A formalized hierarchy of probabilistic system types: Proof pearl

Abstract

Numerous models of probabilistic systems are studied in the literature. Coalgebra has been used to classify them into system types and compare their expressiveness. In this work, we formalize the resulting hierarchy of probabilistic system types in Isabelle/HOL by modeling the semantics of the different systems as codatatypes. This approach yields simple and concise proofs, as bisimilarity coincides with equality for codatatypes. On the way, we develop libraries of bounded sets and discrete probability distributions and integrate them with the facility for (co)datatype definitions.