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Pure type systems with more liberal rules

Published online by Cambridge University Press:  12 March 2014

Martin Bunder  and
Wil Dekkers
Martin Bunder
Affiliation:
Faculty of Informatics, Department of Mathematics, University of Wollongong NSW 2522, Australia, E-Mail: martin_bunder@uow.edu.au
Wil Dekkers
Affiliation:
Faculty of Mathematics and Computer Science, Catholic University, Nijmegen. The Netherlands E-mail:, wil@cs.kun.nl
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Abstract.

Pure Type Systems. PTSs, introduced as a generalisation of the type systems of Barendregt's lambda-cube, provide a foundation for actual proof assistants, aiming at the mechanic verification of formal proofs. In this paper we consider simplifications of some of the rules of PTSs. This is of independent interest for PTSs as this produces more flexible PTS-like systems, but it will also help, in a later paper, to bridge the gap between PTSs and systems of Illative Combinatory Logic.

First we consider a simplification of the start and weakening rules of PTSs. which allows contexts to be sets of statements, and a generalisation of the conversion rule. The resulting Set-modified PTSs or SPTSs, though essentially equivalent to PTSs, are closer to standard logical systems.

A simplification of the abstraction rule results in Abstraction-modified PTSs or APTSs. These turn out to be equivalent to standard PTSs if and only if a condition (*) holds. Finally we consider SAPTSs which have both modifications.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 4 , December 2001 , pp. 1561 - 1580
Copyright
Copyright © Association for Symbolic Logic 2001

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