First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly ï¬ nitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is inï¬ nitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by inï¬ nitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a ï¬ nitely branching transition system. This result is achieved by considering an encoding of lambda- calculus in Combinatory Logic.
