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Logic Journal of IGPL Advance Access originally published online on May 15, 2007
Logic Journal of IGPL 2007 15(2):121-147; doi:10.1093/jigpal/jzm003
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

The Weak Normalization of the Simply Typed {lambda}se-calculus

Ariel Arbiser

Department of Computer Science, University of Buenos Aires, Pabellón I - Ciudad Universitaria (1428) Buenos Aires, Argentina. E-mail: arbiser{at}dc.uba.ar

Fairouz Kamareddine

School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland. E-mail: fairouz{at}macs.hw.ac.uk

Alejandro Rios

Department of Computer Science, University of Buenos Aires, Pabellón I - Ciudad Universitaria (1428) Buenos Aires, Argentina. E-mail: rios{at}dc.uba.ar


  Abstract

In this paper, we show the weak normalization (WN) of the simply-typed {lambda}se-calculus with open terms where abstractions are decorated with types, and metavariables, de Bruijn indices and updating operators are decorated with environments. We show a proof of WN using the {lambda}{omega}e-calculus, a calculus isomorphic to Formula. This proof is strongly influenced by Goubault-Larrecq's proof of WN for the {lambda} {sigma}-calculus but with subtle differences which show that the two styles require different attention. Furthermore, we give a new calculus {lambda} {omega}'e which works like {lambda}se but which is closer to {lambda} {sigma} than {lambda}{omega}e. For both {lambda}{omega}e and {lambda} {omega}'e we prove WN for typed semi-open terms (i.e. those which allow term variables but no substitution variables), unlike the result of Goubault-Larrecq which covered all {lambda} {sigma} open terms.

Key Words: lambda calculus • explicit substitution • weak normalization • simple typing

Received for publication 24 November 2004. Revision received 23 December 2005.
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