text: r"""
  Convex rational polyhedral cones
  
  This module was designed as a part of framework for toric varieties
  (:mod:`~sage.schemes.toric.variety`,
  :mod:`~sage.schemes.toric.fano_variety`). While the emphasis is on
  strictly convex cones, non-strictly convex cones are supported as well. Work
  with distinct lattices (in the sense of discrete subgroups spanning vector
  spaces) is supported. The default lattice is :class:`ToricLattice
  <sage.geometry.toric_lattice.ToricLatticeFactory>` `N` of the appropriate
  dimension. The only case when you must specify lattice explicitly is creation
  of a 0-dimensional cone, where dimension of the ambient space cannot be
  guessed.
  
  AUTHORS:
  
  - Andrey Novoseltsev (2010-05-13): initial version.
  
  - Andrey Novoseltsev (2010-06-17): substantial improvement during review by
    Volker Braun.
  
  - Volker Braun (2010-06-21): various spanned/quotient/dual lattice
    computations added.
  
  - Volker Braun (2010-12-28): Hilbert basis for cones.
  
  - Andrey Novoseltsev (2012-02-23): switch to PointCollection container.
  
  EXAMPLES:
  
  Use :func:`Cone` to construct cones::
  
      sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
      sage: halfspace = Cone([(1,0,0), (0,1,0), (-1,-1,0), (0,0,1)])
      sage: positive_xy = Cone([(1,0,0), (0,1,0)])
      sage: four_rays = Cone([(1,1,1), (1,-1,1), (-1,-1,1), (-1,1,1)])
  
  For all of the cones above we have provided primitive generating rays, but in
  fact this is not necessary - a cone can be constructed from any collection of
  rays (from the same space, of course). If there are non-primitive (or even
  non-integral) rays, they will be replaced with primitive ones. If there are
  extra rays, they will be discarded. Of course, this means that :func:`Cone`
  has to do some work before actually constructing the cone and sometimes it is
  not desirable, if you know for sure that your input is already "good". In this
  case you can use options ``check=False`` to force :func:`Cone` to use
  exactly the directions that you have specified and ``normalize=False`` to
  force it to use exactly the rays that you have specified. However, it is
  better not to use these possibilities without necessity, since cones are
  assumed to be represented by a minimal set of primitive generating rays.
  See :func:`Cone` for further documentation on construction.
  
  Once you have a cone, you can perform numerous operations on it. The most
  important ones are, probably, ray accessing methods::
  
      sage: rays = halfspace.rays()
      sage: rays
      N( 0,  0, 1),
      N( 0,  1, 0),
      N( 0, -1, 0),
      N( 1,  0, 0),
      N(-1,  0, 0)
      in 3-d lattice N
      sage: rays.set()
      frozenset({N(-1, 0, 0), N(0, -1, 0), N(0, 0, 1), N(0, 1, 0), N(1, 0, 0)})
      sage: rays.matrix()
      [ 0  0  1]
      [ 0  1  0]
      [ 0 -1  0]
      [ 1  0  0]
      [-1  0  0]
      sage: rays.column_matrix()
      [ 0  0  0  1 -1]
      [ 0  1 -1  0  0]
      [ 1  0  0  0  0]
      sage: rays(3)
      N(1, 0, 0)
      in 3-d lattice N
      sage: rays[3]
      N(1, 0, 0)
      sage: halfspace.ray(3)
      N(1, 0, 0)
  
  The method :meth:`~IntegralRayCollection.rays` returns a
  :class:`~sage.geometry.point_collection.PointCollection` with the
  `i`-th element being the primitive integral generator of the `i`-th
  ray. It is possible to convert this collection to a matrix with either
  rows or columns corresponding to these generators. You may also change
  the default
  :meth:`~sage.geometry.point_collection.PointCollection.output_format`
  of all point collections to be such a matrix.
  
  If you want to do something with each ray of a cone, you can write ::
  
      sage: for ray in positive_xy: print(ray)
      N(1, 0, 0)
      N(0, 1, 0)
  
  There are two dimensions associated to each cone - the dimension of the
  subspace spanned by the cone and the dimension of the space where it lives::
  
      sage: positive_xy.dim()
      2
      sage: positive_xy.lattice_dim()
      3
  
  You also may be interested in this dimension::
  
      sage: dim(positive_xy.linear_subspace())
      0
      sage: dim(halfspace.linear_subspace())
      2
  
  Or, perhaps, all you care about is whether it is zero or not::
  
      sage: positive_xy.is_strictly_convex()
      True
      sage: halfspace.is_strictly_convex()
      False
  
  You can also perform these checks::
  
      sage: positive_xy.is_simplicial()
      True
      sage: four_rays.is_simplicial()
      False
      sage: positive_xy.is_smooth()
      True
  
  You can work with subcones that form faces of other cones::
  
      sage: face = four_rays.faces(dim=2)[0]
      sage: face
      2-d face of 3-d cone in 3-d lattice N
      sage: face.rays()
      N(-1, -1, 1),
      N(-1,  1, 1)
      in 3-d lattice N
      sage: face.ambient_ray_indices()
      (2, 3)
      sage: four_rays.rays(face.ambient_ray_indices())
      N(-1, -1, 1),
      N(-1,  1, 1)
      in 3-d lattice N
  
  If you need to know inclusion relations between faces, you can use ::
  
      sage: L = four_rays.face_lattice()
      sage: [len(s) for s in L.level_sets()]
      [1, 4, 4, 1]
      sage: face = L.level_sets()[2][0]
      sage: face.rays()
      N(1,  1, 1),
      N(1, -1, 1)
      in 3-d lattice N
      sage: L.hasse_diagram().neighbors_in(face)
      [1-d face of 3-d cone in 3-d lattice N,
       1-d face of 3-d cone in 3-d lattice N]
  
  .. WARNING::
  
      The order of faces in level sets of
      the face lattice may differ from the order of faces returned by
      :meth:`~ConvexRationalPolyhedralCone.faces`. While the first order is
      random, the latter one ensures that one-dimensional faces are listed in
      the same order as generating rays.
  
  When all the functionality provided by cones is not enough, you may want to
  check if you can do necessary things using polyhedra corresponding to cones::
  
      sage: four_rays.polyhedron()
      A 3-dimensional polyhedron in ZZ^3 defined as
      the convex hull of 1 vertex and 4 rays
  
  And of course you are always welcome to suggest new features that should be
  added to cones!
  
  REFERENCES:
  
  - [Ful1993]_
  """
  
  # ****************************************************************************
  #       Copyright (C) 2010 Volker Braun <vbraun.name@gmail.com>
  #       Copyright (C) 2012 Andrey Novoseltsev <novoselt@gmail.com>
  #       Copyright (C) 2010 William Stein <wstein@gmail.com>
  #
  #  Distributed under the terms of the GNU General Public License (GPL)
  #  as published by the Free Software Foundation; either version 2 of
  #  the License, or (at your option) any later version.
  #                  https://www.gnu.org/licenses/
  # ****************************************************************************
  
  # Use python-3.x versions of print() and range().
  from __future__ import print_function
  
  import collections
  import copy
  import warnings
  
  from sage.arith.all import gcd, lcm
  from sage.combinat.posets.posets import FinitePoset
  from sage.geometry.point_collection import PointCollection
  from sage.geometry.polyhedron.constructor import Polyhedron
  from sage.geometry.polyhedron.base import is_Polyhedron
  from sage.geometry.hasse_diagram import lattice_from_incidences
  from sage.geometry.toric_lattice import (ToricLattice, is_ToricLattice,
                                           is_ToricLatticeQuotient)
  from sage.geometry.toric_plotter import ToricPlotter, label_list
  from sage.graphs.digraph import DiGraph
  from sage.matrix.all import column_matrix, matrix, MatrixSpace
  from sage.misc.all import cached_method, flatten, latex
  from sage.modules.all import span, vector, VectorSpace
  from sage.rings.all import QQ, RR, ZZ
  from sage.structure.all import SageObject, parent
  from sage.structure.richcmp import richcmp_method, richcmp
  from ppl import (C_Polyhedron, Generator_System, Constraint_System,
                   Linear_Expression, ray as PPL_ray, point as PPL_point,
                   Poly_Con_Relation)
  from sage.geometry.integral_points import parallelotope_points
  
  
  def is_Cone(x):
      r"""
      Check if ``x`` is a cone.
  
      INPUT:
  
      - ``x`` -- anything.
  
      OUTPUT:
  
      - ``True`` if ``x`` is a cone and ``False`` otherwise.
  
      EXAMPLES::
  
          sage: from sage.geometry.cone import is_Cone
          sage: is_Cone(1)
          False
          sage: quadrant = Cone([(1,0), (0,1)])
          sage: quadrant
          2-d cone in 2-d lattice N
          sage: is_Cone(quadrant)
          True
      """
      return isinstance(x, ConvexRationalPolyhedralCone)
  
  
  def Cone(rays, lattice=None, check=True, normalize=True):
      r"""
      Construct a (not necessarily strictly) convex rational polyhedral cone.
  
      INPUT:
  
      - ``rays`` -- a list of rays. Each ray should be given as a list
        or a vector convertible to the rational extension of the given
        ``lattice``. May also be specified by a
        :class:`~sage.geometry.polyhedron.base.Polyhedron_base` object;
  
      - ``lattice`` -- :class:`ToricLattice
        <sage.geometry.toric_lattice.ToricLatticeFactory>`, `\ZZ^n`, or any
        other object that behaves like these. If not specified, an attempt will
        be made to determine an appropriate toric lattice automatically;
  
      - ``check`` -- by default the input data will be checked for
        correctness (e.g. that all rays have the same number of
        components) and generating rays will be constructed from
        ``rays``. If you know that the input is a minimal set of
        generators of a valid cone, you may significantly decrease
        construction time using ``check=False`` option;
  
      - ``normalize`` -- you can further speed up construction using
        ``normalize=False`` option. In this case ``rays`` must be a list of
        immutable primitive rays in ``lattice``. In general, you should not use
        this option, it is designed for code optimization and does not give as
        drastic improvement in speed as the previous one.
  
      OUTPUT:
  
      - convex rational polyhedral cone determined by ``rays``.
  
      EXAMPLES:
  
      Let's define a cone corresponding to the first quadrant of the plane
      (note, you can even mix objects of different types to represent rays, as
      long as you let this function to perform all the checks and necessary
      conversions!)::
  
          sage: quadrant = Cone([(1,0), [0,1]])
          sage: quadrant
          2-d cone in 2-d lattice N
          sage: quadrant.rays()
          N(1, 0),
          N(0, 1)
          in 2-d lattice N
  
      If you give more rays than necessary, the extra ones will be discarded::
  
          sage: Cone([(1,0), (0,1), (1,1), (0,1)]).rays()
          N(0, 1),
          N(1, 0)
          in 2-d lattice N
  
      However, this work is not done with ``check=False`` option, so use it
      carefully! ::
  
          sage: Cone([(1,0), (0,1), (1,1), (0,1)], check=False).rays()
          N(1, 0),
          N(0, 1),
          N(1, 1),
          N(0, 1)
          in 2-d lattice N
  
      Even worse things can happen with ``normalize=False`` option::
  
          sage: Cone([(1,0), (0,1)], check=False, normalize=False)
          Traceback (most recent call last):
          ...
          AttributeError: 'tuple' object has no attribute 'parent'
  
      You can construct different "not" cones: not full-dimensional, not
      strictly convex, not containing any rays::
  
          sage: one_dimensional_cone = Cone([(1,0)])
          sage: one_dimensional_cone.dim()
          1
          sage: half_plane = Cone([(1,0), (0,1), (-1,0)])
          sage: half_plane.rays()
          N( 0, 1),
          N( 1, 0),
          N(-1, 0)
          in 2-d lattice N
          sage: half_plane.is_strictly_convex()
          False
          sage: origin = Cone([(0,0)])
          sage: origin.rays()
          Empty collection
          in 2-d lattice N
          sage: origin.dim()
          0
          sage: origin.lattice_dim()
          2
  
      You may construct the cone above without giving any rays, but in this case
      you must provide ``lattice`` explicitly::
  
          sage: origin = Cone([])
          Traceback (most recent call last):
          ...
          ValueError: lattice must be given explicitly if there are no rays!
          sage: origin = Cone([], lattice=ToricLattice(2))
          sage: origin.dim()
          0
          sage: origin.lattice_dim()
          2
          sage: origin.lattice()
          2-d lattice N
  
      However, the trivial cone in ``n`` dimensions has a predefined
      constructor for you to use::
  
          sage: origin = cones.trivial(2)
          sage: origin.rays()
          Empty collection
          in 2-d lattice N
  
      Of course, you can also provide ``lattice`` in other cases::
  
          sage: L = ToricLattice(3, "L")
          sage: c1 = Cone([(1,0,0),(1,1,1)], lattice=L)
          sage: c1.rays()
          L(1, 0, 0),
          L(1, 1, 1)
          in 3-d lattice L
  
      Or you can construct cones from rays of a particular lattice::
  
          sage: ray1 = L(1,0,0)
          sage: ray2 = L(1,1,1)
          sage: c2 = Cone([ray1, ray2])
          sage: c2.rays()
          L(1, 0, 0),
          L(1, 1, 1)
          in 3-d lattice L
          sage: c1 == c2
          True
  
      When the cone in question is not strictly convex, the standard form for
      the "generating rays" of the linear subspace is "basis vectors and their
      negatives", as in the following example::
  
          sage: plane = Cone([(1,0), (0,1), (-1,-1)])
          sage: plane.rays()
          N( 0,  1),
          N( 0, -1),
          N( 1,  0),
          N(-1,  0)
          in 2-d lattice N
  
      The cone can also be specified by a
      :class:`~sage.geometry.polyhedron.base.Polyhedron_base`::
  
          sage: p = plane.polyhedron()
          sage: Cone(p)
          2-d cone in 2-d lattice N
          sage: Cone(p) == plane
          True
  
      TESTS::
  
          sage: N = ToricLattice(2)
          sage: Nsub = N.span([ N(1,2) ])
          sage: Cone(Nsub.basis())
          1-d cone in Sublattice <N(1, 2)>
          sage: Cone([N(0)])
          0-d cone in 2-d lattice N
      """
      # Cone from Polyhedron
      if is_Polyhedron(rays):
          polyhedron = rays
          if lattice is None:
              lattice = ToricLattice(polyhedron.ambient_dim())
          if polyhedron.n_vertices() > 1:
              raise ValueError("%s is not a cone!" % polyhedron)
          apex = polyhedron.vertices()[0]
          if apex.count(0) != len(apex):
              raise ValueError("the apex of %s is not at the origin!"
                               % polyhedron)
          rays = normalize_rays(polyhedron.rays(), lattice)
          for line in normalize_rays(polyhedron.lines(), lattice):
              rays.append(line)
              rays.append(-line)
              rays[-1].set_immutable()
          return ConvexRationalPolyhedralCone(rays, lattice)
      # Cone from rays
      if check or normalize:
          rays = normalize_rays(rays, lattice)
      if lattice is None:
          if rays:
              lattice = rays[0].parent()
          else:
              raise ValueError(
                  "lattice must be given explicitly if there are no rays!")
      if not check or not rays:
          return ConvexRationalPolyhedralCone(rays, lattice)
      # Any set of rays forms a cone, but we want to keep only generators
      if is_ToricLatticeQuotient(lattice):
          gs = Generator_System(
                          PPL_point(Linear_Expression(lattice(0).vector(), 0)))
          for r in rays:
              if not r.is_zero():
                  gs.insert(PPL_ray(Linear_Expression(r.vector(), 0)))
      else:
          gs = Generator_System( PPL_point(Linear_Expression(lattice(0),0)) )
          for r in rays:
              if not r.is_zero():
                  gs.insert( PPL_ray(Linear_Expression(r,0)) )
      cone = C_Polyhedron(gs)
      return _Cone_from_PPL(cone, lattice, rays)
  
  
  def _Cone_from_PPL(cone, lattice, original_rays=None):
      r"""
      Construct a cone from a :class:`~ppl.polyhedron.Polyhedron`.
  
      This is a private function and not intended to be exposed to the
      end user. It is used internally by :func:`Cone` and in
      :meth:`ConvexRationalPolyhedralCone.intersection`.
  
      INPUT:
  
      - ``cone`` -- a :class:`~ppl.polyhedron.Polyhedron` having the
        origin as its single point.
  
      - ``lattice`` -- :class:`ToricLattice
        <sage.geometry.toric_lattice.ToricLatticeFactory>`, `\ZZ^n`, or any
        other object that behaves like these.
  
      - ``original_rays`` -- (default: ``None``) if given, must be a minimal list
        of normalized generating rays of ``cone``. If ``cone`` is strictly convex
        and ``original_rays`` were given, they will be used as internal rays of
        the constructed cone, in the given order.
  
      OUTPUT:
  
      A :class:`ConvexRationalPolyhedralCone`.
  
      TESTS::
  
          sage: Cone([(1,0), (0,1), (1,1), (0,1)]).rays()   # indirect doctest
          N(0, 1),
          N(1, 0)
          in 2-d lattice N
      """
      rays = []
      lines = []
      for g in cone.minimized_generators():
          if g.is_ray():
              rays.append(g)
          if g.is_line():
              lines.append(g)
      if (original_rays is not None and not lines and
          len(rays) == len(original_rays)):
          return ConvexRationalPolyhedralCone(original_rays, lattice, PPL=cone)
      else:
          rays = [ray.coefficients() for ray in rays]
          for line in lines:
              rays.append(line.coefficients())
              rays.append(-vector(ZZ, rays[-1]))
          try:
              for i, ray in enumerate(rays):
                  rays[i] = lattice(ray)
                  rays[i].set_immutable()
          except TypeError:
              rays = normalize_rays(rays, lattice)
          return ConvexRationalPolyhedralCone(rays, lattice, PPL=cone)
  
  
  def _ambient_space_point(body, data):
      r"""
      Try to convert ``data`` to a point of the ambient space of ``body``.
  
      INPUT:
  
      - ``body`` -- a cone, fan, or lattice polytope with ``lattice()`` method
  
      - ``data`` -- anything
  
      OUTPUT:
  
      - integral, rational or numeric point of the ambient space of ``body``
        if ``data`` were successfully interpreted in such a way, otherwise a
        ``TypeError`` exception is raised
  
      TESTS::
  
          sage: from sage.geometry.cone import _ambient_space_point
          sage: c = Cone([(1,0), (0,1)])
          sage: _ambient_space_point(c, [1,1])
          N(1, 1)
          sage: _ambient_space_point(c, c.dual_lattice()([1,1]))
          Traceback (most recent call last):
          ...
          TypeError: the point M(1, 1) and
           2-d cone in 2-d lattice N have incompatible lattices
          sage: _ambient_space_point(c, [1,1/3])
          (1, 1/3)
          sage: _ambient_space_point(c, [1/2,1/sqrt(3)])
          (0.500000000000000, 0.577350269189626)
          sage: _ambient_space_point(c, [1,1,3])
          Traceback (most recent call last):
          ...
          TypeError: [1, 1, 3] does not represent a valid point
           in the ambient space of 2-d cone in 2-d lattice N
      """
      L = body.lattice()
      try: # to make a lattice element...
          return L(data)
      except TypeError:
          # Special treatment for toric lattice elements
          if is_ToricLattice(parent(data)):
              raise TypeError("the point %s and %s have incompatible "
                              "lattices" % (data, body))
      try: # ... or an exact point...
          return L.base_extend(QQ)(data)
      except TypeError:
          pass
      try: # ... or at least a numeric one
          return L.base_extend(RR)(data)
      except TypeError:
          pass
      # Raise TypeError with our own message
      raise TypeError("%s does not represent a valid point in the ambient "
                      "space of %s" % (data, body))
  
  
  def integral_length(v):
      """
      Compute the integral length of a given rational vector.
  
      INPUT:
  
      - ``v`` -- any object which can be converted to a list of rationals
  
      OUTPUT:
  
      Rational number `r`` such that ``v = r * u``, where ``u`` is the
      primitive integral vector in the direction of ``v``.
  
      EXAMPLES::
  
          sage: from sage.geometry.cone import integral_length
          sage: integral_length([1, 2, 4])
          1
          sage: integral_length([2, 2, 4])
          2
          sage: integral_length([2/3, 2, 4])
          2/3
      """
      data = [QQ(e) for e in list(v)]
      ns = [e.numerator() for e in data]
      ds = [e.denominator() for e in data]
      return gcd(ns) / lcm(ds)
  
  
  def normalize_rays(rays, lattice):
      r"""
      Normalize a list of rational rays: make them primitive and immutable.
  
      INPUT:
  
      - ``rays`` -- list of rays which can be converted to the rational
        extension of ``lattice``;
  
      - ``lattice`` -- :class:`ToricLattice
        <sage.geometry.toric_lattice.ToricLatticeFactory>`, `\ZZ^n`, or any
        other object that behaves like these. If ``None``, an attempt will
        be made to determine an appropriate toric lattice automatically.
  
      OUTPUT:
  
      - list of immutable primitive vectors of the ``lattice`` in the same
        directions as original ``rays``.
  
      EXAMPLES::
  
          sage: from sage.geometry.cone import normalize_rays
          sage: normalize_rays([(0, 1), (0, 2), (3, 2), (5/7, 10/3)], None)
          [N(0, 1), N(0, 1), N(3, 2), N(3, 14)]
          sage: L = ToricLattice(2, "L")
          sage: normalize_rays([(0, 1), (0, 2), (3, 2), (5/7, 10/3)], L.dual())
          [L*(0, 1), L*(0, 1), L*(3, 2), L*(3, 14)]
          sage: ray_in_L = L(0,1)
          sage: normalize_rays([ray_in_L, (0, 2), (3, 2), (5/7, 10/3)], None)
          [L(0, 1), L(0, 1), L(3, 2), L(3, 14)]
          sage: normalize_rays([(0, 1), (0, 2), (3, 2), (5/7, 10/3)], ZZ^2)
          [(0, 1), (0, 1), (3, 2), (3, 14)]
          sage: normalize_rays([(0, 1), (0, 2), (3, 2), (5/7, 10/3)], ZZ^3)
          Traceback (most recent call last):
          ...
          TypeError: cannot convert (0, 1) to
          Vector space of dimension 3 over Rational Field!
          sage: normalize_rays([], ZZ^3)
          []
      """
      if rays is None:
          rays = []
      try:
          rays = list(rays)
      except TypeError:
          raise TypeError(
                      "rays must be given as a list or a compatible structure!"
                      "\nGot: %s" % rays)
      if rays:
          if lattice is None:
              ray_parent = parent(rays[0])
              lattice = (ray_parent if is_ToricLattice(ray_parent)
                                    else ToricLattice(len(rays[0])))
          if lattice.base_ring() is not ZZ:
              raise TypeError("lattice must be a free module over ZZ")
          # Are we dealing with a quotient lattice?
          try:
              if not lattice.is_torsion_free():
                  raise ValueError("cannot normalize rays of torsion quotients!")
          except AttributeError:
              pass
          V = None
          try:
              if lattice.is_ambient():
                  # Handle the most common case efficiently.
                  V = lattice.base_extend(QQ)
                  length = lambda ray: integral_length(ray)
          except AttributeError:
              pass
          if V is None:
              # Use a more general, but slower way.
              V = lattice.vector_space_span_of_basis(lattice.basis())
              length = lambda ray: integral_length(V.coordinate_vector(ray))
          for n, ray in enumerate(rays):
              try:
                  if isinstance(ray, (list, tuple, V.element_class)):
                      ray = V(ray)
                  else:
                      ray = V(list(ray))
              except TypeError:
                  raise TypeError("cannot convert %s to %s!" % (ray, V))
              if ray.is_zero():
                  ray = lattice(0)
              else:
                  ray = lattice(ray / length(ray))
              ray.set_immutable()
              rays[n] = ray
      return rays
  
  
  @richcmp_method
  class IntegralRayCollection(SageObject,
                              collections.Hashable,
                              collections.Iterable):
      r"""
      Create a collection of integral rays.
  
      .. WARNING::
  
          No correctness check or normalization is performed on the input data.
          This class is designed for internal operations and you probably should
          not use it directly.
  
      This is a base class for :class:`convex rational polyhedral cones
      <ConvexRationalPolyhedralCone>` and :class:`fans
      <sage.geometry.fan.RationalPolyhedralFan>`.
  
      Ray collections are immutable, but they cache most of the returned values.
  
      INPUT:
  
      - ``rays`` -- list of immutable vectors in ``lattice``;
  
      - ``lattice`` -- :class:`ToricLattice
        <sage.geometry.toric_lattice.ToricLatticeFactory>`, `\ZZ^n`, or any
        other object that behaves like these. If ``None``, it will be determined
        as :func:`parent` of the first ray. Of course, this cannot be done if
        there are no rays, so in this case you must give an appropriate
        ``lattice`` directly. Note that ``None`` is *not* the default value -
        you always *must* give this argument explicitly, even if it is ``None``.
  
      OUTPUT:
  
      - collection of given integral rays.
      """
  
      def __init__(self, rays, lattice):
          r"""
          See :class:`IntegralRayCollection` for documentation.
  
          TESTS::
  
              sage: from sage.geometry.cone import (
              ....:           IntegralRayCollection)
              sage: v = vector([1,0])
              sage: v.set_immutable()
              sage: c = IntegralRayCollection([v], ZZ^2)
              sage: c = IntegralRayCollection([v], None)
              sage: c.lattice()  # Determined automatically
              Ambient free module of rank 2
              over the principal ideal domain Integer Ring
              sage: c.rays()
              (1, 0)
              in Ambient free module of rank 2
              over the principal ideal domain Integer Ring
              sage: TestSuite(c).run()
          """
          if lattice is None:
              lattice = rays[0].parent()
          self._rays = PointCollection(rays, lattice)
          self._lattice = lattice
  
      def __richcmp__(self, right, op):
          r"""
          Compare ``self`` and ``right``.
  
