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

from sage.all import *

def nonnegative_orthant(n):
    r"""
    The nonnegative orthant in ``n`` dimensions.

    INPUT:

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

    OUTPUT:

    The convex cone having ``n`` standard basis vectors as its
    generators. Each generating ray will have the integer ring as its
    base ring.

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

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