Stream processing coalgebraically

We study various operations for splitting, partitioning, projecting and merging streams of data. These operations are motivated by their use in dataflow programming and stream processing languages. We use the framework of stream calculus and stream circuits for defining and proving properties of such operations using behavioural differential equations and coinduction proof principles. As a featured example we give proofs of results, observed by Moessner, from elementary number theory using our framework. We study the invariance of certain well patterned classes of streams, namely rational and algebraic streams, under splitting and merging. Finally we show that stream circuits extended with gates for dyadic split and merge are expressive enough to realise some non-rational algebraic streams, thereby going beyond ordinary stream circuits.