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...

This paper will concentrate on a new feature related to reshuffling terms so that more redexes become visible. The idea is explained as follows: Assume a redex is a `[' next to a `]'. What will happen in a term of the form `[ [ ] [ ] ]' ? We know that the two internal `[ ]' are redexes, but classical notation does not allow us to say that the outside `[' and `]' form a redex. In [BKN 9x], we generalised the notion of a redex from a pair of adjacent matching parentheses to simply a pair of matching parentheses. Hence, with generalised reduction all the three redexes are visible in `[ [ ] [ ] ]'. In this paper, we propose to reshuffle `[ [ ] [ ] ]' to `[ ] [ ] [ ]' where the first `[' has been moved next to the last `]'. The item notation enables us to see the matching parentheses and to reshuffle terms so that all matching paretheses become adjacent