Proving Behavioural Apartness

Bisimilarity is a central notion for coalgebras. In recent work, Geuvers and Jacobs suggest to focus on apartness, which they define by dualising coalgebraic bisimulations. This yields the possibility of finite proofs of distinguishability for a wide variety of state-based systems. We propose behavioural apartness, defined by dualising behavioural equivalence rather than bisimulations. A motivating example is the subdistribution functor, where the proof system based on bisimilarity requires an infinite quantification over couplings, whereas behavioural apartness instantiates to a finite rule. In addition, we provide optimised proof rules for behavioural apartness and show their use in several examples.