this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of -terms in . Quite apart from the new light it sheds on fi-reduction, such an analysis turns out to have several other benefits

We embed the standard #-calculus, denoted #, into two larger #-calculi, denoted # # and &# # . The standard notion of #-reduction for # corresponds to two new notions of reduction, # # for # # and &# # for &# # . A distinctive feature of our new calculus # # (resp., &# # ) is that, in every function application, an argument is used at most once (resp., exactly once) in the body of the function. We establish various connections between the three notions of reduction, #, # # and &# # . As a consequence, we provide an alternative framework to study the relationship between #-weak normalization and #-strong normalization, and give a new proof of the oft-mentioned equivalence between #-strong normalization of standard #-terms and typability in a system of "intersection types".