;
  for (Congruence_System::const_iterator i = cgs.begin(),
         cgs_end = cgs.end(); i != cgs_end; ++i) {
    if (i->is_equality()) {
      Linear_Expression le(*i);
      Constraint c(le, Constraint::EQUALITY, NECESSARILY_CLOSED);
      // Enforce normalization.
      c.strong_normalize();
      // TODO: Consider stealing the row in c when adding it to cs.
      cs.insert(c);
      inserted = true;
    }
    else if (i->is_inconsistent()) {
      set_empty();
      return;
    }
  }
  // Only add cgs if congruences were inserted into cgs, as the
  // dimension of cs must be at most that of the polyhedron.
  if (inserted)
    add_recycled_constraints(cs);
}

void
PPL::Polyhedron::unconstrain(const Variable var) {
  // Dimension-compatibility check.
  if (space_dim < var.space_dimension())
    throw_dimension_incompatible("unconstrain(var)", var.space_dimension());

  // Do something only if the polyhedron is non-empty.
  if (marked_empty()
      || (has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators()))
    // Empty: do nothing.
    return;

  PPL_ASSERT(generators_are_up_to_date());
  // Since `gen_sys' is not empty, the topology and space dimension
  // of the inserted generator are automatically adjusted.
  if (can_have_something_pending()) {
    gen_sys.insert_pending(Generator::line(var));
    set_generators_pending();
  }
  else {
    gen_sys.insert(Generator::line(var));
    // After adding the new generator,
    // constraints are no longer up-to-date.
    clear_generators_minimized();
    clear_constraints_up_to_date();
  }
  PPL_ASSERT_HEAVY(OK(true));
}

void
PPL::Polyhedron::unconstrain(const Variables_Set& vars) {
  // The cylindrification with respect to no dimensions is a no-op.
  // This case also captures the only legal cylindrification
  // of a polyhedron in a 0-dim space.
  if (vars.empty())
    return;

  // Dimension-compatibility check.
  const dimension_type min_space_dim = vars.space_dimension();
  if (space_dim < min_space_dim)
    throw_dimension_incompatible("unconstrain(vs)", min_space_dim);

  // Do something only if the polyhedron is non-empty.
  if (marked_empty()
      || (has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators()))
    // Empty: do nothing.
    return;

  PPL_ASSERT(generators_are_up_to_date());
  // Since `gen_sys' is not empty, the topology and space dimension
  // of the inserted generators are automatically adjusted.
  Variables_Set::const_iterator vsi = vars.begin();
  Variables_Set::const_iterator vsi_end = vars.end();
  if (can_have_something_pending()) {
    for ( ; vsi != vsi_end; ++vsi)
      gen_sys.insert_pending(Generator::line(Variable(*vsi)));
    set_generators_pending();
  }
  else {
    for ( ; vsi != vsi_end; ++vsi)
      gen_sys.insert(Generator::line(Variable(*vsi)));
    // After adding the new generators,
    // constraints are no longer up-to-date.
    clear_generators_minimized();
    clear_constraints_up_to_date();
  }
  PPL_ASSERT_HEAVY(OK(true));
}

void
PPL::Polyhedron::intersection_assign(const Polyhedron& y) {
  Polyhedron& x = *this;
  // Topology compatibility check.
  if (x.topology() != y.topology())
    throw_topology_incompatible("intersection_assign(y)", "y", y);
  // Dimension-compatibility check.
  if (x.space_dim != y.space_dim)
    throw_dimension_incompatible("intersection_assign(y)", "y", y);

  // If one of the two polyhedra is empty, the intersection is empty.
  if (x.marked_empty())
    return;
  if (y.marked_empty()) {
    x.set_empty();
    return;
  }

  // If both polyhedra are zero-dimensional,
  // then at this point they are necessarily non-empty,
  // so that their intersection is non-empty too.
  if (x.space_dim == 0)
    return;

  // Both systems of constraints have to be up-to-date,
  // possibly having pending constraints.
  if (x.has_pending_generators())
    x.process_pending_generators();
  else if (!x.constraints_are_up_to_date())
    x.update_constraints();

  if (y.has_pending_generators())
    y.process_pending_generators();
  else if (!y.constraints_are_up_to_date())
    y.update_constraints();

  // Here both systems are up-to-date and possibly have pending constraints
  // (but they cannot have pending generators).
  PPL_ASSERT(!x.has_pending_generators() && x.constraints_are_up_to_date());
  PPL_ASSERT(!y.has_pending_generators() && y.constraints_are_up_to_date());

  // If `x' can support pending constraints,
  // the constraints of `y' are added as pending constraints of `x'.
  if (x.can_have_something_pending()) {
    x.con_sys.add_pending_rows(y.con_sys);
    x.set_constraints_pending();
  }
  else {
    // `x' cannot support pending constraints.
    // If both constraint systems are (fully) sorted, then we can
    // merge them; otherwise we simply add the second to the first.
    if (x.con_sys.is_sorted()
        && y.con_sys.is_sorted() && !y.has_pending_constraints())
      x.con_sys.merge_rows_assign(y.con_sys);
    else
      x.con_sys.add_rows(y.con_sys);
    // Generators are no longer up-to-date and constraints are no
    // longer minimized.
    x.clear_generators_up_to_date();
    x.clear_constraints_minimized();
  }
  PPL_ASSERT_HEAVY(x.OK() && y.OK());
}

namespace {

struct Ruled_Out_Pair {
  PPL::dimension_type constraint_index;
  PPL::dimension_type num_ruled_out;
};

struct Ruled_Out_Less_Than {
  bool operator()(const Ruled_Out_Pair& x,
                  const Ruled_Out_Pair& y) const {
    return x.num_ruled_out > y.num_ruled_out;
  }
};

bool
add_to_system_and_check_independence(PPL::Linear_System& eq_sys,
                                     const PPL::Linear_Row& eq) {
  // Check if eq is linearly independent from eq_sys.
  PPL_ASSERT(eq.is_line_or_equality());
  eq_sys.insert(eq);
  const PPL::dimension_type eq_sys_num_rows = eq_sys.num_rows();
  const PPL::dimension_type rank = eq_sys.gauss(eq_sys_num_rows);
  if (rank == eq_sys_num_rows)
    // eq is linearly independent from eq_sys.
    return true;
  else {
    // eq is not linearly independent from eq_sys.
    PPL_ASSERT(rank == eq_sys_num_rows - 1);
    eq_sys.remove_trailing_rows(1);
    return false;
  }
}

/*
  Modifies the vector of pointers \p ineqs_p, setting to 0 those entries
  that point to redundant inequalities or masked equalities.
  The redundancy test is based on saturation matrix \p sat and
  on knowing that there exists \p rank non-redundant equalities
  (they are implicit, i.e., not explicitly listed in \p ineqs_p).
*/
void
drop_redundant_inequalities(std::vector<const PPL::Constraint*>& ineqs_p,
                            const PPL::Topology topology,
                            const PPL::Bit_Matrix& sat,
                            const PPL::dimension_type rank) {
  using namespace Parma_Polyhedra_Library;
  const dimension_type num_rows = ineqs_p.size();
  PPL_ASSERT(num_rows > 0);
  // `rank' is the rank of the (implicit) system of equalities.
  const dimension_type space_dim = ineqs_p[0]->space_dimension();
  PPL_ASSERT(space_dim > 0 && space_dim >= rank);
  const dimension_type num_coefficients
    = space_dim + ((topology == NECESSARILY_CLOSED) ? 0U : 1U);
  const dimension_type min_sat = num_coefficients - rank;
  const dimension_type num_cols_sat = sat.num_columns();

  // Perform quick redundancy test based on the number of saturators.
  for (dimension_type i = num_rows; i-- > 0; ) {
    if (sat[i].empty())
      // Masked equalities are redundant.
      ineqs_p[i] = 0;
    else {
      const dimension_type num_sat = num_cols_sat - sat[i].count_ones();
      if (num_sat < min_sat)
        ineqs_p[i] = 0;
    }
  }

  // Re-examine remaining inequalities.
  // Iteration index `i' is _intentionally_ increasing.
  for (dimension_type i = 0; i < num_rows; ++i) {
    if (ineqs_p[i] != 0) {
      for (dimension_type j = 0; j < num_rows; ++j) {
        bool strict_subset;
        if (ineqs_p[j] != 0 && i != j
            && subset_or_equal(sat[j], sat[i], strict_subset)) {
          if (strict_subset) {
            ineqs_p[i] = 0;
            break;
          }
          else
            // Here `sat[j] == sat[i]'.
            ineqs_p[j] = 0;
        }
      }
    }
  }
}

