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From Types to Sets by Local Type Definition in Higher-Order Logic

From Types to Sets by Local Type Definition in Higher-Order Logic Types in higher-order logic (HOL) are naturally interpreted as nonempty sets. This intuition is reflected in the type definition rule for the HOL-based systems (including Isabelle/HOL), where a new type can be defined whenever a nonempty set is exhibited. However, in HOL this definition mechanism cannot be applied inside proof contexts. We propose a more expressive type definition rule that addresses the limitation and we prove its consistency. This higher expressive power opens the opportunity for a HOL tool that relativizes type-based statements to more flexible set-based variants in a principled way. We also address particularities of Isabelle/HOL and show how to perform the relativization in the presence of type classes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Automated Reasoning Springer Journals

From Types to Sets by Local Type Definition in Higher-Order Logic

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Computer Science; Mathematical Logic and Formal Languages; Artificial Intelligence; Mathematical Logic and Foundations; Symbolic and Algebraic Manipulation
ISSN
0168-7433
eISSN
1573-0670
DOI
10.1007/s10817-018-9464-6
Publisher site
See Article on Publisher Site

Abstract

Types in higher-order logic (HOL) are naturally interpreted as nonempty sets. This intuition is reflected in the type definition rule for the HOL-based systems (including Isabelle/HOL), where a new type can be defined whenever a nonempty set is exhibited. However, in HOL this definition mechanism cannot be applied inside proof contexts. We propose a more expressive type definition rule that addresses the limitation and we prove its consistency. This higher expressive power opens the opportunity for a HOL tool that relativizes type-based statements to more flexible set-based variants in a principled way. We also address particularities of Isabelle/HOL and show how to perform the relativization in the presence of type classes.

Journal

Journal of Automated ReasoningSpringer Journals

Published: Jun 4, 2018

References