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Isabelle Formalization of Set Theoretic Structures and Set Comprehensions

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Mathematical Aspects of Computer and Information Sciences (MACIS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10693))

Abstract

Reasoning about computers and programming languages on paper is most often done with set theory, while most proof assistant formalizations of languages and programs use alternative mathematical foundations. One of the few exceptions has been Mizar where the Simple Concrete Model of computers has been used to verify programs expressed as abstract programming language instruction sequences. The model uses extended set theory features including structures and Fraenkel set comprehension operators. In this paper we show how to formally specify such objects in the Isabelle object logic implementing the Mizar foundations as definitional extensions. To show the adequacy and usability of the mechanisms, we reformalize a number of Mizar definitions and theorems related to structures and set comprehensions, including both mathematical and programming language examples: groups, machines and properties of computer memory states.

The paper has been supported by the resources of the Polish National Science Center granted by decision n\(^\circ \)DEC-2015/19/D/ST6/01473.

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Acknowledgements

This work has been supported by the European Research Council (ERC) grant no. 714034 SMART and the Polish National Science Center granted by decision n\(^\circ \)DEC-2015/19/D/ST6/01473.

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Authors and Affiliations

  1. Universität Innsbruck, Innsbruck, Austria

    Cezary Kaliszyk

  2. Uniwersytet w Białymstoku, Białystok, Poland

    Karol Pąk

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  1. Cezary Kaliszyk
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  2. Karol Pąk
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Correspondence to Karol Pąk .

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Editors and Affiliations

  1. University of Paderborn, Paderborn, Germany

    Johannes Blömer

  2. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  3. Johannes Kepler University Linz, Linz, Austria

    Temur Kutsia

  4. SBA Research, Vienna, Austria

    Dimitris E. Simos

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Kaliszyk, C., Pąk, K. (2017). Isabelle Formalization of Set Theoretic Structures and Set Comprehensions. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-72453-9_12

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  • Online ISBN: 978-3-319-72453-9

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