+def is_cross_positive(L,K):
+ r"""
+ Determine whether or not ``L`` is cross-positive on ``K``.
+
+ We say that ``L`` is cross-positive 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 that this property need only be
+ checked for generators of ``K`` and its dual.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
+
+ ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is cross-positive
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ cross-positive 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 nonnegative.
+
+ EXAMPLES:
+
+ The identity is always cross-positive 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_cross_positive(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_cross_positive(L,K)
+ True
+
+ Everything in ``K.cross_positive_operator_gens()`` should be
+ cross-positive on ``K``::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_cross_positive(L,K)
+ ....: for L in K.cross_positive_operator_gens() ])
+ True
+
+ """
+ if L.base_ring().is_exact() or L.base_ring() is SR:
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
+
+