          INPUT:
  
          - ``right`` -- anything.
  
          OUTPUT:
  
          boolean
  
          There is equality if ``right`` is of the same type as
          ``self``, they have the same ambient lattices, and their
          rays are the same and listed in the same order.
  
          TESTS::
  
              sage: c1 = Cone([(1,0), (0,1)])
              sage: c2 = Cone([(0,1), (1,0)])
              sage: c3 = Cone([(0,1), (1,0)])
              sage: c1 > c2
              True
              sage: c2 < c1
              True
              sage: c2 == c3
              True
              sage: c2 is c3
              False
          """
          if type(self) != type(right):
              return NotImplemented
  
          # We probably do need to have explicit comparison of lattices here
          # since if one of the collections does not live in a toric lattice,
          # comparison of rays may miss the difference.
          return richcmp((self.lattice(), self.rays()),
                         (right.lattice(), right.rays()), op)
  
      def __hash__(self):
          r"""
          Return the hash of ``self`` computed from rays.
  
          OUTPUT:
  
          - integer.
  
          TESTS::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: hash(c) == hash(c)
              True
          """
          if "_hash" not in self.__dict__:
              self._hash = hash(self._rays)
          return self._hash
  
      def __iter__(self):
          r"""
          Return an iterator over rays of ``self``.
  
          OUTPUT:
  
          -  iterator.
  
          TESTS::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: for ray in c: print(ray)
              N(1, 0)
              N(0, 1)
          """
          return iter(self._rays)
  
      def cartesian_product(self, other, lattice=None):
          r"""
          Return the Cartesian product of ``self`` with ``other``.
  
          INPUT:
  
          - ``other`` -- an :class:`IntegralRayCollection`;
  
          - ``lattice`` -- (optional) the ambient lattice for the result. By
            default, the direct sum of the ambient lattices of ``self`` and
            ``other`` is constructed.
  
          OUTPUT:
  
          - an :class:`IntegralRayCollection`.
  
          By the Cartesian product of ray collections `(r_0, \dots, r_{n-1})` and
          `(s_0, \dots, s_{m-1})` we understand the ray collection of the form
          `((r_0, 0), \dots, (r_{n-1}, 0), (0, s_0), \dots, (0, s_{m-1}))`, which
          is suitable for Cartesian products of cones and fans. The ray order is
          guaranteed to be as described.
  
          EXAMPLES::
  
              sage: c = Cone([(1,)])
              sage: c.cartesian_product(c)    # indirect doctest
              2-d cone in 2-d lattice N+N
              sage: _.rays()
              N+N(1, 0),
              N+N(0, 1)
              in 2-d lattice N+N
          """
          assert isinstance(other, IntegralRayCollection)
          if lattice is None:
              lattice = self.lattice().direct_sum(other.lattice())
          suffix = [0] * other.lattice_dim()
          rays = [lattice(list(r1) + suffix) for r1 in self.rays()]
          prefix = [0] * self.lattice_dim()
          rays.extend(lattice(prefix + list(r2)) for r2 in other.rays())
          for r in rays:
              r.set_immutable()
          return IntegralRayCollection(rays, lattice)
  
      def __neg__(self):
          """
          Return the collection with opposite rays.
  
          EXAMPLES::
  
              sage: c = Cone([(1,1),(0,1)]); c
              2-d cone in 2-d lattice N
              sage: d = -c  # indirect doctest
              sage: d.rays()
              N(-1, -1),
              N( 0, -1)
              in 2-d lattice N
          """
          lattice = self.lattice()
          rays = [-r1 for r1 in self.rays()]
          for r in rays:
              r.set_immutable()
          return IntegralRayCollection(rays, lattice)
  
      def dim(self):
          r"""
          Return the dimension of the subspace spanned by rays of ``self``.
  
          OUTPUT:
  
          - integer.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0)])
              sage: c.lattice_dim()
              2
              sage: c.dim()
              1
          """
          if "_dim" not in self.__dict__:
              self._dim = self.rays().matrix().rank()
          return self._dim
  
      def lattice(self):
          r"""
          Return the ambient lattice of ``self``.
  
          OUTPUT:
  
          - lattice.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0)])
              sage: c.lattice()
              2-d lattice N
              sage: Cone([], ZZ^3).lattice()
              Ambient free module of rank 3
              over the principal ideal domain Integer Ring
          """
          return self._lattice
  
      @cached_method
      def dual_lattice(self):
          r"""
          Return the dual of the ambient lattice of ``self``.
  
          OUTPUT:
  
          - lattice. If possible (that is, if :meth:`lattice` has a
            ``dual()`` method), the dual lattice is returned. Otherwise,
            `\ZZ^n` is returned, where `n` is the dimension of :meth:`lattice`.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0)])
              sage: c.dual_lattice()
              2-d lattice M
              sage: Cone([], ZZ^3).dual_lattice()
              Ambient free module of rank 3
              over the principal ideal domain Integer Ring
  
          TESTS:
  
          The dual lattice of the dual lattice of a random cone should be
          the original lattice::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8, max_rays=10)
              sage: K.dual_lattice().dual() is K.lattice()
              True
          """
          try:
              return self.lattice().dual()
          except AttributeError:
              return ZZ**self.lattice_dim()
  
      def lattice_dim(self):
          r"""
          Return the dimension of the ambient lattice of ``self``.
  
          OUTPUT:
  
          - integer.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0)])
              sage: c.lattice_dim()
              2
              sage: c.dim()
              1
          """
          return self.lattice().dimension()
  
      def nrays(self):
          r"""
          Return the number of rays of ``self``.
  
          OUTPUT:
  
          - integer.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: c.nrays()
              2
          """
          return len(self._rays)
  
      def plot(self, **options):
          r"""
          Plot ``self``.
  
          INPUT:
  
          - any options for toric plots (see :func:`toric_plotter.options
            <sage.geometry.toric_plotter.options>`), none are mandatory.
  
          OUTPUT:
  
          - a plot.
  
          EXAMPLES::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant.plot()
              Graphics object consisting of 9 graphics primitives
          """
          tp = ToricPlotter(options, self.lattice().degree(), self.rays())
          return tp.plot_lattice() + tp.plot_rays() + tp.plot_generators()
  
      def ray(self, n):
          r"""
          Return the ``n``-th ray of ``self``.
  
          INPUT:
  
          - ``n`` -- integer, an index of a ray of ``self``. Enumeration of rays
            starts with zero.
  
          OUTPUT:
  
          - ray, an element of the lattice of ``self``.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: c.ray(0)
              N(1, 0)
          """
          return self._rays[n]
  
      def rays(self, *args):
          r"""
          Return (some of the) rays of ``self``.
  
          INPUT:
  
          - ``ray_list`` -- a list of integers, the indices of the requested
            rays. If not specified, all rays of ``self`` will be returned.
  
          OUTPUT:
  
          - a :class:`~sage.geometry.point_collection.PointCollection`
            of primitive integral ray generators.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0), (0,1), (-1, 0)])
              sage: c.rays()
              N( 0, 1),
              N( 1, 0),
              N(-1, 0)
              in 2-d lattice N
              sage: c.rays([0, 2])
              N( 0, 1),
              N(-1, 0)
              in 2-d lattice N
  
          You can also give ray indices directly, without packing them into a
          list::
  
              sage: c.rays(0, 2)
              N( 0, 1),
              N(-1, 0)
              in 2-d lattice N
          """
          return self._rays if not args else self._rays(*args)
  
      def codim(self):
          r"""
          Return the codimension of ``self``.
  
          The codimension of a collection of rays (of a cone/fan) is the
          difference between the dimension of the ambient space and the
          dimension of the subspace spanned by those rays (of the cone/fan).
  
          OUTPUT:
  
          A nonnegative integer representing the codimension of ``self``.
  
          .. SEEALSO::
  
              :meth:`dim`, :meth:`lattice_dim`
  
          EXAMPLES:
  
          The codimension of the nonnegative orthant is zero, since the
          span of its generators equals the entire ambient space::
  
              sage: K = cones.nonnegative_orthant(3)
              sage: K.codim()
              0
  
          However, if we remove a ray so that the entire cone is contained
          within the `x`-`y` plane, then the resulting cone will have
          codimension one, because the `z`-axis is perpendicular to every
          element of the cone::
  
              sage: K = Cone([(1,0,0), (0,1,0)])
              sage: K.codim()
              1
  
          If our cone is all of `\mathbb{R}^{2}`, then its codimension is
          zero::
  
              sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
              sage: K.is_full_space()
              True
              sage: K.codim()
              0
  
          And if the cone is trivial in any space, then its codimension is
          equal to the dimension of the ambient space::
  
              sage: K = cones.trivial(0)
              sage: K.lattice_dim()
              0
              sage: K.codim()
              0
  
              sage: K = cones.trivial(1)
              sage: K.lattice_dim()
              1
              sage: K.codim()
              1
  
              sage: K = cones.trivial(2)
              sage: K.lattice_dim()
              2
              sage: K.codim()
              2
  
          TESTS:
  
          The codimension of a cone should be an integer between zero and
          the dimension of the ambient space, inclusive::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8)
              sage: c = K.codim()
              sage: c in ZZ
              True
              sage: 0 <= c <= K.lattice_dim()
              True
  
          A solid cone should have codimension zero::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8, solid = True)
              sage: K.codim()
              0
  
          The codimension of a cone is equal to the lineality of its dual::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8)
              sage: K.codim() == K.dual().lineality()
              True
          """
          return (self.lattice_dim() - self.dim())
  
  
      def span(self, base_ring=None):
          r"""
          Return the span of ``self``.
  
          INPUT:
  
          - ``base_ring`` -- (default: from lattice) the base ring to use
                             for the generated module.
  
          OUTPUT:
  
          A module spanned by the generators of ``self``.
  
          EXAMPLES:
  
          The span of a single ray is a one-dimensional sublattice::
  
              sage: K1 = Cone([(1,)])
              sage: K1.span()
              Sublattice <N(1)>
              sage: K2 = Cone([(1,0)])
              sage: K2.span()
              Sublattice <N(1, 0)>
  
          The span of the nonnegative orthant is the entire ambient lattice::
  
              sage: K = cones.nonnegative_orthant(3)
              sage: K.span() == K.lattice()
              True
  
          By specifying a ``base_ring``, we can obtain a vector space::
  
              sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
              sage: K.span(base_ring=QQ)
              Vector space of degree 3 and dimension 3 over Rational Field
              Basis matrix:
              [1 0 0]
              [0 1 0]
              [0 0 1]
  
          TESTS:
  
          We can take the span of the trivial cone::
  
              sage: cones.trivial(0).span()
              Sublattice <>
  
          The span of a solid cone is the entire ambient space::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6, max_rays=8, solid=True)
              sage: K.span().vector_space() == K.lattice().vector_space()
              True
          """
          L = self.lattice()
  
          if base_ring is None:
              base_ring = L.base_ring()
  
          return L.span(self, base_ring)
  
  
  def classify_cone_2d(ray0, ray1, check=True):
      r"""
      Return `(d,k)` classifying the lattice cone spanned by the two rays.
  
      INPUT:
  
      - ``ray0``, ``ray1`` -- two primitive integer vectors. The
        generators of the two rays generating the two-dimensional cone.
  
      - ``check`` -- boolean (default: ``True``). Whether to check the
        input rays for consistency.
  
      OUTPUT:
  
      A pair `(d,k)` of integers classifying the cone up to `GL(2, \ZZ)`
      equivalence. See Proposition 10.1.1 of [CLS2011]_ for the
      definition. We return the unique `(d,k)` with minimal `k`, see
      Proposition 10.1.3 of [CLS2011]_.
  
      EXAMPLES::
  
          sage: ray0 = vector([1,0])
          sage: ray1 = vector([2,3])
          sage: from sage.geometry.cone import classify_cone_2d
          sage: classify_cone_2d(ray0, ray1)
          (3, 2)
  
          sage: ray0 = vector([2,4,5])
          sage: ray1 = vector([5,19,11])
          sage: classify_cone_2d(ray0, ray1)
          (3, 1)
  
          sage: m = matrix(ZZ, [(19, -14, -115), (-2, 5, 25), (43, -42, -298)])
          sage: m.det()   # check that it is in GL(3,ZZ)
          -1
          sage: classify_cone_2d(m*ray0, m*ray1)
          (3, 1)
  
      TESTS:
  
      Check using the connection between the Hilbert basis of the cone
      spanned by the two rays (in arbitrary dimension) and the
      Hirzebruch-Jung continued fraction expansion, see Chapter 10 of
      [CLS2011]_ ::
  
          sage: from sage.geometry.cone import normalize_rays
          sage: for i in range(10):
          ....:     ray0 = random_vector(ZZ, 3)
          ....:     ray1 = random_vector(ZZ, 3)
          ....:     if ray0.is_zero() or ray1.is_zero(): continue
          ....:     ray0, ray1 = normalize_rays([ray0, ray1], ZZ^3)
          ....:     d, k = classify_cone_2d(ray0, ray1, check=True)
          ....:     assert (d,k) == classify_cone_2d(ray1, ray0)
          ....:     if d == 0: continue
          ....:     frac = (k/d).continued_fraction_list("hj")
          ....:     if len(frac)>100: continue   # avoid expensive computation
          ....:     hilb = Cone([ray0, ray1]).Hilbert_basis()
          ....:     assert len(hilb) == len(frac) + 1
      """
      if check:
          assert ray0.parent() is ray1.parent()
          assert ray0.base_ring() is ZZ
          assert gcd(ray0) == 1
          assert gcd(ray1) == 1
          assert not ray0.is_zero() and not ray1.is_zero()
  
      m = matrix([ray0, ray1])              # dim(ray) x 2 matrix
      basis = m.saturation().solve_left(m)  # 2-d basis for the span of the cone
      basis = basis.change_ring(ZZ).transpose()
      if basis.nrows() < 2:
          d = 0
          k = basis[0,1]
      else:
          basis.echelonize()                    # columns are the "cone normal form"
          d = basis[1,1]
          k = basis[0,1]
  
      if check:
          if d == 0:  # degenerate cone
              assert basis[0,0] == 1
              assert k == -1 or k == +1
          else:       # non-degenerate cone
              assert basis[0,0] == 1 and basis[1,0] == 0
              assert d > 0
              assert 0 <= k < d
              assert gcd(d,k) == 1
  
      # compute unique k, see Proposition 10.1.3 of [CLS2011]
      if d > 0:
          for ktilde in range(k):
              if (k*ktilde) % d == 1:
                  k = ktilde
                  break
      return (d,k)
  
  
  # Derived classes MUST allow construction of their objects using ``ambient``
  # and ``ambient_ray_indices`` keyword parameters. See ``intersection`` method
  # for an example why this is needed.
  @richcmp_method
  class ConvexRationalPolyhedralCone(IntegralRayCollection,
                                     collections.Container):
      r"""
      Create a convex rational polyhedral cone.
  
      .. WARNING::
  
          This class does not perform any checks of correctness of input nor
          does it convert input into the standard representation. Use
          :func:`Cone` to construct cones.
  
      Cones are immutable, but they cache most of the returned values.
  
      INPUT:
  
      The input can be either:
  
      - ``rays`` -- list of immutable primitive vectors in ``lattice``;
  
      - ``lattice`` -- :class:`ToricLattice
        <sage.geometry.toric_lattice.ToricLatticeFactory>`, `\ZZ^n`, or any
        other object that behaves like these. If ``None``, it will be determined
        as :func:`parent` of the first ray. Of course, this cannot be done if
        there are no rays, so in this case you must give an appropriate
        ``lattice`` directly.
  
      or (these parameters must be given as keywords):
  
      - ``ambient`` -- ambient structure of this cone, a bigger :class:`cone
        <ConvexRationalPolyhedralCone>` or a :class:`fan
        <sage.geometry.fan.RationalPolyhedralFan>`, this cone *must be a face
        of* ``ambient``;
  
      - ``ambient_ray_indices`` -- increasing list or tuple of integers, indices
        of rays of ``ambient`` generating this cone.
  
      In both cases, the following keyword parameter may be specified in addition:
  
      - ``PPL`` -- either ``None`` (default) or a
        :class:`~ppl.polyhedron.C_Polyhedron` representing the cone. This
        serves only to cache the polyhedral data if you know it
        already. The constructor does not make a copy so the ``PPL`` object
        should not be modified afterwards.
  
      OUTPUT:
  
      - convex rational polyhedral cone.
  
      .. NOTE::
  
          Every cone has its ambient structure. If it was not specified, it is
          this cone itself.
      """
  
      def __init__(self, rays=None, lattice=None,
                   ambient=None, ambient_ray_indices=None, PPL=None):
          r"""
          See :class:`ConvexRationalPolyhedralCone` for documentation.
  
          TESTS::
  
              sage: from sage.geometry.cone import (
              ....:       ConvexRationalPolyhedralCone)
              sage: v1 = vector([1,0])
              sage: v2 = vector([0,1])
              sage: v1.set_immutable()
              sage: v2.set_immutable()
              sage: ac = ConvexRationalPolyhedralCone([v1, v2], ZZ^2)
              sage: ac = ConvexRationalPolyhedralCone([v1, v2], None)
              sage: ac.lattice()  # Determined automatically
              Ambient free module of rank 2
              over the principal ideal domain Integer Ring
              sage: ac.rays()
              (1, 0),
              (0, 1)
              in Ambient free module of rank 2
              over the principal ideal domain Integer Ring
              sage: ac.ambient() is ac
              True
              sage: TestSuite(ac).run()
              sage: sc = ConvexRationalPolyhedralCone(ambient=ac,
              ....:                       ambient_ray_indices=[1])
              sage: sc.rays()
              (0, 1)
              in Ambient free module of rank 2
              over the principal ideal domain Integer Ring
              sage: sc.ambient() is ac
              True
              sage: TestSuite(sc).run()
          """
          superinit = super(ConvexRationalPolyhedralCone, self).__init__
          if ambient is None:
              superinit(rays, lattice)
              self._ambient = self
              self._ambient_ray_indices = tuple(range(self.nrays()))
          else:
              self._ambient = ambient
              self._ambient_ray_indices = tuple(ambient_ray_indices)
              superinit(ambient.rays(self._ambient_ray_indices),
                        ambient.lattice())
          if not PPL is None:
              self._PPL_C_Polyhedron = PPL
  
      def _sage_input_(self, sib, coerced):
          """
          Return Sage command to reconstruct ``self``.
  
          See :mod:`sage.misc.sage_input` for details.
  
          EXAMPLES::
  
              sage: cone = Cone([(1,0), (1,1)])
              sage: sage_input(cone)
              Cone([(1, 0), (1, 1)])
          """
          return sib.name('Cone')([sib(tuple(r)) for r in self.rays()])
  
      def _PPL_cone(self):
          r"""
          Returns the Parma Polyhedra Library (PPL) representation of the cone.
  
          OUTPUT:
  
          A :class:`~ppl.polyhedron.C_Polyhedron` representing the cone.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0), (1,1), (0,1)])
              sage: c._PPL_cone()
              A 2-dimensional polyhedron in QQ^2 defined as
              the convex hull of 1 point, 2 rays
              sage: c._PPL_cone().minimized_generators()
              Generator_System {point(0/1, 0/1), ray(0, 1), ray(1, 0)}
              sage: c._PPL_cone().minimized_constraints()
              Constraint_System {x1>=0, x0>=0}
  
          TESTS:
  
          There are no empty cones, the origin always belongs to them::
  
              sage: Cone([(0,0)])._PPL_cone()
              A 0-dimensional polyhedron in QQ^2
              defined as the convex hull of 1 point
              sage: cones.trivial(2)._PPL_cone()
              A 0-dimensional polyhedron in QQ^2
              defined as the convex hull of 1 point
          """
          if "_PPL_C_Polyhedron" not in self.__dict__:
              gs = Generator_System(
                              PPL_point(Linear_Expression(self._lattice(0), 0)))
              for r in self.rays():
                  gs.insert( PPL_ray(Linear_Expression(r,0)) )
              self._PPL_C_Polyhedron = C_Polyhedron(gs)
          return self._PPL_C_Polyhedron
  
      def __contains__(self, point):
          r"""
          Check if ``point`` is contained in ``self``.
  
          See :meth:`_contains` (which is called by this function) for
          documentation.
  
          TESTS::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: (1,1) in c
              True
              sage: [1,1] in c
              True
              sage: (-1,0) in c
              False
          """
          return self._contains(point)
  
      def __getstate__(self):
          r"""
          Return the dictionary that should be pickled.
  
          OUTPUT:
  
          - :class:`dict`.
  
          TESTS::
  
              sage: C = Cone([(1,0)])
              sage: C.face_lattice()
              Finite lattice containing 2 elements with distinguished linear extension
              sage: C._test_pickling()
              sage: C2 = loads(dumps(C)); C2
              1-d cone in 2-d lattice N
              sage: C2 == C
              True
              sage: C2 is C      # Is this desirable?
              False
          """
          state = copy.copy(self.__dict__)
          state.pop("_PPL_C_Polyhedron", None) # PPL is not picklable.
  
          # TODO: do we want to keep the face lattice in the pickle?
          # Currently there is an unpickling loop if do:
          # Unpickling a cone C requires first to unpickle its face lattice.
          # The latter is a Poset which takes C among its arguments. Due
          # to UniqueRepresentation, this triggers a call to hash(C) which
          # itself depends on the attribute C._rays which have not yet
          # been unpickled.  See ``explain_pickle(dumps(C))``.
          state.pop("_face_lattice", None)
          return state
  
      def _contains(self, point, region='whole cone'):
          r"""
          Check if ``point`` is contained in ``self``.
  
          This function is called by :meth:`__contains__` and :meth:`contains`
          to ensure the same call depth for warning messages.
  
          INPUT:
  
          - ``point`` -- anything. An attempt will be made to convert it into a
            single element of the ambient space of ``self``. If it fails,
            ``False`` is returned;
  
          - ``region`` -- string. Can be either 'whole cone' (default),
            'interior', or 'relative interior'. By default, a point on
            the boundary of the cone is considered part of the cone. If
            you want to test whether the **interior** of the cone
            contains the point, you need to pass the optional argument
            ``'interior'``.  If you want to test whether the **relative
            interior** of the cone contains the point, you need to pass
            the optional argument ``'relative_interior'``.
  
          OUTPUT:
  
          - ``True`` if ``point`` is contained in the specified ``region`` of
            ``self``, ``False`` otherwise.
  
          Raises a ``ValueError`` if ``region`` is not one of the
          three allowed values.
  
          TESTS::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: c._contains((1,1))
              True
          """
          try:
              point = _ambient_space_point(self, point)
          except TypeError as ex:
              if str(ex).endswith("have incompatible lattices!"):
                  warnings.warn("you have checked if a cone contains a point "
                                "from an incompatible lattice, this is False!",
                                stacklevel=3)
              return False
  
          if region not in ("whole cone", "relative interior", "interior"):
              raise ValueError("%s is an unknown region of the cone!" % region)
          if region == "interior" and self.dim() < self.lattice_dim():
              return False
          need_strict = region.endswith("interior")
          M = self.dual_lattice()
          for c in self._PPL_cone().minimized_constraints():
              pr = M(c.coefficients()) * point
              if c.is_equality():
                  if pr != 0:
                      return False
              elif pr < 0 or need_strict and pr == 0:
                  return False
          return True
  
      def interior_contains(self, *args):
          r"""
          Check if a given point is contained in the interior of ``self``.
  
          For a cone of strictly lower-dimension than the ambient space,
          the interior is always empty. You probably want to use
          :meth:`relative_interior_contains` in this case.
  
          INPUT:
  
          - anything. An attempt will be made to convert all arguments into a
            single element of the ambient space of ``self``. If it fails,
            ``False`` will be returned.
  
          OUTPUT:
  
          - ``True`` if the given point is contained in the interior of
            ``self``, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: c.contains((1,1))
              True
              sage: c.interior_contains((1,1))
              True
              sage: c.contains((1,0))
              True
              sage: c.interior_contains((1,0))
              False
          """
          point = flatten(args)
          if len(point) == 1:
             point = point[0]
          return self._contains(point, 'interior')
  
      def relative_interior_contains(self, *args):
          r"""
          Check if a given point is contained in the relative interior of ``self``.
  
          For a full-dimensional cone the relative interior is simply
          the interior, so this method will do the same check as
          :meth:`interior_contains`. For a strictly lower-dimensional cone, the
          relative interior is the cone without its facets.
  
          INPUT:
  
          - anything. An attempt will be made to convert all arguments into a
            single element of the ambient space of ``self``. If it fails,
            ``False`` will be returned.
  
          OUTPUT:
  
          - ``True`` if the given point is contained in the relative
            interior of ``self``, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0,0), (0,1,0)])
              sage: c.contains((1,1,0))
              True
              sage: c.relative_interior_contains((1,1,0))
              True
              sage: c.interior_contains((1,1,0))
              False
              sage: c.contains((1,0,0))
              True
              sage: c.relative_interior_contains((1,0,0))
              False
              sage: c.interior_contains((1,0,0))
              False
          """
          point = flatten(args)
          if len(point) == 1:
             point = point[0]
          return self._contains(point, 'relative interior')
  
      def cartesian_product(self, other, lattice=None):
          r"""
          Return the Cartesian product of ``self`` with ``other``.
  
          INPUT:
  
          - ``other`` -- a :class:`cone <ConvexRationalPolyhedralCone>`;
  
          - ``lattice`` -- (optional) the ambient lattice for the
            Cartesian product cone. By default, the direct sum of the
            ambient lattices of ``self`` and ``other`` is constructed.
  
          OUTPUT:
  
          - a :class:`cone <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES::
  
              sage: c = Cone([(1,)])
              sage: c.cartesian_product(c)
              2-d cone in 2-d lattice N+N
              sage: _.rays()
              N+N(1, 0),
              N+N(0, 1)
              in 2-d lattice N+N
          """
          assert is_Cone(other)
          rc = super(ConvexRationalPolyhedralCone, self).cartesian_product(
                                                                  other, lattice)
          return ConvexRationalPolyhedralCone(rc.rays(), rc.lattice())
  
      def __neg__(self):
          """
          Return the cone with opposite rays.
  