} // namespace

bool
PPL::Polyhedron::simplify_using_context_assign(const Polyhedron& y) {
  Polyhedron& x = *this;
  // Topology compatibility check.
  if (x.topology() != y.topology())
    throw_topology_incompatible("simplify_using_context_assign(y)", "y", y);
  // Dimension-compatibility check.
  if (x.space_dim != y.space_dim)
    throw_dimension_incompatible("simplify_using_context_assign(y)", "y", y);

  // Filter away the zero-dimensional case.
  if (x.space_dim == 0) {
    if (y.is_empty()) {
      x.set_zero_dim_univ();
      return false;
    }
    else
      return !x.is_empty();
  }

  // If `y' is empty, the biggest enlargement for `x' is the universe.
  if (!y.minimize()) {
    Polyhedron ph(x.topology(), x.space_dim, UNIVERSE);
    m_swap(ph);
    return false;
  }

  // If `x' is empty, the intersection is empty.
  if (!x.minimize()) {
    // Search for a constraint of `y' that is not a tautology.
    PPL_ASSERT(!y.has_pending_generators() && y.constraints_are_up_to_date());
    for (dimension_type i = y.con_sys.num_rows(); i-- > 0; ) {
      const Constraint& y_con_sys_i = y.con_sys[i];
      if (!y_con_sys_i.is_tautological()) {
        // Found: we obtain a constraint `c' contradicting the one we
        // found, and assign to `x' the polyhedron `ph' with `c' as
        // the only constraint.
        Polyhedron ph(x.topology(), x.space_dim, UNIVERSE);
        Linear_Expression le(y_con_sys_i);
        switch (y_con_sys_i.type()) {
        case Constraint::EQUALITY:
          ph.refine_no_check(le == 1);
          break;
        case Constraint::NONSTRICT_INEQUALITY:
          ph.refine_no_check(le <= -1);
          break;
        case Constraint::STRICT_INEQUALITY:
          ph.refine_no_check(le == 0);
          break;
        }
        m_swap(ph);
        PPL_ASSERT_HEAVY(OK());
        return false;
      }
    }
    // `y' is the universe: `x' cannot be enlarged.
    return false;
  }

  PPL_ASSERT(x.constraints_are_minimized()
         && !x.has_something_pending()
         && y.generators_are_minimized()
         && !y.has_something_pending());
  const Constraint_System& x_cs = x.con_sys;
  const dimension_type x_cs_num_rows = x_cs.num_rows();
  const Generator_System& y_gs = y.gen_sys;

  // Record into `redundant_by_y' the info about which constraints of
  // `x' are redundant in the context `y'.  Count the number of
  // redundancies found.
  std::vector<bool> redundant_by_y(x_cs_num_rows, false);
  dimension_type num_redundant_by_y = 0;
  for (dimension_type i = 0; i < x_cs_num_rows; ++i)
    if (y_gs.satisfied_by_all_generators(x_cs[i])) {
      redundant_by_y[i] = true;
      ++num_redundant_by_y;
    }

  Constraint_System result_cs;

  if (num_redundant_by_y < x_cs_num_rows) {
    // Some constraints were not identified as redundant (yet?).
    const Constraint_System& y_cs = y.con_sys;
    const dimension_type y_cs_num_rows = y_cs.num_rows();
    // Compute into `z' the minimized intersection of `x' and `y'.
    const bool x_first = (x_cs_num_rows > y_cs_num_rows);
    Polyhedron z(x_first ? x : y);
    if (x_first)
      z.add_constraints(y_cs);
    else {
      // Only copy (and then recycle) the non-redundant constraints.
      Constraint_System tmp_cs;
      for (dimension_type i = 0; i < x_cs_num_rows; ++i) {
        if (!redundant_by_y[i])
          tmp_cs.insert(x_cs[i]);
      }
      z.add_recycled_constraints(tmp_cs);
    }
    if (!z.minimize()) {
      // The objective function is the default, zero.
      // We do not care about minimization or maximization, since
      // we are only interested in satisfiability.
      MIP_Problem lp;
      if (x.is_necessarily_closed()) {
        lp.add_space_dimensions_and_embed(x.space_dim);
        lp.add_constraints(y_cs);
      }
      else {
        // KLUDGE: temporarily mark `y_cs' if it was necessarily
        // closed, so that we can interpret the epsilon dimension as a
        // standard dimension. Be careful to reset the topology of `cs'
        // even on exceptional execution path.
        const_cast<Constraint_System&>(y_cs).set_necessarily_closed();
        try {
          lp.add_space_dimensions_and_embed(x.space_dim+1);
          lp.add_constraints(y_cs);
          const_cast<Constraint_System&>(y_cs).set_not_necessarily_closed();
        }
        catch (...) {
          const_cast<Constraint_System&>(y_cs).set_not_necessarily_closed();
          throw;
        }
      }
      // We apply the following heuristics here: constraints of `x' that
      // are not made redundant by `y' are added to `lp' depending on
      // the number of generators of `y' they rule out (the more generators
      // they rule out, the sooner they are added).  Of course, as soon
      // as `lp' becomes unsatisfiable, we stop adding.
      std::vector<Ruled_Out_Pair>
        ruled_out_vec(x_cs_num_rows - num_redundant_by_y);
      for (dimension_type i = 0, j = 0; i < x_cs_num_rows; ++i) {
        if (!redundant_by_y[i]) {
          const Constraint& c = x_cs[i];
          Topology_Adjusted_Scalar_Product_Sign sps(c);
          dimension_type num_ruled_out_generators = 0;
          for (Generator_System::const_iterator k = y_gs.begin(),
                 y_gs_end = y_gs.end(); k != y_gs_end; ++k) {
            const Generator& g = *k;
            const int sp_sign = sps(g, c);
            if (x.is_necessarily_closed()) {
              if (g.is_line()) {
                // Lines must saturate the constraint.
                if (sp_sign != 0)
                  goto ruled_out;
              }
              else {
                // `g' is either a ray, a point or a closure point.
                if (c.is_inequality()) {
                  // `c' is a non-strict inequality.
                  if (sp_sign < 0)
                    goto ruled_out;
                }
                else
                  // `c' is an equality.
                  if (sp_sign != 0)
                    goto ruled_out;
              }
            }
            else
              // The topology is not necessarily closed.
              switch (g.type()) {
              case Generator::LINE:
                // Lines must saturate the constraint.
                if (sp_sign != 0)
                  goto ruled_out;
                break;
              case Generator::POINT:
                // Have to perform the special test when dealing with
                // a strict inequality.
                switch (c.type()) {
                case Constraint::EQUALITY:
                  if (sp_sign != 0)
                    goto ruled_out;
                  break;
                case Constraint::NONSTRICT_INEQUALITY:
                  if (sp_sign < 0)
                    goto ruled_out;
                  break;
                case Constraint::STRICT_INEQUALITY:
                  if (sp_sign <= 0)
                    goto ruled_out;
                  break;
                }
                break;
              case Generator::RAY:
                // Intentionally fall through.
              case Generator::CLOSURE_POINT:
                if (c.is_inequality()) {
                  // Constraint `c' is either a strict or a non-strict
                  // inequality.
                  if (sp_sign < 0)
                    goto ruled_out;
                }
                else
                  // Constraint `c' is an equality.
                  if (sp_sign != 0)
                    goto ruled_out;
                break;
              }

            // If we reach this point, `g' satisfies `c'.
            continue;
          ruled_out:
            ++num_ruled_out_generators;
          }
          ruled_out_vec[j].constraint_index = i;
          ruled_out_vec[j].num_ruled_out = num_ruled_out_generators;
          ++j;
        }
      }
      std::sort(ruled_out_vec.begin(), ruled_out_vec.end(),
                Ruled_Out_Less_Than());

      for (std::vector<Ruled_Out_Pair>::const_iterator
             j = ruled_out_vec.begin(),
             ruled_out_vec_end = ruled_out_vec.end();
           j != ruled_out_vec_end;
           ++j) {
        const Constraint& c = x_cs[j->constraint_index];
        result_cs.insert(c);
        lp.add_constraint(c);
        MIP_Problem_Status status = lp.solve();
        if (status == UNFEASIBLE_MIP_PROBLEM) {
          Polyhedron result_ph(x.topology(), x.space_dim, UNIVERSE);
          result_ph.add_constraints(result_cs);
          x.m_swap(result_ph);
          PPL_ASSERT_HEAVY(x.OK());
          return false;
        }
      }
      // Cannot exit from here.
      PPL_UNREACHABLE;
    }
    else {
      // Here `z' is not empty and minimized.
      PPL_ASSERT(z.constraints_are_minimized()
             && z.generators_are_minimized()
             && !z.has_something_pending());
      const Constraint_System& z_cs = z.con_sys;
      const Generator_System& z_gs = z.gen_sys;
      const dimension_type z_gs_num_rows = z_gs.num_rows();

