from sage.all import *

def face_generated_by(K,S):
    r"""
    Return the intersection of all faces of ``K`` that contain ``S``.

    REFERENCES:

    .. [Tam] Bit-Shun Tam. On the duality operator of a convex cone. Linear
       Algebra and its Applications, 64:33-56, 1985, doi:10.1016/0024-3795(85)
       90265-4.

    SETUP::

        sage: from mjo.cone.faces import face_generated_by

    EXAMPLES:

    The face generated by a standard basis vector in the nonnegative
    orthant should be the cone consisting of all nonnegative multiples
    of that standard basis vector::

        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
        sage: F = Cone([(1,0,0)])
        sage: face_generated_by(K,F) == F
        True

    If we take a nontrivial combination of standard basis vectors in the
    nonnegative orthant, then the face that the combination generates
    should be the cone generated by those two basis vectors::

        sage: e1 = vector(QQ, [1,0,0])
        sage: e2 = vector(QQ, [0,1,0])
        sage: e3 = vector(QQ, [0,0,1])
        sage: K = Cone([e1,e2,e2])
        sage: F = [e1/2 + e2/2]
        sage: face_generated_by(K,F).is_equivalent(Cone([e1,e2]))
        True

     An error is raised if ``S`` is not contained in ``K``::

        sage: K = Cone([(1,0),(0,1)])
        sage: S = [(1,1), (-1,3)]
        sage: face_generated_by(K,S)
        Traceback (most recent call last):
        ...
        ValueError: S is not a subset of the cone

    TESTS:

    The face generated by <whatever> should be a face::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
        sage: S = [K.random_element() for idx in range(0,5)]
        sage: F = face_generated_by(K, S)
        sage: F.is_face_of(K)
        True

    The face generated by a set should always contain that set::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
        sage: S = [K.random_element() for idx in range(0,5)]
        sage: F = face_generated_by(K, S)
        sage: all([F.contains(x) for x in S])
        True

    The generators of a proper cone are all extreme vectors of the cone,
    and therefore generate their own faces::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8,
        ....:                 max_rays=10,
        ....:                 strictly_convex=True,
        ....:                 solid=True)
        sage: all([face_generated_by(K, [r]) == Cone([r]) for r in K])
        True

    For any point ``x`` in ``K`` and any face ``F`` of ``K``, we have
    that ``x`` is in the relative interior of ``F`` if and only if
    ``F`` is the face generated by ``x`` [Tam]_::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
        sage: x = K.random_element()
        sage: S = [x]
        sage: F = K.face_lattice().random_element()
        sage: expected = F.relative_interior_contains(x)
        sage: actual = (F == face_generated_by(K, S))
        sage: actual == expected
        True

    If ``F`` and ``G`` are two faces of ``K``, then the join of ``F``
    and ``G`` in the face lattice is equal to the face generated by
    ``F + G`` (in the Minkowski sense) [Tam]_::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
        sage: L = K.face_lattice()
        sage: F = L.random_element()
        sage: G = L.random_element()
        sage: expected = L.join(F,G)
        sage: actual = face_generated_by(K, F.rays() + G.rays())
        sage: actual == expected
        True

    Combining Proposition 3.1 and Corollary 3.9 in [Tam]_ gives the
    following equality for any ``y`` in ``K``::

        sage: set_random_seed()
        sage: K = random_cone(max_ambient_dim=8, max_rays=10)
        sage: y = K.random_element()
        sage: S = [y]
        sage: phi_y = face_generated_by(K,S)
        sage: points_cone_gens = list(K.rays()) + [-z for z in phi_y.rays()]
        sage: points_cone = Cone(points_cone_gens, K.lattice())
        sage: actual = phi_y.span(QQ)
        sage: expected = points_cone.linear_subspace()
        sage: actual == expected
        True

    """
    face_lattice = K.face_lattice()
    candidates = [F for F in face_lattice if all([F.contains(x) for x in S])]

    # K itself is a face of K, so unless we were given a set S that
    # isn't a subset of K, the candidates list will be nonempty.
    if len(candidates) == 0:
        raise ValueError('S is not a subset of the cone')
    else:
        return face_lattice.sorted(candidates)[0]