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...

In the area of foundations of mathematics and computer science, three related topics dominate. These are -calculus, type theory and logic.

We introduce a -calculus notation which enables us to detect in a term, more fi- redexes than in the usual notation. On this basis, we define an extended fi-reduction which is yet a subrelation of conversion. The Church Rosser property holds for this extended reduction. Moreover, we show that we can transform generalised redexes into usual ones by a process called "term reshuffling". Keywords: Item notation, Redexes, Church Rosser. Contents 1 Introduction 3 1.1 The item notation and visible redexes . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Generalising redexes and fi-reduction 7 2.1 Extending redexes from segments to couples . . . . . . . . . . . . . . . . . . . 7 2.2 Extending fi-reduction and the Church Rosser theorem . . . . . . . . . . . . . 8 3 Term reshuffling 10 3.1 Partitioning terms into bachelor and well-balanced segments . . . . . . . . . . 11 3.2 The reshuffling procedure...

In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend fi-reduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satisfies all its original properties (such as Church Rosser, Subject Reduction and Strong Normalisation). The notation itself is very simple: if I translates classical terms to our notation, then I(t 1 t 2 ) j (I(t 2 )ffi)I(t 1 ) and I( v:ae :t) j (ae v )I(t). For example, t j (( x7 :X4 :( x6 :X3 : x5 :X1!X2 :x 5 x 4 )x 3 )x 2 )x 1 , can be written in our item notation as I(t) j (x 1 ffi)(x 2 ffi)(X 4 x7 )(x 3 ffi)(X 3 x6 )((X 1 ! X 2 ) x5 )(x 4 ffi)x 5 where the visible redexes are based on all the matching ffi-couples. So here, the redexes are based on (x 2 ffi)(X 4 x7 ), (x 3 ffi)(X 3 x6 ) and (x 1 ffi)((X 1 ! X 2 ) x5 ). In classical notation however, only the redexes based on ( x7 :X4 : \Gamma \Gamma)x 2 and ( x6 :X3 : \Gamm...

We study lambda calculus and refine the notions of fi-reduction and fi-equality. In particular, we define the operation TS (term reshuffling) on -terms which reshuffles a term in such a way that more redexes become visible. Two terms are called shuffleequivalent if they have syntactically equivalent TS-images. The shuffle-equivalence classes are shown to divide the classes of fi-equal terms into smaller classes consisting of terms with similar reduction behaviour. The refinement of fi-reduction from a relation on terms to a relation on shuffle classes, called shuffle-reduction, allows one to make more redexes visible and to contract these newly visible redexes. This enables one to have more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. Moreover, this gain in reductional breadth is not at the expense of reductional length. We show that the -cube of [Barendregt 92] extended with shuffle...

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