Rewriting Modulo β in the λΠ-Calculus Modulo

The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduc...

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Main Author: Ronan Saillard
Format: Article
Language:English
Published: Open Publishing Association 2015-07-01
Series:Electronic Proceedings in Theoretical Computer Science
Online Access:http://arxiv.org/pdf/1507.08055v1
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spelling doaj-f7062a97f1ff423d9beb2ebb2da99d2c2020-11-24T23:01:18ZengOpen Publishing AssociationElectronic Proceedings in Theoretical Computer Science2075-21802015-07-01185Proc. LFMTP 20158710110.4204/EPTCS.185.6:4Rewriting Modulo β in the λΠ-Calculus ModuloRonan Saillard0 MINES ParisTech, PSL Research University, France The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with beta-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference between the beta-reduction and rewrite rules with lambda-abstraction on their left-hand side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus Modulo. We prove that confluence of rewriting modulo beta is enough to ensure subject reduction and uniqueness of types. We achieve our goal by encoding the lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a consequence, we also make the confluence results for HRSs available for the lambda-Pi-calculus Modulo.http://arxiv.org/pdf/1507.08055v1
collection DOAJ
language English
format Article
sources DOAJ
author Ronan Saillard
spellingShingle Ronan Saillard
Rewriting Modulo β in the λΠ-Calculus Modulo
Electronic Proceedings in Theoretical Computer Science
author_facet Ronan Saillard
author_sort Ronan Saillard
title Rewriting Modulo β in the λΠ-Calculus Modulo
title_short Rewriting Modulo β in the λΠ-Calculus Modulo
title_full Rewriting Modulo β in the λΠ-Calculus Modulo
title_fullStr Rewriting Modulo β in the λΠ-Calculus Modulo
title_full_unstemmed Rewriting Modulo β in the λΠ-Calculus Modulo
title_sort rewriting modulo β in the λπ-calculus modulo
publisher Open Publishing Association
series Electronic Proceedings in Theoretical Computer Science
issn 2075-2180
publishDate 2015-07-01
description The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with beta-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference between the beta-reduction and rewrite rules with lambda-abstraction on their left-hand side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus Modulo. We prove that confluence of rewriting modulo beta is enough to ensure subject reduction and uniqueness of types. We achieve our goal by encoding the lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a consequence, we also make the confluence results for HRSs available for the lambda-Pi-calculus Modulo.
url http://arxiv.org/pdf/1507.08055v1
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