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

Conference paper

DOI: 10.1007/978-3-319-43144-4_13

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9807)
Cite this paper as:
Kunčar O., Popescu A. (2016) From Types to Sets by Local Type Definitions in Higher-Order Logic. In: Blanchette J., Merz S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science, vol 9807. Springer, Cham

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 soundness. 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.

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Fakultät für InformatikTechnische Universität MünchenMunichGermany
  2. 2.Department of Computer Science, School of Science and TechnologyMiddlesex UniversityLondonUK
  3. 3.Institute of Mathematics Simion Stoilow of the Romanian AcademyBucharestRomania

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