Rank 2 intersection types for local definitions and conditional expressions

We propose a rank 2 intersection type system with new typing rules for local definitions (let-expressions and letrec-expressions) and conditional expressions (if-expressions and match-expressions). This is a further step towards the use of intersection types in "real" programming languages.The technique for typing local definitions relies entirely on the principal typing property (i.e. it does not depend on particulars of rank 2 intersection), so it can be applied to any system with principal typings. The technique for typing conditional expressions, which is based on the idea of introducing metrics on types to "limit the use" of the intersection type constructor in the types assigned to the branches of the conditionals, is instead tailored to rank 2 intersection. However, the underlying idea might also be useful for other type systems.