      // Compute the number of equalities in x_cs, y_cs and z_cs
      // (exploiting minimal form knowledge).
      dimension_type x_cs_num_eq = 0;
      while (x_cs[x_cs_num_eq].is_equality())
        ++x_cs_num_eq;
      dimension_type y_cs_num_eq = 0;
      while (y_cs[y_cs_num_eq].is_equality())
        ++y_cs_num_eq;
      dimension_type z_cs_num_eq = 0;
      while (z_cs[z_cs_num_eq].is_equality())
        ++z_cs_num_eq;
      PPL_ASSERT(x_cs_num_eq <= z_cs_num_eq && y_cs_num_eq <= z_cs_num_eq);

      // Identify non-redundant equalities.
      Constraint_System non_redundant_eq;
      dimension_type num_non_redundant_eq = 0;
      const dimension_type needed_non_redundant_eq = z_cs_num_eq - y_cs_num_eq;
      Linear_System eqs(x.topology());
      if (needed_non_redundant_eq > 0) {
        // Populate eqs with the equalities from y.
        for (dimension_type i = 0; i < y_cs_num_eq; ++i)
          eqs.insert(y_cs[i]);
        // Try to find another `needed_non_redundant_eq' linear independent
        // equalities among those from x.
        for (dimension_type i = 0; i < x_cs_num_eq; ++i) {
          const Constraint& x_cs_i = x_cs[i];
          if (add_to_system_and_check_independence(eqs, x_cs_i)) {
            // x_cs_i is linear independent.
            non_redundant_eq.insert(x_cs_i);
            ++num_non_redundant_eq;
            if (num_non_redundant_eq == needed_non_redundant_eq)
              // Already found all the needed equalities.
              break;
          }
        }
        // NOTE: if num_non_redundant_eq < needed_non_redundant_eq
        // then we haven't found all the needed equalities yet:
        // this means that some inequalities from x actually holds
        // as "masked" equalities in the context of y.
        PPL_ASSERT(eqs.num_rows() <= z_cs_num_eq);
        PPL_ASSERT(num_non_redundant_eq <= needed_non_redundant_eq);
        PPL_ASSERT(z_cs_num_eq - eqs.num_rows()
               == needed_non_redundant_eq - num_non_redundant_eq);
      }

      // Identify non-redundant inequalities.
      // Avoid useless copies (no modifications are needed).
      std::vector<const Constraint*> non_redundant_ineq_p;
      // Fill non_redundant_ineq_p with (pointers to) inequalities
      // from y_cs ...
      for (dimension_type i = y_cs_num_eq; i < y_cs_num_rows; ++i)
        non_redundant_ineq_p.push_back(&y_cs[i]);
      // ... and (pointers to) non-redundant inequalities from x_cs.
      for (dimension_type i = x_cs_num_eq; i < x_cs_num_rows; ++i)
        if (!redundant_by_y[i])
          non_redundant_ineq_p.push_back(&x_cs[i]);

      const dimension_type non_redundant_ineq_p_size
        = non_redundant_ineq_p.size();
      const dimension_type y_cs_num_ineq = y_cs_num_rows - y_cs_num_eq;

      // Compute saturation info.
      const dimension_type sat_num_rows = non_redundant_ineq_p_size;
      Bit_Matrix sat(sat_num_rows, z_gs_num_rows);
      for (dimension_type i = sat_num_rows; i-- > 0; ) {
        const Constraint& non_redundant_ineq_i = *(non_redundant_ineq_p[i]);
        Bit_Row& sat_i = sat[i];
        for (dimension_type j = z_gs_num_rows; j-- > 0; )
          if (Scalar_Products::sign(non_redundant_ineq_i, z_gs[j]) != 0)
            sat_i.set(j);
        if (sat_i.empty() && num_non_redundant_eq < needed_non_redundant_eq) {
          // `non_redundant_ineq_i' is actually masking an equality
          // and we are still looking for some masked inequalities.
          // Iteration goes downwards, so the inequality comes from x_cs.
          PPL_ASSERT(i >= y_cs_num_ineq);
          // Check if the equality is independent in eqs.
          Linear_Row masked_eq = Linear_Row(non_redundant_ineq_i);
          masked_eq.set_is_line_or_equality();
          masked_eq.sign_normalize();
          if (add_to_system_and_check_independence(eqs, masked_eq)) {
            // It is independent: add the _inequality_ to non_redundant_eq.
            non_redundant_eq.insert(non_redundant_ineq_i);
            ++num_non_redundant_eq;
          }
        }
      }
      // Here we have already found all the needed (masked) equalities.
      PPL_ASSERT(num_non_redundant_eq == needed_non_redundant_eq);

      drop_redundant_inequalities(non_redundant_ineq_p, x.topology(),
                                  sat, z_cs_num_eq);

      // Place the non-redundant (masked) equalities into result_cs.
      result_cs.m_swap(non_redundant_eq);
      // Add to result_cs the non-redundant inequalities from x_cs,
      // i.e., those having indices no smaller than y_cs_num_ineq.
      for (dimension_type i = y_cs_num_ineq;
           i < non_redundant_ineq_p_size;
           ++i)
        if (non_redundant_ineq_p[i] != 0)
          result_cs.insert(*non_redundant_ineq_p[i]);
    }
  }

  Polyhedron result_ph(x.topology(), x.space_dim, UNIVERSE);
  result_ph.add_recycled_constraints(result_cs);
  x.m_swap(result_ph);
  PPL_ASSERT_HEAVY(x.OK());
  return true;
}

void
PPL::Polyhedron::poly_hull_assign(const Polyhedron& y) {
  Polyhedron& x = *this;
  // Topology compatibility check.
  if (x.topology() != y.topology())
    throw_topology_incompatible("poly_hull_assign(y)", "y", y);
  // Dimension-compatibility check.
  if (x.space_dim != y.space_dim)
    throw_dimension_incompatible("poly_hull_assign(y)", "y", y);

  // The poly-hull of a polyhedron `p' with an empty polyhedron is `p'.
  if (y.marked_empty())
    return;
  if (x.marked_empty()) {
    x = y;
    return;
  }

  // If both polyhedra are zero-dimensional,
  // then at this point they are necessarily universe polyhedra,
  // so that their poly-hull is the universe polyhedron too.
  if (x.space_dim == 0)
    return;

  // Both systems of generators have to be up-to-date,
  // possibly having pending generators.
  if ((x.has_pending_constraints() && !x.process_pending_constraints())
      || (!x.generators_are_up_to_date() && !x.update_generators())) {
    // Discovered `x' empty when updating generators.
    x = y;
    return;
  }
  if ((y.has_pending_constraints() && !y.process_pending_constraints())
      || (!y.generators_are_up_to_date() && !y.update_generators()))
    // Discovered `y' empty when updating generators.
    return;

  // Here both systems are up-to-date and possibly have pending generators
  // (but they cannot have pending constraints).
  PPL_ASSERT(!x.has_pending_constraints() && x.generators_are_up_to_date());
  PPL_ASSERT(!y.has_pending_constraints() && y.generators_are_up_to_date());

  // If `x' can support pending generators,
  // the generators of `y' are added as pending generators of `x'.
  if (x.can_have_something_pending()) {
    x.gen_sys.add_pending_rows(y.gen_sys);
    x.set_generators_pending();
  }
  else {
    // `x' cannot support pending generators.
    // If both generator systems are (fully) sorted, then we can merge
    // them; otherwise we simply add the second to the first.
    if (x.gen_sys.is_sorted()
        && y.gen_sys.is_sorted() && !y.has_pending_generators())
      x.gen_sys.merge_rows_assign(y.gen_sys);
    else
      x.gen_sys.add_rows(y.gen_sys);
    // Constraints are no longer up-to-date
    // and generators are no longer minimized.
    x.clear_constraints_up_to_date();
    x.clear_generators_minimized();
  }
  // At this point both `x' and `y' are not empty.
  PPL_ASSERT_HEAVY(x.OK(true) && y.OK(true));
}

void
PPL::Polyhedron::poly_difference_assign(const Polyhedron& y) {
  Polyhedron& x = *this;
  // Topology compatibility check.
  if (x.topology() != y.topology())
    throw_topology_incompatible("poly_difference_assign(y)", "y", y);
  // Dimension-compatibility check.
  if (x.space_dim != y.space_dim)
    throw_dimension_incompatible("poly_difference_assign(y)", "y", y);

  // The difference of a polyhedron `p' and an empty polyhedron is `p'.
  if (y.marked_empty())
    return;
  // The difference of an empty polyhedron and of a polyhedron `p' is empty.
  if (x.marked_empty())
    return;

  // If both polyhedra are zero-dimensional,
  // then at this point they are necessarily universe polyhedra,
  // so that their difference is empty.
  if (x.space_dim == 0) {
    x.set_empty();
    return;
  }