          OUTPUT:
  
          - a :class:`cone <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES::
  
              sage: c = Cone([(1,1),(0,1)]); c
              2-d cone in 2-d lattice N
              sage: d = -c; d  # indirect doctest
              2-d cone in 2-d lattice N
              sage: -d == c
              True
              sage: d.rays()
              N(-1, -1),
              N( 0, -1)
              in 2-d lattice N
          """
          rc = super(ConvexRationalPolyhedralCone, self).__neg__()
          return ConvexRationalPolyhedralCone(rc.rays(), rc.lattice())
  
      def __richcmp__(self, right, op):
          r"""
          Compare ``self`` and ``right``.
  
          INPUT:
  
          - ``right`` -- anything.
  
          OUTPUT:
  
          boolean
  
          There is equality if ``self`` and ``right`` are cones of any
          kind in the same lattice with the same rays listed in the
          same order.
  
          TESTS::
  
              sage: c1 = Cone([(1,0), (0,1)])
              sage: c2 = Cone([(0,1), (1,0)])
              sage: c3 = Cone([(0,1), (1,0)])
              sage: c1 > c2
              True
              sage: c2 < c1
              True
              sage: c2 == c3
              True
              sage: c2 is c3
              False
          """
          if is_Cone(right):
              # We don't care about particular type of right in this case
              return richcmp((self.lattice(), self.rays()),
                             (right.lattice(), right.rays()), op)
          else:
              return NotImplemented
  
      def _latex_(self):
          r"""
          Return a LaTeX representation of ``self``.
  
          OUTPUT:
  
          - string.
  
          TESTS::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant._latex_()
              '\\sigma^{2}'
              sage: quadrant.facets()[0]._latex_()
              '\\sigma^{1} \\subset \\sigma^{2}'
          """
          if self.ambient() is self:
              return r"\sigma^{%d}" % self.dim()
          else:
              return r"\sigma^{%d} \subset %s" % (self.dim(),
                                                  latex(self.ambient()))
  
      def _repr_(self):
          r"""
          Return a string representation of ``self``.
  
          OUTPUT:
  
          - string.
  
          TESTS::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant._repr_()
              '2-d cone in 2-d lattice N'
              sage: quadrant
              2-d cone in 2-d lattice N
              sage: quadrant.facets()[0]
              1-d face of 2-d cone in 2-d lattice N
          """
          result = "%d-d" % self.dim()
          if self.ambient() is self:
              result += " cone in"
              if is_ToricLattice(self.lattice()):
                  result += " %s" % self.lattice()
              else:
                  result += " %d-d lattice" % self.lattice_dim()
          else:
              result += " face of %s" % self.ambient()
          return result
  
      def _sort_faces(self,  faces):
          r"""
          Return sorted (if necessary) ``faces`` as a tuple.
  
          This function ensures that one-dimensional faces are listed in
          agreement with the order of corresponding rays and facets with
          facet normals.
  
          INPUT:
  
          - ``faces`` -- iterable of :class:`cones
            <ConvexRationalPolyhedralCone>`.
  
          OUTPUT:
  
          - :class:`tuple` of :class:`cones <ConvexRationalPolyhedralCone>`.
  
          TESTS::
  
              sage: octant = Cone(identity_matrix(3).columns())
              sage: # indirect doctest
              sage: for i, face in enumerate(octant.faces(1)):
              ....:     if face.ray(0) != octant.ray(i):
              ....:         print("Wrong order!")
          """
          faces = tuple(faces)
          if len(faces) > 1: # Otherwise there is nothing to sort
              if faces[0].nrays() == 1:
                  faces = tuple(sorted(faces,
                                       key=lambda f: f._ambient_ray_indices))
              elif faces[0].dim() == self.dim() - 1 and \
                      self.facet_normals.is_in_cache():
                  # If we already have facet normals, sort according to them
                  faces = set(faces)
                  sorted_faces = [None] * len(faces)
                  for i, n in enumerate(self.facet_normals()):
                      for f in faces:
                          if n*f.rays() == 0:
                              sorted_faces[i] = f
                              faces.remove(f)
                              break
                  faces = tuple(sorted_faces)
          return faces
  
      @cached_method
      def adjacent(self):
          r"""
          Return faces adjacent to ``self`` in the ambient face lattice.
  
          Two *distinct* faces `F_1` and `F_2` of the same face lattice are
          **adjacent** if all of the following conditions hold:
  
          * `F_1` and `F_2` have the same dimension `d`;
  
          * `F_1` and `F_2` share a facet of dimension `d-1`;
  
          * `F_1` and `F_2` are facets of some face of dimension `d+1`, unless
            `d` is the dimension of the ambient structure.
  
          OUTPUT:
  
          - :class:`tuple` of :class:`cones <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES::
  
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: octant.adjacent()
              ()
              sage: one_face = octant.faces(1)[0]
              sage: len(one_face.adjacent())
              2
              sage: one_face.adjacent()[1]
              1-d face of 3-d cone in 3-d lattice N
  
          Things are a little bit subtle with fans, as we illustrate below.
  
          First, we create a fan from two cones in the plane::
  
              sage: fan = Fan(cones=[(0,1), (1,2)],
              ....:           rays=[(1,0), (0,1), (-1,0)])
              sage: cone = fan.generating_cone(0)
              sage: len(cone.adjacent())
              1
  
          The second generating cone is adjacent to this one. Now we create the
          same fan, but embedded into the 3-dimensional space::
  
              sage: fan = Fan(cones=[(0,1), (1,2)],
              ....:           rays=[(1,0,0), (0,1,0), (-1,0,0)])
              sage: cone = fan.generating_cone(0)
              sage: len(cone.adjacent())
              1
  
          The result is as before, since we still have::
  
              sage: fan.dim()
              2
  
          Now we add another cone to make the fan 3-dimensional::
  
              sage: fan = Fan(cones=[(0,1), (1,2), (3,)],
              ....:           rays=[(1,0,0), (0,1,0), (-1,0,0), (0,0,1)])
              sage: cone = fan.generating_cone(0)
              sage: len(cone.adjacent())
              0
  
          Since now ``cone`` has smaller dimension than ``fan``, it and its
          adjacent cones must be facets of a bigger one, but since ``cone``
          in this example is generating, it is not contained in any other.
          """
          L = self._ambient._face_lattice_function()
          adjacent = set()
          facets = self.facets()
          superfaces = self.facet_of()
          if superfaces:
              for superface in superfaces:
                  for facet in facets:
                      adjacent.update(L.open_interval(facet,  superface))
              if adjacent:
                  adjacent.remove(L(self))
              return self._sort_faces(adjacent)
          elif self.dim() == self._ambient.dim():
              # Special treatment relevant for fans
              for facet in facets:
                  adjacent.update(facet.facet_of())
              if adjacent:
                  adjacent.remove(self)
              return self._sort_faces(adjacent)
          else:
              return ()
  
      def ambient(self):
          r"""
          Return the ambient structure of ``self``.
  
          OUTPUT:
  
          - cone or fan containing ``self`` as a face.
  
          EXAMPLES::
  
              sage: cone = Cone([(1,2,3), (4,6,5), (9,8,7)])
              sage: cone.ambient()
              3-d cone in 3-d lattice N
              sage: cone.ambient() is cone
              True
              sage: face = cone.faces(1)[0]
              sage: face
              1-d face of 3-d cone in 3-d lattice N
              sage: face.ambient()
              3-d cone in 3-d lattice N
              sage: face.ambient() is cone
              True
          """
          return self._ambient
  
      def ambient_ray_indices(self):
          r"""
          Return indices of rays of the ambient structure generating ``self``.
  
          OUTPUT:
  
          - increasing :class:`tuple` of integers.
  
          EXAMPLES::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant.ambient_ray_indices()
              (0, 1)
              sage: quadrant.facets()[1].ambient_ray_indices()
              (1,)
          """
          return self._ambient_ray_indices
  
      def contains(self, *args):
          r"""
          Check if a given point is contained in ``self``.
  
          INPUT:
  
          - anything. An attempt will be made to convert all arguments into a
            single element of the ambient space of ``self``. If it fails,
            ``False`` will be returned.
  
          OUTPUT:
  
          - ``True`` if the given point is contained in ``self``, ``False``
            otherwise.
  
          EXAMPLES::
  
              sage: c = Cone([(1,0), (0,1)])
              sage: c.contains(c.lattice()(1,0))
              True
              sage: c.contains((1,0))
              True
              sage: c.contains((1,1))
              True
              sage: c.contains(1,1)
              True
              sage: c.contains((-1,0))
              False
              sage: c.contains(c.dual_lattice()(1,0)) #random output (warning)
              False
              sage: c.contains(c.dual_lattice()(1,0))
              False
              sage: c.contains(1)
              False
              sage: c.contains(1/2, sqrt(3))
              True
              sage: c.contains(-1/2, sqrt(3))
              False
          """
          point = flatten(args)
          if len(point) == 1:
             point = point[0]
          return self._contains(point)
  
      def dual(self):
          r"""
          Return the dual cone of ``self``.
  
          OUTPUT:
  
          - :class:`cone <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES::
  
              sage: cone = Cone([(1,0), (-1,3)])
              sage: cone.dual().rays()
              M(0, 1),
              M(3, 1)
              in 2-d lattice M
  
          Now let's look at a more complicated case::
  
              sage: cone = Cone([(-2,-1,2), (4,1,0), (-4,-1,-5), (4,1,5)])
              sage: cone.is_strictly_convex()
              False
              sage: cone.dim()
              3
              sage: cone.dual().rays()
              M(7, -18, -2),
              M(1,  -4,  0)
              in 3-d lattice M
              sage: cone.dual().dual() is cone
              True
  
          We correctly handle the degenerate cases::
  
              sage: N = ToricLattice(2)
              sage: Cone([], lattice=N).dual().rays()  # empty cone
              M( 1,  0),
              M(-1,  0),
              M( 0,  1),
              M( 0, -1)
              in 2-d lattice M
              sage: Cone([(1,0)], lattice=N).dual().rays()  # ray in 2d
              M(1,  0),
              M(0,  1),
              M(0, -1)
              in 2-d lattice M
              sage: Cone([(1,0),(-1,0)], lattice=N).dual().rays()  # line in 2d
              M(0,  1),
              M(0, -1)
              in 2-d lattice M
              sage: Cone([(1,0),(0,1)], lattice=N).dual().rays()  # strictly convex cone
              M(0, 1),
              M(1, 0)
              in 2-d lattice M
              sage: Cone([(1,0),(-1,0),(0,1)], lattice=N).dual().rays()  # half space
              M(0, 1)
              in 2-d lattice M
              sage: Cone([(1,0),(0,1),(-1,-1)], lattice=N).dual().rays()  # whole space
              Empty collection
              in 2-d lattice M
  
          TESTS:
  
          The dual cone of a (random) dual cone is the original cone::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8, max_rays=10)
              sage: K.dual().dual() is K
              True
          """
          if "_dual" not in self.__dict__:
              rays = list(self.facet_normals())
              for ray in self.orthogonal_sublattice().gens():
                  rays.append(ray)
                  rays.append(-ray)
              self._dual = Cone(rays, lattice=self.dual_lattice(), check=False)
              self._dual._dual = self
          return self._dual
  
      def embed(self, cone):
          r"""
          Return the cone equivalent to the given one, but sitting in ``self`` as
          a face.
  
          You may need to use this method before calling methods of ``cone`` that
          depend on the ambient structure, such as
          :meth:`~sage.geometry.cone.ConvexRationalPolyhedralCone.ambient_ray_indices`
          or
          :meth:`~sage.geometry.cone.ConvexRationalPolyhedralCone.facet_of`. The
          cone returned by this method will have ``self`` as ambient. If ``cone``
          does not represent a valid cone of ``self``, ``ValueError`` exception
          is raised.
  
          .. NOTE::
  
              This method is very quick if ``self`` is already the ambient
              structure of ``cone``, so you can use without extra checks and
              performance hit even if ``cone`` is likely to sit in ``self`` but
              in principle may not.
  
          INPUT:
  
          - ``cone`` -- a :class:`cone
            <sage.geometry.cone.ConvexRationalPolyhedralCone>`.
  
          OUTPUT:
  
          - a :class:`cone <sage.geometry.cone.ConvexRationalPolyhedralCone>`,
            equivalent to ``cone`` but sitting inside ``self``.
  
          EXAMPLES:
  
          Let's take a 3-d cone on 4 rays::
  
              sage: c = Cone([(1,0,1), (0,1,1), (-1,0,1), (0,-1,1)])
  
          Then any ray generates a 1-d face of this cone, but if you construct
          such a face directly, it will not "sit" inside the cone::
  
              sage: ray = Cone([(0,-1,1)])
              sage: ray
              1-d cone in 3-d lattice N
              sage: ray.ambient_ray_indices()
              (0,)
              sage: ray.adjacent()
              ()
              sage: ray.ambient()
              1-d cone in 3-d lattice N
  
          If we want to operate with this ray as a face of the cone, we need to
          embed it first::
  
              sage: e_ray = c.embed(ray)
              sage: e_ray
              1-d face of 3-d cone in 3-d lattice N
              sage: e_ray.rays()
              N(0, -1, 1)
              in 3-d lattice N
              sage: e_ray is ray
              False
              sage: e_ray.is_equivalent(ray)
              True
              sage: e_ray.ambient_ray_indices()
              (3,)
              sage: e_ray.adjacent()
              (1-d face of 3-d cone in 3-d lattice N,
               1-d face of 3-d cone in 3-d lattice N)
              sage: e_ray.ambient()
              3-d cone in 3-d lattice N
  
          Not every cone can be embedded into a fixed ambient cone::
  
              sage: c.embed(Cone([(0,0,1)]))
              Traceback (most recent call last):
              ...
              ValueError: 1-d cone in 3-d lattice N is not a face
              of 3-d cone in 3-d lattice N!
              sage: c.embed(Cone([(1,0,1), (-1,0,1)]))
              Traceback (most recent call last):
              ...
              ValueError: 2-d cone in 3-d lattice N is not a face
              of 3-d cone in 3-d lattice N!
          """
          assert is_Cone(cone)
          if cone.ambient() is self:
              return cone
          if self.is_strictly_convex():
              rays = self.rays()
              try:
                  ray_indices = tuple(sorted(rays.index(ray)
                                             for ray in cone.rays()))
                  for face in self.faces(cone.dim()):
                      if face.ambient_ray_indices() == ray_indices:
                          return face
              except ValueError:
                  pass
          else:
              # We cannot use the trick with indices since rays are not unique.
              for face in self.faces(cone.dim()):
                  if cone.is_equivalent(face):
                      return face
          # If we are here, then either ValueError was raised or we went through
          # all faces and didn't find the matching one.
          raise ValueError("%s is not a face of %s!" % (cone, self))
  
      def face_lattice(self):
          r"""
          Return the face lattice of ``self``.
  
          This lattice will have the origin as the bottom (we do not include the
          empty set as a face) and this cone itself as the top.
  
          OUTPUT:
  
          - :class:`finite poset <sage.combinat.posets.posets.FinitePoset>` of
            :class:`cones <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES:
  
          Let's take a look at the face lattice of the first quadrant::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: L = quadrant.face_lattice()
              sage: L
              Finite lattice containing 4 elements with distinguished linear extension
  
          To see all faces arranged by dimension, you can do this::
  
              sage: for level in L.level_sets(): print(level)
              [0-d face of 2-d cone in 2-d lattice N]
              [1-d face of 2-d cone in 2-d lattice N,
               1-d face of 2-d cone in 2-d lattice N]
              [2-d cone in 2-d lattice N]
  
          For a particular face you can look at its actual rays... ::
  
              sage: face = L.level_sets()[1][0]
              sage: face.rays()
              N(1, 0)
              in 2-d lattice N
  
          ... or you can see the index of the ray of the original cone that
          corresponds to the above one::
  
              sage: face.ambient_ray_indices()
              (0,)
              sage: quadrant.ray(0)
              N(1, 0)
  
          An alternative to extracting faces from the face lattice is to use
          :meth:`faces` method::
  
              sage: face is quadrant.faces(dim=1)[0]
              True
  
          The advantage of working with the face lattice directly is that you
          can (relatively easily) get faces that are related to the given one::
  
              sage: face = L.level_sets()[1][0]
              sage: D = L.hasse_diagram()
              sage: D.neighbors(face)
              [2-d cone in 2-d lattice N,
               0-d face of 2-d cone in 2-d lattice N]
  
          However, you can achieve some of this functionality using
          :meth:`facets`, :meth:`facet_of`, and :meth:`adjacent` methods::
  
              sage: face = quadrant.faces(1)[0]
              sage: face
              1-d face of 2-d cone in 2-d lattice N
              sage: face.rays()
              N(1, 0)
              in 2-d lattice N
              sage: face.facets()
              (0-d face of 2-d cone in 2-d lattice N,)
              sage: face.facet_of()
              (2-d cone in 2-d lattice N,)
              sage: face.adjacent()
              (1-d face of 2-d cone in 2-d lattice N,)
              sage: face.adjacent()[0].rays()
              N(0, 1)
              in 2-d lattice N
  
          Note that if ``cone`` is a face of ``supercone``, then the face
          lattice of ``cone`` consists of (appropriate) faces of ``supercone``::
  
              sage: supercone = Cone([(1,2,3,4), (5,6,7,8),
              ....:                   (1,2,4,8), (1,3,9,7)])
              sage: supercone.face_lattice()
              Finite lattice containing 16 elements with distinguished linear extension
              sage: supercone.face_lattice().top()
              4-d cone in 4-d lattice N
              sage: cone = supercone.facets()[0]
              sage: cone
              3-d face of 4-d cone in 4-d lattice N
              sage: cone.face_lattice()
              Finite poset containing 8 elements with distinguished linear extension
              sage: cone.face_lattice().bottom()
              0-d face of 4-d cone in 4-d lattice N
              sage: cone.face_lattice().top()
              3-d face of 4-d cone in 4-d lattice N
              sage: cone.face_lattice().top() == cone
              True
  
          TESTS::
  
              sage: C1 = Cone([(0,1)])
              sage: C2 = Cone([(0,1)])
              sage: C1 == C2
              True
              sage: C1 is C2
              False
  
          C1 and C2 are equal, but not identical. We currently want them
          to have non identical face lattices, even if the faces
          themselves are equal (see :trac:`10998`)::
  
              sage: C1.face_lattice() is C2.face_lattice()
              False
  
              sage: C1.facets()[0]
              0-d face of 1-d cone in 2-d lattice N
              sage: C2.facets()[0]
              0-d face of 1-d cone in 2-d lattice N
  
              sage: C1.facets()[0].ambient() is C1
              True
              sage: C2.facets()[0].ambient() is C1
              False
              sage: C2.facets()[0].ambient() is C2
              True
          """
          if "_face_lattice" not in self.__dict__:
              if self._ambient is self:
                  # We need to compute face lattice on our own. To accommodate
                  # non-strictly convex cones we split rays (or rather their
                  # indices) into those in the linear subspace and others, which
                  # we refer to as atoms.
                  S = self.linear_subspace()
                  subspace_rays = []
                  atom_to_ray = []
                  for i, ray in enumerate(self):
                      # This try...except tests whether ray lies in S;
                      # "ray in S" does not work because ray lies in a
                      # toric lattice and S is a "plain" vector space,
                      # and there is only a conversion (no coercion)
                      # between them as of Trac ticket #10513.
                      try:
                          S(ray)
                          subspace_rays.append(i)
                      except (TypeError, ValueError):
                          atom_to_ray.append(i)
  
                  def ConeFace(atoms, facets):
                      if facets:
                          rays = sorted([atom_to_ray[a] for a in atoms]
                                        + subspace_rays)
                          face = ConvexRationalPolyhedralCone(
                                      ambient=self, ambient_ray_indices=rays)
                          # It may be nice if this functionality is exposed,
                          # however it makes sense only for cones which are
                          # thought of as faces of a single cone, not of a fan.
                          face._containing_cone_facets = facets
                          return face
                      else:
                          return self
  
                  # Obtain a modified version of the incidence matrix,
                  # with rows corresponding to rays in subspace removed.
                  mod_incidence_matrix = self.incidence_matrix()[atom_to_ray]
  
                  atom_to_facets = [row.nonzero_positions()
                                    for row in mod_incidence_matrix.rows()]
                  facet_to_atoms = [column.nonzero_positions()
                                    for column in mod_incidence_matrix.columns()]
  
                  self._face_lattice = lattice_from_incidences(
                                      atom_to_facets, facet_to_atoms, ConeFace,
                                      key = id(self))
              else:
                  # Get face lattice as a sublattice of the ambient one
                  allowed_indices = frozenset(self._ambient_ray_indices)
                  L = DiGraph()
                  origin = \
                      self._ambient._face_lattice_function().bottom()
                  L.add_vertex(0) # In case it is the only one
                  dfaces = [origin]
                  faces = [origin]
                  face_to_index = {origin:0}
                  next_index = 1
                  next_d = 1 # Dimension of faces to be considered next.
                  while next_d < self.dim():
                      ndfaces = []
                      for face in dfaces:
                          face_index = face_to_index[face]
                          for new_face in face.facet_of():
                              if not allowed_indices.issuperset(
                                              new_face._ambient_ray_indices):
                                  continue
                              if new_face in ndfaces:
                                  new_face_index = face_to_index[new_face]
                              else:
                                  ndfaces.append(new_face)
                                  face_to_index[new_face] = next_index
                                  new_face_index = next_index
                                  next_index += 1
                              L.add_edge(face_index, new_face_index)
                      faces.extend(ndfaces)
                      dfaces = ndfaces
                      next_d += 1
                  if self.dim() > 0:
                      # Last level is very easy to build, so we do it separately
                      # even though the above cycle could do it too.
                      faces.append(self)
                      for face in dfaces:
                          L.add_edge(face_to_index[face], next_index)
                  D = {i:f for i,f in enumerate(faces)}
                  L.relabel(D)
                  self._face_lattice = FinitePoset(L, faces, key = id(self))
          return self._face_lattice
  
      # Internally we use this name for a uniform behaviour of cones and fans.
      _face_lattice_function = face_lattice
  
      def faces(self, dim=None, codim=None):
          r"""
          Return faces of ``self`` of specified (co)dimension.
  
          INPUT:
  
          - ``dim`` -- integer, dimension of the requested faces;
  
          - ``codim`` -- integer, codimension of the requested faces.
  
          .. NOTE::
  
              You can specify at most one parameter. If you don't give any, then
              all faces will be returned.
  
          OUTPUT:
  
          - if either ``dim`` or ``codim`` is given, the output will be a
            :class:`tuple` of :class:`cones <ConvexRationalPolyhedralCone>`;
  
          - if neither ``dim`` nor ``codim`` is given, the output will be the
            :class:`tuple` of tuples as above, giving faces of all existing
            dimensions. If you care about inclusion relations between faces,
            consider using :meth:`face_lattice` or :meth:`adjacent`,
            :meth:`facet_of`, and :meth:`facets`.
  
          EXAMPLES:
  
          Let's take a look at the faces of the first quadrant::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant.faces()
              ((0-d face of 2-d cone in 2-d lattice N,),
               (1-d face of 2-d cone in 2-d lattice N,
                1-d face of 2-d cone in 2-d lattice N),
               (2-d cone in 2-d lattice N,))
              sage: quadrant.faces(dim=1)
              (1-d face of 2-d cone in 2-d lattice N,
               1-d face of 2-d cone in 2-d lattice N)
              sage: face = quadrant.faces(dim=1)[0]
  
          Now you can look at the actual rays of this face... ::
  
              sage: face.rays()
              N(1, 0)
              in 2-d lattice N
  
          ... or you can see indices of the rays of the original cone that
          correspond to the above ray::
  
              sage: face.ambient_ray_indices()
              (0,)
              sage: quadrant.ray(0)
              N(1, 0)
  
          Note that it is OK to ask for faces of too small or high dimension::
  
              sage: quadrant.faces(-1)
              ()
              sage: quadrant.faces(3)
              ()
  
          In the case of non-strictly convex cones even faces of small
          non-negative dimension may be missing::
  
              sage: halfplane = Cone([(1,0), (0,1), (-1,0)])
              sage: halfplane.faces(0)
              ()
              sage: halfplane.faces()
              ((1-d face of 2-d cone in 2-d lattice N,),
               (2-d cone in 2-d lattice N,))
              sage: plane = Cone([(1,0), (0,1), (-1,-1)])
              sage: plane.faces(1)
              ()
              sage: plane.faces()
              ((2-d cone in 2-d lattice N,),)
  
          TESTS:
  
          Now we check that "general" cones whose dimension is smaller than the
          dimension of the ambient space work as expected (see :trac:`9188`)::
  
              sage: c = Cone([(1,1,1,3),(1,-1,1,3),(-1,-1,1,3)])
              sage: c.faces()
              ((0-d face of 3-d cone in 4-d lattice N,),
               (1-d face of 3-d cone in 4-d lattice N,
                1-d face of 3-d cone in 4-d lattice N,
                1-d face of 3-d cone in 4-d lattice N),
               (2-d face of 3-d cone in 4-d lattice N,
                2-d face of 3-d cone in 4-d lattice N,
                2-d face of 3-d cone in 4-d lattice N),
               (3-d cone in 4-d lattice N,))
  
          We also ensure that a call to this function does not break
          :meth:`facets` method (see :trac:`9780`)::
  
              sage: cone = toric_varieties.dP8().fan().generating_cone(0)
              sage: cone
              2-d cone of Rational polyhedral fan in 2-d lattice N
              sage: for f in cone.facets(): print(f.rays())
              N(1, 1)
              in 2-d lattice N
              N(0, 1)
              in 2-d lattice N
              sage: len(cone.faces())
              3
              sage: for f in cone.facets(): print(f.rays())
              N(1, 1)
              in 2-d lattice N
              N(0, 1)
              in 2-d lattice N
          """
          if dim is not None and codim is not None:
              raise ValueError(
                      "dimension and codimension cannot be specified together!")
          dim = self.dim() - codim if codim is not None else dim
          if "_faces" not in self.__dict__:
              self._faces = tuple(map(self._sort_faces,
                                      self.face_lattice().level_sets()))
          if dim is None:
              return self._faces
          else:
              lsd = self.linear_subspace().dimension()
              return self._faces[dim - lsd] if lsd <= dim <= self.dim() else ()
  
      @cached_method
      def facet_normals(self):
          r"""
          Return inward normals to facets of ``self``.
  
