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Journal of Logic and Computation 2003 13(5):689-706; doi:10.1093/logcom/13.5.689
© 2003 by Oxford University Press
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Original Article

Strong Normalization of Herbelin's Explicit Substitution Calculus with Substitution Propagation

Roy Dyckhoff1 and Christian Urban2

1 School of Computer Science, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland. E-mail: rd{at}dcs.st-and.ac.uk 2 Corpus Christi College, Cambridge CB2 1RH, UK. cu200{at}dpmms.cam.ac.uk

Herbelin presented (at CSL'94) a simple sequent calculus for minimal implicational logic, extensible to full firstorder intuitionistic logic, with a complete system of cut-reduction rules which is both confluent and strongly normalizing. Some of the cut rules may be regarded as rules to construct explicit substitutions. He observed that the addition of a cut permutation rule, for propagation of such substitutions, breaks the proof of strong normalization; the implicit conjecture is that the rule may be added without breaking strong normalization. We prove this conjecture, thusshowing how to model beta-reduction in his calculus (extended with rules toallow cut permutations).

Keywords: Explicit substitution, lambda-calculus, strong normalization.

Received 5 February 2002.


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