Coinductive properties of Lipschitz functions on streams

A simple hierarchical structure is imposed on the set of Lipschitz functions on streams (i.e. sequences over a fixed alphabet set) under the standard metric. We prove that sets of non-expanding and contractive functions are closed under a certain coiterative construction. The closure property is used to construct new final stream coalgebras over finite alphabets. For an example, we show that the 2-adic extension of the Collatz function and certain variants yield final bitstream coalgebras.