A Proof Pearl with the Fan Theorem and Bar Induction

Walking through Infinite Trees with Mixed Induction and Coinduction
  • Keiko Nakata
  • Tarmo Uustalu
  • Marc Bezem
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7078)


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.


Binary Tree Weak Continuity Constructive Type Theory Blue Tree Basic Constructive Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Keiko Nakata
    • 1
  • Tarmo Uustalu
    • 1
  • Marc Bezem
    • 2
  1. 1.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia
  2. 2.Institutt for InformatikkUniversitet i BergenBergenNorway

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