          .. NOTE::
  
              #. For a not full-dimensional cone facet normals will specify
                 hyperplanes whose intersections with the space spanned by
                 ``self`` give facets of ``self``.
  
              #. For a not strictly convex cone facet normals will be orthogonal
                 to the linear subspace of ``self``, i.e. they always will be
                 elements of the dual cone of ``self``.
  
              #. The order of normals is random, but consistent with
                 :meth:`facets`.
  
          OUTPUT:
  
          - a :class:`~sage.geometry.point_collection.PointCollection`.
  
          If the ambient :meth:`~IntegralRayCollection.lattice` of ``self`` is a
          :class:`toric lattice
          <sage.geometry.toric_lattice.ToricLatticeFactory>`, the facet normals
          will be elements of the dual lattice. If it is a general lattice (like
          ``ZZ^n``) that does not have a ``dual()`` method, the facet normals
          will be returned as integral vectors.
  
          EXAMPLES::
  
              sage: cone = Cone([(1,0), (-1,3)])
              sage: cone.facet_normals()
              M(0, 1),
              M(3, 1)
              in 2-d lattice M
  
          Now let's look at a more complicated case::
  
              sage: cone = Cone([(-2,-1,2), (4,1,0), (-4,-1,-5), (4,1,5)])
              sage: cone.is_strictly_convex()
              False
              sage: cone.dim()
              3
              sage: cone.linear_subspace().dimension()
              1
              sage: lsg = (QQ^3)(cone.linear_subspace().gen(0)); lsg
              (1, 1/4, 5/4)
              sage: cone.facet_normals()
              M(7, -18, -2),
              M(1,  -4,  0)
              in 3-d lattice M
              sage: [lsg*normal for normal in cone.facet_normals()]
              [0, 0]
  
          A lattice that does not have a ``dual()`` method::
  
              sage: Cone([(1,1),(0,1)], lattice=ZZ^2).facet_normals()
              (-1, 1),
              ( 1, 0)
              in Ambient free module of rank 2
              over the principal ideal domain Integer Ring
  
          We correctly handle the degenerate cases::
  
              sage: N = ToricLattice(2)
              sage: Cone([], lattice=N).facet_normals()  # empty cone
              Empty collection
              in 2-d lattice M
              sage: Cone([(1,0)], lattice=N).facet_normals()  # ray in 2d
              M(1, 0)
              in 2-d lattice M
              sage: Cone([(1,0),(-1,0)], lattice=N).facet_normals()  # line in 2d
              Empty collection
              in 2-d lattice M
              sage: Cone([(1,0),(0,1)], lattice=N).facet_normals()  # strictly convex cone
              M(0, 1),
              M(1, 0)
              in 2-d lattice M
              sage: Cone([(1,0),(-1,0),(0,1)], lattice=N).facet_normals()  # half space
              M(0, 1)
              in 2-d lattice M
              sage: Cone([(1,0),(0,1),(-1,-1)], lattice=N).facet_normals()  # whole space
              Empty collection
              in 2-d lattice M
          """
          cone = self._PPL_cone()
          normals = []
          for c in cone.minimized_constraints():
              assert c.inhomogeneous_term() == 0
              if c.is_inequality():
                  normals.append(c.coefficients())
          M = self.dual_lattice()
          normals = tuple(map(M, normals))
          for n in normals:
              n.set_immutable()
          if len(normals) > 1:
              # Sort normals if they are rays
              if self.dim() == 2 and normals[0]*self.ray(0) != 0:
                  normals = (normals[1], normals[0])
              else:
                  try:    # or if we have combinatorial faces already
                      facets = self._faces[-2]
                      normals = set(normals)
                      sorted_normals = [None] * len(normals)
                      for i, f in enumerate(facets):
                          for n in normals:
                              if n*f.rays() == 0:
                                  sorted_normals[i] = n
                                  normals.remove(n)
                                  break
                      normals = tuple(sorted_normals)
                  except AttributeError:
                      pass
          return PointCollection(normals, M)
  
      @cached_method
      def facet_of(self):
          r"""
          Return *cones* of the ambient face lattice having ``self`` as a facet.
  
          OUTPUT:
  
          - :class:`tuple` of :class:`cones <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES::
  
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: octant.facet_of()
              ()
              sage: one_face = octant.faces(1)[0]
              sage: len(one_face.facet_of())
              2
              sage: one_face.facet_of()[1]
              2-d face of 3-d cone in 3-d lattice N
  
          While fan is the top element of its own cone lattice, which is a
          variant of a face lattice, we do not refer to cones as its facets::
  
              sage: fan = Fan([octant])
              sage: fan.generating_cone(0).facet_of()
              ()
  
          Subcones of generating cones work as before::
  
              sage: one_cone = fan(1)[0]
              sage: len(one_cone.facet_of())
              2
          """
          L = self._ambient._face_lattice_function()
          H = L.hasse_diagram()
          return self._sort_faces(
              f for f in H.neighbors_out(L(self)) if is_Cone(f))
  
      def facets(self):
          r"""
          Return facets (faces of codimension 1) of ``self``.
  
          OUTPUT:
  
          - :class:`tuple` of :class:`cones <ConvexRationalPolyhedralCone>`.
  
          EXAMPLES::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant.facets()
              (1-d face of 2-d cone in 2-d lattice N,
               1-d face of 2-d cone in 2-d lattice N)
          """
          return self.faces(codim=1)
  
      @cached_method
      def incidence_matrix(self):
          r"""
          Return the incidence matrix.
  
          .. NOTE::
  
             The columns correspond to facets/facet normals
             in the order of :meth:`facet_normals`, the rows
             correspond to the rays in the order of
             :meth:`rays`.
  
          EXAMPLES::
  
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: octant.incidence_matrix()
              [0 1 1]
              [1 0 1]
              [1 1 0]
  
              sage: halfspace = Cone([(1,0,0), (0,1,0), (-1,-1,0), (0,0,1)])
              sage: halfspace.incidence_matrix()
              [0]
              [1]
              [1]
              [1]
              [1]
  
          TESTS::
  
              sage: halfspace.incidence_matrix().is_immutable()
              True
  
          Check that the base ring is ``ZZ``, see :trac:`29840`::
  
              sage: halfspace.incidence_matrix().base_ring()
              Integer Ring
          """
          normals = self.facet_normals()
          incidence_matrix = matrix(ZZ, self.nrays(),
                                    len(normals), 0)
  
          for Hindex, normal in enumerate(self.facet_normals()):
              for Vindex, ray in enumerate(self.rays()):
                  if normal*ray == 0:
                      incidence_matrix[Vindex, Hindex] = 1
  
          incidence_matrix.set_immutable()
          return incidence_matrix
  
      def intersection(self, other):
          r"""
          Compute the intersection of two cones.
  
          INPUT:
  
          - ``other`` - :class:`cone <ConvexRationalPolyhedralCone>`.
  
          OUTPUT:
  
          - :class:`cone <ConvexRationalPolyhedralCone>`.
  
          Raises ``ValueError`` if the ambient space dimensions are not
          compatible.
  
          EXAMPLES::
  
              sage: cone1 = Cone([(1,0), (-1, 3)])
              sage: cone2 = Cone([(-1,0), (2, 5)])
              sage: cone1.intersection(cone2).rays()
              N(-1, 3),
              N( 2, 5)
              in 2-d lattice N
  
          It is OK to intersect cones living in sublattices of the same ambient
          lattice::
  
              sage: N = cone1.lattice()
              sage: Ns = N.submodule([(1,1)])
              sage: cone3 = Cone([(1,1)], lattice=Ns)
              sage: I = cone1.intersection(cone3)
              sage: I.rays()
              N(1, 1)
              in Sublattice <N(1, 1)>
              sage: I.lattice()
              Sublattice <N(1, 1)>
  
          But you cannot intersect cones from incompatible lattices without
          explicit conversion::
  
              sage: cone1.intersection(cone1.dual())
              Traceback (most recent call last):
              ...
              ValueError: 2-d lattice N and 2-d lattice M
              have different ambient lattices!
              sage: cone1.intersection(Cone(cone1.dual().rays(), N)).rays()
              N(3, 1),
              N(0, 1)
              in 2-d lattice N
          """
          if self._ambient is other._ambient:
              # Cones of the same ambient cone or fan intersect nicely/quickly.
              # Can we maybe even return an element of the cone lattice?..
              # But currently it can be done only for strictly convex cones.
              ambient_ray_indices = tuple(r for r in self._ambient_ray_indices
                                            if r in other._ambient_ray_indices)
              # type(self) allows this code to work nicely for derived classes,
              # although it forces all of them to accept such input
              return type(self)(ambient=self._ambient,
                                ambient_ray_indices=ambient_ray_indices)
          # Generic (slow) intersection, returning a generic cone.
          p = C_Polyhedron(self._PPL_cone())
          p.add_constraints(other._PPL_cone().constraints())
          return _Cone_from_PPL(p, self.lattice().intersection(other.lattice()))
  
      def is_equivalent(self, other):
          r"""
          Check if ``self`` is "mathematically" the same as ``other``.
  
          INPUT:
  
          - ``other`` - cone.
  
          OUTPUT:
  
          - ``True`` if ``self`` and ``other`` define the same cones as sets of
            points in the same lattice, ``False`` otherwise.
  
          There are three different equivalences between cones `C_1` and `C_2`
          in the same lattice:
  
          #. They have the same generating rays in the same order.
             This is tested by ``C1 == C2``.
          #. They describe the same sets of points.
             This is tested by ``C1.is_equivalent(C2)``.
          #. They are in the same orbit of `GL(n,\ZZ)` (and, therefore,
             correspond to isomorphic affine toric varieties).
             This is tested by ``C1.is_isomorphic(C2)``.
  
          EXAMPLES::
  
              sage: cone1 = Cone([(1,0), (-1, 3)])
              sage: cone2 = Cone([(-1,3), (1, 0)])
              sage: cone1.rays()
              N( 1, 0),
              N(-1, 3)
              in 2-d lattice N
              sage: cone2.rays()
              N(-1, 3),
              N( 1, 0)
              in 2-d lattice N
              sage: cone1 == cone2
              False
              sage: cone1.is_equivalent(cone2)
              True
  
          TESTS:
  
          A random cone is equivalent to itself::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8, max_rays=10)
              sage: K.is_equivalent(K)
              True
  
          """
          if self is other:
              return True
          # TODO: Next check is pointless if cones and fans are made to be unique
          if self.ambient() is other.ambient() and self.is_strictly_convex():
              return self.ambient_ray_indices() == other.ambient_ray_indices()
          if self.lattice() != other.lattice():
              return False
          return self._PPL_cone() == other._PPL_cone()
  
      def is_face_of(self, cone):
          r"""
          Check if ``self`` forms a face of another ``cone``.
  
          INPUT:
  
          - ``cone`` -- cone.
  
          OUTPUT:
  
          - ``True`` if ``self`` is a face of ``cone``, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: cone1 = Cone([(1,0)])
              sage: cone2 = Cone([(1,2)])
              sage: quadrant.is_face_of(quadrant)
              True
              sage: cone1.is_face_of(quadrant)
              True
              sage: cone2.is_face_of(quadrant)
              False
  
          Being a face means more than just saturating a facet
          inequality::
  
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: cone = Cone([(2,1,0),(1,2,0)])
              sage: cone.is_face_of(octant)
              False
  
          TESTS:
  
          Any cone is a face of itself::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8, max_rays=10)
              sage: K.is_face_of(K)
              True
  
          """
          if self.lattice() != cone.lattice():
              return False
          if self._ambient is cone._ambient:
              # Cones are always faces of their ambient structure, so
              return self.rays().set().issubset(cone.rays().set())
          if self.is_equivalent(cone):
              return True
          # Obviously False case
          if self.dim() >= cone.dim(): # if == and face, we return True above
              return False
  
          # It remains to test whether self is a proper face of cone:
          # 1) self must saturate at least one facet inequality
          saturates = Poly_Con_Relation.saturates()
          supporting_hyperplanes = Constraint_System()
          for c in cone._PPL_cone().minimized_constraints():
              rel = self._PPL_cone().relation_with(c)
              if c.is_equality() and not rel.implies(saturates):
                  return False
              if c.is_inequality() and rel.implies(saturates):
                  c_eq = (Linear_Expression(c.coefficients(),
                                            c.inhomogeneous_term()) == 0)
                  supporting_hyperplanes.insert(c_eq)
          if supporting_hyperplanes.empty():
              return False
          # 2) self must be a whole face, and not just a part of one
          cone_face = C_Polyhedron(cone._PPL_cone())
          cone_face.add_constraints(supporting_hyperplanes)
          return cone_face == self._PPL_cone()
  
      def is_isomorphic(self, other):
          r"""
          Check if ``self`` is in the same `GL(n, \ZZ)`-orbit as ``other``.
  
          INPUT:
  
          - ``other`` - cone.
  
          OUTPUT:
  
          - ``True`` if ``self`` and ``other`` are in the same
            `GL(n, \ZZ)`-orbit, ``False`` otherwise.
  
          There are three different equivalences between cones `C_1` and `C_2`
          in the same lattice:
  
          #. They have the same generating rays in the same order.
             This is tested by ``C1 == C2``.
          #. They describe the same sets of points.
             This is tested by ``C1.is_equivalent(C2)``.
          #. They are in the same orbit of `GL(n,\ZZ)` (and, therefore,
             correspond to isomorphic affine toric varieties).
             This is tested by ``C1.is_isomorphic(C2)``.
  
          EXAMPLES::
  
              sage: cone1 = Cone([(1,0), (0, 3)])
              sage: m = matrix(ZZ, [(1, -5), (-1, 4)]) # a GL(2,ZZ)-matrix
              sage: cone2 = Cone( m*r for r in cone1.rays() )
              sage: cone1.is_isomorphic(cone2)
              True
  
              sage: cone1 = Cone([(1,0), (0, 3)])
              sage: cone2 = Cone([(-1,3), (1, 0)])
              sage: cone1.is_isomorphic(cone2)
              False
  
          TESTS::
  
              sage: from sage.geometry.cone import classify_cone_2d
              sage: classify_cone_2d(*cone1.rays())
              (1, 0)
              sage: classify_cone_2d(*cone2.rays())
              (3, 2)
  
          We check that :trac:`18613` is fixed::
  
              sage: K = cones.trivial(0)
              sage: K.is_isomorphic(K)
              True
              sage: K = cones.trivial(1)
              sage: K.is_isomorphic(K)
              True
              sage: K = cones.trivial(2)
              sage: K.is_isomorphic(K)
              True
  
          A random (strictly convex) cone is isomorphic to itself::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6, strictly_convex=True)
              sage: K.is_isomorphic(K)
              True
          """
          if self.is_strictly_convex() and other.is_strictly_convex():
              from sage.geometry.fan import Fan
              return Fan([self]).is_isomorphic(Fan([other]))
          if self.is_strictly_convex() ^ other.is_strictly_convex():
              return False
          raise NotImplementedError("isomorphism check for not strictly convex "
                                    "cones is not implemented")
  
      def is_simplicial(self):
          r"""
          Check if ``self`` is simplicial.
  
          A cone is called **simplicial** if primitive vectors along its
          generating rays form a part of a *rational* basis of the ambient
          space.
  
          OUTPUT:
  
          - ``True`` if ``self`` is simplicial, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: cone1 = Cone([(1,0), (0, 3)])
              sage: cone2 = Cone([(1,0), (0, 3), (-1,-1)])
              sage: cone1.is_simplicial()
              True
              sage: cone2.is_simplicial()
              False
          """
          return self.nrays() == self.dim()
  
      @cached_method
      def is_smooth(self):
          r"""
          Check if ``self`` is smooth.
  
          A cone is called **smooth** if primitive vectors along its
          generating rays form a part of an *integral* basis of the
          ambient space. Equivalently, they generate the whole lattice
          on the linear subspace spanned by the rays.
  
          OUTPUT:
  
          - ``True`` if ``self`` is smooth, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: cone1 = Cone([(1,0), (0, 1)])
              sage: cone2 = Cone([(1,0), (-1, 3)])
              sage: cone1.is_smooth()
              True
              sage: cone2.is_smooth()
              False
  
          The following cones are the same up to a `SL(2,\ZZ)`
          coordinate transformation::
  
              sage: Cone([(1,0,0), (2,1,-1)]).is_smooth()
              True
              sage: Cone([(1,0,0), (2,1,1)]).is_smooth()
              True
              sage: Cone([(1,0,0), (2,1,2)]).is_smooth()
              True
          """
          if not self.is_simplicial():
              return False
          return self.rays().matrix().elementary_divisors() == [1] * self.nrays()
  
      def is_trivial(self):
          """
          Checks if the cone has no rays.
  
          OUTPUT:
  
          - ``True`` if the cone has no rays, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: c0 = cones.trivial(3)
              sage: c0.is_trivial()
              True
              sage: c0.nrays()
              0
          """
          return self.nrays() == 0
  
      def is_strictly_convex(self):
          r"""
          Check if ``self`` is strictly convex.
  
          A cone is called **strictly convex** if it does not contain any lines.
  
          OUTPUT:
  
          - ``True`` if ``self`` is strictly convex, ``False`` otherwise.
  
          EXAMPLES::
  
              sage: cone1 = Cone([(1,0), (0, 1)])
              sage: cone2 = Cone([(1,0), (-1, 0)])
              sage: cone1.is_strictly_convex()
              True
              sage: cone2.is_strictly_convex()
              False
          """
          if "_is_strictly_convex" not in self.__dict__:
              convex = True
              for gs in self._PPL_cone().minimized_generators():
                  if gs.is_line():
                      convex = False
                      break
              self._is_strictly_convex = convex
          return self._is_strictly_convex
  
      @cached_method
      def linear_subspace(self):
          r"""
          Return the largest linear subspace contained inside of ``self``.
  
          OUTPUT:
  
          - subspace of the ambient space of ``self``.
  
          EXAMPLES::
  
              sage: halfplane = Cone([(1,0), (0,1), (-1,0)])
              sage: halfplane.linear_subspace()
              Vector space of degree 2 and dimension 1 over Rational Field
              Basis matrix:
              [1 0]
  
          TESTS:
  
          The linear subspace of any closed convex cone can be identified
          with the orthogonal complement of the span of its dual::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8)
              sage: expected = K.dual().span().vector_space().complement()
              sage: K.linear_subspace() == expected
              True
          """
          if self.is_strictly_convex():
              return span([vector(QQ, self.lattice_dim())], QQ)
          return span(self.lines(), QQ)
  
      @cached_method
      def lines(self):
          r"""
          Return lines generating the linear subspace of ``self``.
  
          OUTPUT:
  
          - :class:`tuple` of primitive vectors in the lattice of ``self``
            giving directions of lines that span the linear subspace of
            ``self``. These lines are arbitrary, but fixed. If you do not care
            about the order, see also :meth:`line_set`.
  
          EXAMPLES::
  
              sage: halfplane = Cone([(1,0), (0,1), (-1,0)])
              sage: halfplane.lines()
              N(1, 0)
              in 2-d lattice N
              sage: fullplane = Cone([(1,0), (0,1), (-1,-1)])
              sage: fullplane.lines()
              N(0, 1),
              N(1, 0)
              in 2-d lattice N
          """
          lines = []
          for g in self._PPL_cone().minimized_generators():
              if g.is_line():
                  lines.append(g.coefficients())
          N = self.lattice()
          lines = tuple(map(N, lines))
          for l in lines:
              l.set_immutable()
          return PointCollection(lines, N)
  
      def plot(self, **options):
          r"""
          Plot ``self``.
  
          INPUT:
  
          - any options for toric plots (see :func:`toric_plotter.options
            <sage.geometry.toric_plotter.options>`), none are mandatory.
  
          OUTPUT:
  
          - a plot.
  
          EXAMPLES::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant.plot()
              Graphics object consisting of 9 graphics primitives
          """
          # What to do with 3-d cones in 5-d? Use some projection method?
          deg = self.lattice().degree()
          tp = ToricPlotter(options, deg, self.rays())
          # Modify ray labels to match the ambient cone or fan.
          tp.ray_label = label_list(tp.ray_label, self.nrays(), deg <= 2,
                                     self.ambient_ray_indices())
          result = tp.plot_lattice() + tp.plot_generators()
          # To deal with non-strictly convex cones we separate rays and labels.
          result += tp.plot_ray_labels()
          tp.ray_label = None
          lsd = self.linear_subspace().dimension()
          if lsd == 1:
              # Plot only rays of the line
              v = self.lines()[0]
              tp.set_rays([v, -v])
          if lsd <= 1:
              result += tp.plot_rays()
          # Modify wall labels to match the ambient cone or fan too.
          walls = self.faces(2)
          try:
              ambient_walls = self.ambient().faces(2)
          except AttributeError:
              ambient_walls = self.ambient().cones(2)
          tp.wall_label = label_list(tp.wall_label, len(walls), deg <= 2,
                              [ambient_walls.index(wall) for wall in walls])
          tp.set_rays(self.ambient().rays())
          result += tp.plot_walls(walls)
          return result
  
      def polyhedron(self):
          r"""
          Return the polyhedron associated to ``self``.
  
          Mathematically this polyhedron is the same as ``self``.
  
          OUTPUT:
  
          - :class:`~sage.geometry.polyhedron.base.Polyhedron_base`.
  
          EXAMPLES::
  
              sage: quadrant = Cone([(1,0), (0,1)])
              sage: quadrant.polyhedron()
              A 2-dimensional polyhedron in ZZ^2 defined as the convex hull
              of 1 vertex and 2 rays
              sage: line = Cone([(1,0), (-1,0)])
              sage: line.polyhedron()
              A 1-dimensional polyhedron in ZZ^2 defined as the convex hull
              of 1 vertex and 1 line
  
          Here is an example of a trivial cone (see :trac:`10237`)::
  
              sage: origin = Cone([], lattice=ZZ^2)
              sage: origin.polyhedron()
              A 0-dimensional polyhedron in ZZ^2 defined as the convex hull
              of 1 vertex
          """
          return Polyhedron(rays=self.rays(), vertices=[self.lattice()(0)])
  
      @cached_method
      def strict_quotient(self):
          r"""
          Return the quotient of ``self`` by the linear subspace.
  
          We define the **strict quotient** of a cone to be the image of this
          cone in the quotient of the ambient space by the linear subspace of
          the cone, i.e. it is the "complementary part" to the linear subspace.
  
          OUTPUT:
  
          - cone.
  
          EXAMPLES::
  
              sage: halfplane = Cone([(1,0), (0,1), (-1,0)])
              sage: ssc = halfplane.strict_quotient()
              sage: ssc
              1-d cone in 1-d lattice N
              sage: ssc.rays()
              N(1)
              in 1-d lattice N
              sage: line = Cone([(1,0), (-1,0)])
              sage: ssc = line.strict_quotient()
              sage: ssc
              0-d cone in 1-d lattice N
              sage: ssc.rays()
              Empty collection
              in 1-d lattice N
  
          The quotient of the trivial cone is trivial::
  
              sage: K = cones.trivial(0)
              sage: K.strict_quotient()
              0-d cone in 0-d lattice N
              sage: K = Cone([(0,0,0,0)])
              sage: K.strict_quotient()
              0-d cone in 4-d lattice N
  
          TESTS:
  
          The strict quotient of any cone should be strictly convex::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K.strict_quotient().is_strictly_convex()
              True
  
          If the original cone is solid, then its strict quotient is proper::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6, solid=True)
              sage: K.strict_quotient().is_proper()
              True
  
          The strict quotient of a strictly convex cone is itself::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6, strictly_convex=True)
              sage: K.strict_quotient() is K
              True
  
          The complement of our linear subspace has the same dimension as
          our dual, so the strict quotient cannot have a larger dimension
          than our dual::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K.strict_quotient().dim() <= K.dual().dim()
              True
  
          The strict quotient is idempotent::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K1 = K.strict_quotient()
              sage: K2 = K1.strict_quotient()
              sage: K1 is K2
              True
          """
          if self.is_strictly_convex():
              return self
          L = self.lattice()
          Q = L.base_extend(QQ) / self.linear_subspace()
          # Maybe we can improve this one if we create something special
          # for sublattices. But it seems to be the most natural choice
          # for names. If many subcones land in the same lattice -
          # that's just how it goes.
          if is_ToricLattice(L):
              S = ToricLattice(Q.dimension(), L._name, L._dual_name,
                               L._latex_name, L._latex_dual_name)
          else:
              S = ZZ**Q.dimension()
          rays = ( Q(ray) for ray in self if not Q(ray).is_zero() )
          quotient = Cone(rays, S, check=False)
          quotient._is_strictly_convex = True
          return quotient
  
      @cached_method
      def solid_restriction(self):
          r"""
          Return a solid representation of this cone in terms of a basis
          of its :meth:`sublattice`.
  
          We define the **solid restriction** of a cone to be a
          representation of that cone in a basis of its own
          sublattice. Since a cone's sublattice is just large enough to
          hold the cone (by definition), the resulting solid restriction
          :meth:`is_solid`. For convenience, the solid restriction lives
          in a new lattice (of the appropriate dimension) and not actually
          in the sublattice object returned by :meth:`sublattice`.
  