  // TODO: This is just an executable specification.
  //       Have to find a more efficient method.

  if (y.contains(x)) {
    x.set_empty();
    return;
  }

  // Being lazy here is only harmful.
  // `minimize()' will process any pending constraints or generators.
  if (!y.minimize())
    return;
  x.minimize();

  Polyhedron new_polyhedron(topology(), x.space_dim, EMPTY);

  const Constraint_System& y_cs = y.constraints();
  for (Constraint_System::const_iterator i = y_cs.begin(),
         y_cs_end = y_cs.end(); i != y_cs_end; ++i) {
    const Constraint& c = *i;
    PPL_ASSERT(!c.is_tautological());
    PPL_ASSERT(!c.is_inconsistent());
    // If the polyhedron `x' is included in the polyhedron defined by
    // `c', then `c' can be skipped, as adding its complement to `x'
    // would result in the empty polyhedron.  Moreover, if we operate
    // on C-polyhedra and `c' is a non-strict inequality, c _must_ be
    // skipped for otherwise we would obtain a result that is less
    // precise than the poly-difference.
    if (x.relation_with(c).implies(Poly_Con_Relation::is_included()))
      continue;
    Polyhedron z = x;
    const Linear_Expression e = Linear_Expression(c);
    switch (c.type()) {
    case Constraint::NONSTRICT_INEQUALITY:
      if (is_necessarily_closed())
        z.refine_no_check(e <= 0);
      else
        z.refine_no_check(e < 0);
      break;
    case Constraint::STRICT_INEQUALITY:
      z.refine_no_check(e <= 0);
      break;
    case Constraint::EQUALITY:
      if (is_necessarily_closed())
        // We have already filtered out the case
        // when `x' is included in `y': the result is `x'.
        return;
      else {
        Polyhedron w = x;
        w.refine_no_check(e < 0);
        new_polyhedron.poly_hull_assign(w);
        z.refine_no_check(e > 0);
      }
      break;
    }
    new_polyhedron.poly_hull_assign(z);
  }
  *this = new_polyhedron;

  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::
affine_image(const Variable var,
             const Linear_Expression& expr,
             Coefficient_traits::const_reference denominator) {
  // The denominator cannot be zero.
  if (denominator == 0)
    throw_invalid_argument("affine_image(v, e, d)", "d == 0");

  // Dimension-compatibility checks.
  // The dimension of `expr' should not be greater than the dimension
  // of `*this'.
  if (space_dim < expr.space_dimension())
    throw_dimension_incompatible("affine_image(v, e, d)", "e", expr);
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("affine_image(v, e, d)", "v", var);

  if (marked_empty())
    return;

  if (expr.coefficient(var) != 0) {
    // The transformation is invertible:
    // minimality and saturators are preserved, so that
    // pending rows, if present, are correctly handled.
    if (generators_are_up_to_date()) {
      // Generator_System::affine_image() requires the third argument
      // to be a positive Coefficient.
      if (denominator > 0)
        gen_sys.affine_image(var_space_dim, expr, denominator);
      else
        gen_sys.affine_image(var_space_dim, -expr, -denominator);
    }
    if (constraints_are_up_to_date()) {
      // To build the inverse transformation,
      // after copying and negating `expr',
      // we exchange the roles of `expr[var_space_dim]' and `denominator'.
      Linear_Expression inverse;
      if (expr[var_space_dim] > 0) {
        inverse = -expr;
        inverse[var_space_dim] = denominator;
        con_sys.affine_preimage(var_space_dim, inverse, expr[var_space_dim]);
      }
      else {
        // The new denominator is negative: we negate everything once
        // more, as Constraint_System::affine_preimage() requires the
        // third argument to be positive.
        inverse = expr;
        inverse[var_space_dim] = denominator;
        neg_assign(inverse[var_space_dim]);
        con_sys.affine_preimage(var_space_dim, inverse, -expr[var_space_dim]);
      }
    }
  }
  else {
    // The transformation is not invertible.
    // We need an up-to-date system of generators.
    if (has_something_pending())
      remove_pending_to_obtain_generators();
    else if (!generators_are_up_to_date())
      minimize();
    if (!marked_empty()) {
      // Generator_System::affine_image() requires the third argument
      // to be a positive Coefficient.
      if (denominator > 0)
        gen_sys.affine_image(var_space_dim, expr, denominator);
      else
        gen_sys.affine_image(var_space_dim, -expr, -denominator);

      clear_constraints_up_to_date();
      clear_generators_minimized();
      clear_sat_c_up_to_date();
      clear_sat_g_up_to_date();
    }
  }
  PPL_ASSERT_HEAVY(OK());
}


void
PPL::Polyhedron::
affine_preimage(const Variable var,
                const Linear_Expression& expr,
                Coefficient_traits::const_reference denominator) {
  // The denominator cannot be zero.
  if (denominator == 0)
    throw_invalid_argument("affine_preimage(v, e, d)", "d == 0");

  // Dimension-compatibility checks.
  // The dimension of `expr' should not be greater than the dimension
  // of `*this'.
  if (space_dim < expr.space_dimension())
    throw_dimension_incompatible("affine_preimage(v, e, d)", "e", expr);
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("affine_preimage(v, e, d)", "v", var);

  if (marked_empty())
    return;

  if (expr.coefficient(var) != 0) {
    // The transformation is invertible:
    // minimality and saturators are preserved.
    if (constraints_are_up_to_date()) {
      // Constraint_System::affine_preimage() requires the third argument
      // to be a positive Coefficient.
      if (denominator > 0)
        con_sys.affine_preimage(var_space_dim, expr, denominator);
      else
        con_sys.affine_preimage(var_space_dim, -expr, -denominator);
    }
    if (generators_are_up_to_date()) {
      // To build the inverse transformation,
      // after copying and negating `expr',
      // we exchange the roles of `expr[var_space_dim]' and `denominator'.
      Linear_Expression inverse;
      if (expr[var_space_dim] > 0) {
        inverse = -expr;
        inverse[var_space_dim] = denominator;
        gen_sys.affine_image(var_space_dim, inverse, expr[var_space_dim]);
      }
      else {
        // The new denominator is negative:
        // we negate everything once more, as Generator_System::affine_image()
        // requires the third argument to be positive.
        inverse = expr;
        inverse[var_space_dim] = denominator;
        neg_assign(inverse[var_space_dim]);
        gen_sys.affine_image(var_space_dim, inverse, -expr[var_space_dim]);
      }
    }
  }
  else {
    // The transformation is not invertible.
    // We need an up-to-date system of constraints.
    if (has_something_pending())
      remove_pending_to_obtain_constraints();
    else if (!constraints_are_up_to_date())
      minimize();
    // Constraint_System::affine_preimage() requires the third argument
    // to be a positive Coefficient.
    if (denominator > 0)
      con_sys.affine_preimage(var_space_dim, expr, denominator);
    else
      con_sys.affine_preimage(var_space_dim, -expr, -denominator);
    // Generators, minimality and saturators are no longer valid.
    clear_generators_up_to_date();
    clear_constraints_minimized();
    clear_sat_c_up_to_date();
    clear_sat_g_up_to_date();
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::
bounded_affine_image(const Variable var,
                     const Linear_Expression& lb_expr,
                     const Linear_Expression& ub_expr,
                     Coefficient_traits::const_reference denominator) {
  // The denominator cannot be zero.
  if (denominator == 0)
    throw_invalid_argument("bounded_affine_image(v, lb, ub, d)", "d == 0");

  // Dimension-compatibility checks.
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("bounded_affine_image(v, lb, ub, d)",
                                 "v", var);
  // The dimension of `lb_expr' and `ub_expr' should not be
  // greater than the dimension of `*this'.
  const dimension_type lb_space_dim = lb_expr.space_dimension();
  if (space_dim < lb_space_dim)
    throw_dimension_incompatible("bounded_affine_image(v, lb, ub, d)",
                                 "lb", lb_expr);
  const dimension_type ub_space_dim = ub_expr.space_dimension();
  if (space_dim < ub_space_dim)
    throw_dimension_incompatible("bounded_affine_image(v, lb, ub, d)",
                                 "ub", ub_expr);

  // Any image of an empty polyhedron is empty.
  if (marked_empty())
    return;

