Common use of Coalgebra and bisimulation Clause in Contracts

Coalgebra and bisimulation. We recall coalgebras and bisimulations, and make explicit the underlying notion of progression, which we need in the sequel. A coalgebra for a functor F : Set → Set is a pair (X, α) consisting of a set X and a function α : X → FX. A function f : X → Y is an (F-coalgebra) homomorphism between (X, α) and (Y, β) if Ff ◦ α = β ◦ f. Definition 1. For a coalgebra α : X → FX and relations R, S ⊆ X × X, we say R progresses to S , denoted R > S , if there exists a γ : R → FS making the following diagram commute: π1 X ,r R π2 ,zX ,. Fπ1 ., Fπ2 ,. FX ,r FS A bisimulation is a relation R such that R > R. z,FX