          OUTPUT:
  
          A solid cone in a new lattice having the same dimension as this
          cone's :meth:`sublattice`.
  
          EXAMPLES:
  
          The nonnegative quadrant in the plane is left after we take its
          solid restriction in space::
  
              sage: K = Cone([(1,0,0), (0,1,0)])
              sage: K.solid_restriction().rays()
              N(1, 0),
              N(0, 1)
              in 2-d lattice N
  
          The solid restriction of a single ray has the same
          representation regardless of the ambient space::
  
              sage: K = Cone([(1,0)])
              sage: K.solid_restriction().rays()
              N(1)
              in 1-d lattice N
              sage: K = Cone([(1,1,1)])
              sage: K.solid_restriction().rays()
              N(1)
              in 1-d lattice N
  
          The solid restriction of the trivial cone lives in a trivial space::
  
              sage: K = cones.trivial(0)
              sage: K.solid_restriction()
              0-d cone in 0-d lattice N
              sage: K = cones.trivial(4)
              sage: K.solid_restriction()
              0-d cone in 0-d lattice N
  
          The solid restriction of a solid cone is itself::
  
              sage: K = Cone([(1,1),(1,2)])
              sage: K.solid_restriction() is K
              True
  
          TESTS:
  
          The solid restriction of any cone is solid::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K.solid_restriction().is_solid()
              True
  
          If a cone :meth:`is_strictly_convex`, then its solid restriction
          :meth:`is_proper`::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6, strictly_convex=True)
              sage: K.solid_restriction().is_proper()
              True
  
          The solid restriction of a cone has the same dimension as the
          original::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K.solid_restriction().dim() == K.dim()
              True
  
          The solid restriction of a cone has the same number of rays as
          the original::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K.solid_restriction().nrays() == K.nrays()
              True
  
          The solid restriction of a cone has the same lineality as the
          original::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: K.solid_restriction().lineality() == K.lineality()
              True
  
          The solid restriction of a cone has the same number of facets as
          the original::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: len(K.solid_restriction().facets()) == len(K.facets())
              True
          """
          if self.is_solid():
              return self
          # Construct a NEW lattice ``S`` (of the appropriate dimension)
          # to use. This works around the fact that it's difficult to
          # work with sublattice objects. There are naming issues here
          # similar to those in the strict_quotient() method.
          L = self.lattice()
          subL = self.sublattice()
          S = ToricLattice(subL.dimension(), L._name,
                           L._dual_name, L._latex_name, L._latex_dual_name)
  
          # We don't need to check if these rays are zero: they will all
          # have at least one non-zero coordinate; otherwise they would
          # lie outside of the span of our cone. And they don't, because
          # they generate the cone.
          rays = ( S(subL.coordinates(ray)) for ray in self )
          return Cone(rays, lattice=S, check=False)
  
      def _split_ambient_lattice(self):
          r"""
          Compute a decomposition of the ``N``-lattice into `N_\sigma`
          and its complement isomorphic to `N(\sigma)`.
  
          You should not call this function directly, but call
          :meth:`sublattice` and :meth:`sublattice_complement` instead.
  
          EXAMPLES::
  
              sage: c = Cone([ (1,2) ])
              sage: c._split_ambient_lattice()
              sage: c._sublattice
              Sublattice <N(1, 2)>
              sage: c._sublattice_complement
              Sublattice <N(0, 1)>
  
          Degenerate cases::
  
              sage: C2_Z2 = Cone([(1,0),(1,2)])
              sage: C2_Z2._split_ambient_lattice()
              sage: C2_Z2._sublattice
              Sublattice <N(1, 2), N(0, -1)>
  
          Trivial cone::
  
              sage: trivial_cone = cones.trivial(3)
              sage: trivial_cone._split_ambient_lattice()
              sage: trivial_cone._sublattice
              Sublattice <>
              sage: trivial_cone._sublattice_complement
              Sublattice <N(1, 0, 0), N(0, 1, 0), N(0, 0, 1)>
          """
          N = self.lattice()
          n = N.dimension()
          basis = self.rays().basis()
          r = len(basis)
          Nsigma = matrix(ZZ, r, n, ( N.coordinates(v) for v in basis ))
          D, U, V = Nsigma.smith_form()  # D = U*N*V <=> N = Uinv*D*Vinv
          basis = (V.inverse() * N.basis_matrix()).rows()
          # spanned lattice N_sigma
          self._sublattice = N.submodule_with_basis(basis[:r])
          # complement to the spanned lattice, isomorphic to N(sigma)
          self._sublattice_complement = N.submodule_with_basis(basis[r:])
  
      def sublattice(self, *args, **kwds):
          r"""
          The sublattice spanned by the cone.
  
          Let `\sigma` be the given cone and `N=` ``self.lattice()`` the
          ambient lattice. Then, in the notation of [Ful1993]_, this
          method returns the sublattice
  
          .. MATH::
  
              N_\sigma \stackrel{\text{def}}{=} \mathop{span}( N\cap \sigma )
  
          INPUT:
  
          - either nothing or something that can be turned into an element of
            this lattice.
  
          OUTPUT:
  
          - if no arguments were given, a :class:`toric sublattice
            <sage.geometry.toric_lattice.ToricLattice_sublattice_with_basis>`,
            otherwise the corresponding element of it.
  
          .. NOTE::
  
              * The sublattice spanned by the cone is the saturation of
                the sublattice generated by the rays of the cone.
  
              * If you only need a `\QQ`-basis, you may want to try the
                :meth:`~sage.geometry.point_collection.PointCollection.basis`
                method on the result of :meth:`~IntegralRayCollection.rays`.
  
              * The returned lattice points are usually not rays of the
                cone. In fact, for a non-smooth cone the rays do not
                generate the sublattice `N_\sigma`, but only a finite
                index sublattice.
  
          EXAMPLES::
  
              sage: cone = Cone([(1, 1, 1), (1, -1, 1), (-1, -1, 1), (-1, 1, 1)])
              sage: cone.rays().basis()
              N( 1,  1, 1),
              N( 1, -1, 1),
              N(-1, -1, 1)
              in 3-d lattice N
              sage: cone.rays().basis().matrix().det()
              -4
              sage: cone.sublattice()
              Sublattice <N(-1, -1, 1), N(1, 0, 0), N(1, 1, 0)>
              sage: matrix( cone.sublattice().gens() ).det()
              1
  
          Another example::
  
              sage: c = Cone([(1,2,3), (4,-5,1)])
              sage: c
              2-d cone in 3-d lattice N
              sage: c.rays()
              N(1,  2, 3),
              N(4, -5, 1)
              in 3-d lattice N
              sage: c.sublattice()
              Sublattice <N(1, 2, 3), N(4, -5, 1)>
              sage: c.sublattice(5, -3, 4)
              N(5, -3, 4)
              sage: c.sublattice(1, 0, 0)
              Traceback (most recent call last):
              ...
              TypeError: element [1, 0, 0] is not in free module
          """
          if "_sublattice" not in self.__dict__:
              self._split_ambient_lattice()
          if args or kwds:
              return self._sublattice(*args, **kwds)
          else:
              return self._sublattice
  
      def sublattice_quotient(self, *args, **kwds):
          r"""
          The quotient of the ambient lattice by the sublattice spanned
          by the cone.
  
          INPUT:
  
          - either nothing or something that can be turned into an element of
            this lattice.
  
          OUTPUT:
  
          - if no arguments were given, a :class:`quotient of a toric lattice
            <sage.geometry.toric_lattice.ToricLattice_quotient>`,
            otherwise the corresponding element of it.
  
          EXAMPLES::
  
              sage: C2_Z2 = Cone([(1,0),(1,2)])     # C^2/Z_2
              sage: c1, c2 = C2_Z2.facets()
              sage: c2.sublattice_quotient()
              1-d lattice, quotient of 2-d lattice N by Sublattice <N(1, 2)>
              sage: N = C2_Z2.lattice()
              sage: n = N(1,1)
              sage: n_bar = c2.sublattice_quotient(n); n_bar
              N[1, 1]
              sage: n_bar.lift()
              N(1, 1)
              sage: vector(n_bar)
              (-1)
          """
          if "_sublattice_quotient" not in self.__dict__:
              self._sublattice_quotient = self.lattice() / self.sublattice()
          if args or kwds:
              return self._sublattice_quotient(*args, **kwds)
          else:
              return self._sublattice_quotient
  
      def sublattice_complement(self, *args, **kwds):
          r"""
          A complement of the sublattice spanned by the cone.
  
          In other words, :meth:`sublattice` and
          :meth:`sublattice_complement` together form a
          `\ZZ`-basis for the ambient :meth:`lattice()
          <sage.geometry.cone.IntegralRayCollection.lattice>`.
  
          In the notation of [Ful1993]_, let `\sigma` be the given cone
          and `N=` ``self.lattice()`` the ambient lattice. Then this
          method returns
  
          .. MATH::
  
              N(\sigma) \stackrel{\text{def}}{=} N / N_\sigma
  
          lifted (non-canonically) to a sublattice of `N`.
  
          INPUT:
  
          - either nothing or something that can be turned into an element of
            this lattice.
  
          OUTPUT:
  
          - if no arguments were given, a :class:`toric sublattice
            <sage.geometry.toric_lattice.ToricLattice_sublattice_with_basis>`,
            otherwise the corresponding element of it.
  
          EXAMPLES::
  
              sage: C2_Z2 = Cone([(1,0),(1,2)])     # C^2/Z_2
              sage: c1, c2 = C2_Z2.facets()
              sage: c2.sublattice()
              Sublattice <N(1, 2)>
              sage: c2.sublattice_complement()
              Sublattice <N(0, 1)>
  
          A more complicated example::
  
              sage: c = Cone([(1,2,3), (4,-5,1)])
              sage: c.sublattice()
              Sublattice <N(1, 2, 3), N(4, -5, 1)>
              sage: c.sublattice_complement()
              Sublattice <N(0, -6, -5)>
              sage: m = matrix( c.sublattice().gens() + c.sublattice_complement().gens() )
              sage: m
              [ 1  2  3]
              [ 4 -5  1]
              [ 0 -6 -5]
              sage: m.det()
              -1
              """
          if "_sublattice_complement" not in self.__dict__:
              self._split_ambient_lattice()
          if args or kwds:
              return self._sublattice_complement(*args, **kwds)
          else:
              return self._sublattice_complement
  
      def orthogonal_sublattice(self, *args, **kwds):
          r"""
          The sublattice (in the dual lattice) orthogonal to the
          sublattice spanned by the cone.
  
          Let `M=` ``self.dual_lattice()`` be the lattice dual to the
          ambient lattice of the given cone `\sigma`. Then, in the
          notation of [Ful1993]_, this method returns the sublattice
  
          .. MATH::
  
              M(\sigma) \stackrel{\text{def}}{=}
              \sigma^\perp \cap M
              \subset M
  
          INPUT:
  
          - either nothing or something that can be turned into an element of
            this lattice.
  
          OUTPUT:
  
          - if no arguments were given, a :class:`toric sublattice
            <sage.geometry.toric_lattice.ToricLattice_sublattice_with_basis>`,
            otherwise the corresponding element of it.
  
          EXAMPLES::
  
              sage: c = Cone([(1,1,1), (1,-1,1), (-1,-1,1), (-1,1,1)])
              sage: c.orthogonal_sublattice()
              Sublattice <>
              sage: c12 = Cone([(1,1,1), (1,-1,1)])
              sage: c12.sublattice()
              Sublattice <N(1, -1, 1), N(0, 1, 0)>
              sage: c12.orthogonal_sublattice()
              Sublattice <M(1, 0, -1)>
  
          TESTS:
  
          We check that :trac:`24541` is fixed::
  
              sage: c = Cone([(1,0)], lattice=ZZ^2)
              sage: c.orthogonal_sublattice()
              Free module of degree 2 and rank 1 over Integer Ring
              User basis matrix:
              [0 1]
              sage: c.dual()
              2-d cone in 2-d lattice
          """
          if "_orthogonal_sublattice" not in self.__dict__:
              try:
                  self._orthogonal_sublattice = self.sublattice_quotient().dual()
              except AttributeError:
                  N = self.lattice()
                  basis = self.rays().basis()
                  Nsigma = column_matrix(ZZ, (N.coordinates(v) for v in basis))
                  D, U, V = Nsigma.smith_form()  # D = U * Nsigma * V
                  M = self.dual_lattice()
                  self._orthogonal_sublattice = M.submodule_with_basis(
                      U.rows()[len(basis):])
          if args or kwds:
              return self._orthogonal_sublattice(*args, **kwds)
          else:
              return self._orthogonal_sublattice
  
      def relative_quotient(self, subcone):
          r"""
          The quotient of the spanned lattice by the lattice spanned by
          a subcone.
  
          In the notation of [Ful1993]_, let `N` be the ambient lattice
          and `N_\sigma` the sublattice spanned by the given cone
          `\sigma`. If `\rho < \sigma` is a subcone, then `N_\rho` =
          ``rho.sublattice()`` is a saturated sublattice of `N_\sigma` =
          ``self.sublattice()``. This method returns the quotient
          lattice. The lifts of the quotient generators are
          `\dim(\sigma)-\dim(\rho)` linearly independent primitive
          lattice points that, together with `N_\rho`, generate
          `N_\sigma`.
  
          OUTPUT:
  
          - :class:`toric lattice quotient
            <sage.geometry.toric_lattice.ToricLattice_quotient>`.
  
          .. NOTE::
  
              * The quotient `N_\sigma / N_\rho` of spanned sublattices
                has no torsion since the sublattice `N_\rho` is saturated.
  
              * In the codimension one case, the generator of
                `N_\sigma / N_\rho` is chosen to be in the same direction as the
                image `\sigma / N_\rho`
  
          EXAMPLES::
  
              sage: sigma = Cone([(1,1,1,3),(1,-1,1,3),(-1,-1,1,3),(-1,1,1,3)])
              sage: rho   = Cone([(-1, -1, 1, 3), (-1, 1, 1, 3)])
              sage: sigma.sublattice()
              Sublattice <N(-1, -1, 1, 3), N(1, 0, 0, 0), N(1, 1, 0, 0)>
              sage: rho.sublattice()
              Sublattice <N(-1, 1, 1, 3), N(0, -1, 0, 0)>
              sage: sigma.relative_quotient(rho)
              1-d lattice, quotient
              of Sublattice <N(-1, -1, 1, 3), N(1, 0, 0, 0), N(1, 1, 0, 0)>
              by Sublattice <N(1, 0, -1, -3), N(0, 1, 0, 0)>
              sage: sigma.relative_quotient(rho).gens()
              (N[1, 1, 0, 0],)
  
          More complicated example::
  
              sage: rho = Cone([(1, 2, 3), (1, -1, 1)])
              sage: sigma = Cone([(1, 2, 3), (1, -1, 1), (-1, 1, 1), (-1, -1, 1)])
              sage: N_sigma = sigma.sublattice()
              sage: N_sigma
              Sublattice <N(-1, 1, 1), N(1, 2, 3), N(0, 1, 1)>
              sage: N_rho = rho.sublattice()
              sage: N_rho
              Sublattice <N(1, -1, 1), N(1, 2, 3)>
              sage: sigma.relative_quotient(rho).gens()
              (N[0, 1, 1],)
              sage: N = rho.lattice()
              sage: N_sigma == N.span(N_rho.gens() + tuple(q.lift()
              ....:            for q in sigma.relative_quotient(rho).gens()))
              True
  
          Sign choice in the codimension one case::
  
              sage: sigma1 = Cone([(1, 2, 3), (1, -1, 1), (-1, 1, 1), (-1, -1, 1)])  # 3d
              sage: sigma2 = Cone([(1, 1, -1), (1, 2, 3), (1, -1, 1), (1, -1, -1)])  # 3d
              sage: rho = sigma1.intersection(sigma2)
              sage: rho.sublattice()
              Sublattice <N(1, -1, 1), N(1, 2, 3)>
              sage: sigma1.relative_quotient(rho)
              1-d lattice, quotient
              of Sublattice <N(-1, 1, 1), N(1, 2, 3), N(0, 1, 1)>
              by Sublattice <N(1, 2, 3), N(0, 3, 2)>
              sage: sigma1.relative_quotient(rho).gens()
              (N[0, 1, 1],)
              sage: sigma2.relative_quotient(rho).gens()
              (N[-1, 0, -2],)
          """
          try:
              cached_values = self._relative_quotient
          except AttributeError:
              self._relative_quotient = {}
              cached_values = self._relative_quotient
  
          try:
              return cached_values[subcone]
          except KeyError:
              pass
  
          Ncone = self.sublattice()
          Nsubcone = subcone.sublattice()
  
          extra_ray = None
          if Ncone.dimension()-Nsubcone.dimension()==1:
              extra_ray = set(self.rays().set() - subcone.rays().set()).pop()
  
          Q = Ncone.quotient(Nsubcone, positive_point=extra_ray)
          assert Q.is_torsion_free()
          cached_values[subcone] = Q
          return Q
  
      def relative_orthogonal_quotient(self, supercone):
          r"""
          The quotient of the dual spanned lattice by the dual of the
          supercone's spanned lattice.
  
          In the notation of [Ful1993]_, if ``supercone`` = `\rho >
          \sigma` = ``self`` is a cone that contains `\sigma` as a face,
          then `M(\rho)` = ``supercone.orthogonal_sublattice()`` is a
          saturated sublattice of `M(\sigma)` =
          ``self.orthogonal_sublattice()``. This method returns the
          quotient lattice. The lifts of the quotient generators are
          `\dim(\rho)-\dim(\sigma)` linearly independent M-lattice
          lattice points that, together with `M(\rho)`, generate
          `M(\sigma)`.
  
          OUTPUT:
  
          - :class:`toric lattice quotient
            <sage.geometry.toric_lattice.ToricLattice_quotient>`.
  
          If we call the output ``Mrho``, then
  
              - ``Mrho.cover() == self.orthogonal_sublattice()``, and
  
              - ``Mrho.relations() == supercone.orthogonal_sublattice()``.
  
          .. NOTE::
  
              * `M(\sigma) / M(\rho)` has no torsion since the sublattice
                `M(\rho)` is saturated.
  
              * In the codimension one case, (a lift of) the generator of
                `M(\sigma) / M(\rho)` is chosen to be positive on `\sigma`.
  
          EXAMPLES::
  
              sage: rho = Cone([(1,1,1,3),(1,-1,1,3),(-1,-1,1,3),(-1,1,1,3)])
              sage: rho.orthogonal_sublattice()
              Sublattice <M(0, 0, 3, -1)>
              sage: sigma = rho.facets()[1]
              sage: sigma.orthogonal_sublattice()
              Sublattice <M(0, 1, 1, 0), M(0, 0, 3, -1)>
              sage: sigma.is_face_of(rho)
              True
              sage: Q = sigma.relative_orthogonal_quotient(rho); Q
              1-d lattice, quotient
              of Sublattice <M(0, 1, 1, 0), M(0, 0, 3, -1)>
              by Sublattice <M(0, 0, 3, -1)>
              sage: Q.gens()
              (M[0, 1, 1, 0],)
  
          Different codimension::
  
              sage: rho = Cone([[1,-1,1,3],[-1,-1,1,3]])
              sage: sigma = rho.facets()[0]
              sage: sigma.orthogonal_sublattice()
              Sublattice <M(1, 0, 2, -1), M(0, 1, 1, 0), M(0, 0, 3, -1)>
              sage: rho.orthogonal_sublattice()
              Sublattice <M(0, 1, 1, 0), M(0, 0, 3, -1)>
              sage: sigma.relative_orthogonal_quotient(rho).gens()
              (M[-1, 0, -2, 1],)
  
          Sign choice in the codimension one case::
  
              sage: sigma1 = Cone([(1, 2, 3), (1, -1, 1), (-1, 1, 1), (-1, -1, 1)])  # 3d
              sage: sigma2 = Cone([(1, 1, -1), (1, 2, 3), (1, -1, 1), (1, -1, -1)])  # 3d
              sage: rho = sigma1.intersection(sigma2)
              sage: rho.relative_orthogonal_quotient(sigma1).gens()
              (M[-5, -2, 3],)
              sage: rho.relative_orthogonal_quotient(sigma2).gens()
              (M[5, 2, -3],)
          """
          try:
              cached_values = self._relative_orthogonal_quotient
          except AttributeError:
              self._relative_orthogonal_quotient = {}
              cached_values = self._relative_orthogonal_quotient
  
          try:
              return cached_values[supercone]
          except KeyError:
              pass
  
          Mcone = self.orthogonal_sublattice()
          Msupercone = supercone.orthogonal_sublattice()
  
          extra_ray = None
          if Mcone.dimension()-Msupercone.dimension()==1:
              extra_ray = set(supercone.rays().set() - self.rays().set()).pop()
  
          Q = Mcone.quotient(Msupercone, positive_dual_point=extra_ray)
          assert Q.is_torsion_free()
          cached_values[supercone] = Q
          return Q
  
      def semigroup_generators(self):
          r"""
          Return generators for the semigroup of lattice points of ``self``.
  
          OUTPUT:
  
          - a :class:`~sage.geometry.point_collection.PointCollection`
            of lattice points generating the semigroup of lattice points
            contained in ``self``.
  
          .. note::
  
              No attempt is made to return a minimal set of generators, see
              :meth:`Hilbert_basis` for that.
  
          EXAMPLES:
  
          The following command ensures that the output ordering in the examples
          below is independent of TOPCOM, you don't have to use it::
  
              sage: PointConfiguration.set_engine('internal')
  
          We start with a simple case of a non-smooth 2-dimensional cone::
  
              sage: Cone([ (1,0), (1,2) ]).semigroup_generators()
              N(1, 1),
              N(1, 0),
              N(1, 2)
              in 2-d lattice N
  
          A non-simplicial cone works, too::
  
              sage: cone = Cone([(3,0,-1), (1,-1,0), (0,1,0), (0,0,1)])
              sage: sorted(cone.semigroup_generators())
              [N(0, 0, 1), N(0, 1, 0), N(1, -1, 0), N(1, 0, 0), N(3, 0, -1)]
  
          GAP's toric package thinks this is challenging::
  
              sage: cone = Cone([[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]).dual()
              sage: len( cone.semigroup_generators() )
              2806
  
          The cone need not be strictly convex::
  
              sage: halfplane = Cone([(1,0),(2,1),(-1,0)])
              sage: halfplane.semigroup_generators()
              (N(0, 1), N(1, 0), N(-1, 0))
              sage: line = Cone([(1,1,1),(-1,-1,-1)])
              sage: line.semigroup_generators()
              (N(1, 1, 1), N(-1, -1, -1))
              sage: wedge = Cone([ (1,0,0), (1,2,0), (0,0,1), (0,0,-1) ])
              sage: sorted(wedge.semigroup_generators())
              [N(0, 0, -1), N(0, 0, 1), N(1, 0, 0), N(1, 1, 0), N(1, 2, 0)]
  
          Nor does it have to be full-dimensional (see :trac:`11312`)::
  
              sage: Cone([(1,1,0), (-1,1,0)]).semigroup_generators()
              N( 0, 1, 0),
              N( 1, 1, 0),
              N(-1, 1, 0)
              in 3-d lattice N
  
          Neither full-dimensional nor simplicial::
  
              sage: A = matrix([(1, 3, 0), (-1, 0, 1), (1, 1, -2), (15, -2, 0)])
              sage: A.elementary_divisors()
              [1, 1, 1, 0]
              sage: cone3d = Cone([(3,0,-1), (1,-1,0), (0,1,0), (0,0,1)])
              sage: rays = ( A*vector(v) for v in cone3d.rays() )
              sage: gens = Cone(rays).semigroup_generators(); sorted(gens)
              [N(-2, -1, 0, 17),
               N(0, 1, -2, 0),
               N(1, -1, 1, 15),
               N(3, -4, 5, 45),
               N(3, 0, 1, -2)]
              sage: set(map(tuple,gens)) == set( tuple(A*r) for r in cone3d.semigroup_generators() )
              True
  
          TESTS::
  
              sage: len(Cone(identity_matrix(10).rows()).semigroup_generators())
              10
  
              sage: trivial_cone = cones.trivial(3)
              sage: trivial_cone.semigroup_generators()
              Empty collection
              in 3-d lattice N
  
          ALGORITHM:
  
          If the cone is not simplicial, it is first triangulated. Each
          simplicial subcone has the integral points of the spaned
          parallelotope as generators. This is the first step of the
          primal Normaliz algorithm, see [Normaliz]_. For each
          simplicial cone (of dimension `d`), the integral points of the
          open parallelotope
  
          .. MATH::
  
              par \langle x_1, \dots, x_d \rangle =
              \ZZ^n \cap
              \left\{
              q_1 x_1 + \cdots +q_d x_d
              :~
              0 \leq q_i < 1
              \right\}
  
          are then computed [BK2001]_.
  
          Finally, the union of the generators of all simplicial
          subcones is returned.
          """
          # if the cone is not simplicial, triangulate and run
          # recursively
          N = self.lattice()
          if not self.is_simplicial():
              from sage.geometry.triangulation.point_configuration \
                      import PointConfiguration
              origin = self.nrays() # last one in pc
              pc = PointConfiguration(tuple(self.rays()) + (N(0),), star=origin)
              triangulation = pc.triangulate()
              subcones = ( Cone(( self.ray(i) for i in simplex if i!=origin ),
                                lattice=N, check=False)
                           for simplex in triangulation )
              gens = set()
              for cone in subcones:
                  gens.update(cone.semigroup_generators())
              return tuple(gens)
  
          gens = list(parallelotope_points(self.rays(), N)) + list(self.rays())
          gens = ( v for v in gens if gcd(v) == 1 )
          return PointCollection(gens, N)
  
      @cached_method
      def Hilbert_basis(self):
          r"""
          Return the Hilbert basis of the cone.
  