  // Check whether `var' occurs in `lb_expr' and/or `ub_expr'.
  if (lb_expr.coefficient(var) == 0) {
    // Here `var' may only occur in `ub_expr'.
    generalized_affine_image(var,
                             LESS_OR_EQUAL,
                             ub_expr,
                             denominator);
    if (denominator > 0)
      refine_no_check(lb_expr <= denominator*var);
    else
      refine_no_check(denominator*var <= lb_expr);
  }
  else if (ub_expr.coefficient(var) == 0) {
    // Here `var' only occurs in `lb_expr'.
    generalized_affine_image(var,
                             GREATER_OR_EQUAL,
                             lb_expr,
                             denominator);
    if (denominator > 0)
      refine_no_check(denominator*var <= ub_expr);
    else
      refine_no_check(ub_expr <= denominator*var);
  }
  else {
    // Here `var' occurs in both `lb_expr' and `ub_expr'.
    // To ease the computation, we add an additional dimension.
    const Variable new_var(space_dim);
    add_space_dimensions_and_embed(1);
    // Constrain the new dimension to be equal to `ub_expr'.
    refine_no_check(denominator*new_var == ub_expr);
    // Apply the affine lower bound.
    generalized_affine_image(var,
                             GREATER_OR_EQUAL,
                             lb_expr,
                             denominator);
    if (!marked_empty())
      // Now apply the affine upper bound, as recorded in `new_var'.
      refine_no_check(new_var >= var);
    // Remove the temporarily added dimension.
    remove_higher_space_dimensions(space_dim-1);
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::
bounded_affine_preimage(const Variable var,
                        const Linear_Expression& lb_expr,
                        const Linear_Expression& ub_expr,
                        Coefficient_traits::const_reference denominator) {
  // The denominator cannot be zero.
  if (denominator == 0)
    throw_invalid_argument("bounded_affine_preimage(v, lb, ub, d)", "d == 0");

  // Dimension-compatibility checks.
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("bounded_affine_preimage(v, lb, ub, d)",
                                 "v", var);
  // The dimension of `lb_expr' and `ub_expr' should not be
  // greater than the dimension of `*this'.
  const dimension_type lb_space_dim = lb_expr.space_dimension();
  if (space_dim < lb_space_dim)
    throw_dimension_incompatible("bounded_affine_preimage(v, lb, ub, d)",
                                 "lb", lb_expr);
  const dimension_type ub_space_dim = ub_expr.space_dimension();
  if (space_dim < ub_space_dim)
    throw_dimension_incompatible("bounded_affine_preimage(v, lb, ub, d)",
                                 "ub", ub_expr);

  // Any preimage of an empty polyhedron is empty.
  if (marked_empty())
    return;

  // Check whether `var' occurs in neither `lb_expr' nor `ub_expr'.
  if (lb_expr.coefficient(var) == 0 && ub_expr.coefficient(var) == 0) {
    if (denominator > 0) {
      refine_no_check(lb_expr <= denominator*var);
      refine_no_check(denominator*var <= ub_expr);
    }
    else {
      refine_no_check(ub_expr <= denominator*var);
      refine_no_check(denominator*var <= lb_expr);
    }
    unconstrain(var);
  }
  else {
    // Here `var' occurs in `lb_expr' or `ub_expr'.
    // To ease the computation, add an additional dimension.
    const Variable new_var(space_dim);
    add_space_dimensions_and_embed(1);
    // Swap dimensions `var' and `new_var'.
    std::vector<dimension_type> swapping_cycle;
    swapping_cycle.push_back(var_space_dim);
    swapping_cycle.push_back(space_dim);
    swapping_cycle.push_back(0);
    if (constraints_are_up_to_date())
      con_sys.permute_columns(swapping_cycle);
    if (generators_are_up_to_date())
      gen_sys.permute_columns(swapping_cycle);
    // Constrain the new dimension as dictated by `lb_expr' and `ub_expr'.
    // (we force minimization because we will need the generators).
    if (denominator > 0) {
      refine_no_check(lb_expr <= denominator*new_var);
      refine_no_check(denominator*new_var <= ub_expr);
    }
    else {
      refine_no_check(ub_expr <= denominator*new_var);
      refine_no_check(denominator*new_var <= lb_expr);
    }
    // Remove the temporarily added dimension.
    remove_higher_space_dimensions(space_dim-1);
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::
generalized_affine_image(const Variable var,
                         const Relation_Symbol relsym,
                         const Linear_Expression& expr,
                         Coefficient_traits::const_reference denominator) {
  // The denominator cannot be zero.
  if (denominator == 0)
    throw_invalid_argument("generalized_affine_image(v, r, e, d)", "d == 0");

  // Dimension-compatibility checks.
  // The dimension of `expr' should not be greater than the dimension
  // of `*this'.
  if (space_dim < expr.space_dimension())
    throw_dimension_incompatible("generalized_affine_image(v, r, e, d)",
                                 "e", expr);
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("generalized_affine_image(v, r, e, d)",
                                 "v", var);

  // Strict relation symbols are only admitted for NNC polyhedra.
  if (is_necessarily_closed()
      && (relsym == LESS_THAN || relsym == GREATER_THAN))
    throw_invalid_argument("generalized_affine_image(v, r, e, d)",
                           "r is a strict relation symbol");
  // The relation symbol cannot be a disequality.
  if (relsym == NOT_EQUAL)
    throw_invalid_argument("generalized_affine_image(v, r, e, d)",
                           "r is the disequality relation symbol");

  // First compute the affine image.
  affine_image(var, expr, denominator);

  if (relsym == EQUAL)
    // The affine relation is indeed an affine function.
    return;

  // Any image of an empty polyhedron is empty.
  // Note: DO check for emptiness here, as we will later add a ray.
  if (is_empty())
    return;

  switch (relsym) {
  case LESS_OR_EQUAL:
    add_generator(ray(-var));
    break;
  case GREATER_OR_EQUAL:
    add_generator(ray(var));
    break;
  case LESS_THAN:
  // Intentionally fall through.
  case GREATER_THAN:
    {
      // The relation symbol is strict.
      PPL_ASSERT(!is_necessarily_closed());
      // While adding the ray, we minimize the generators
      // in order to avoid adding too many redundant generators later.
      add_generator(ray((relsym == GREATER_THAN) ? var : -var));
      minimize();
      // We split each point of the generator system into two generators:
      // a closure point, having the same coordinates of the given point,
      // and another point, having the same coordinates for all but the
      // `var' dimension, which is displaced along the direction of the
      // newly introduced ray.
      const dimension_type eps_index = space_dim + 1;
      for (dimension_type i =  gen_sys.num_rows(); i-- > 0; )
        if (gen_sys[i].is_point()) {
          // Add a `var'-displaced copy of `gen_sys[i]' to the generator system.
          gen_sys.add_row(gen_sys[i]);
          if (relsym == GREATER_THAN)
            ++gen_sys[gen_sys.num_rows()-1][var_space_dim];
          else
            --gen_sys[gen_sys.num_rows()-1][var_space_dim];
          // Transform `gen_sys[i]' into a closure point.
          gen_sys[i][eps_index] = 0;
        }
      clear_constraints_up_to_date();
      clear_generators_minimized();
      gen_sys.set_sorted(false);
      clear_sat_c_up_to_date();
      clear_sat_g_up_to_date();
    }
    break;
  default:
    // The EQUAL and NOT_EQUAL cases have been already dealt with.
    PPL_UNREACHABLE;
    break;
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::
generalized_affine_preimage(const Variable var,
                            const Relation_Symbol relsym,
                            const Linear_Expression& expr,
                            Coefficient_traits::const_reference denominator) {
  // The denominator cannot be zero.
  if (denominator == 0)
    throw_invalid_argument("generalized_affine_preimage(v, r, e, d)",
                           "d == 0");

  // Dimension-compatibility checks.
  // The dimension of `expr' should not be greater than the dimension
  // of `*this'.
  if (space_dim < expr.space_dimension())
    throw_dimension_incompatible("generalized_affine_preimage(v, r, e, d)",
                                 "e", expr);
  // `var' should be one of the dimensions of the polyhedron.
  const dimension_type var_space_dim = var.space_dimension();
  if (space_dim < var_space_dim)
    throw_dimension_incompatible("generalized_affine_preimage(v, r, e, d)",
                                 "v", var);

  // Strict relation symbols are only admitted for NNC polyhedra.
  if (is_necessarily_closed()
      && (relsym == LESS_THAN || relsym == GREATER_THAN))
    throw_invalid_argument("generalized_affine_preimage(v, r, e, d)",
                           "r is a strict relation symbol");
  // The relation symbol cannot be a disequality.
  if (relsym == NOT_EQUAL)
    throw_invalid_argument("generalized_affine_preimage(v, r, e, d)",
                           "r is the disequality relation symbol");

  // Check whether the affine relation is indeed an affine function.
  if (relsym == EQUAL) {
    affine_preimage(var, expr, denominator);
    return;
  }

