Principal typings in a restricted intersection type system for beta normal forms with de Bruijn indices

Abstract

The λ -calculus with de Bruijn indices assembles each α-class of λ -terms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable λ -terms through the property that a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in β -contractions, several calculi of explicit substitution were developed mostly with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborate type systems such as intersection types. In a previous work, we introduced a de Bruijn version of the λ-calculus with an intersection type system and proved that it preserves subject reduction, a basic property of type systems. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings forβ -normal forms is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss the failure of the subject reduction property and some possible solutions for it. © D. Ventura & M. Ayala-Rinćon & F. Kamareddine.