          Given a strictly convex cone `C\subset \RR^d`, the Hilbert
          basis of `C` is the set of all irreducible elements in the
          semigroup `C\cap \ZZ^d`. It is the unique minimal generating
          set over `\ZZ` for the integral points `C\cap \ZZ^d`.
  
          If the cone `C` is not strictly convex, this method finds the
          (unique) minimal set of lattice points that need to be added
          to the defining rays of the cone to generate the whole
          semigroup `C\cap \ZZ^d`. But because the rays of the cone are
          not unique nor necessarily minimal in this case, neither is
          the returned generating set (consisting of the rays plus
          additional generators).
  
          See also :meth:`semigroup_generators` if you are not
          interested in a minimal set of generators.
  
          OUTPUT:
  
          - a
            :class:`~sage.geometry.point_collection.PointCollection`. The
            rays of ``self`` are the first ``self.nrays()`` entries.
  
          EXAMPLES:
  
          The following command ensures that the output ordering in the examples
          below is independent of TOPCOM, you don't have to use it::
  
              sage: PointConfiguration.set_engine('internal')
  
          We start with a simple case of a non-smooth 2-dimensional cone::
  
              sage: Cone([ (1,0), (1,2) ]).Hilbert_basis()
              N(1, 0),
              N(1, 2),
              N(1, 1)
              in 2-d lattice N
  
          Two more complicated example from GAP/toric::
  
              sage: Cone([[1,0],[3,4]]).dual().Hilbert_basis()
              M(0,  1),
              M(4, -3),
              M(3, -2),
              M(2, -1),
              M(1,  0)
              in 2-d lattice M
              sage: cone = Cone([[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]).dual()
              sage: cone.Hilbert_basis()           # long time
              M(10,  -7,  0,  1),
              M(-5,  21,  0, -3),
              M( 0,  -2,  0,  1),
              M(15, -63, 25,  9),
              M( 2,  -3,  0,  1),
              M( 1,  -4,  1,  1),
              M(-1,   3,  0,  0),
              M( 4,  -4,  0,  1),
              M( 1,  -5,  2,  1),
              M( 3,  -5,  1,  1),
              M( 6,  -5,  0,  1),
              M( 3, -13,  5,  2),
              M( 2,  -6,  2,  1),
              M( 5,  -6,  1,  1),
              M( 0,   1,  0,  0),
              M( 8,  -6,  0,  1),
              M(-2,   8,  0, -1),
              M(10, -42, 17,  6),
              M( 7, -28, 11,  4),
              M( 5, -21,  9,  3),
              M( 6, -21,  8,  3),
              M( 5, -14,  5,  2),
              M( 2,  -7,  3,  1),
              M( 4,  -7,  2,  1),
              M( 7,  -7,  1,  1),
              M( 0,   0,  1,  0),
              M(-3,  14,  0, -2),
              M(-1,   7,  0, -1),
              M( 1,   0,  0,  0)
              in 4-d lattice M
  
          Not a strictly convex cone::
  
              sage: wedge = Cone([ (1,0,0), (1,2,0), (0,0,1), (0,0,-1) ])
              sage: sorted(wedge.semigroup_generators())
              [N(0, 0, -1), N(0, 0, 1), N(1, 0, 0), N(1, 1, 0), N(1, 2, 0)]
              sage: wedge.Hilbert_basis()
              N(1, 2,  0),
              N(1, 0,  0),
              N(0, 0,  1),
              N(0, 0, -1),
              N(1, 1,  0)
              in 3-d lattice N
  
          Not full-dimensional cones are ok, too (see :trac:`11312`)::
  
              sage: Cone([(1,1,0), (-1,1,0)]).Hilbert_basis()
              N( 1, 1, 0),
              N(-1, 1, 0),
              N( 0, 1, 0)
              in 3-d lattice N
  
          ALGORITHM:
  
          The primal Normaliz algorithm, see [Normaliz]_.
          """
          if self.is_strictly_convex():
              def not_in_linear_subspace(x): return True
          else:
              linear_subspace = self.linear_subspace()
              def not_in_linear_subspace(x):
                  # "x in linear_subspace" does not work, due to absence
                  # of coercion maps as of Trac ticket #10513.
                  try:
                      linear_subspace(x)
                      return False
                  except (TypeError, ValueError):
                      return True
  
          irreducible = list(self.rays())  # these are irreducible for sure
          gens = list(self.semigroup_generators())
          for x in irreducible:
              try:
                  gens.remove(x)
              except ValueError:
                  pass
  
          while gens:
              x = gens.pop()
              if any(not_in_linear_subspace(y) and x-y in self
                     for y in irreducible+gens):
                  continue
              irreducible.append(x)
          if len(irreducible) == self.nrays():
              return self.rays()
          else:
              return PointCollection(irreducible, self.lattice())
  
      def Hilbert_coefficients(self, point, solver=None, verbose=0):
          r"""
          Return the expansion coefficients of ``point`` with respect to
          :meth:`Hilbert_basis`.
  
          INPUT:
  
          - ``point`` -- a :meth:`~IntegralRayCollection.lattice` point
            in the cone, or something that can be converted to a
            point. For example, a list or tuple of integers.
  
          - ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver
            to be used. If set to ``None``, the default one is used. For more
            information on LP solvers and which default solver is used, see the
            method :meth:`~sage.numerical.mip.MixedIntegerLinearProgram.solve` of
            the class :class:`~sage.numerical.mip.MixedIntegerLinearProgram`.
  
          - ``verbose`` -- integer (default: ``0``). Sets the level of verbosity
            of the LP solver. Set to 0 by default, which means quiet.
  
          OUTPUT:
  
          A `\ZZ`-vector of length ``len(self.Hilbert_basis())`` with nonnegative
          components.
  
          .. note::
  
              Since the Hilbert basis elements are not necessarily linearly
              independent, the expansion coefficients are not unique. However,
              this method will always return the same expansion coefficients when
              invoked with the same argument.
  
          EXAMPLES::
  
              sage: cone = Cone([(1,0),(0,1)])
              sage: cone.rays()
              N(1, 0),
              N(0, 1)
              in 2-d lattice N
              sage: cone.Hilbert_coefficients([3,2])
              (3, 2)
  
          A more complicated example::
  
              sage: N = ToricLattice(2)
              sage: cone = Cone([N(1,0),N(1,2)])
              sage: cone.Hilbert_basis()
              N(1, 0),
              N(1, 2),
              N(1, 1)
              in 2-d lattice N
              sage: cone.Hilbert_coefficients( N(1,1) )
              (0, 0, 1)
  
          The cone need not be strictly convex::
  
              sage: N = ToricLattice(3)
              sage: cone = Cone([N(1,0,0),N(1,2,0),N(0,0,1),N(0,0,-1)])
              sage: cone.Hilbert_basis()
              N(1, 2,  0),
              N(1, 0,  0),
              N(0, 0,  1),
              N(0, 0, -1),
              N(1, 1,  0)
              in 3-d lattice N
              sage: cone.Hilbert_coefficients( N(1,1,3) )
              (0, 0, 3, 0, 1)
          """
          point = self.lattice()(point)
          if point not in self:
              raise ValueError('The given point is not in the cone!')
          basis = self.Hilbert_basis()
  
          from sage.numerical.mip import MixedIntegerLinearProgram
          p = MixedIntegerLinearProgram(maximization=False, solver=solver)
          p.set_objective(None)
          x = p.new_variable(integer=True, nonnegative=True)
          for i in range(self.lattice_dim()):
              p.add_constraint(p.sum(b[i]*x[j] for j,b in enumerate(basis)) == point[i])
          p.solve(log=verbose)
  
          return vector(ZZ, p.get_values(x))
  
      def is_solid(self):
          r"""
          Check if this cone is solid.
  
          A cone is said to be solid if it has nonempty interior. That
          is, if its extreme rays span the entire ambient space.
  
          OUTPUT:
  
          ``True`` if this cone is solid, and ``False`` otherwise.
  
          .. SEEALSO::
  
              :meth:`is_proper`
  
          EXAMPLES:
  
          The nonnegative orthant is always solid::
  
              sage: quadrant = cones.nonnegative_orthant(2)
              sage: quadrant.is_solid()
              True
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: octant.is_solid()
              True
  
          However, if we embed the two-dimensional nonnegative quadrant
          into three-dimensional space, then the resulting cone no longer
          has interior, so it is not solid::
  
              sage: quadrant = Cone([(1,0,0), (0,1,0)])
              sage: quadrant.is_solid()
              False
  
          TESTS:
  
          A closed convex cone is solid if and only if its dual is
          strictly convex::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8)
              sage: K.is_solid() == K.dual().is_strictly_convex()
              True
          """
          return (self.dim() == self.lattice_dim())
  
      def is_proper(self):
          r"""
          Check if this cone is proper.
  
          A cone is said to be proper if it is closed, convex, solid,
          and contains no lines. This cone is assumed to be closed and
          convex; therefore it is proper if it is solid and contains no
          lines.
  
          OUTPUT:
  
          ``True`` if this cone is proper, and ``False`` otherwise.
  
          .. SEEALSO::
  
              :meth:`is_strictly_convex`, :meth:`is_solid`
  
          EXAMPLES:
  
          The nonnegative orthant is always proper::
  
              sage: quadrant = cones.nonnegative_orthant(2)
              sage: quadrant.is_proper()
              True
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: octant.is_proper()
              True
  
          However, if we embed the two-dimensional nonnegative quadrant
          into three-dimensional space, then the resulting cone no longer
          has interior, so it is not solid, and thus not proper::
  
              sage: quadrant = Cone([(1,0,0), (0,1,0)])
              sage: quadrant.is_proper()
              False
  
          Likewise, a half-space contains at least one line, so it is not
          proper::
  
              sage: halfspace = Cone([(1,0),(0,1),(-1,0)])
              sage: halfspace.is_proper()
              False
  
          """
          return (self.is_strictly_convex() and self.is_solid())
  
      def is_full_space(self):
          r"""
          Check if this cone is equal to its ambient vector space.
  
          OUTPUT:
  
          ``True`` if this cone equals its entire ambient vector
          space and ``False`` otherwise.
  
          EXAMPLES:
  
          A single ray in two dimensions is not equal to the entire
          space::
  
              sage: K = Cone([(1,0)])
              sage: K.is_full_space()
              False
  
          Neither is the nonnegative orthant::
  
              sage: K = cones.nonnegative_orthant(2)
              sage: K.is_full_space()
              False
  
          The right half-space contains a vector subspace, but it is
          still not equal to the entire space::
  
              sage: K = Cone([(1,0),(-1,0),(0,1)])
              sage: K.is_full_space()
              False
  
          However, if we allow conic combinations of both axes, then
          the resulting cone is the entire two-dimensional space::
  
              sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
              sage: K.is_full_space()
              True
  
          """
          return self.linear_subspace() == self.lattice().vector_space()
  
      def lineality(self):
          r"""
          Return the lineality of this cone.
  
          The lineality of a cone is the dimension of the largest linear
          subspace contained in that cone.
  
          OUTPUT:
  
          A nonnegative integer; the dimension of the largest subspace
          contained within this cone.
  
          REFERENCES:
  
          - [Roc1970]_
  
          EXAMPLES:
  
          The lineality of the nonnegative orthant is zero, since it clearly
          contains no lines::
  
              sage: K = cones.nonnegative_orthant(3)
              sage: K.lineality()
              0
  
          However, if we add another ray so that the entire `x`-axis belongs
          to the cone, then the resulting cone will have lineality one::
  
              sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
              sage: K.lineality()
              1
  
          If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
          to the dimension of the ambient space (i.e. two)::
  
              sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
              sage: K.is_full_space()
              True
              sage: K.lineality()
              2
              sage: K.lattice_dim()
              2
  
          Per the definition, the lineality of the trivial cone in a trivial
          space is zero::
  
              sage: K = cones.trivial(0)
              sage: K.lineality()
              0
  
          TESTS:
  
          The lineality of a cone should be an integer between zero and the
          dimension of the ambient space, inclusive::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8)
              sage: l = K.lineality()
              sage: l in ZZ
              True
              sage: 0 <= l <= K.lattice_dim()
              True
  
          A strictly convex cone should have lineality zero::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim = 8, strictly_convex = True)
              sage: K.lineality()
              0
          """
          return self.linear_subspace().dimension()
  
      @cached_method
      def discrete_complementarity_set(self):
          r"""
          Compute a discrete complementarity set of this cone.
  
          A discrete complementarity set of a cone is the set of all
          orthogonal pairs `(x,s)` where `x` is in some fixed generating
          set of the cone, and `s` is in some fixed generating set of its
          dual. The generators chosen for this cone and its dual are
          simply their :meth:`~IntegralRayCollection.rays`.
  
          OUTPUT:
  
          A tuple of pairs `(x,s)` such that,
  
            * `x` and `s` are nonzero.
            * `s(x)` is zero.
            * `x` is one of this cone's :meth:`~IntegralRayCollection.rays`.
            * `s` is one of the :meth:`~IntegralRayCollection.rays` of this
              cone's :meth:`dual`.
  
          REFERENCES:
  
          - [Or2017]_
  
          EXAMPLES:
  
          Pairs of standard basis elements form a discrete complementarity
          set for the nonnegative orthant::
  
              sage: K = cones.nonnegative_orthant(2)
              sage: K.discrete_complementarity_set()
              ((N(1, 0), M(0, 1)), (N(0, 1), M(1, 0)))
  
          If a cone consists of a single ray, then the second components
          of a discrete complementarity set for that cone should generate
          the orthogonal complement of the ray::
  
              sage: K = Cone([(1,0)])
              sage: K.discrete_complementarity_set()
              ((N(1, 0), M(0, 1)), (N(1, 0), M(0, -1)))
              sage: K = Cone([(1,0,0)])
              sage: K.discrete_complementarity_set()
              ((N(1, 0, 0), M(0, 1, 0)),
               (N(1, 0, 0), M(0, -1, 0)),
               (N(1, 0, 0), M(0, 0, 1)),
               (N(1, 0, 0), M(0, 0, -1)))
  
          When a cone is the entire space, its dual is the trivial cone,
          so the only discrete complementarity set for it is empty::
  
              sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
              sage: K.is_full_space()
              True
              sage: K.discrete_complementarity_set()
              ()
  
          Likewise for trivial cones, whose duals are the entire space::
  
              sage: cones.trivial(0).discrete_complementarity_set()
              ()
  
          TESTS:
  
          A discrete complementarity set for the dual can be obtained by
          switching components in a discrete complementarity set of the
          original cone::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: dcs_dual = K.dual().discrete_complementarity_set()
              sage: expected = tuple( (x,s) for (s,x) in dcs_dual )
              sage: actual = K.discrete_complementarity_set()
              sage: sorted(actual) == sorted(expected)
              True
  
          The pairs in a discrete complementarity set are in fact
          complementary::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=6)
              sage: dcs = K.discrete_complementarity_set()
              sage: sum( (s*x).abs() for (x,s) in dcs )
              0
          """
          # Return an immutable tuple instead of a mutable list because
          # the result will be cached.
          return tuple( (x,s) for x in self
                              for s in self.dual()
                              if s*x == 0 )
  
      def lyapunov_like_basis(self):
          r"""
          Compute a basis of Lyapunov-like transformations on this cone.
  
          A linear transformation `L` is said to be Lyapunov-like on this
          cone if `L(x)` and `s` are orthogonal for every pair `(x,s)` in
          its :meth:`discrete_complementarity_set`. The set of all such
          transformations forms a vector space, namely the Lie algebra of
          the automorphism group of this cone.
  
          OUTPUT:
  
          A list of matrices forming a basis for the space of all
          Lyapunov-like transformations on this cone.
  
          .. SEEALSO::
  
              :meth:`cross_positive_operators_gens`,
              :meth:`positive_operators_gens`,
              :meth:`Z_operators_gens`
  
          REFERENCES:
  
          - [Or2017]_
          - [RNPA2011]_
  
          EXAMPLES:
  
          Every transformation is Lyapunov-like on the trivial cone::
  
              sage: K = cones.trivial(2)
              sage: M = MatrixSpace(K.lattice().base_field(), K.lattice_dim())
              sage: list(M.basis()) == K.lyapunov_like_basis()
              True
  
          And by duality, every transformation is Lyapunov-like on the
          ambient space::
  
              sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
              sage: K.is_full_space()
              True
              sage: M = MatrixSpace(K.lattice().base_field(), K.lattice_dim())
              sage: list(M.basis()) == K.lyapunov_like_basis()
              True
  
          However, in a trivial space, there are no non-trivial linear maps,
          so there can be no Lyapunov-like basis::
  
              sage: K = cones.trivial(0)
              sage: K.lyapunov_like_basis()
              []
  
          The Lyapunov-like transformations on the nonnegative orthant are
          diagonal matrices::
  
              sage: K = cones.nonnegative_orthant(1)
              sage: K.lyapunov_like_basis()
              [[1]]
  
              sage: K = cones.nonnegative_orthant(2)
              sage: K.lyapunov_like_basis()
              [
              [1 0]  [0 0]
              [0 0], [0 1]
              ]
  
              sage: K = cones.nonnegative_orthant(3)
              sage: K.lyapunov_like_basis()
              [
              [1 0 0]  [0 0 0]  [0 0 0]
              [0 0 0]  [0 1 0]  [0 0 0]
              [0 0 0], [0 0 0], [0 0 1]
              ]
  
          Only the identity matrix is Lyapunov-like on the pyramids
          defined by the one- and infinity-norms [RNPA2011]_::
  
              sage: l31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
              sage: l31.lyapunov_like_basis()
              [
              [1 0 0]
              [0 1 0]
              [0 0 1]
              ]
  
              sage: l3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
              sage: l3infty.lyapunov_like_basis()
              [
              [1 0 0]
              [0 1 0]
              [0 0 1]
              ]
  
          TESTS:
  
          Every operator in a :meth:`lyapunov_like_basis` is Lyapunov-like
          on the cone::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: LL = K.lyapunov_like_basis()
              sage: all( L.is_lyapunov_like_on(K) for L in LL )
              True
  
          The Lyapunov-like transformations on a cone and its dual are
          transposes of one another. However, there's no reason to expect
          that one basis will consist of transposes of the other::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: LL1 = K.lyapunov_like_basis()
              sage: LL2 = (L.transpose() for L in K.dual().lyapunov_like_basis())
              sage: V = VectorSpace(K.lattice().base_field(), K.lattice_dim()^2)
              sage: LL1_vecs = ( V(m.list()) for m in LL1 )
              sage: LL2_vecs = ( V(m.list()) for m in LL2 )
              sage: V.span(LL1_vecs) == V.span(LL2_vecs)
              True
  
          The space of all Lyapunov-like transformations is a Lie algebra
          and should therefore be closed under the lie bracket::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=4)
              sage: LL = K.lyapunov_like_basis()
              sage: W = VectorSpace(K.lattice().base_field(), K.lattice_dim()**2)
              sage: LL_W = W.span( W(m.list()) for m in LL )
              sage: brackets = ( W((L1*L2 - L2*L1).list()) for L1 in LL
              ....:                                        for L2 in LL )
              sage: all( b in LL_W for b in brackets )
              True
          """
          # Matrices are not vectors in Sage, so we have to convert them
          # to vectors explicitly before we can find a basis. We need these
          # two values to construct the appropriate "long vector" space.
          F = self.lattice().base_field()
          n = self.lattice_dim()
  
          # These tensor products contain a basis for the orthogonal
          # complement of the Lyapunov-like transformations on this cone.
          tensor_products = ( s.tensor_product(x)
                              for (x,s) in self.discrete_complementarity_set() )
  
          # Convert those tensor products to long vectors.
          W = VectorSpace(F, n**2)
          perp_vectors = ( W(tp.list()) for tp in tensor_products )
  
          # Now find the Lyapunov-like transformations (as long vectors).
          LL_vectors = W.span(perp_vectors).complement()
  
          # And finally convert the long vectors back to matrices.
          M = MatrixSpace(F, n, n)
          return [ M(v.list()) for v in LL_vectors.basis() ]
  
      def lyapunov_rank(self):
          r"""
          Compute the Lyapunov rank of this cone.
  
          The Lyapunov rank of a cone is the dimension of the space of its
          Lyapunov-like transformations --- that is, the length of a
          :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is
          the dimension of the Lie algebra of the automorphism group of
          the cone.
  
          OUTPUT:
  
          A nonnegative integer representing the Lyapunov rank of this cone.
  
          If the ambient space is trivial, then the Lyapunov rank will be
          zero. On the other hand, if the dimension of the ambient vector
          space is `n > 0`, then the resulting Lyapunov rank will be
          between `1` and `n^2` inclusive. If this cone :meth:`is_proper`,
          then that upper bound reduces from `n^2` to `n`. A Lyapunov rank
          of `n-1` is not possible (by Lemma 6 [Or2017]_) in either case.
  
          ALGORITHM:
  
          Algorithm 3 [Or2017]_ is used. Every closed convex cone is
          isomorphic to a Cartesian product of a proper cone, a subspace,
          and a trivial cone. The Lyapunov ranks of the subspace and
          trivial cone are easy to compute. Essentially, we "peel off"
          those easy parts of the cone and compute their Lyapunov ranks
          separately. We then compute the rank of the proper cone by
          counting a :meth:`lyapunov_like_basis` for it. Summing the
          individual ranks gives the Lyapunov rank of the original cone.
  
          REFERENCES:
  
          - [GT2014]_
          - [Or2017]_
          - [RNPA2011]_
  
          EXAMPLES:
  
          The Lyapunov rank of the nonnegative orthant is the same as the
          dimension of the ambient space [RNPA2011]_::
  
              sage: positives = cones.nonnegative_orthant(1)
              sage: positives.lyapunov_rank()
              1
              sage: quadrant = cones.nonnegative_orthant(2)
              sage: quadrant.lyapunov_rank()
              2
              sage: octant = cones.nonnegative_orthant(3)
              sage: octant.lyapunov_rank()
              3
  
          A vector space of dimension `n` has Lyapunov rank `n^{2}`
          [Or2017]_::
  
              sage: Q5 = VectorSpace(QQ, 5)
              sage: gs = Q5.basis() + [ -r for r in Q5.basis() ]
              sage: K = Cone(gs)
              sage: K.lyapunov_rank()
              25
  
          A pyramid in three dimensions has Lyapunov rank one [RNPA2011]_::
  
              sage: l31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
              sage: l31.lyapunov_rank()
              1
              sage: l3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
              sage: l3infty.lyapunov_rank()
              1
  
          A ray in `n` dimensions has Lyapunov rank `n^{2} - n + 1`
          [Or2017]_::
  
              sage: K = Cone([(1,0,0,0,0)])
              sage: K.lyapunov_rank()
              21
              sage: K.lattice_dim()**2 - K.lattice_dim() + 1
              21
  
          A subspace of dimension `m` in an `n`-dimensional ambient space
          has Lyapunov rank `n^{2} - m(n - m)` [Or2017]_::
  
              sage: e1 = vector(QQ, [1,0,0,0,0])
              sage: e2 = vector(QQ, [0,1,0,0,0])
              sage: z = (0,0,0,0,0)
              sage: K = Cone([e1, -e1, e2, -e2, z, z, z])
              sage: K.lyapunov_rank()
              19
              sage: K.lattice_dim()**2 - K.dim()*K.codim()
              19
  
          Lyapunov rank is additive on a product of proper cones [RNPA2011]_::
  
              sage: l31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
              sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
              sage: K = l31.cartesian_product(octant)
              sage: K.lyapunov_rank()
              4
              sage: l31.lyapunov_rank() + octant.lyapunov_rank()
              4
  
          Two linearly-isomorphic cones have the same Lyapunov rank
          [RNPA2011]_. A cone linearly-isomorphic to the nonnegative octant
          will have Lyapunov rank ``3``::
  
              sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
              sage: K.lyapunov_rank()
              3
  
          Lyapunov rank is invariant under :meth:`dual` [RNPA2011]_::
  
              sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
              sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
              True
  
          TESTS:
  
          Lyapunov rank should be additive on a product of proper cones
          [RNPA2011]_::
  
              sage: set_random_seed()
              sage: K1 = random_cone(max_ambient_dim=6,
              ....:                  strictly_convex=True,
              ....:                  solid=True)
              sage: K2 = random_cone(max_ambient_dim=6,
              ....:                  strictly_convex=True,
              ....:                  solid=True)
              sage: K = K1.cartesian_product(K2)
              sage: K.lyapunov_rank() == K1.lyapunov_rank() + K2.lyapunov_rank()
              True
  
          Lyapunov rank should be invariant under a linear isomorphism
          [Or2017]_::
  
              sage: set_random_seed()
              sage: K1 = random_cone(max_ambient_dim=8)
              sage: n = K1.lattice_dim()
              sage: A = random_matrix(QQ, n, algorithm='unimodular')
              sage: K2 = Cone( ( A*r for r in K1 ), lattice=K1.lattice())
              sage: K1.lyapunov_rank() == K2.lyapunov_rank()
              True
  
          Lyapunov rank should be invariant under :meth:`dual` [RNPA2011]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
              True
  
          The Lyapunov rank of a proper polyhedral cone in a non-trivial
          `n`-dimensional space can be any number between `1` and `n`
          inclusive, excluding `n-1` [GT2014]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8,
              ....:                 min_rays=1,
              ....:                 strictly_convex=True,
              ....:                 solid=True)
              sage: b = K.lyapunov_rank()
              sage: n = K.lattice_dim()
              sage: 1 <= b <= n
              True
              sage: b == n-1
              False
  
          No polyhedral closed convex cone in `n` dimensions has Lyapunov
          rank `n-1` [Or2017]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.lyapunov_rank() == K.lattice_dim() - 1
              False
  
          The calculation of the Lyapunov rank of an improper cone can
          be reduced to that of a proper cone [Or2017]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K_SP = K.solid_restriction().strict_quotient()
              sage: l = K.lineality()
              sage: c = K.codim()
              sage: actual = K.lyapunov_rank()
              sage: expected = K_SP.lyapunov_rank() + K.dim()*(l + c) + c**2
              sage: actual == expected
              True
  
          The Lyapunov rank of a cone is the length of a
          :meth:`lyapunov_like_basis` for it::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
              True
  
          A "perfect" cone has Lyapunov rank `n` or more in `n`
          dimensions. We can make any cone perfect by adding a slack
          variable::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: L = ToricLattice(K.lattice_dim() + 1)
              sage: K = Cone([ r.list() + [0] for r in K ], lattice=L)
              sage: K.lyapunov_rank() >= K.lattice_dim()
              True
          """
          # The solid_restriction() and strict_quotient() methods
          # already check if the cone is solid or strictly convex, so we
          # can't save any additional time here by seeing if those
          # methods would be no-ops.
          #
          # The call to solid_restriction() restricts K to its own span,
          # resulting in the cone K_S from the paper. The call to
          # strict_quotient() then restricts K_S to the span of its dual.
          K_SP = self.solid_restriction().strict_quotient()
  
          # K_SP is proper, so we have to compute its Lyapunov rank the
          # hard way -- by counting a Lyapunov-like basis for it.
          m = self.dim()
          n = self.lattice_dim()
          l = self.lineality()
  
          # cf. Theorem 2
          return len(K_SP.lyapunov_like_basis()) + l*m + (n - m)*n
  
      def random_element(self, ring=ZZ):
          r"""
          Return a random element of this cone.
  
