def has_IP_property(self):
        """
        Whether the lattice polytope has the IP property.

        That is, the polytope is full-dimensional and the origin is a
        interior point not on the boundary.

        OUTPUT:

        Boolean.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: LatticePolytope_PPL((-1,-1),(0,1),(1,0)).has_IP_property()
            True
            sage: LatticePolytope_PPL((-1,-1),(1,1)).has_IP_property()
            False
        """
        origin = C_Polyhedron(point(0*Variable(self.space_dimension())))
        is_included = Poly_Con_Relation.is_included()
        saturates = Poly_Con_Relation.saturates()
        for c in self.constraints():
            rel = origin.relation_with(c)
            if (not rel.implies(is_included)) or rel.implies(saturates):
                return False
        return True

    def vertices_saturating(self, constraint):
        """
        Return the vertices saturating the constraint

        INPUT:

        - ``constraint`` -- a constraint (inequality or equation) of
          the polytope.

        OUTPUT:

        The tuple of vertices saturating the constraint. The vertices
        are returned as `\ZZ`-vectors, as in :meth:`vertices`.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: p = LatticePolytope_PPL((0,0),(0,1),(1,0))
            sage: ieq = next(iter(p.constraints()));  ieq
            x0>=0
            sage: p.vertices_saturating(ieq)
            ((0, 0), (0, 1))
        """
        from sage.libs.ppl import C_Polyhedron, Poly_Con_Relation
        result = []
        for i,v in enumerate(self.minimized_generators()):
            v = C_Polyhedron(v)
            if v.relation_with(constraint).implies(Poly_Con_Relation.saturates()):
                result.append(self.vertices()[i])
        return tuple(result)

    def contains(self, point_coordinates):
        r"""
        Test whether point is contained in the polytope.

        INPUT:

        - ``point_coordinates`` -- a list/tuple/iterable of rational
          numbers. The coordinates of the point.

        OUTPUT:

        Boolean.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: line = LatticePolytope_PPL((1,2,3), (-1,-2,-3))
            sage: line.contains([0,0,0])
            True
            sage: line.contains([1,0,0])
            False
        """
        p = C_Polyhedron(point(Linear_Expression(list(point_coordinates), 1)))
        is_included = Poly_Con_Relation.is_included()
        for c in self.constraints():
            if not p.relation_with(c).implies(is_included):
                return False
        return True

    def _init_from_Vrepresentation(self, vertices, rays, lines, minimize=True, verbose=False):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``verbose`` -- boolean (default: ``False``). Whether to print
          verbose output for debugging purposes.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_ppl
            sage: Polyhedron_ppl._init_from_Vrepresentation(p, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = LCM_list([denominator(v_i) for v_i in v])
            if d.is_one():
                gs.insert(point(Linear_Expression(v, 0)))
            else:
                dv = [ d*v_i for v_i in v ]
                gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = LCM_list([denominator(r_i) for r_i in r])
            if d.is_one():
                gs.insert(ray(Linear_Expression(r, 0)))
            else:
                dr = [ d*r_i for r_i in r ]
                gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = LCM_list([denominator(l_i) for l_i in l])
            if d.is_one():
                gs.insert(line(Linear_Expression(l, 0)))
            else:
                dl = [ d*l_i for l_i in l ]
                gs.insert(line(Linear_Expression(dl, 0)))
        if gs.empty():
            self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'empty')
        else:
            self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)

    def _init_from_Hrepresentation(self, ieqs, eqns, minimize=True, verbose=False):
        """
        Construct polyhedron from H-representation data.

