r"""
The nonnegative orthant in `\mathbb{Z}^{n}`. I'm sick and tired of
typing it.
"""

from sage.all import *

def nonnegative_orthant(n, lattice=None):
    r"""
    The nonnegative orthant 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 convex cone having ``n`` standard basis vectors as its
    generators. Each generating ray will have the integer ring as its
    base ring.

    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.nonnegative_orthant import nonnegative_orthant

    EXAMPLES::

        sage: nonnegative_orthant(3).rays()
        N(1, 0, 0),
        N(0, 1, 0),
        N(0, 0, 1)
        in 3-d lattice N

    TESTS:

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

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

    The nonnegative orthant is a proper cone::

        sage: n = ZZ.random_element(10)
        sage: K = nonnegative_orthant(n)
        sage: K.is_proper()
        True

    If a ``lattice`` was given, it is actually used::

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

    Unless the rank of the lattice disagrees with ``n``::

        sage: L = ToricLattice(1, 'M')
        sage: nonnegative_orthant(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))

    I = identity_matrix(ZZ,n)
    return Cone(I.rows(), lattice)