+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+ sage: K_S = rho(K)
+ sage: K_SP = rho(K_S.dual()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = rho(K_S, K_S.dual())
+ sage: K_SP.is_proper()
+ True
+
+ Test the proposition in our paper concerning the duals and
+ restrictions. Generate a random cone, then create a subcone of
+ it. The operation of dual-taking should then commute with rho::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W = rho(K, J)
+ sage: K_star_W_star = rho(K.dual(), J).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W = rho(K, J)
+ sage: K_star_W_star = rho(K.dual(), J).dual()
+ sage: basically_the_same(K_W, K_star_W_star)
+ True