        INPUT:

        - ``ieqs`` -- list of inequalities. Each line can be specified
          as any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``eqns`` -- list of equalities. Each line can be specified
          as any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``verbose`` -- boolean (default: ``False``). Whether to print
          verbose output for debugging purposes.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_ppl
            sage: Polyhedron_ppl._init_from_Hrepresentation(p, [], [])
        """
        cs = Constraint_System()
        if ieqs is None: ieqs = []
        for ieq in ieqs:
            d = LCM_list([denominator(ieq_i) for ieq_i in ieq])
            dieq = [ ZZ(d*ieq_i) for ieq_i in ieq ]
            b = dieq[0]
            A = dieq[1:]
            cs.insert(Linear_Expression(A, b) >= 0)
        if eqns is None: eqns = []
        for eqn in eqns:
            d = LCM_list([denominator(eqn_i) for eqn_i in eqn])
            deqn = [ ZZ(d*eqn_i) for eqn_i in eqn ]
            b = deqn[0]
            A = deqn[1:]
            cs.insert(Linear_Expression(A, b) == 0)
        if cs.empty():
            self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'universe')
        else:
            self._ppl_polyhedron = C_Polyhedron(cs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)

    def _init_from_Vrepresentation(self, ambient_dim, vertices, rays, lines, minimize=True):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.
        
        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
            sage: Polyhedron_QQ_ppl._init_from_Vrepresentation(p, 2, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = lcm([denominator(v_i) for v_i in v])
            dv = [ ZZ(d*v_i) for v_i in v ]
            gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = lcm([denominator(r_i) for r_i in r])
            dr = [ ZZ(d*r_i) for r_i in r ]
            gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = lcm([denominator(l_i) for l_i in l])
            dl = [ ZZ(d*l_i) for l_i in l ]
            gs.insert(line(Linear_Expression(dl, 0)))
        self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)

    def _init_empty_polyhedron(self):
        """
        Initializes an empty polyhedron.

        TESTS::

            sage: empty = Polyhedron(backend='ppl'); empty
            The empty polyhedron in ZZ^0
            sage: empty.Vrepresentation()
            ()
            sage: empty.Hrepresentation()
            (An equation -1 == 0,)
            sage: Polyhedron(vertices = [], backend='ppl')
            The empty polyhedron in ZZ^0
            sage: Polyhedron(backend='ppl')._init_empty_polyhedron()
        """
        super(Polyhedron_ppl, self)._init_empty_polyhedron()
        self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'empty')

    def _init_from_Hrepresentation(self, ambient_dim, ieqs, eqns, minimize=True):
        """
        Construct polyhedron from H-representation data.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.
        
        - ``ieqs`` -- list of inequalities. Each line can be specified
          as any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``eqns`` -- list of equalities. Each line can be specified
          as any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
            sage: Polyhedron_QQ_ppl._init_from_Hrepresentation(p, 2, [], [])
        """
        cs = Constraint_System()
        if ieqs is None: ieqs = []
        for ieq in ieqs:
            d = lcm([denominator(ieq_i) for ieq_i in ieq])
            dieq = [ ZZ(d*ieq_i) for ieq_i in ieq ]
            b = dieq[0]
            A = dieq[1:]
            cs.insert(Linear_Expression(A, b) >= 0)
        if eqns is None: eqns = []
        for eqn in eqns:
            d = lcm([denominator(eqn_i) for eqn_i in eqn])
            deqn = [ ZZ(d*eqn_i) for eqn_i in eqn ]
            b = deqn[0]
            A = deqn[1:]
            cs.insert(Linear_Expression(A, b) == 0)
        self._ppl_polyhedron = C_Polyhedron(cs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)

    def _init_empty_polyhedron(self, ambient_dim):
        """
        Initializes an empty polyhedron.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.

        TESTS::

            sage: empty = Polyhedron(backend='ppl'); empty
            The empty polyhedron in QQ^0
            sage: empty.Vrepresentation()
            ()
            sage: empty.Hrepresentation()
            (An equation -1 == 0,)
            sage: Polyhedron(vertices = [], backend='ppl')
            The empty polyhedron in QQ^0
            sage: Polyhedron(backend='ppl')._init_empty_polyhedron(0)
        """
        super(Polyhedron_QQ_ppl, self)._init_empty_polyhedron(ambient_dim)
        self._ppl_polyhedron = C_Polyhedron(ambient_dim, 'empty')

    def fibration_generator(self, dim):
        """
        Generate the lattice polytope fibrations.

        For the purposes of this function, a lattice polytope fiber is
        a sub-lattice polytope. Projecting the plane spanned by the
        subpolytope to a point yields another lattice polytope, the
        base of the fibration.

