+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
+ sage: actual == 3
+ True
+
+ The lineality of the cone of positive operators follows from the
+ description of its generators::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: n = K.lattice_dim()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.lineality()
+ sage: expected = n**2 - K.dim()*K.dual().dim()
+ sage: actual == expected
+ True
+
+ The trivial cone, full space, and half-plane all give rise to the
+ expected linealities::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.lineality()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: pi_cone.lineality() == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False)
+ sage: actual = pi_cone.lineality()
+ sage: actual == 2
+ True
+
+ A cone is proper if and only if its cone of positive operators
+ is proper::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: K.is_proper() == pi_cone.is_proper()
+ True
+
+ The positive operators of a permuted cone can be obtained by
+ conjugation::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
+ sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
+ sage: pi_of_pK = positive_operator_gens(pK)
+ sage: actual = Cone([t.list() for t in pi_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True