from sage.all import *

def is_lyapunov_like(L,K):
    r"""
    Determine whether or not ``L`` is Lyapunov-like on ``K``.

    We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
    L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
    `\left\langle x,s \right\rangle` in the complementarity set of
    ``K``. It is known [Orlitzky]_ that this property need only be
    checked for generators of ``K`` and its dual.

    There are faster ways of checking this property. For example, we
    could compute a `lyapunov_like_basis` of the cone, and then test
    whether or not the given matrix is contained in the span of that
    basis. The value of this function is that it works on symbolic
    matrices.

    INPUT:

    - ``L`` -- A linear transformation or matrix.

    - ``K`` -- A polyhedral closed convex cone.

    OUTPUT:

    ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
    and ``False`` otherwise.

    .. WARNING::

        If this function returns ``True``, then ``L`` is Lyapunov-like
        on ``K``. However, if ``False`` is returned, that could mean one
        of two things. The first is that ``L`` is definitely not
        Lyapunov-like on ``K``. The second is more of an "I don't know"
        answer, returned (for example) if we cannot prove that an inner
        product is zero.

    REFERENCES:

    M. Orlitzky. The Lyapunov rank of an improper cone.
    http://www.optimization-online.org/DB_HTML/2015/10/5135.html

    EXAMPLES:

    The identity is always Lyapunov-like in a nontrivial space::

        sage: set_random_seed()
        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
        sage: L = identity_matrix(K.lattice_dim())
        sage: is_lyapunov_like(L,K)
        True

    As is the "zero" transformation::

        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
        sage: R = K.lattice().vector_space().base_ring()
        sage: L = zero_matrix(R, K.lattice_dim())
        sage: is_lyapunov_like(L,K)
        True

        Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
        on ``K``::

        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
        sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
        True

    """
    return all([(L*x).inner_product(s) == 0
                for (x,s) in K.discrete_complementarity_set()])

def LL_cone(K):
    gens = K.lyapunov_like_basis()
    L = ToricLattice(K.lattice_dim()**2)
    return Cone([ g.list() for g in gens ], lattice=L, check=False)

def Sigma_cone(K):
    gens = K.cross_positive_operator_gens()
    L = ToricLattice(K.lattice_dim()**2)
    return Cone([ g.list() for g in gens ], lattice=L, check=False)

def Z_cone(K):
    gens = K.Z_operator_gens()
    L = ToricLattice(K.lattice_dim()**2)
    return Cone([ g.list() for g in gens ], lattice=L, check=False)

def pi_cone(K1, K2=None):
    if K2 is None:
        K2 = K1
    gens = K1.positive_operator_gens(K2)
    L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
    return Cone([ g.list() for g in gens ], lattice=L, check=False)