In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, Definitions, Barendregt Cube, Church Rosser, Subject Reduction, Strong Normalisation. Contents 1 Introduction 3 1.1 Why generalised reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why definition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The item notation for definitions and generalised reduction . . . . . . . . . . 4 2 The item notation 7 3 The ordinary typing relation and its properties 10 3.1 The typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Properties of the ordinary typing relation . . . . . . . . . . . . . . . . . . . . 13 4 Generalising reduction in the Cube 15 4.1 The generalised...

We show the soundness of a -calculus B where de Bruijn indices are used, substitution is explicit, and reduction is step-wise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and step-wise reduction and the first clear account of exactly when ff-reduction is needed. Keywords: Explicit Substitution, de Bruijn indices, Variable names, Soundness. 1. Introduction Variables play a very demanding role in the reduction and substitution of the -calculus. This has lead in many cases to using explicit rather than implicit substitution. Implementations of the -calculus provide their own explicit substitution procedures as in Nuprl 9 and Automath 23 . Furthermore, research on theories of explicit substitution has been striving lately 5;12;13;22;4;18 . In this paper, we extend the calculus of [13] (which is influenced by Automath...

3.1 Presentation of the λ-Calculus..................... 5