# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
# have to explicitly mangle our sitedir here so that "mjo.cone"
# resolves.
from os.path import abspath
from site import addsitedir
addsitedir(abspath('../../'))

from sage.all import *
from mjo.cone.cone import lyapunov_rank

def rearrangement_cone(p,n):
    r"""
    Return the rearrangement cone of order ``p`` in ``n`` dimensions.

    The rearrangement cone in ``n`` dimensions has as its elements
    vectors of length ``n``. For inclusion in the cone, the smallest
    ``p`` components of a vector must sum to a nonnegative number.

    For example, the rearrangement cone of order ``p == 1`` has its
    single smallest component nonnegative. This implies that all
    components are nonnegative, and that therefore the rearrangement
    cone of order one is the nonnegative orthant.

    When ``p == n``, the sum of all components of a vector must be
    nonnegative for inclusion in the cone. That is, the cone is a
    half-space in ``n`` dimensions.

    INPUT:

    - ``p`` -- The number of components to "rearrange."

    - ``n`` -- The dimension of the ambient space for the resulting cone.

    OUTPUT:

    A polyhedral closed convex cone object representing a rearrangement
    cone of order ``p`` in ``n`` dimensions.

    EXAMPLES:

    The rearrangement cones of order one are nonnegative orthants::

        sage: rearrangement_cone(1,1) == Cone([(1,)])
        True
        sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
        True
        sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
        True

    When ``p == n``, the resulting cone will be a half-space, so we
    expect its lineality to be one less than ``n`` because it will
    contain a hyperplane but is not the entire space::

        sage: rearrangement_cone(5,5).lineality()
        4

    All rearrangement cones are proper::

        sage: all([ rearrangement_cone(p,n).is_proper()
        ....:               for n in range(10)
        ....:               for p in range(n) ])
        True

    The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
    dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise::

        sage: all([ lyapunov_rank(rearrangement_cone(p,n)) == n
        ....:               for n in range(2, 10)
        ....:               for p in [1, n-1] ])
        True
        sage: all([ lyapunov_rank(rearrangement_cone(p,n)) == 1
        ....:               for n in range(3, 10)
        ....:               for p in range(2, n-1) ])
        True

    TESTS:

    The rearrangement cone is permutation-invariant::

        sage: n = ZZ.random_element(2,10).abs()
        sage: p = ZZ.random_element(1,n)
        sage: K = rearrangement_cone(p,n)
        sage: P = SymmetricGroup(n).random_element().matrix()
        sage: all([ K.contains(P*r) for r in K.rays() ])
        True

    """

    def d(j):
        v = [1]*n    # Create the list of all ones...
        v[j] = 1 - p # Now "fix" the ``j``th entry.
        return v

    V = VectorSpace(QQ, n)
    G = V.basis() + [ d(j) for j in range(n) ]
    return Cone(G)