Pointfree factorization of operation refinement

Abstract

The standard operation refinement ordering is a kind of “meet of op- posites”: non-determinism reduction suggests “smaller” behaviour while increase of definition suggests “larger” behaviour. Groves’ factorization of this ordering into two simpler relations, one per refinement concern, makes it more mathe- matically tractable but is far from fully exploited in the literature. We present a pointfree theory for this factorization which is more agile and calculational than the standard set-theoretic approach. In particular, we show that factorization leads to a simple proof of structural refinement for arbitrary parametric types and ex- ploit factor instantiation across different subclasses of (relational) operation. The prospect of generalizing the factorization to coalgebraic refinement is discussed