          All elements of a convex cone can be represented as a
          nonnegative linear combination of its generators. A random
          element is thus constructed by assigning random nonnegative
          weights to the generators of this cone.  By default, these
          weights are integral and the resulting random element will live
          in the same lattice as the cone.
  
          The random nonnegative weights are chosen from ``ring`` which
          defaults to ``ZZ``. When ``ring`` is not ``ZZ``, the random
          element returned will be a vector. Only the rings ``ZZ`` and
          ``QQ`` are currently supported.
  
          INPUT:
  
            - ``ring`` -- (default: ``ZZ``) the ring from which the random
              generator weights are chosen; either ``ZZ`` or ``QQ``.
  
          OUTPUT:
  
          Either a lattice element or vector contained in both this cone
          and its ambient vector space. If ``ring`` is ``ZZ``, a lattice
          element is returned; otherwise a vector is returned. If ``ring``
          is neither ``ZZ`` nor ``QQ``, then a ``NotImplementedError`` is
          raised.
  
          EXAMPLES:
  
          The trivial element ``()`` is always returned in a trivial space::
  
              sage: set_random_seed()
              sage: K = cones.trivial(0)
              sage: K.random_element()
              N()
              sage: K.random_element(ring=QQ)
              ()
  
          A random element of the trivial cone in a nontrivial space is zero::
  
              sage: set_random_seed()
              sage: K = cones.trivial(3)
              sage: K.random_element()
              N(0, 0, 0)
              sage: K.random_element(ring=QQ)
              (0, 0, 0)
  
          A random element of the nonnegative orthant should have all
          components nonnegative::
  
              sage: set_random_seed()
              sage: K = cones.nonnegative_orthant(3)
              sage: all( x >= 0 for x in K.random_element() )
              True
              sage: all( x >= 0 for x in K.random_element(ring=QQ) )
              True
  
          If ``ring`` is not ``ZZ`` or ``QQ``, an error is raised::
  
              sage: set_random_seed()
              sage: K = Cone([(1,0),(0,1)])
              sage: K.random_element(ring=RR)
              Traceback (most recent call last):
              ...
              NotImplementedError: ring must be either ZZ or QQ.
  
          TESTS:
  
          Any cone should contain a random element of itself::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.contains(K.random_element())
              True
              sage: K.contains(K.random_element(ring=QQ))
              True
  
          The ambient vector space of the cone should contain a random
          element of the cone::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.random_element() in K.lattice().vector_space()
              True
              sage: K.random_element(ring=QQ) in K.lattice().vector_space()
              True
  
          By default, the random element should live in this cone's lattice::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.random_element() in K.lattice()
              True
  
          A strictly convex cone contains no lines, and thus no negative
          multiples of any of its elements besides zero::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
              sage: x = K.random_element()
              sage: x.is_zero() or not K.contains(-x)
              True
  
          The sum of random elements of a cone lies in the cone::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: K.contains(sum(K.random_element() for i in range(10)))
              True
              sage: K.contains(sum(K.random_element(QQ) for i in range(10)))
              True
  
          The sum of random elements of a cone belongs to its ambient
          vector space::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: V = K.lattice().vector_space()
              sage: sum(K.random_element() for i in range(10)) in V
              True
              sage: sum(K.random_element(ring=QQ) for i in range(10)) in V
              True
  
          By default, the sum of random elements of the cone should live
          in the cone's lattice::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=8)
              sage: sum(K.random_element() for i in range(10)) in K.lattice()
              True
          """
          if not ring in [ZZ, QQ]:
              # This cone theoretically lives in a real vector space,
              # but in Sage, we work over the rationals to avoid
              # numerical issues. Thus ``ring`` must consist of
              # rationals so that the ambient vector space will contain
              # the resulting random element.
              raise NotImplementedError('ring must be either ZZ or QQ.')
  
          # The lattice or vector space in which the return value will live.
          L = self.lattice()
          if ring is not ZZ:
              L = L.vector_space()
  
          # Scale each generator by a random nonnegative factor.
          terms = ( ring.random_element().abs()*L(g) for g in self )
  
          # Make sure we return a lattice element or vector. Without the
          # explicit conversion, we return ``0`` when we have no rays.
          return L(sum(terms))
  
      def positive_operators_gens(self, K2=None):
          r"""
          Compute minimal generators of the positive operators on this cone.
  
          A linear operator on a cone is positive if the image of
          the cone under the operator is a subset of the cone. This
          concept can be extended to two cones: the image of the
          first cone under a positive operator is a subset of the
          second cone, which may live in a different space.
  
          The positive operators (on one or two fixed cones) themselves
          form a closed convex cone. This method computes and returns
          the generators of that cone as a list of matrices.
  
          INPUT:
  
          - ``K2`` -- (default: ``self``) the codomain cone; the image of
            this cone under the returned generators is a subset of ``K2``.
  
          OUTPUT:
  
          A list of `m`-by-`n` matrices where `m` is the ambient dimension
          of ``K2`` and `n` is the ambient dimension of this cone. Each
          matrix `P` in the list has the property that `P(x)` is an
          element of ``K2`` whenever `x` is an element of this cone.
  
          The returned matrices generate the cone of positive operators
          from this cone to ``K2``; that is,
  
          - Any nonnegative linear combination of the returned matrices
            sends elements of this cone to ``K2``.
  
          - Every positive operator on this cone (with respect to ``K2``)
            is some nonnegative linear combination of the returned matrices.
  
          ALGORITHM:
  
          Computing positive operators directly is difficult, but
          computing their dual is straightforward using the generators of
          Berman and Gaiha. We construct the dual of the positive
          operators, and then return the dual of that, which is guaranteed
          to be the desired positive operators because everything is
          closed, convex, and polyhedral.
  
          .. SEEALSO::
  
              :meth:`cross_positive_operators_gens`,
              :meth:`lyapunov_like_basis`,
              :meth:`Z_operators_gens`
  
          REFERENCES:
  
          - [BG1972]_
          - [BP1994]_
          - [Or2018b]_
  
          EXAMPLES:
  
          Positive operators on the nonnegative orthant are nonnegative
          matrices::
  
              sage: K = Cone([(1,)])
              sage: K.positive_operators_gens()
              [[1]]
  
              sage: K = Cone([(1,0),(0,1)])
              sage: K.positive_operators_gens()
              [
              [1 0]  [0 1]  [0 0]  [0 0]
              [0 0], [0 0], [1 0], [0 1]
              ]
  
          The trivial cone in a trivial space has no positive operators::
  
              sage: K = cones.trivial(0)
              sage: K.positive_operators_gens()
              []
  
          Every operator is positive on the trivial cone::
  
              sage: K = cones.trivial(1)
              sage: K.positive_operators_gens()
              [[1], [-1]]
  
              sage: K = cones.trivial(2)
              sage: K.is_trivial()
              True
              sage: K.positive_operators_gens()
              [
              [1 0]  [-1  0]  [0 1]  [ 0 -1]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
              [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
              ]
  
          Every operator is positive on the ambient vector space::
  
              sage: K = Cone([(1,),(-1,)])
              sage: K.is_full_space()
              True
              sage: K.positive_operators_gens()
              [[1], [-1]]
  
              sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
              sage: K.is_full_space()
              True
              sage: K.positive_operators_gens()
              [
              [1 0]  [-1  0]  [0 1]  [ 0 -1]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
              [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
              ]
  
          A non-obvious application is to find the positive operators on the
          right half-plane [Or2018b]_::
  
              sage: K = Cone([(1,0),(0,1),(0,-1)])
              sage: K.positive_operators_gens()
              [
              [1 0]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
              [0 0], [1 0], [-1  0], [0 1], [ 0 -1]
              ]
  
          TESTS:
  
          A random positive operator should send a random element of one
          cone into the other cone::
  
              sage: set_random_seed()
              sage: K1 = random_cone(max_ambient_dim=3)
              sage: K2 = random_cone(max_ambient_dim=3)
              sage: pi_gens = K1.positive_operators_gens(K2)
              sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
              sage: pi_cone = Cone(( g.list() for g in pi_gens ),
              ....:                lattice=L,
              ....:                check=False)
              sage: P = matrix(K2.lattice_dim(),
              ....:            K1.lattice_dim(),
              ....:            pi_cone.random_element(QQ).list())
              sage: K2.contains(P*K1.random_element(ring=QQ))
              True
  
          The lineality space of the dual of the positive operators
          can be computed from the lineality spaces of the cone and
          its dual [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: pi_gens = K.positive_operators_gens()
              sage: L = ToricLattice(K.lattice_dim()**2)
              sage: pi_cone = Cone(( g.list() for g in pi_gens ),
              ....:                lattice=L,
              ....:                check=False)
              sage: actual = pi_cone.dual().linear_subspace()
              sage: U1 = [ vector((s.tensor_product(x)).list())
              ....:        for x in K.lines()
              ....:        for s in K.dual() ]
              sage: U2 = [ vector((s.tensor_product(x)).list())
              ....:        for x in K
              ....:        for s in K.dual().lines() ]
              sage: expected = pi_cone.lattice().vector_space().span(U1+U2)
              sage: actual == expected
              True
  
          The lineality of the dual of the positive operators is known
          from its lineality space [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: n = K.lattice_dim()
              sage: m = K.dim()
              sage: l = K.lineality()
              sage: pi_gens = K.positive_operators_gens()
              sage: L = ToricLattice(n**2)
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: actual = pi_cone.dual().lineality()
              sage: expected = l*(m - l) + m*(n - m)
              sage: actual == expected
              True
  
          The dimension of the positive operators on a cone depends on the
          dimension and lineality of that cone [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: n = K.lattice_dim()
              sage: m = K.dim()
              sage: l = K.lineality()
              sage: pi_gens = K.positive_operators_gens()
              sage: L = ToricLattice(n**2)
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: actual = pi_cone.dim()
              sage: expected = n**2 - l*(m - l) - (n - m)*m
              sage: actual == expected
              True
  
          The trivial cone, full space, and half-plane all give rise to the
          expected dimensions [Or2018b]_::
  
              sage: n = ZZ.random_element(5)
              sage: K = cones.trivial(n)
              sage: L = ToricLattice(n^2)
              sage: pi_gens = K.positive_operators_gens()
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: pi_cone.dim() == n^2
              True
  
              sage: K = K.dual()
              sage: K.is_full_space()
              True
              sage: pi_gens = K.positive_operators_gens()
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: pi_cone.dim() == n^2
              True
  
              sage: K = Cone([(1,0),(0,1),(0,-1)])
              sage: pi_gens = K.positive_operators_gens()
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                check=False)
              sage: pi_cone.dim() == 3
              True
  
          The lineality of the positive operators follows from the
          description of its generators [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: n = K.lattice_dim()
              sage: pi_gens = K.positive_operators_gens()
              sage: L = ToricLattice(n**2)
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: actual = pi_cone.lineality()
              sage: expected = n**2 - K.dim()*K.dual().dim()
              sage: actual == expected
              True
  
          The trivial cone, full space, and half-plane all give rise to
          the expected linealities [Or2018b]_::
  
              sage: n = ZZ.random_element(5)
              sage: K = cones.trivial(n)
              sage: L = ToricLattice(n^2)
              sage: pi_gens = K.positive_operators_gens()
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: pi_cone.lineality() == n^2
              True
  
              sage: K = K.dual()
              sage: K.is_full_space()
              True
              sage: pi_gens = K.positive_operators_gens()
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: pi_cone.lineality() == n^2
              True
  
              sage: K = Cone([(1,0),(0,1),(0,-1)])
              sage: pi_gens = K.positive_operators_gens()
              sage: pi_cone = Cone((g.list() for g in pi_gens), check=False)
              sage: pi_cone.lineality() == 2
              True
  
          A cone is proper if and only if its positive operators form a
          proper cone [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: pi_gens = K.positive_operators_gens()
              sage: L = ToricLattice(K.lattice_dim()**2)
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: K.is_proper() == pi_cone.is_proper()
              True
  
          The positive operators on a permuted cone can be obtained by
          conjugation::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: L = ToricLattice(K.lattice_dim()**2)
              sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
              sage: pK = Cone(( p*k for k in K ), K.lattice(), check=False)
              sage: pi_gens = pK.positive_operators_gens()
              sage: actual = Cone((g.list() for g in pi_gens),
              ....:               lattice=L,
              ....:               check=False)
              sage: pi_gens = K.positive_operators_gens()
              sage: expected = Cone(((p*g*p.inverse()).list() for g in pi_gens),
              ....:                 lattice=L,
              ....:                 check=False)
              sage: actual.is_equivalent(expected)
              True
  
          An operator is positive from one cone to another if and only if
          its adjoint is positive from the dual of the second cone to the
          dual of the first::
  
              sage: set_random_seed()
              sage: K1 = random_cone(max_ambient_dim=3)
              sage: K2 = random_cone(max_ambient_dim=3)
              sage: F = K1.lattice().vector_space().base_field()
              sage: n = K1.lattice_dim()
              sage: m = K2.lattice_dim()
              sage: L = ToricLattice(n*m)
              sage: W = VectorSpace(F, n*m)
              sage: pi_gens = K1.positive_operators_gens(K2)
              sage: pi_fwd = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: pi_gens = K2.dual().positive_operators_gens(K1.dual())
              sage: pi_back = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: M_fwd = MatrixSpace(F, m, n)
              sage: M_back = MatrixSpace(F, n, m)
              sage: L = M_fwd(pi_fwd.random_element(ring=QQ).list())
              sage: pi_back.contains(W(L.transpose().list()))
              True
              sage: L = M_back(pi_back.random_element(ring=QQ).list())
              sage: pi_fwd.contains(W(L.transpose().list()))
              True
  
          The Lyapunov rank of the positive operators is the product of
          the Lyapunov ranks of the associated cones if both are proper
          [Or2018a]_::
  
              sage: set_random_seed()
              sage: K1 = random_cone(max_ambient_dim=3,
              ....:                  strictly_convex=True,
              ....:                  solid=True)
              sage: K2 = random_cone(max_ambient_dim=3,
              ....:                  strictly_convex=True,
              ....:                  solid=True)
              sage: pi_gens = K1.positive_operators_gens(K2)
              sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: beta1 = K1.lyapunov_rank()
              sage: beta2 = K2.lyapunov_rank()
              sage: pi_cone.lyapunov_rank() == beta1*beta2
              True
  
          Lyapunov-like operators on a proper polyhedral positive operator
          cone can be computed from the Lyapunov-like operators on the cones
          with respect to which the operators are positive [Or2018a]_::
  
              sage: set_random_seed()
              sage: K1 = random_cone(max_ambient_dim=3,
              ....:                  strictly_convex=True,
              ....:                  solid=True)
              sage: K2 = random_cone(max_ambient_dim=3,
              ....:                  strictly_convex=True,
              ....:                  solid=True)
              sage: pi_gens = K1.positive_operators_gens(K2)
              sage: F = K1.lattice().base_field()
              sage: m = K1.lattice_dim()
              sage: n = K2.lattice_dim()
              sage: L = ToricLattice(m*n)
              sage: M1 = MatrixSpace(F, m, m)
              sage: M2 = MatrixSpace(F, n, n)
              sage: tps = ( M2(s.list()).tensor_product(M1(x.list()))
              ....:             for x in K1.dual().lyapunov_like_basis()
              ....:             for s in K2.lyapunov_like_basis() )
              sage: W = VectorSpace(F, (m**2)*(n**2))
              sage: expected = span(F, ( W(x.list()) for x in tps ))
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: LL_pi = pi_cone.lyapunov_like_basis()
              sage: actual = span(F, ( W(x.list()) for x in LL_pi ))
              sage: actual == expected
              True
          """
          if K2 is None:
              K2 = self
  
          # Matrices are not vectors in Sage, so we have to convert them
          # to vectors explicitly before we can find a basis. We need these
          # two values to construct the appropriate "long vector" space.
          F = self.lattice().base_field()
          n = self.lattice_dim()
          m = K2.lattice_dim()
  
          tensor_products = ( s.tensor_product(x) for x in self
                                                  for s in K2.dual() )
  
          # Convert those tensor products to long vectors.
          W = VectorSpace(F, n*m)
          vectors = ( W(tp.list()) for tp in tensor_products )
  
          check = True
          if self.is_proper() and K2.is_proper():
              # All of the generators involved are extreme vectors and
              # therefore minimal. If this cone is neither solid nor
              # strictly convex, then the tensor product of ``s`` and ``x``
              # is the same as that of ``-s`` and ``-x``. However, as a
              # /set/, ``tensor_products`` may still be minimal.
              check = False
  
          # Create the dual cone of the positive operators, expressed as
          # long vectors.
          pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check)
  
          # Now compute the desired cone from its dual...
          pi_cone = pi_dual.dual()
  
          # And finally convert its rays back to matrix representations.
          M = MatrixSpace(F, m, n)
          return [ M(v.list()) for v in pi_cone ]
  
      def cross_positive_operators_gens(self):
          r"""
          Compute minimal generators of the cross-positive operators on this
          cone.
  
          Any positive operator `P` on this cone will have `s(P(x)) \ge 0`
          whenever `x` is an element of this cone and `s` is an element of
          its dual. By contrast, the cross-positive operators need only
          satisfy that property on the :meth:`discrete_complementarity_set`;
          that is, when `x` and `s` are "cross" (orthogonal).
  
          The cross-positive operators (on some fixed cone) themselves
          form a closed convex cone. This method computes and returns
          the generators of that cone as a list of matrices.
  
          Cross-positive operators are also called exponentially-positive,
          since they become positive operators when exponentiated. Other
          equivalent names are resolvent-positive, essentially-positive,
          and quasimonotone.
  
          OUTPUT:
  
          A list of `n`-by-`n` matrices where `n` is the ambient dimension
          of this cone. Each matrix `L` in the list has the property that
          `s(L(x)) \ge 0` whenever `(x,s)` is an element of this cone's
          :meth:`discrete_complementarity_set`.
  
          The returned matrices generate the cone of cross-positive operators
          on this cone; that is,
  
          - Any nonnegative linear combination of the returned matrices
            is cross-positive on this cone.
  
          - Every cross-positive operator on this cone is some nonnegative
            linear combination of the returned matrices.
  
          .. SEEALSO::
  
             :meth:`lyapunov_like_basis`,
             :meth:`positive_operators_gens`,
             :meth:`Z_operators_gens`
  
          REFERENCES:
  
          - [SV1970]_
          - [Or2018b]_
  
          EXAMPLES:
  
          Cross-positive operators on the nonnegative orthant are
          negations of Z-matrices; that is, matrices whose off-diagonal
          elements are nonnegative::
  
              sage: K = cones.nonnegative_orthant(2)
              sage: K.cross_positive_operators_gens()
              [
              [0 1]  [0 0]  [1 0]  [-1  0]  [0 0]  [ 0  0]
              [0 0], [1 0], [0 0], [ 0  0], [0 1], [ 0 -1]
              ]
              sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
              sage: all( c[i][j] >= 0 for c in K.cross_positive_operators_gens()
              ....:                   for i in range(c.nrows())
              ....:                   for j in range(c.ncols())
              ....:                   if i != j )
              True
  
          The trivial cone in a trivial space has no cross-positive
          operators::
  
              sage: K = cones.trivial(0)
              sage: K.cross_positive_operators_gens()
              []
  
          Every operator is a cross-positive operator on the ambient
          vector space::
  
              sage: K = Cone([(1,),(-1,)])
              sage: K.is_full_space()
              True
              sage: K.cross_positive_operators_gens()
              [[1], [-1]]
  
              sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
              sage: K.is_full_space()
              True
              sage: K.cross_positive_operators_gens()
              [
              [1 0]  [-1  0]  [0 1]  [ 0 -1]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
              [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
              ]
  
          A non-obvious application is to find the cross-positive
          operators on the right half-plane [Or2018b]_::
  
              sage: K = Cone([(1,0),(0,1),(0,-1)])
              sage: K.cross_positive_operators_gens()
              [
              [1 0]  [-1  0]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
              [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
              ]
  
          Cross-positive operators on a subspace are Lyapunov-like and
          vice-versa::
  
              sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
              sage: K.is_full_space()
              True
              sage: lls = span( vector(l.list())
              ....:             for l in K.lyapunov_like_basis() )
              sage: cs  = span( vector(c.list())
              ....:             for c in K.cross_positive_operators_gens() )
              sage: cs == lls
              True
  
          TESTS:
  
          The cross-positive property is possessed by every cross-positive
          operator::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: all( L.is_cross_positive_on(K) for L in cp_gens )
              True
  
          The lineality space of the cone of cross-positive operators is
          the space of Lyapunov-like operators [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: L = ToricLattice(K.lattice_dim()**2)
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: cp_cone = Cone((g.list() for g in cp_gens),
              ....:                    lattice=L,
              ....:                    check=False)
              sage: ll_basis = ( vector(l.list())
              ....:              for l in K.lyapunov_like_basis() )
              sage: lls = L.vector_space().span(ll_basis)
              sage: cp_cone.linear_subspace() == lls
              True
  
          The lineality spaces of the duals of the positive and cross-
          positive operator cones are equal. From this it follows that
          the dimensions of the cross-positive operator cone and positive
          operator cone are equal [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: pi_gens = K.positive_operators_gens()
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: L = ToricLattice(K.lattice_dim()**2)
              sage: pi_cone = Cone((g.list() for g in pi_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: cp_cone = Cone((g.list() for g in cp_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: pi_cone.dim() == cp_cone.dim()
              True
              sage: pi_star = pi_cone.dual()
              sage: cp_star = cp_cone.dual()
              sage: pi_star.linear_subspace() == cp_star.linear_subspace()
              True
  
          The trivial cone, full space, and half-plane all give rise to
          the expected dimensions [Or2018b]_::
  
              sage: n = ZZ.random_element(5)
              sage: K = cones.trivial(n)
              sage: L = ToricLattice(n^2)
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: cp_cone = Cone((g.list() for g in cp_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: cp_cone.dim() == n^2
              True
  
              sage: K = K.dual()
              sage: K.is_full_space()
              True
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: cp_cone = Cone((g.list() for g in cp_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: cp_cone.dim() == n^2
              True
  
              sage: K = Cone([(1,0),(0,1),(0,-1)])
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: cp_cone = Cone(( g.list() for g in cp_gens ), check=False)
              sage: cp_cone.dim() == 3
              True
  
          The cross-positive operators of a permuted cone can be obtained by
          conjugation::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: L = ToricLattice(K.lattice_dim()**2)
              sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
              sage: pK = Cone(( p*k for k in K ), K.lattice(), check=False)
              sage: cp_gens = pK.cross_positive_operators_gens()
              sage: actual = Cone((g.list() for g in cp_gens),
              ....:               lattice=L,
              ....:               check=False)
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: expected = Cone(((p*g*p.inverse()).list() for g in cp_gens),
              ....:                 lattice=L,
              ....:                 check=False)
              sage: actual.is_equivalent(expected)
              True
  
          An operator is cross-positive on a cone if and only if its
          adjoint is cross-positive on the dual of that cone [Or2018b]_::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: F = K.lattice().vector_space().base_field()
              sage: n = K.lattice_dim()
              sage: L = ToricLattice(n**2)
              sage: W = VectorSpace(F, n**2)
              sage: cp_gens = K.cross_positive_operators_gens()
              sage: cp_cone = Cone((g.list() for g in cp_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: cp_gens = K.dual().cross_positive_operators_gens()
              sage: cp_star = Cone((g.list() for g in cp_gens),
              ....:                lattice=L,
              ....:                check=False)
              sage: M = MatrixSpace(F, n)
              sage: L = M(cp_cone.random_element(ring=QQ).list())
              sage: cp_star.contains(W(L.transpose().list()))
              True
              sage: L = M(cp_star.random_element(ring=QQ).list())
              sage: cp_cone.contains(W(L.transpose().list()))
              True
          """
          # Matrices are not vectors in Sage, so we have to convert them
          # to vectors explicitly before we can find a basis. We need these
          # two values to construct the appropriate "long vector" space.
          F = self.lattice().base_field()
          n = self.lattice_dim()
  
          # These tensor products contain generators for the dual cone of
          # the cross-positive operators.
          tensor_products = ( s.tensor_product(x)
                              for (x,s) in self.discrete_complementarity_set() )
  
          # Turn our matrices into long vectors...
          W = VectorSpace(F, n**2)
          vectors = ( W(m.list()) for m in tensor_products )
  
          check = True
          if self.is_proper():
              # All of the generators involved are extreme vectors and
              # therefore minimal. If this cone is neither solid nor
              # strictly convex, then the tensor product of ``s`` and
              # ``x`` is the same as that of ``-s`` and ``-x``. However,
              # as a /set/, ``tensor_products`` may still be minimal.
              check = False
  
          # Create the dual cone of the cross-positive operators,
          # expressed as long vectors.
          cp_dual = Cone(vectors,
                         lattice=ToricLattice(W.dimension()),
                         check=check)
  
          # Now compute the desired cone from its dual...
          cp_cone = cp_dual.dual()
  
          # And finally convert its rays back to matrix representations.
          M = MatrixSpace(F, n)
          return [ M(v.list()) for v in cp_cone ]
  
      def Z_operators_gens(self):
          r"""
          Compute minimal generators of the Z-operators on this cone.
  
