  We present bipartite Bell-type inequalities which allow the two partners to
use some non-local resource. Such inequality can only be violated if the
parties use a resource which is more non-local than the one permitted by the
inequality. We introduce a family of N-inputs non-local machines, which are
generalizations of the well-known PR-box. Then we construct Bell-type
inequalities that cannot be violated by strategies that use one these new
machines. Finally we discuss implications for the simulation of quantum states.
