Type system for λσ-calculus with all de Bruijn indices

Abstract

The λ-calculus was developed in the 30s as a mechanism of computing numerical functions. Its main operation is the β-conversion that performs the process of substitutions implicitly. In 1990 [1], Abadi, Cardelli, Curien, and Lèvy introduced an extension of λ-calculus, called λσ-calculus, using De Bruijn notation. The λσ-calculus is a calculus with explicit substitutions and its syntax contains just the natural number 1 and all the other natural numbers are encoded from the index 1 using closures with compositions of the shift operator (denoted by ↑). They showed that the λσ-calculus simulates the one-step β-reduction and it is confluent over its ground terms. In this work, we propose a variant of λσ-calculus with all De Bruin indices in its syntax, which we call λσdB-calculus and we adapt the proof of confluence of ground terms of λσ to the λσdB. We prove that λσdB enjoys the main properties of λσ: a) the subcalculus of substitutions is terminating and confluent, b) the full calculus simulates the β-reduction and c) it is confluent over ground terms, which are those without metavariables. Beside that, we define a system type for λσdB and we prove that the system is type checking decidable, soundness, equivalent with the the S1-system and with L1-system on σdB-nf.