        INPUT:

        - ``dim`` -- integer. The dimension of the lattice polytope
          fiber.

        OUTPUT:

        A generator yielding the distinct lattice polytope fibers of
        given dimension.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: p = LatticePolytope_PPL((-9,-6,-1,-1),(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))
            sage: list( p.fibration_generator(2) )
            [A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]
        """
        assert self.is_full_dimensional()
        codim = self.space_dimension() - dim
        # "points" are the potential vertices of the fiber. They are
        # in the $codim$-skeleton of the polytope, which is contained
        # in the points that saturate at least $dim$ equations.
        points = [ p for p in self._integral_points_saturating() if len(p[1])>=dim ]
        points = sorted(points, key=lambda x:len(x[1]))

        # iterate over point combinations subject to all points being on one facet.
        def point_combinations_iterator(n, i0=0, saturated=None):
            for i in range(i0, len(points)):
                p, ieqs = points[i]
                if saturated is None:
                    saturated_ieqs = ieqs
                else:
                    saturated_ieqs = saturated.intersection(ieqs)
                if len(saturated_ieqs)==0:
                    continue
                if n == 1:
                    yield [i]
                else:
                    for c in point_combinations_iterator(n-1, i+1, saturated_ieqs):
                        yield [i] + c

        point_lines = [ line(Linear_Expression(p[0].list(),0)) for p in points ]
        origin = point()
        fibers = set()
        gs = Generator_System()
        for indices in point_combinations_iterator(dim):
            gs.clear()
            gs.insert(origin)
            for i in indices:
                gs.insert(point_lines[i])
            plane = C_Polyhedron(gs)
            if plane.affine_dimension() != dim:
                continue
            plane.intersection_assign(self)
            if (not self.is_full_dimensional()) and (plane.affine_dimension() != dim):
                continue
            try:
                fiber = LatticePolytope_PPL(plane)
            except TypeError:   # not a lattice polytope
                continue
            fiber_vertices = tuple(sorted(fiber.vertices()))
            if fiber_vertices not in fibers:
                yield fiber
                fibers.update([fiber_vertices])

class Polyhedron_QQ_ppl(Polyhedron_QQ):
    """
    Polyhedra over `\QQ` with ppl

    INPUT:

    - ``ambient_dim`` -- integer. The dimension of the ambient space.

    - ``Vrep`` -- a list ``[vertices, rays, lines]``.
        
    - ``Hrep`` -- a list ``[ieqs, eqns]``.

    EXAMPLES::

        sage: p = Polyhedron(vertices=[(0,0),(1,0),(0,1)], rays=[(1,1)], lines=[], backend='ppl')
        sage: TestSuite(p).run()
    """

    def _init_from_Vrepresentation(self, ambient_dim, vertices, rays, lines, minimize=True):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.
        
        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
            sage: Polyhedron_QQ_ppl._init_from_Vrepresentation(p, 2, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = lcm([denominator(v_i) for v_i in v])
            dv = [ ZZ(d*v_i) for v_i in v ]
            gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = lcm([denominator(r_i) for r_i in r])
            dr = [ ZZ(d*r_i) for r_i in r ]
            gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = lcm([denominator(l_i) for l_i in l])
            dl = [ ZZ(d*l_i) for l_i in l ]
            gs.insert(line(Linear_Expression(dl, 0)))
        self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)


    def _init_from_Hrepresentation(self, ambient_dim, ieqs, eqns, minimize=True):
        """
        Construct polyhedron from H-representation data.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.
        
        - ``ieqs`` -- list of inequalities. Each line can be specified
          as any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``eqns`` -- list of equalities. Each line can be specified
          as any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
            sage: Polyhedron_QQ_ppl._init_from_Hrepresentation(p, 2, [], [])
        """
        cs = Constraint_System()
        if ieqs is None: ieqs = []
        for ieq in ieqs:
            d = lcm([denominator(ieq_i) for ieq_i in ieq])
            dieq = [ ZZ(d*ieq_i) for ieq_i in ieq ]
            b = dieq[0]
            A = dieq[1:]
            cs.insert(Linear_Expression(A, b) >= 0)
        if eqns is None: eqns = []
        for eqn in eqns:
            d = lcm([denominator(eqn_i) for eqn_i in eqn])
            deqn = [ ZZ(d*eqn_i) for eqn_i in eqn ]
            b = deqn[0]
#.........这里部分代码省略.........

    def find_isomorphism(self, polytope):
        """
        Return a lattice isomorphism with ``polytope``.

