A proof pearl with the fan theorem and bar induction: walking through infinite trees with mixed induction and coinduction

Authors:
Keiko Nakata;Tarmo Uustalu;Marc Bezem
Affiliations:
Institute of Cybernetics, Tallinn University of Technology, Tallinn, Estonia;Institute of Cybernetics, Tallinn University of Technology, Tallinn, Estonia;Institutt for Informatikk, Universitet i Bergen, Bergen, Norway
Venue:
APLAS'11 Proceedings of the 9th Asian conference on Programming Languages and Systems
Year:
2011

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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 $#956;-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.