  // Compute the reversed relation symbol to simplify later coding.
  Relation_Symbol reversed_relsym;
  switch (relsym) {
  case LESS_THAN:
    reversed_relsym = GREATER_THAN;
    break;
  case LESS_OR_EQUAL:
    reversed_relsym = GREATER_OR_EQUAL;
    break;
  case GREATER_OR_EQUAL:
    reversed_relsym = LESS_OR_EQUAL;
    break;
  case GREATER_THAN:
    reversed_relsym = LESS_THAN;
    break;
  default:
    // The EQUAL and NOT_EQUAL cases have been already dealt with.
    PPL_UNREACHABLE;
    return;
  }

  // Check whether the preimage of this affine relation can be easily
  // computed as the image of its inverse relation.
  const Coefficient& var_coefficient = expr.coefficient(var);
  if (var_coefficient != 0) {
    Linear_Expression inverse_expr
      = expr - (denominator + var_coefficient) * var;
    PPL_DIRTY_TEMP_COEFFICIENT(inverse_denominator);
    neg_assign(inverse_denominator, var_coefficient);
    Relation_Symbol inverse_relsym
      = (sgn(denominator) == sgn(inverse_denominator))
      ? relsym : reversed_relsym;
    generalized_affine_image(var, inverse_relsym, inverse_expr,
                             inverse_denominator);
    return;
  }

  // Here `var_coefficient == 0', so that the preimage cannot
  // be easily computed by inverting the affine relation.
  // Shrink the polyhedron by adding the constraint induced
  // by the affine relation.
  const Relation_Symbol corrected_relsym
    = (denominator > 0) ? relsym : reversed_relsym;
  switch (corrected_relsym) {
  case LESS_THAN:
    refine_no_check(denominator*var < expr);
    break;
  case LESS_OR_EQUAL:
    refine_no_check(denominator*var <= expr);
    break;
  case GREATER_OR_EQUAL:
    refine_no_check(denominator*var >= expr);
    break;
  case GREATER_THAN:
    refine_no_check(denominator*var > expr);
    break;
  default:
    // The EQUAL and NOT_EQUAL cases have been already dealt with.
    PPL_UNREACHABLE;
    break;
  }
  unconstrain(var);
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::generalized_affine_image(const Linear_Expression& lhs,
                                          const Relation_Symbol relsym,
                                          const Linear_Expression& rhs) {
  // Dimension-compatibility checks.
  // The dimension of `lhs' should not be greater than the dimension
  // of `*this'.
  dimension_type lhs_space_dim = lhs.space_dimension();
  if (space_dim < lhs_space_dim)
    throw_dimension_incompatible("generalized_affine_image(e1, r, e2)",
                                 "e1", lhs);
  // The dimension of `rhs' should not be greater than the dimension
  // of `*this'.
  const dimension_type rhs_space_dim = rhs.space_dimension();
  if (space_dim < rhs_space_dim)
    throw_dimension_incompatible("generalized_affine_image(e1, r, e2)",
                                 "e2", rhs);

  // Strict relation symbols are only admitted for NNC polyhedra.
  if (is_necessarily_closed()
      && (relsym == LESS_THAN || relsym == GREATER_THAN))
    throw_invalid_argument("generalized_affine_image(e1, r, e2)",
                           "r is a strict relation symbol");
  // The relation symbol cannot be a disequality.
  if (relsym == NOT_EQUAL)
    throw_invalid_argument("generalized_affine_image(e1, r, e2)",
                           "r is the disequality relation symbol");

  // Any image of an empty polyhedron is empty.
  if (marked_empty())
    return;

  // Compute the actual space dimension of `lhs',
  // i.e., the highest dimension having a non-zero coefficient in `lhs'.
  for ( ; lhs_space_dim > 0; lhs_space_dim--)
    if (lhs.coefficient(Variable(lhs_space_dim - 1)) != 0)
      break;
  // If all variables have a zero coefficient, then `lhs' is a constant:
  // we can simply add the constraint `lhs relsym rhs'.
  if (lhs_space_dim == 0) {
    switch (relsym) {
    case LESS_THAN:
      refine_no_check(lhs < rhs);
      break;
    case LESS_OR_EQUAL:
      refine_no_check(lhs <= rhs);
      break;
    case EQUAL:
      refine_no_check(lhs == rhs);
      break;
    case GREATER_OR_EQUAL:
      refine_no_check(lhs >= rhs);
      break;
    case GREATER_THAN:
      refine_no_check(lhs > rhs);
      break;
    case NOT_EQUAL:
      // The NOT_EQUAL case has been already dealt with.
      PPL_UNREACHABLE;
      break;
    }
    return;
  }

  // Gather in `new_lines' the collections of all the lines having
  // the direction of variables occurring in `lhs'.
  // While at it, check whether or not there exists a variable
  // occurring in both `lhs' and `rhs'.
  Generator_System new_lines;
  bool lhs_vars_intersects_rhs_vars = false;
  for (dimension_type i = lhs_space_dim; i-- > 0; )
    if (lhs.coefficient(Variable(i)) != 0) {
      new_lines.insert(line(Variable(i)));
      if (rhs.coefficient(Variable(i)) != 0)
        lhs_vars_intersects_rhs_vars = true;
    }

  if (lhs_vars_intersects_rhs_vars) {
    // Some variables in `lhs' also occur in `rhs'.
    // To ease the computation, we add an additional dimension.
    const Variable new_var(space_dim);
    add_space_dimensions_and_embed(1);

    // Constrain the new dimension to be equal to the right hand side.
    // (check for emptiness because we will add lines).
    refine_no_check(new_var == rhs);
    if (!is_empty()) {
      // Existentially quantify the variables in the left hand side.
      add_recycled_generators(new_lines);

      // Constrain the new dimension so that it is related to
      // the left hand side as dictated by `relsym'
      // (we force minimization because we will need the generators).
      switch (relsym) {
      case LESS_THAN:
        refine_no_check(lhs < new_var);
        break;
      case LESS_OR_EQUAL:
        refine_no_check(lhs <= new_var);
        break;
      case EQUAL:
        refine_no_check(lhs == new_var);
        break;
      case GREATER_OR_EQUAL:
        refine_no_check(lhs >= new_var);
        break;
      case GREATER_THAN:
        refine_no_check(lhs > new_var);
        break;
      case NOT_EQUAL:
        // The NOT_EQUAL case has been already dealt with.
        PPL_UNREACHABLE;
        break;
      }
    }
    // Remove the temporarily added dimension.
    remove_higher_space_dimensions(space_dim-1);
  }
  else {
    // `lhs' and `rhs' variables are disjoint:
    // there is no need to add a further dimension.

    // Any image of an empty polyhedron is empty.
    // Note: DO check for emptiness here, as we will add lines.
    if (is_empty())
      return;

    // Existentially quantify the variables in the left hand side.
    add_recycled_generators(new_lines);

    // Constrain the left hand side expression so that it is related to
    // the right hand side expression as dictated by `relsym'.
    switch (relsym) {
    case LESS_THAN:
      refine_no_check(lhs < rhs);
      break;
    case LESS_OR_EQUAL:
      refine_no_check(lhs <= rhs);
      break;
    case EQUAL:
      refine_no_check(lhs == rhs);
      break;
    case GREATER_OR_EQUAL:
      refine_no_check(lhs >= rhs);
      break;
    case GREATER_THAN:
      refine_no_check(lhs > rhs);
      break;
    case NOT_EQUAL:
      // The NOT_EQUAL case has been already dealt with.
      PPL_UNREACHABLE;
      break;
    }
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::generalized_affine_preimage(const Linear_Expression& lhs,
                                             const Relation_Symbol relsym,
                                             const Linear_Expression& rhs) {
  // Dimension-compatibility checks.
  // The dimension of `lhs' should not be greater than the dimension
  // of `*this'.
  dimension_type lhs_space_dim = lhs.space_dimension();
  if (space_dim < lhs_space_dim)
    throw_dimension_incompatible("generalized_affine_preimage(e1, r, e2)",
                                 "e1", lhs);
  // The dimension of `rhs' should not be greater than the dimension
  // of `*this'.
  const dimension_type rhs_space_dim = rhs.space_dimension();
  if (space_dim < rhs_space_dim)
    throw_dimension_incompatible("generalized_affine_preimage(e1, r, e2)",
                                 "e2", rhs);

  // Strict relation symbols are only admitted for NNC polyhedra.
  if (is_necessarily_closed()
      && (relsym == LESS_THAN || relsym == GREATER_THAN))
    throw_invalid_argument("generalized_affine_preimage(e1, r, e2)",
                           "r is a strict relation symbol");
  // The relation symbol cannot be a disequality.
  if (relsym == NOT_EQUAL)
    throw_invalid_argument("generalized_affine_preimage(e1, r, e2)",
                           "r is the disequality relation symbol");

  // Any preimage of an empty polyhedron is empty.
  if (marked_empty())
    return;

  // Compute the actual space dimension of `lhs',
  // i.e., the highest dimension having a non-zero coefficient in `lhs'.
  for ( ; lhs_space_dim > 0; lhs_space_dim--)
    if (lhs.coefficient(Variable(lhs_space_dim - 1)) != 0)
      break;