          The Z-operators on a cone generalize the Z-matrices over the
          nonnegative orthant. They are simply negations of the
          :meth:`cross_positive_operators_gens`.
  
          OUTPUT:
  
          A list of `n`-by-`n` matrices where `n` is the ambient dimension
          of this cone. Each matrix `L` in the list has the property that
          `s(L(x)) \le 0` whenever `(x,s)` is an element of this cone's
          :meth:`discrete_complementarity_set`.
  
          The returned matrices generate the cone of Z-operators on this
          cone; that is,
  
          - Any nonnegative linear combination of the returned matrices
            is a Z-operator on this cone.
  
          - Every Z-operator on this cone is some nonnegative linear
            combination of the returned matrices.
  
          .. SEEALSO::
  
             :meth:`cross_positive_operators_gens`,
             :meth:`lyapunov_like_basis`,
             :meth:`positive_operators_gens`
  
          REFERENCES:
  
          - [BP1994]_
          - [Or2018b]_
  
          TESTS:
  
          The Z-property is possessed by every Z-operator::
  
              sage: set_random_seed()
              sage: K = random_cone(max_ambient_dim=3)
              sage: Z_gens = K.Z_operators_gens()
              sage: all( L.is_Z_operator_on(K) for L in Z_gens )
              True
          """
          return [ -cp for cp in self.cross_positive_operators_gens() ]
  
  
  def random_cone(lattice=None, min_ambient_dim=0, max_ambient_dim=None,
                  min_rays=0, max_rays=None, strictly_convex=None, solid=None):
      r"""
      Generate a random convex rational polyhedral cone.
  
      Lower and upper bounds may be provided for both the dimension of the
      ambient space and the number of generating rays of the cone. If a
      lower bound is left unspecified, it defaults to zero. Unspecified
      upper bounds will be chosen randomly, unless you set ``solid``, in
      which case they are chosen a little more wisely.
  
      You may specify the ambient ``lattice`` for the returned cone. In
      that case, the ``min_ambient_dim`` and ``max_ambient_dim``
      parameters are ignored.
  
      You may also request that the returned cone be strictly convex (or
      not). Likewise you may request that it be (non-)solid.
  
      .. WARNING::
  
          If you request a large number of rays in a low-dimensional
          space, you might be waiting for a while. For example, in three
          dimensions, it is possible to obtain an octagon raised up to height
          one (all z-coordinates equal to one). But in practice, we usually
          generate the entire three-dimensional space with six rays before we
          get to the eight rays needed for an octagon. We therefore have to
          throw the cone out and start over from scratch. This process repeats
          until we get lucky.
  
          We also refrain from "adjusting" the min/max parameters given to
          us when a (non-)strictly convex or (non-)solid cone is
          requested. This means that it may take a long time to generate
          such a cone if the parameters are chosen unwisely.
  
          For example, you may want to set ``min_rays`` close to
          ``min_ambient_dim`` if you desire a solid cone. Or, if you desire a
          non-strictly-convex cone, then they all contain at least two
          generating rays. So that might be a good candidate for
          ``min_rays``.
  
      INPUT:
  
      * ``lattice`` (default: random) -- A ``ToricLattice`` object in
        which the returned cone will live. By default a new lattice will
        be constructed with a randomly-chosen rank (subject to
        ``min_ambient_dim`` and ``max_ambient_dim``).
  
      * ``min_ambient_dim`` (default: zero) -- A nonnegative integer
        representing the minimum dimension of the ambient lattice.
  
      * ``max_ambient_dim`` (default: random) -- A nonnegative integer
        representing the maximum dimension of the ambient lattice.
  
      * ``min_rays`` (default: zero) -- A nonnegative integer representing
        the minimum number of generating rays of the cone.
  
      * ``max_rays`` (default: random) -- A nonnegative integer representing
        the maximum number of generating rays of the cone.
  
      * ``strictly_convex`` (default: random) -- Whether or not to make the
        returned cone strictly convex. Specify ``True`` for a strictly convex
        cone, ``False`` for a non-strictly-convex cone, or ``None`` if you
        don't care.
  
      * ``solid`` (default: random) -- Whether or not to make the returned
        cone solid. Specify ``True`` for a solid cone, ``False`` for a
        non-solid cone, or ``None`` if you don't care.
  
      OUTPUT:
  
      A new, randomly generated cone.
  
      A ``ValueError`` will be thrown under the following conditions:
  
      * Any of ``min_ambient_dim``, ``max_ambient_dim``, ``min_rays``, or
        ``max_rays`` are negative.
  
      * ``max_ambient_dim`` is less than ``min_ambient_dim``.
  
      * ``max_rays`` is less than ``min_rays``.
  
      * Both ``max_ambient_dim`` and ``lattice`` are specified.
  
      * ``min_rays`` is greater than four but ``max_ambient_dim`` is less than
        three.
  
      * ``min_rays`` is greater than four but ``lattice`` has dimension
        less than three.
  
      * ``min_rays`` is greater than two but ``max_ambient_dim`` is less than
        two.
  
      * ``min_rays`` is greater than two but ``lattice`` has dimension less
        than two.
  
      * ``min_rays`` is positive but ``max_ambient_dim`` is zero.
  
      * ``min_rays`` is positive but ``lattice`` has dimension zero.
  
      * A trivial lattice is supplied and a non-strictly-convex cone
        is requested.
  
      * A non-strictly-convex cone is requested but ``max_rays`` is less
        than two.
  
      * A solid cone is requested but ``max_rays`` is less than
        ``min_ambient_dim``.
  
      * A solid cone is requested but ``max_rays`` is less than the
        dimension of ``lattice``.
  
      * A non-solid cone is requested but ``max_ambient_dim`` is zero.
  
      * A non-solid cone is requested but ``lattice`` has dimension zero.
  
      * A non-solid cone is requested but ``min_rays`` is so large that
        it guarantees a solid cone.
  
      ALGORITHM:
  
      First, a lattice is determined from ``min_ambient_dim`` and
      ``max_ambient_dim`` (or from the supplied ``lattice``).
  
      Then, lattice elements are generated one at a time and added to a
      cone. This continues until either the cone meets the user's
      requirements, or the cone is equal to the entire space (at which
      point it is futile to generate more).
  
      We check whether or not the resulting cone meets the user's
      requirements; if it does, it is returned. If not, we throw it away
      and start over. This process repeats indefinitely until an
      appropriate cone is generated.
  
      EXAMPLES:
  
      Generate a trivial cone in a trivial space::
  
          sage: set_random_seed()
          sage: random_cone(max_ambient_dim=0, max_rays=0)
          0-d cone in 0-d lattice N
  
      We can predict the ambient dimension when
      ``min_ambient_dim == max_ambient_dim``::
  
          sage: set_random_seed()
          sage: K = random_cone(min_ambient_dim=4, max_ambient_dim=4)
          sage: K.lattice_dim()
          4
  
      Likewise for the number of rays when ``min_rays == max_rays``::
  
          sage: set_random_seed()
          sage: K = random_cone(min_rays=3, max_rays=3)
          sage: K.nrays()
          3
  
      If we specify a lattice, then the returned cone will live in it::
  
          sage: set_random_seed()
          sage: L = ToricLattice(5, "L")
          sage: K = random_cone(lattice=L)
          sage: K.lattice() is L
          True
  
      We can also request a strictly convex cone::
  
          sage: set_random_seed()
          sage: K = random_cone(max_ambient_dim=8, max_rays=10,
          ....:                 strictly_convex=True)
          sage: K.is_strictly_convex()
          True
  
      Or one that isn't strictly convex::
  
          sage: set_random_seed()
          sage: K = random_cone(min_ambient_dim=5, min_rays=2,
          ....:                 strictly_convex=False)
          sage: K.is_strictly_convex()
          False
  
      An example with all parameters set::
  
          sage: set_random_seed()
          sage: K = random_cone(min_ambient_dim=4, max_ambient_dim=7,
          ....:                 min_rays=2, max_rays=10,
          ....:                 strictly_convex=False, solid=True)
          sage: 4 <= K.lattice_dim() and K.lattice_dim() <= 7
          True
          sage: 2 <= K.nrays() and K.nrays() <= 10
          True
          sage: K.is_strictly_convex()
          False
          sage: K.is_solid()
          True
  
      TESTS:
  
      It's hard to test the output of a random process, but we can at
      least make sure that we get a cone back.
  
          sage: set_random_seed()
          sage: from sage.geometry.cone import is_Cone
          sage: K = random_cone(max_ambient_dim=6, max_rays=10)
          sage: is_Cone(K)
          True
  
      The upper/lower bounds are respected::
  
          sage: set_random_seed()
          sage: K = random_cone(min_ambient_dim=5, max_ambient_dim=8,
          ....:                 min_rays=3, max_rays=4)
          sage: 5 <= K.lattice_dim() and K.lattice_dim() <= 8
          True
          sage: 3 <= K.nrays() and K.nrays() <= 4
          True
  
      Ensure that an exception is raised when either lower bound is greater
      than its respective upper bound::
  
          sage: set_random_seed()
          sage: random_cone(min_ambient_dim=5, max_ambient_dim=2)
          Traceback (most recent call last):
          ...
          ValueError: max_ambient_dim cannot be less than min_ambient_dim.
  
          sage: random_cone(min_rays=5, max_rays=2)
          Traceback (most recent call last):
          ...
          ValueError: max_rays cannot be less than min_rays.
  
      Or if we specify both ``max_ambient_dim`` and ``lattice``::
  
          sage: set_random_seed()
          sage: L = ToricLattice(5, "L")
          sage: random_cone(lattice=L, max_ambient_dim=10)
          Traceback (most recent call last):
          ...
          ValueError: max_ambient_dim cannot be specified when a lattice is
          provided.
  
      If the user requests too many rays in zero, one, or two dimensions,
      a ``ValueError`` is thrown::
  
          sage: set_random_seed()
          sage: random_cone(max_ambient_dim=0, min_rays=1)
          Traceback (most recent call last):
          ...
          ValueError: all cones in zero dimensions have no generators.
          Please increase max_ambient_dim to at least 1, or decrease min_rays.
  
          sage: random_cone(max_ambient_dim=1, min_rays=3)
          Traceback (most recent call last):
          ...
          ValueError: all cones in zero/one dimensions have two or fewer
          generators. Please increase max_ambient_dim to at least 2, or decrease
          min_rays.
  
          sage: random_cone(max_ambient_dim=2, min_rays=5)
          Traceback (most recent call last):
          ...
          ValueError: all cones in zero/one/two dimensions have four or fewer
          generators. Please increase max_ambient_dim to at least 3, or decrease
          min_rays.
  
          sage: L = ToricLattice(0)
          sage: random_cone(lattice=L, min_rays=1)
          Traceback (most recent call last):
          ...
          ValueError: all cones in the given lattice have no generators.
          Please decrease min_rays.
  
          sage: L = ToricLattice(1)
          sage: random_cone(lattice=L, min_rays=3)
          Traceback (most recent call last):
          ...
          ValueError: all cones in the given lattice have two or fewer
          generators. Please decrease min_rays.
  
          sage: L = ToricLattice(2)
          sage: random_cone(lattice=L, min_rays=5)
          Traceback (most recent call last):
          ...
          ValueError: all cones in the given lattice have four or fewer
          generators. Please decrease min_rays.
  
      Ensure that we can obtain a cone in three dimensions with a large
      number (in particular, more than 2*dim) rays. The ``max_rays`` is
      not strictly necessary, but it minimizes the number of times that
      we will loop with an absurd, unattainable, number of rays::
  
          sage: set_random_seed()                  # long time
          sage: K = random_cone(min_ambient_dim=3, # long time
          ....:                 max_ambient_dim=3, # long time
          ....:                 min_rays=7,        # long time
          ....:                 max_rays=9)        # long time
          sage: K.nrays() >= 7                     # long time
          True
          sage: K.lattice_dim()                    # long time
          3
  
      We need three dimensions to obtain five rays; we should throw out
      cones in zero/one/two dimensions until we get lucky::
  
          sage: set_random_seed()
          sage: K = random_cone(max_ambient_dim=3, min_rays=5)
          sage: K.nrays() >= 5
          True
          sage: K.lattice_dim()
          3
  
      It is an error to request a non-strictly-convex trivial cone::
  
          sage: set_random_seed()
          sage: L = ToricLattice(0,"L")
          sage: random_cone(lattice=L, strictly_convex=False)
          Traceback (most recent call last):
          ...
          ValueError: all cones in this lattice are strictly convex (trivial).
  
      Or a non-strictly-convex cone with fewer than two rays::
  
          sage: set_random_seed()
          sage: random_cone(max_rays=1, strictly_convex=False)
          Traceback (most recent call last):
          ...
          ValueError: all cones are strictly convex when ``max_rays`` is
          less than two.
  
      But fine to ask for a strictly convex trivial cone::
  
          sage: set_random_seed()
          sage: L = ToricLattice(0,"L")
          sage: random_cone(lattice=L, strictly_convex=True)
          0-d cone in 0-d lattice L
  
      A ``ValueError`` is thrown if a non-solid cone is requested in a
      zero-dimensional lattice::
  
          sage: set_random_seed()
          sage: L = ToricLattice(0)
          sage: random_cone(lattice=L, solid=False)
          Traceback (most recent call last):
          ...
          ValueError: all cones in the given lattice are solid.
  
          sage: random_cone(max_ambient_dim=0, solid=False)
          Traceback (most recent call last):
          ...
          ValueError: all cones are solid when max_ambient_dim is zero.
  
      A ``ValueError`` is thrown if a solid cone is requested but the
      maximum number of rays is too few::
  
          sage: set_random_seed()
          sage: random_cone(min_ambient_dim=4, max_rays=3, solid=True)
          Traceback (most recent call last):
          ...
          ValueError: max_rays must be at least min_ambient_dim for a solid cone.
  
          sage: L = ToricLattice(5)
          sage: random_cone(lattice=L, max_rays=3, solid=True)
          Traceback (most recent call last):
          ...
          ValueError: max_rays must be at least 5 for a solid cone in this
          lattice.
  
      A ``ValueError`` is thrown if a non-solid cone is requested but
      ``min_rays`` guarantees a solid cone::
  
          sage: set_random_seed()
          sage: random_cone(max_ambient_dim=4, min_rays=10, solid=False)
          Traceback (most recent call last):
          ...
          ValueError: every cone is solid when min_rays > 2*(max_ambient_dim - 1).
  
          sage: L = ToricLattice(4)
          sage: random_cone(lattice=L, min_rays=10, solid=False)
          Traceback (most recent call last):
          ...
          ValueError: every cone is solid when min_rays > 2*(d - 1) where d
          is the dimension of the given lattice.
  
      """
  
      # Catch obvious mistakes so that we can generate clear error
      # messages.
  
      if min_ambient_dim < 0:
          raise ValueError('min_ambient_dim must be nonnegative.')
  
      if min_rays < 0:
          raise ValueError('min_rays must be nonnegative.')
  
      if max_ambient_dim is not None:
          if max_ambient_dim < 0:
              raise ValueError('max_ambient_dim must be nonnegative.')
          if (max_ambient_dim < min_ambient_dim):
              msg = 'max_ambient_dim cannot be less than min_ambient_dim.'
              raise ValueError(msg)
          if lattice is not None:
              msg = 'max_ambient_dim cannot be specified when a lattice is '
              msg += 'provided.'
              raise ValueError(msg)
  
          # The next three checks prevent an infinite loop (a futile
          # search for more rays) in zero, one, or two dimensions.
          if min_rays > 4 and max_ambient_dim < 3:
              msg  = 'all cones in zero/one/two dimensions have four or fewer '
              msg += 'generators. Please increase max_ambient_dim to at least '
              msg += '3, or decrease min_rays.'
              raise ValueError(msg)
  
          if min_rays > 2 and max_ambient_dim < 2:
              msg  = 'all cones in zero/one dimensions have two or fewer '
              msg += 'generators. Please increase max_ambient_dim to at least '
              msg += '2, or decrease min_rays.'
              raise ValueError(msg)
  
          if min_rays > 0 and max_ambient_dim == 0:
              msg  = 'all cones in zero dimensions have no generators. '
              msg += 'Please increase max_ambient_dim to at least 1, or '
              msg += 'decrease min_rays.'
              raise ValueError(msg)
  
      if max_rays is not None:
          if max_rays < 0:
              raise ValueError('max_rays must be nonnegative.')
          if (max_rays < min_rays):
              raise ValueError('max_rays cannot be less than min_rays.')
  
      # Also perform the "futile search" checks when a lattice is given,
      # using its dimension rather than max_ambient_dim as the indicator.
      if lattice is not None:
          if min_rays > 4 and lattice.dimension() < 3:
              msg  = 'all cones in the given lattice have four or fewer '
              msg += 'generators. Please decrease min_rays.'
              raise ValueError(msg)
  
          if min_rays > 2 and lattice.dimension() < 2:
              msg  = 'all cones in the given lattice have two or fewer '
              msg += 'generators. Please decrease min_rays.'
              raise ValueError(msg)
  
          if min_rays > 0 and lattice.dimension() == 0:
              msg  = 'all cones in the given lattice have no generators. '
              msg += 'Please decrease min_rays.'
              raise ValueError(msg)
  
      # Sanity checks for strictly_convex.
      if strictly_convex is not None and not strictly_convex:
          if lattice is not None and lattice.dimension() == 0:
              msg = 'all cones in this lattice are strictly convex (trivial).'
              raise ValueError(msg)
          if max_rays is not None and max_rays < 2:
              msg  = 'all cones are strictly convex when ``max_rays`` is '
              msg += 'less than two.'
              raise ValueError(msg)
  
      # Sanity checks for solid cones.
      if solid is not None and solid:
          # The user wants a solid cone.
          if lattice is None:
              if max_rays is not None:
                  if max_rays < min_ambient_dim:
                      msg = 'max_rays must be at least min_ambient_dim for '
                      msg += 'a solid cone.'
                      raise ValueError(msg)
          else:
              # Repeat the checks above when a lattice is given.
              if max_rays is not None and max_rays < lattice.dimension():
                  msg = "max_rays must be at least {0} for a solid cone "
                  msg += "in this lattice."
                  raise ValueError(msg.format(lattice.dimension()))
  
      # Sanity checks for non-solid cones.
      if solid is not None and not solid:
          if lattice is None:
              if max_ambient_dim is not None and max_ambient_dim == 0:
                  msg = 'all cones are solid when max_ambient_dim is zero.'
                  raise ValueError(msg)
              if (max_ambient_dim is not None and
                      min_rays > 2*(max_ambient_dim - 1)):
                  msg = 'every cone is solid when '
                  msg += 'min_rays > 2*(max_ambient_dim - 1).'
                  raise ValueError(msg)
          else:
              if lattice.dimension() == 0:
                  msg = 'all cones in the given lattice are solid.'
                  raise ValueError(msg)
              if  min_rays > 2*(lattice.dimension() - 1):
                  msg = 'every cone is solid when min_rays > 2*(d - 1) '
                  msg += 'where d is the dimension of the given lattice.'
                  raise ValueError(msg)
  
  
      # Now that we've sanity-checked our parameters, we can massage the
      # min/maxes for (non-)solid cones. It doesn't violate the user's
      # expectation to increase a minimum, decrease a maximum, or fix an
      # "I don't care" parameter.
      if solid is not None:
          if solid:
              # If max_ambient_dim is "I don't care", we can set it so that we
              # guaranteed to generate a solid cone.
              if max_rays is not None and max_ambient_dim is None:
                  # We won't make max_ambient_dim less than min_ambient_dim,
                  # since we already checked that
                  # min_ambient_dim <= min_rays = max_ambient_dim.
                  max_ambient_dim = min_rays
          else:
              if max_rays is None and max_ambient_dim is not None:
                  # This is an upper limit on the number of rays in a
                  # non-solid cone.
                  max_rays = 2*(max_ambient_dim - 1)
              if max_rays is None and lattice is not None:
                  # Same thing, except when we're given a lattice.
                  max_rays = 2*(lattice.dimension() - 1)
  
      def random_min_max(l,u):
          r"""
          We need to handle two cases for the upper bounds, and we need
          to do the same thing for max_ambient_dim/max_rays. So we consolidate
          the logic here.
          """
          if u is None:
              # The upper bound is unspecified; return a random integer
              # in [l,infinity).
              return l + ZZ.random_element().abs()
          else:
              # We have an upper bound, and it's greater than or equal
              # to our lower bound. So we generate a random integer in
              # [0,u-l], and then add it to l to get something in
              # [l,u]. To understand the "+1", check the
              # ZZ.random_element() docs.
              return l + ZZ.random_element(u - l + 1)
  
      def is_valid(K):
          r"""
          Check if the given cone is valid; that is, if its ambient
          dimension and number of rays meet the upper and lower bounds
          provided by the user.
          """
          if lattice is None:
              # We only care about min/max_ambient_dim when no lattice is given.
              if K.lattice_dim() < min_ambient_dim:
                  return False
              if (max_ambient_dim is not None and
                      K.lattice_dim() > max_ambient_dim):
                  return False
          else:
              if K.lattice() is not lattice:
                  return False
          return all([K.nrays() >= min_rays,
                      max_rays is None or K.nrays() <= max_rays,
                      solid is None or K.is_solid() == solid,
                      strictly_convex is None or
                      K.is_strictly_convex() == strictly_convex])
  
      # Now we actually compute the thing. To avoid recursion (and the
      # associated "maximum recursion depth exceeded" error), we loop
      # until we have a valid cone and occasionally throw everything out
      # and start over from scratch.
      while True:
          L = lattice
  
          if lattice is None:
              # No lattice given, make our own.
              d = random_min_max(min_ambient_dim, max_ambient_dim)
              L = ToricLattice(d)
          else:
              d = L.dimension()
  
          # The number of rays that we will try to attain in this iteration.
          r = random_min_max(min_rays, max_rays)
  
          # The rays are trickier to generate, since we could generate v and
          # 2*v as our "two rays." In that case, the resulting cone would
          # have only one generating ray -- not what we want.
          #
          # Let's begin with an easier question: how many rays should we
          # start with? If we want to attain r rays in this iteration,
          # then surely r is a good number to start with, even if some
          # of them will be redundant?
          #
          # Not quite, because after 2*d rays, there is a greater
          # tendency for them to be redundant. If, for example, the
          # maximum number of rays is unbounded, then r could be enormous
          # Ultimately that won't be a problem, because almost all of
          # those rays will be thrown out. However, as we discovered in
          # Trac #24517, simply generating the random rays in the first
          # place (and storing them in a list) is problematic.
          #
          # Since the returns fall off around 2*d, we start with the
          # smaller of the two numbers 2*d or r to ensure that we don't
          # pay a huge performance penalty for things we're going to
          # throw out anyway. This has a side effect, namely that if you
          # ask for more than 2*d rays, then you'll probably get the
          # minimum amount, because we'll start with 2*d and add them
          # one-at-a-time (see below).
          rays = [L.random_element() for i in range(min(r,2*d))]
  
          # The lattice parameter is required when no rays are given, so
          # we pass it in case r == 0 or d == 0 (or d == 1 but we're
          # making a strictly convex cone).
          K = Cone(rays, lattice=L)
  
          # Now, some of the rays that we generated were probably redundant,
          # so we need to come up with more. We can obviously stop if K
          # becomes the entire ambient vector space.
          #
          # We're still not guaranteed to have the correct number of
          # rays after this! Since we normalize the generators in the
          # loop above, we can jump from two to four generators by
          # adding e.g. (1,1) to [(0,1), (0,-1)]. Rather than trying to
          # mangle what we have, we just start over if we get a cone
          # that won't work.
          #
          while r > K.nrays() and not K.is_full_space():
              rays.append(L.random_element())
              K = Cone(rays, lattice=L)
              rays = list(K.rays()) # Avoid re-normalizing next time around
  
  
          if strictly_convex is not None:
              if strictly_convex:
                  if not K.is_strictly_convex():
                      # The user wants a strictly convex cone, but
                      # didn't get one. So let's take our rays, and give
                      # them all either (strictly) positive or negative
                      # leading coordinates. This makes the resulting
                      # cone strictly convex. Whether or not those
                      # coordinates become positive/negative is chosen
                      # randomly.
                      from random import choice
                      pm = choice([-1,1])
  
                      # rays has immutable elements
                      rays = [copy.copy(ray) for ray in rays]
  
                      for i, ray in enumerate(rays):
                          rays[i][0] = pm * (ray[0].abs() + 1)
  
                      K = Cone(rays, lattice=L)
              else:
                  # The user requested that the cone be NOT strictly
                  # convex. So it should contain some line...
                  if K.is_strictly_convex():
                      # ...but it doesn't. If K has at least two rays,
                      # we can just make the second one a multiple of
                      # the first -- then K will contain a line. If K
                      # has fewer than two rays, we punt.
                      if len(rays) >= 2:
                          rays[1] = -rays[0]
                          K = Cone(rays, lattice=L)
  
          if is_valid(K):
              # Loop if we don't have a valid cone.
              return K
  