from sage.all import *

def trivial_cone(n, lattice=None):
    r"""
    The trivial cone with no generators in ``n`` dimensions.

    INPUT:

      - ``n`` -- the dimension of the ambient space.

      - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
                       to be passed to the :func:`Cone` constructor.

    OUTPUT:

    The trivial cone with no generators.

    If a ``lattice`` was specified, then the resulting cone will live in
    that lattice unless its rank is incompatible with the dimension
    ``n`` (in which case a ``ValueError`` is raised).

    SETUP::

        sage: from mjo.cone.trivial_cone import trivial_cone

    EXAMPLES::

    We can construct the trivial cone as the nonnegative orthant in a
    trivial ambient space::

        sage: trivial_cone(0)
        0-d cone in 0-d lattice N

    Or in a nontrivial ambient space::

        sage: trivial_cone(3)
        0-d cone in 3-d lattice N

    If a ``lattice`` is given, the trivial cone will live in that lattice::

        sage: L = ToricLattice(3, 'M')
        sage: trivial_cone(3, lattice=L)
        0-d cone in 3-d lattice M

    TESTS:

    An error is raised if the rank of the lattice disagrees with ``n``::

        sage: L = ToricLattice(1, 'M')
        sage: trivial_cone(3, lattice=L)
        Traceback (most recent call last):
        ...
        ValueError: lattice rank=1 and dimension n=3 are incompatible

    """
    if lattice is None:
        lattice = ToricLattice(n)

    if lattice.rank() != n:
        raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
                         %
                         (lattice.rank(), n))

    return Cone([], lattice)