  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 finitely 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 infinitely branching, due to the fact that, in
standard context categories, parametric rules such as the beta-rule can be
represented only by infinitely 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 finitely branching transition system. This result
is achieved by considering an encoding of lambda-calculus in Combinatory Logic.