        INPUT:

        - ``polytope`` -- a polytope, potentially higher-dimensional.

        OUTPUT:

        A
        :class:`~sage.geometry.polyhedron.lattice_euclidean_group_element.LatticeEuclideanGroupElement`. It
        is not necessarily invertible if the affine dimension of
        ``self`` or ``polytope`` is not two. A
        :class:`~sage.geometry.polyhedron.lattice_euclidean_group_element.LatticePolytopesNotIsomorphicError`
        is raised if no such isomorphism exists.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: L1 = LatticePolytope_PPL((1,0),(0,1),(0,0))
            sage: L2 = LatticePolytope_PPL((1,0,3),(0,1,0),(0,0,1))
            sage: iso = L1.find_isomorphism(L2)
            sage: iso(L1) == L2
            True

            sage: L1 = LatticePolytope_PPL((0, 1), (3, 0), (0, 3), (1, 0))
            sage: L2 = LatticePolytope_PPL((0,0,2,1),(0,1,2,0),(2,0,0,3),(2,3,0,0))
            sage: iso = L1.find_isomorphism(L2)
            sage: iso(L1) == L2
            True

        The following polygons are isomorphic over `\QQ`, but not as
        lattice polytopes::

            sage: L1 = LatticePolytope_PPL((1,0),(0,1),(-1,-1))
            sage: L2 = LatticePolytope_PPL((0, 0), (0, 1), (1, 0))
            sage: L1.find_isomorphism(L2)
            Traceback (most recent call last):
            ...
            LatticePolytopesNotIsomorphicError: different number of integral points
            sage: L2.find_isomorphism(L1)
            Traceback (most recent call last):
            ...
            LatticePolytopesNotIsomorphicError: different number of integral points
        """
        from sage.geometry.polyhedron.lattice_euclidean_group_element import \
            LatticePolytopesNotIsomorphicError
        if polytope.affine_dimension() != self.affine_dimension():
            raise LatticePolytopesNotIsomorphicError('different dimension')
        polytope_vertices = polytope.vertices()
        if len(polytope_vertices) != self.n_vertices():
            raise LatticePolytopesNotIsomorphicError('different number of vertices')
        self_vertices = self.ordered_vertices()
        if len(polytope.integral_points()) != len(self.integral_points()):
            raise LatticePolytopesNotIsomorphicError('different number of integral points')

        if len(self_vertices) < 3:
            return self._find_isomorphism_degenerate(polytope)

        polytope_origin = polytope_vertices[0]
        origin_P = C_Polyhedron(next(Generator_System_iterator(
            polytope.minimized_generators())))

        neighbors = []
        for c in polytope.minimized_constraints():
            if not c.is_inequality():
                continue
            if origin_P.relation_with(c).implies(Poly_Con_Relation.saturates()):
                for i, g in enumerate(polytope.minimized_generators()):
                    if i == 0:
                        continue
                    g = C_Polyhedron(g)
                    if g.relation_with(c).implies(Poly_Con_Relation.saturates()):
                        neighbors.append(polytope_vertices[i])
                        break

        p_ray_left = neighbors[0] - polytope_origin
        p_ray_right = neighbors[1] - polytope_origin
        try:
            return self._find_cyclic_isomorphism_matching_edge(polytope, polytope_origin,
                                                               p_ray_left, p_ray_right)
        except LatticePolytopesNotIsomorphicError:
            pass
        try:
            return self._find_cyclic_isomorphism_matching_edge(polytope, polytope_origin,
                                                               p_ray_right, p_ray_left)
        except LatticePolytopesNotIsomorphicError:
            pass
        raise LatticePolytopesNotIsomorphicError('different polygons')