  // If all variables have a zero coefficient, then `lhs' is a constant:
  // in this case, preimage and image happen to be the same.
  if (lhs_space_dim == 0) {
    generalized_affine_image(lhs, relsym, rhs);
    return;
  }

  // Gather in `new_lines' the collections of all the lines having
  // the direction of variables occurring in `lhs'.
  // While at it, check whether or not there exists a variable
  // occurring in both `lhs' and `rhs'.
  Generator_System new_lines;
  bool lhs_vars_intersects_rhs_vars = false;
  for (dimension_type i = lhs_space_dim; i-- > 0; )
    if (lhs.coefficient(Variable(i)) != 0) {
      new_lines.insert(line(Variable(i)));
      if (rhs.coefficient(Variable(i)) != 0)
        lhs_vars_intersects_rhs_vars = true;
    }

  if (lhs_vars_intersects_rhs_vars) {
    // Some variables in `lhs' also occur in `rhs'.
    // To ease the computation, we add an additional dimension.
    const Variable new_var(space_dim);
    add_space_dimensions_and_embed(1);

    // Constrain the new dimension to be equal to `lhs'
    // (also check for emptiness because we have to add lines).
    refine_no_check(new_var == lhs);
    if (!is_empty()) {
      // Existentially quantify the variables in the left hand side.
      add_recycled_generators(new_lines);

      // Constrain the new dimension so that it is related to
      // the right hand side as dictated by `relsym'.
      switch (relsym) {
      case LESS_THAN:
        refine_no_check(new_var < rhs);
        break;
      case LESS_OR_EQUAL:
        refine_no_check(new_var <= rhs);
        break;
      case EQUAL:
        refine_no_check(new_var == rhs);
        break;
      case GREATER_OR_EQUAL:
        refine_no_check(new_var >= rhs);
        break;
      case GREATER_THAN:
        refine_no_check(new_var > rhs);
        break;
      case NOT_EQUAL:
        // The NOT_EQUAL case has been already dealt with.
        PPL_UNREACHABLE;
        break;
      }
    }
    // Remove the temporarily added dimension.
    remove_higher_space_dimensions(space_dim-1);
  }
  else {
    // `lhs' and `rhs' variables are disjoint:
    // there is no need to add a further dimension.

    // Constrain the left hand side expression so that it is related to
    // the right hand side expression as dictated by `relsym'.
    switch (relsym) {
    case LESS_THAN:
      refine_no_check(lhs < rhs);
      break;
    case LESS_OR_EQUAL:
      refine_no_check(lhs <= rhs);
      break;
    case EQUAL:
      refine_no_check(lhs == rhs);
      break;
    case GREATER_OR_EQUAL:
      refine_no_check(lhs >= rhs);
      break;
    case GREATER_THAN:
      refine_no_check(lhs > rhs);
      break;
    case NOT_EQUAL:
      // The NOT_EQUAL case has been already dealt with.
      PPL_UNREACHABLE;
      break;
    }
    // Any image of an empty polyhedron is empty.
    // Note: DO check for emptiness here, as we will add lines.
    if (is_empty())
      return;
    // Existentially quantify all the variables occurring in `lhs'.
    add_recycled_generators(new_lines);
  }
  PPL_ASSERT_HEAVY(OK());
}

void
PPL::Polyhedron::time_elapse_assign(const Polyhedron& y) {
  Polyhedron& x = *this;
  // Topology compatibility check.
  if (x.topology() != y.topology())
    throw_topology_incompatible("time_elapse_assign(y)", "y", y);
  // Dimension-compatibility checks.
  if (x.space_dim != y.space_dim)
    throw_dimension_incompatible("time_elapse_assign(y)", "y", y);

  // Dealing with the zero-dimensional case.
  if (x.space_dim == 0) {
    if (y.marked_empty())
      x.set_empty();
    return;
  }

  // If either one of `x' or `y' is empty, the result is empty too.
  if (x.marked_empty() || y.marked_empty()
      || (x.has_pending_constraints() && !x.process_pending_constraints())
      || (!x.generators_are_up_to_date() && !x.update_generators())
      || (y.has_pending_constraints() && !y.process_pending_constraints())
      || (!y.generators_are_up_to_date() && !y.update_generators())) {
    x.set_empty();
    return;
  }

  // At this point both generator systems are up-to-date,
  // possibly containing pending generators.
  Generator_System gs = y.gen_sys;
  const dimension_type old_gs_num_rows = gs.num_rows();
  dimension_type gs_num_rows = old_gs_num_rows;

  if (!x.is_necessarily_closed()) {
    // `x' and `y' are NNC polyhedra.
    for (dimension_type i = gs_num_rows; i-- > 0; )
      switch (gs[i].type()) {
      case Generator::POINT:
        // The points of `gs' can be erased,
        // since their role can be played by closure points.
        --gs_num_rows;
        using std::swap;
        swap(gs[i], gs[gs_num_rows]);
        break;
      case Generator::CLOSURE_POINT:
        {
          Generator& cp = gs[i];
          // If it is the origin, erase it.
          if (cp.all_homogeneous_terms_are_zero()) {
            --gs_num_rows;
            using std::swap;
            swap(cp, gs[gs_num_rows]);
          }
          // Otherwise, transform the closure point into a ray.
          else {
            cp[0] = 0;
            // Enforce normalization.
            cp.normalize();
          }
        }
        break;
      case Generator::RAY:
      case Generator::LINE:
        // For rays and lines, nothing to be done.
        break;
      }
  }
  else {
    // `x' and `y' are C polyhedra.
    for (dimension_type i = gs_num_rows; i-- > 0; ) {
      // For rays and lines, nothing to be done.
      if (gs[i].is_point()) {
        Generator& p = gs[i];
        // If it is the origin, erase it.
        if (p.all_homogeneous_terms_are_zero()) {
          --gs_num_rows;
          using std::swap;
          swap(p, gs[gs_num_rows]);
        }
        // Otherwise, transform the point into a ray.
        else {
          p[0] = 0;
          // Enforce normalization.
          p.normalize();
        }
      }
    }
  }
  // If it was present, erase the origin point or closure point,
  // which cannot be transformed into a valid ray or line.
  // For NNC polyhedra, also erase all the points of `gs',
  // whose role can be played by the closure points.
  // These have been previously moved to the end of `gs'.
  gs.remove_trailing_rows(old_gs_num_rows - gs_num_rows);
  gs.unset_pending_rows();

  // `gs' may now have no rows.
  // Namely, this happens when `y' was the singleton polyhedron
  // having the origin as the one and only point.
  // In such a case, the resulting polyhedron is equal to `x'.
  if (gs_num_rows == 0)
    return;

  // If the polyhedron can have something pending, we add `gs'
  // to `gen_sys' as pending rows
  if (x.can_have_something_pending()) {
    x.gen_sys.add_pending_rows(gs);
    x.set_generators_pending();
  }
  // Otherwise, the two systems are merged.
  // `Linear_System::merge_rows_assign()' requires both systems to be sorted.
  else {
    if (!x.gen_sys.is_sorted())
      x.gen_sys.sort_rows();
    gs.sort_rows();
    x.gen_sys.merge_rows_assign(gs);
    // Only the system of generators is up-to-date.
    x.clear_constraints_up_to_date();
    x.clear_generators_minimized();
  }
  PPL_ASSERT_HEAVY(x.OK(true) && y.OK(true));
}

bool
PPL::Polyhedron::frequency(const Linear_Expression& expr,
                           Coefficient& freq_n, Coefficient& freq_d,
                           Coefficient& val_n, Coefficient& val_d) const {
  // The dimension of `expr' must be at most the dimension of *this.
  if (space_dim < expr.space_dimension())
    throw_dimension_incompatible("frequency(e, ...)", "e", expr);

  // If the `expr' has a constant value, then the frequency
  // `freq_n' is 0. Otherwise the values for \p expr are not discrete
  // and we return false.

