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