Invariants, Bisimulations and the Correctness of Coalgebraic Reenements

Coalgebraic speciications are used to formally describe the behaviour of classes in object-oriented languages. In this paper, a general notion of reenement between two such coalgebraic speciications is deened, capturing the idea that one \concrete" class speciication realises the behaviour of the other, \abstract" class speciication. Two (complete) proof-techniques are given to establish such reenements: one involving an invariant (a predicate that is closed under transitions) on the concrete class, and one involving a bisimula-tion (a relation that is closed under transitions) between the concrete and the abstract class. The latter can only be used if the abstract class is what we call totally speciied. Parts of the underlying theory of invariants and bisimulations in a coalgebraic setting are included, involving least and greatest invariants and connections between invariants and bisimulations. Also, the proof-principles are illustrated in examples (which are fully formalised and veriied in pvs).