+def rename_lattice(L,s):
+ r"""
+ Change all names of the given lattice to ``s``.
+ """
+ L._name = s
+ L._dual_name = s
+ L._latex_name = s
+ L._latex_dual_name = s
+
+def span_iso(K):
+ r"""
+ Return an isomorphism (and its inverse) that will send ``K`` into a
+ lower-dimensional space isomorphic to its span (and back).
+
+ EXAMPLES:
+
+ The inverse composed with the isomorphism should be the identity::
+
+ sage: K = random_cone(max_dim=10)
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: phi_inv(phi(K)) == K
+ True
+
+ The image of ``K`` under the isomorphism should have full dimension::
+
+ sage: K = random_cone(max_dim=10)
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: phi(K).dim() == phi(K).lattice_dim()
+ True
+
+ The isomorphism should be an inner product space isomorphism, and
+ thus it should preserve dual cones (and commute with the "dual"
+ operation). But beware the automatic renaming of the dual lattice.
+ OH AND YOU HAVE TO SORT THE CONES::
+
+ sage: K = random_cone(max_dim=10, strictly_convex=False, solid=True)
+ sage: L = K.lattice()
+ sage: rename_lattice(L, 'L')
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: sorted(phi_inv( phi(K).dual() )) == sorted(K.dual())
+ True
+
+ We may need to isomorph twice to make sure we stop moving down to
+ smaller spaces. (Once you've done this on a cone and its dual, the
+ result should be proper.) OH AND YOU HAVE TO SORT THE CONES::
+
+ sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
+ sage: L = K.lattice()
+ sage: rename_lattice(L, 'L')
+ sage: (phi, phi_inv) = span_iso(K)
+ sage: K_S = phi(K)
+ sage: (phi_dual, phi_dual_inv) = span_iso(K_S.dual())
+ sage: J_T = phi_dual(K_S.dual()).dual()
+ sage: phi_inv(phi_dual_inv(J_T)) == K
+ True
+
+ """
+ phi_domain = K.sublattice().vector_space()
+ phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
+
+ # S goes from the new space to the cone space.
+ S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
+
+ # phi goes from the cone space to the new space.
+ def phi(J_orig):
+ r"""
+ Takes a cone ``J`` and sends it into the new space.
+ """
+ newrays = map(S.inverse(), J_orig.rays())
+ L = None
+ if len(newrays) == 0:
+ L = ToricLattice(0)
+
+ return Cone(newrays, lattice=L)
+
+ def phi_inverse(J_sub):
+ r"""
+ The inverse to phi which goes from the new space to the cone space.
+ """
+ newrays = map(S, J_sub.rays())
+ return Cone(newrays, lattice=K.lattice())
+
+
+ return (phi, phi_inverse)
+
+