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

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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.

Keywords

HOL Isabelle Local typedef Type definition Relativization Type classes Overloading Dependent types Model Consistency Transfer Type-based theorems Set-based theorems 

Notes

Acknowledgements

We thank reviewers for useful comments and suggestions. The ITP 2016 reviewers helped us to improve the previous conference version of the paper. We thank Fabian Immler and Dmitriy Traytel for interesting discussions on Types to Terms and HOL dependent typing, respectively. We are indebted to Johannes Hölzl to introduce us to HOL-Algebra and to remind us that not every locale can be translated into a corresponding type class in Isabelle. We gratefully acknowledge support from DFG through grant NI 491/13-3 and from EPSRC through grant EP/N019547/1.

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Authors and Affiliations

  1. 1.Fakultät für InformatikTechnische Universität MünchenMunichGermany
  2. 2.Department of Computer Science, School of Science and TechnologyMiddlesex UniversityLondonUK

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