Add tests for Corollary 2 in the paper (lineality of pi-star).

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 8b07f86b329e191e105994e83676f3e30d8c4220..332d4b24ed93ce9daac2a5779df94068d6479838 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -118,7 +118,7 @@ def motzkin_decomposition(K):
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
         sage: (P,S) = motzkin_decomposition(K)
-        sage: x = K.random_element()
+        sage: x = K.random_element(ring=QQ)
         sage: P.contains(x) or S.contains(x)
         True
         sage: x.is_zero() or (P.contains(x) != S.contains(x))
@@ -257,7 +257,7 @@ def positive_operator_gens(K):
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
         sage: pi_of_K = positive_operator_gens(K)
-        sage: all([ K.contains(P*K.random_element()) for P in pi_of_K ])
+        sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
         True
 
     A random element of the positive operator cone should send the
@@ -268,7 +268,7 @@ def positive_operator_gens(K):
         sage: pi_of_K = positive_operator_gens(K)
         sage: L = ToricLattice(K.lattice_dim()**2)
         sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
-        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
         sage: all([ K.contains(P*x) for x in K ])
         True
 
@@ -280,8 +280,8 @@ def positive_operator_gens(K):
         sage: pi_of_K = positive_operator_gens(K)
         sage: L = ToricLattice(K.lattice_dim()**2)
         sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
-        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
-        sage: K.contains(P*K.random_element())
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+        sage: K.contains(P*K.random_element(ring=QQ))
         True
 
     The lineality space of the dual of the cone of positive operators
@@ -303,6 +303,22 @@ def positive_operator_gens(K):
         sage: actual == expected
         True
 
+    The lineality of the dual of the cone of positive operators
+    is known from its lineality space::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: n = K.lattice_dim()
+        sage: m = K.dim()
+        sage: l = K.lineality()
+        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)
+        sage: actual = pi_cone.dual().lineality()
+        sage: expected = l*(m - l) + m*(n - m)
+        sage: actual == expected
+        True
+
     The dimension of the cone of positive operators is given by the
     corollary in my paper::