Explicit substitution calculi with de Bruijn indices and intersection type systems

Explicit substitution calculi propose solutions to the main drawback of the λ-calculus: substitutiondefined as a meta-operation in the system. By making explicit the process of substitution, thetheoretical system gets closer to an eventual implementation. Furthermore, for implementationpurposes, many explicit substitution systems are written with de Bruijn indices. The λ-calculuswith de Bruijn indices, called λdB , assembles each α-class of λ-terms in a unique term, which ismore “machine-friendly” than the classical version with variables. Intersection types (IT) providefinitary type polymorphism satisfying important properties like principal typing (PT), which allowsthe type system to include features such as data abstraction (modularity) and separate compilation.Although some explicit substitution calculi with simple type systems are well investigated, providingnice applications such as specialised implementations of higher order unification, more elaboratedtype systems such as IT have not been proposed/studied for these calculi. In an earlier work,we introduced IT systems for two explicit substitution calculi, λσ and λse, conjecturing them tosatisfy the basic property of subject reduction, which guarantees the preservation of types duringcomputations. In this paper, we take a deeper look at these systems, providing an insight intotheir development which helps us construct for the first time the proofs of subject reduction omittedbefore. This new result also 1) enables us to prove another new result: subject reduction for an ITsystem for λdB , and 2) allows us to introduce for the first time an IT system for the λυ-calculus.