  // Space dimension is 0: if empty, then return false;
  // otherwise the frequency is 1 and the value is 0.
  if (space_dim == 0) {
    if (is_empty())
      return false;
    freq_n = 0;
    freq_d = 1;
    val_n = expr.inhomogeneous_term();
    val_d = 1;
    return true;
  }

  // For an empty polyhedron, we simply return false.
  if (marked_empty()
      || (has_pending_constraints() && !process_pending_constraints())
      || (!generators_are_up_to_date() && !update_generators()))
    return false;

  // The polyhedron has updated, possibly pending generators.
  // The following loop will iterate through the generator
  // to see if `expr' has a constant value.
  PPL_DIRTY_TEMP(mpq_class, value);

  // True if we have no other candidate value to compare with.
  bool first_candidate = true;

  PPL_DIRTY_TEMP_COEFFICIENT(sp);
  PPL_DIRTY_TEMP(mpq_class, candidate);
  for (dimension_type i = gen_sys.num_rows(); i-- > 0; ) {
    const Generator& gen_sys_i = gen_sys[i];
    Scalar_Products::homogeneous_assign(sp, expr, gen_sys_i);
    // Lines and rays in `*this' can cause `expr' to be non-constant.
    if (gen_sys_i.is_line_or_ray()) {
      const int sp_sign = sgn(sp);
      if (sp_sign != 0)
        // `expr' is unbounded in `*this'.
        return false;
    }
    else {
      // We have a point or a closure point.
      PPL_ASSERT(gen_sys_i.is_point() || gen_sys_i.is_closure_point());
      // Notice that we are ignoring the constant term in `expr' here.
      // We will add it to the value if there is a constant value.
      assign_r(candidate.get_num(), sp, ROUND_NOT_NEEDED);
      assign_r(candidate.get_den(), gen_sys_i[0], ROUND_NOT_NEEDED);
      candidate.canonicalize();
      if (first_candidate) {
        // We have a (new) candidate value.
        first_candidate = false;
        value = candidate;
      }
      else if (candidate != value)
        return false;
    }
  }

  // Add in the constant term in `expr'.
  PPL_DIRTY_TEMP(mpz_class, n);
  assign_r(n, expr.inhomogeneous_term(), ROUND_NOT_NEEDED);
  value += n;
  // FIXME: avoid these temporaries, if possible.
  // This can be done adding an `assign' function working on native
  // and checked or an operator= that have on one side a checked and
  // on the other a native or checked.
  // The reason why now we can't use operator= is the fact that we
  // still can have Coefficient defined to mpz_class (and not
  // Checked_Number<mpz_class>).
  val_n = Coefficient(value.get_num());
  val_d = Coefficient(value.get_den());

  freq_n = 0;
  freq_d = 1;
  return true;
}

void
PPL::Polyhedron::topological_closure_assign() {
  // Necessarily closed polyhedra are trivially closed.
  if (is_necessarily_closed())
    return;
  // Any empty or zero-dimensional polyhedron is closed.
  if (marked_empty() || space_dim == 0)
    return;

  // The computation can be done using constraints or generators.
  // If we use constraints, we will change them, so that having pending
  // constraints would be useless. If we use generators, we add generators,
  // so that having pending generators still makes sense.

  // Process any pending constraints.
  if (has_pending_constraints() && !process_pending_constraints())
    return;

  // Use constraints only if they are available and
  // there are no pending generators.
  if (!has_pending_generators() && constraints_are_up_to_date()) {
    const dimension_type eps_index = space_dim + 1;
    bool changed = false;
    // Transform all strict inequalities into non-strict ones.
    for (dimension_type i = con_sys.num_rows(); i-- > 0; ) {
      Constraint& c = con_sys[i];
      if (c[eps_index] < 0 && !c.is_tautological()) {
        c[eps_index] = 0;
        // Enforce normalization.
        c.normalize();
        changed = true;
      }
    }
    if (changed) {
      con_sys.insert(Constraint::epsilon_leq_one());
      con_sys.set_sorted(false);
      // After changing the system of constraints, the generators
      // are no longer up-to-date and the constraints are no longer
      // minimized.
      clear_generators_up_to_date();
      clear_constraints_minimized();
    }
  }
  else {
    // Here we use generators, possibly keeping constraints.
    PPL_ASSERT(generators_are_up_to_date());
    // Add the corresponding point to each closure point.
    gen_sys.add_corresponding_points();
    if (can_have_something_pending())
      set_generators_pending();
    else {
      // We cannot have pending generators; this also implies
      // that generators may have lost their sortedness.
      gen_sys.unset_pending_rows();
      gen_sys.set_sorted(false);
      // Constraints are not up-to-date and generators are not minimized.
      clear_constraints_up_to_date();
      clear_generators_minimized();
    }
  }
  PPL_ASSERT_HEAVY(OK());
}

bool
PPL::operator==(const Polyhedron& x, const Polyhedron& y) {
  // If the two polyhedra are topology-incompatible or dimension-incompatible,
  // then they cannot be the same polyhedron.
  if (x.topology() != y.topology() || x.space_dim != y.space_dim)
    return false;

  if (x.marked_empty())
    return y.is_empty();
  else if (y.marked_empty())
    return x.is_empty();
  else if (x.space_dim == 0)
    return true;

  switch (x.quick_equivalence_test(y)) {
  case Polyhedron::TVB_TRUE:
    return true;

  case Polyhedron::TVB_FALSE:
    return false;

  default:
    if (x.is_included_in(y))
      if (x.marked_empty())
        return y.is_empty();
      else
        return y.is_included_in(x);
    else
      return false;
  }
}

bool
PPL::Polyhedron::contains(const Polyhedron& y) const {
  const Polyhedron& x = *this;

  // Topology compatibility check.
  if (x.topology() != y.topology())
    throw_topology_incompatible("contains(y)", "y", y);

  // Dimension-compatibility check.
  if (x.space_dim != y.space_dim)
    throw_dimension_incompatible("contains(y)", "y", y);

  if (y.marked_empty())
    return true;
  else if (x.marked_empty())
    return y.is_empty();
  else if (y.space_dim == 0)
    return true;
  else if (x.quick_equivalence_test(y) == Polyhedron::TVB_TRUE)
    return true;
  else
    return y.is_included_in(x);
}

bool
PPL::Polyhedron::is_disjoint_from(const Polyhedron& y) const {
  Polyhedron z = *this;
  z.intersection_assign(y);
  return z.is_empty();
}

void
PPL::Polyhedron::ascii_dump(std::ostream& s) const {
  s << "space_dim " << space_dim << "\n";
  status.ascii_dump(s);
  s << "\ncon_sys ("
    << (constraints_are_up_to_date() ? "" : "not_")
    << "up-to-date)"
    << "\n";
  con_sys.ascii_dump(s);
  s << "\ngen_sys ("
    << (generators_are_up_to_date() ? "" : "not_")
    << "up-to-date)"
    << "\n";
  gen_sys.ascii_dump(s);
  s << "\nsat_c\n";
  sat_c.ascii_dump(s);
  s << "\nsat_g\n";
  sat_g.ascii_dump(s);
  s << "\n";
}

PPL_OUTPUT_DEFINITIONS(Polyhedron)

bool
PPL::Polyhedron::ascii_load(std::istream& s) {
  std::string str;

  if (!(s >> str) || str != "space_dim")
    return false;

  if (!(s >> space_dim))
    return false;

  if (!status.ascii_load(s))
    return false;

  if (!(s >> str) || str != "con_sys")
    return false;

  if (!(s >> str) || (str != "(not_up-to-date)" && str != "(up-to-date)"))
    return false;

  if (!con_sys.ascii_load(s))
    return false;

  if (!(s >> str) || str != "gen_sys")
    return false;

  if (!(s >> str) || (str != "(not_up-to-date)" && str != "(up-to-date)"))
    return false;

  if (!gen_sys.ascii_load(s))
    return false;

  if (!(s >> str) || str != "sat_c")
    return false;

  if (!sat_c.ascii_load(s))
    return false;

  if (!(s >> str) || str != "sat_g")
    return false;

  if (!sat_g.ascii_load(s))
    return false;

  // Check invariants.
  PPL_ASSERT_HEAVY(OK());
  return true;
}

PPL::memory_size_type
PPL::Polyhedron::external_memory_in_bytes() const {
  return
    con_sys.external_memory_in_bytes()
    + gen_sys.external_memory_in_bytes()
    + sat_c.external_memory_in_bytes()
    + sat_g.external_memory_in_bytes();
}

void
PPL::Polyhedron::wrap_assign(const Variables_Set& vars,
                             Bounded_Integer_Type_Width w,
                             Bounded_Integer_Type_Representation r,
                             Bounded_Integer_Type_Overflow o,
                             const Constraint_System* cs_p,
                             unsigned complexity_threshold,
                             bool wrap_individually) {
  if (is_necessarily_closed())
    Implementation::wrap_assign(static_cast<C_Polyhedron&>(*this),
                                vars, w, r, o, cs_p,
                                complexity_threshold, wrap_individually,
                                "C_Polyhedron");
  else
    Implementation::wrap_assign(static_cast<NNC_Polyhedron&>(*this),
                                vars, w, r, o, cs_p,
                                complexity_threshold, wrap_individually,
                                "NNC_Polyhedron");
}

std::ostream&
PPL::IO_Operators::operator<<(std::ostream& s, const Polyhedron& ph) {
  if (ph.is_empty())
    s << "false";
  else
    s << ph.minimized_constraints();
  return s;
}