Concrete Stream Calculus—An extended study

Abstract

This paper shows how to reason about streams concisely and precisely. Streams, infinite sequences of elements, live in a coworld: they are given by a coinductive datatype, operations on streams are implemented by corecursive programs, and proofs are typically concocted using coinduction. This paper offers an alternative to coinduction. Suitably restricted, stream equations possess unique solutions. This property gives rise to a simple and attractive proof technique, essentially bringing equational reasoning to the coworld. We redevelop the theory of recurrences, finite calculus and generating functions using streams and stream operators, building on the cornerstone of unique solutions. The paper contains a smörgåsbord of examples: we study recursion elimination, investigate the binary carry sequence, explore Sprague-Grundy numbers and present two proofs of Moessner's Theorem. The calculations benefit from the rich structure of streams. As the type of streams is an applicative functor we can effortlessly lift operations and their properties to streams. In combination with Haskell's facilities for overloading, this greatly contributes to conciseness of notation. The development is indeed constructive: streams and stream operators are implemented in Haskell, usually by one-liners. The resulting calculus or library, if you wish, is elegant and fun to use.