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The Barendregt Cube with Definitions and Generalised Reduction
, 1997
"... 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, ..."
Abstract

Cited by 37 (17 self)
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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...
The Soundness of Explicit Substitution with Nameless Variables
, 1995
"... We show the soundness of a λcalculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. 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 fl ..."
Abstract

Cited by 2 (1 self)
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We show the soundness of a λcalculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. 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 stepwise reduction and the first clear account of exactly when ffreduction is needed.
A Semantics for stepwise substitution and reduction
, 1995
"... We show the soundness of a calculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. 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 fla ..."
Abstract
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We show the soundness of a calculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. 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 stepwise reduction and the first clear account of exactly when ffreduction is needed. Keywords: De Bruijn's indices, variable updating, substitution, reduction, 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 implicit rather than explicit substitution. Implementations of the calculus provide their own explicit substitution procedures as in HOL [GM 93], Nuprl [Con 86] and Authomath [NGdV 94]. Furthermore, research on theories of explicit substitution has been striving lately ([HL 89], [ACCL 91], [KN 93], [Mel 95], [BBLR 95] and [KR 95]). In this...