+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K.dim() == _restrict_to_space(K,K.span()).dim()
+ True
+
+ Nor should it affect the lineality of a cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
+ True
+
+ No matter which space we restrict to, the lineality should not
+ increase::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: S = K.span(); P = K.dual().span()
+ sage: K.lineality() >= _restrict_to_space(K,S).lineality()
+ True
+ sage: K.lineality() >= _restrict_to_space(K,P).lineality()
+ True
+
+ If we do this according to our paper, then the result is proper::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
+ sage: K_SP.is_proper()
+ True
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
+ 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
+ _restrict_to_space::
+
+ sage: set_random_seed()
+ sage: J = random_cone(max_ambient_dim = 8)
+ sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
+ sage: K_W_star = _restrict_to_space(K, J.span()).dual()
+ sage: K_star_W = _restrict_to_space(K.dual(), J.span())
+ sage: _basically_the_same(K_W_star, K_star_W)
+ True
+
+ """
+ # First we want to intersect ``K`` with ``W``. The easiest way to
+ # do this is via cone intersection, so we turn the subspace ``W``
+ # into a cone.
+ W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice())
+ K = K.intersection(W_cone)
+
+ # We've already intersected K with the span of K2, so every
+ # generator of K should belong to W now.
+ K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
+
+ L = ToricLattice(W.dimension())
+ return Cone(K_W_rays, lattice=L)