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Interacting with Modal Logics in the Coq Proof Assistant

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Book cover Computer Science -- Theory and Applications (CSR 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9139))

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Abstract

This paper describes an embedding of higher-order modal logics in the Coq proof assistant. Coq’s capabilities are used to implement modal logics in a minimalistic manner, which is nevertheless sufficient for the formalization of significant, non-trivial modal logic proofs. The elegance, flexibility and convenience of this approach, from a user perspective, are illustrated here with the successful formalization of Gödel’s ontological argument.

Christoph Benzmüller—Supported by the German Research Foundation (DFG) under grant BE 2501/9-1.

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Notes

  1. 1.

    The Coq proof assistant was chosen because of the authors’ greater familiarity with the tactic language of this system. Nevertheless, the techniques presented here are likely to be useful for other proof assistants (e.g. Isabelle [15], HOL-Light [14]).

  2. 2.

    The keyword indicates a lambda abstraction: denotes the function \(\lambda x:t.p\), which takes an argument x (of type t) and returns p.

  3. 3.

    The underlying proof system of Coq (the Calculus of Inductive Constructions (CIC) [18]) is actually more sophisticated and minimalistic than the calculus shown in Fig. 1. But the calculus shown here suffices for the purposes of this paper. This calculus is classical, because of the double negation elimination rule. Although CIC is intuitionistic, it can be made classical by importing Coq ’s classical library, which adds the axiom of the excluded middle and the double negation elimination lemma.

  4. 4.

    The ND calculus with the rules from Figs. 1 and 2 is sound and complete relatively to the calculus of Fig.  1 extended with a necessitation rule and the modal axiom K [22]. Starting from a sound and Henkin-complete ND calculus for classical higher-order logic (cf. Fig. 1), the additional modal rules in Fig. 2 make it sound and Henkin-complete for the rigid higher-order modal logic K.

  5. 5.

    The proofs found automatically by the above provers indeed differ from the one presented here: e.g., the strong S5 principle used below (and by Scott) is not needed; the ATP proofs only rely on axiom B.

References

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Acknowledgements

We thank Cedric Auger and Laurent Théry, for their answers to our questions about Ltac in the Coq-Club mailing-list.

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Authors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  2. Vienna University of Technology, Vienna, Austria

    Bruno Woltzenlogel Paleo

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  1. Christoph Benzmüller
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  2. Bruno Woltzenlogel Paleo
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Correspondence to Bruno Woltzenlogel Paleo .

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Editors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Lomonosov Moscow State University and National Research University Higher School of Economics, Moscow, Russia

    Lev D. Beklemishev

  2. Moscow Institute of Physics and Technology, Moscow, Russia and Kazan (Volga Region) Federal University, Kazan, Kabardino-Balkar.Rep, Russia

    Daniil V. Musatov

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Benzmüller, C., Woltzenlogel Paleo, B. (2015). Interacting with Modal Logics in the Coq Proof Assistant. In: Beklemishev, L., Musatov, D. (eds) Computer Science -- Theory and Applications. CSR 2015. Lecture Notes in Computer Science(), vol 9139. Springer, Cham. https://doi.org/10.1007/978-3-319-20297-6_25

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  • DOI: https://doi.org/10.1007/978-3-319-20297-6_25

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