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λν, a calculus of explicit substitutions which preserves strong normalisation

Published online by Cambridge University Press:  07 November 2008

Zine-El-Abidine Benaissa
Affiliation:
Centre de Recherche en Informatique de Nancy (CNRS), France e-mail: Zine-El-Abidine.Benaissa@loria.fr
Daniel Briaud
Affiliation:
Centre de Recherche en Informatique de Nancy (CNRS), France e-mail: Daniel.Briaud.Benaissa@loria.fr
Pierre Lescanne
Affiliation:
INRIA-Lorraine, Campus Scientifique, BP 239, F54506 Vandœuvre-lès-Nancy, France e-mail: Pierre.Lescanne@loria.fr
Jocelyne Rouyer-Degli
Affiliation:
INRIA-Lorraine, Campus Scientifique, BP 239, F54506 Vandœuvre-lès-Nancy, France e-mail: Jocelyne.Rouyer@loria.fr

Abstract

Explicit substitutions were proposed by Abadi, Cardelli, Curien, Hardin and Lévy to internalise substitutions into λ-calculus and to propose a mechanism for computing on substitutions. λν is another view of the same concept which aims to explain the process of substitution and to decompose it in small steps. It favours simplicity and preservation of strong normalisation. This way, another important property is missed, namely confluence on open terms. In spirit, λν is closely related to another calculus of explicit substitutions proposed by de Bruijn and called CλξΦ. In this paper, we introduce λν, we present CλξΦ in the same framework as λν and we compare both calculi. Moreover, we prove properties of λν; namely λν correctly implements β reduction, λν is confluent on closed terms, i.e. on terms of classical λ-calculus and on all terms that are derived from those terms, and finally λν preserves strong normalisation in the following sense: strongly β normalising terms are strongly λν normalising.

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Articles
Copyright
Copyright © Cambridge University Press 1996

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References

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