  The minimal set of Shannon-type inequalities (referred to as elemental
inequalities), plays a central role in determining whether a given inequality
is Shannon-type. Often, there arises a situation where one needs to check
whether a given inequality is a constrained Shannon-type inequality. Another
important application of elemental inequalities is to formulate and compute the
Shannon outer bound for multi-source multi-sink network coding capacity. Under
this formulation, it is the region of feasible source rates subject to the
elemental inequalities and network coding constraints that is of interest.
Hence it is of fundamental interest to identify the redundancies induced
amongst elemental inequalities when given a set of functional dependence
constraints. In this paper, we characterize a minimal set of Shannon-type
inequalities when functional dependence constraints are present.
