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A Proof Pearl with the Fan Theorem and Bar Induction

Walking through Infinite Trees with Mixed Induction and Coinduction

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Programming Languages and Systems (APLAS 2011)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7078))

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Abstract

We study temporal properties over infinite binary red-blue trees in the setting of constructive type theory. We consider several familiar path-based properties, typical to linear-time and branching-time temporal logics like LTL and CTL*, and the corresponding tree-based properties, in the spirit of the modal μ-calculus. We conduct a systematic study of the relationships of the path-based and tree-based versions of “eventually always blueness” and mixed inductive-coinductive “almost always blueness” and arrive at a diagram relating these properties to each other in terms of implications that hold either unconditionally or under specific assumptions (Weak Continuity for Numbers, the Fan Theorem, Lesser Principle of Omniscience, Bar Induction).

We have fully formalized our development with the Coq proof assistant.

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Author information

Authors and Affiliations

  1. Institute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, 12618, Tallinn, Estonia

    Keiko Nakata & Tarmo Uustalu

  2. Institutt for Informatikk, Universitet i Bergen, Postboks 7800, 5020, Bergen, Norway

    Marc Bezem

Authors
  1. Keiko Nakata
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  2. Tarmo Uustalu
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  3. Marc Bezem
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Parks Road, OX1 3QD, Oxford, UK

    Hongseok Yang

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Nakata, K., Uustalu, T., Bezem, M. (2011). A Proof Pearl with the Fan Theorem and Bar Induction. In: Yang, H. (eds) Programming Languages and Systems. APLAS 2011. Lecture Notes in Computer Science, vol 7078. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25318-8_26

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  • DOI: https://doi.org/10.1007/978-3-642-25318-8_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25317-1

  • Online ISBN: 978-3-642-25318-8

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