Relating Apartness and Bisimulation

A bisimulation for a coalgebra of functor on the category sets can be described via in relations, lifted functor. final then gives rise to coinduction principle, which states that two bisimilar elements are equal. For polynomial functors, this leads well-known descriptions. In present paper we look at dual notion "apartness". Intuitively, apart if there is positive way distinguish them. Phrased differently: and only they not bisimilar. Since apartness an inductive notion, by least fixed point, give proof system, derive apart. This system has derivation rules finite (using rules) fact. We study versus separate ways. First, weak forms labelled transition systems, where silent (tau) steps included, define corresponds another branching bisimulation. The used show bismilar. To support view cast number properties terms prove Next, also more general categorical situation indeed, bisimilarity precise sense: initial algebra induction principle. analogy, include powerset functor, semantics non-deterministic choice process-theory.