Three equivalent ordinal notation systems in cubical Agda

Nordvall Forsberg, Fredrik and Xu, Chuangjie and Ghani, Neil; Blanchette, Jasmin and Hritcu, Catalin, eds. (2020) Three equivalent ordinal notation systems in cubical Agda. In: CPP 2020 : Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs. ACM, New York, 172–185. ISBN 9781450370974 (https://doi.org/10.1145/3372885.3373835)

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Abstract

We present three ordinal notation systems representing ordinals below epsilon zero in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.

ORCID iDs

Nordvall Forsberg, Fredrik ORCID logoORCID: https://orcid.org/0000-0001-6157-9288, Xu, Chuangjie and Ghani, Neil ORCID logoORCID: https://orcid.org/0000-0002-3988-2560; Blanchette, Jasmin and Hritcu, Catalin
  • Item type:Book Section
    ID code:71215
    Dates:
    Date
    Event
    24 January 2020
    Published
    27 November 2019
    Accepted
    Keywords:theory of computation, proof theory, type theory, ordinal notation, cantor normal form, inductive-inductive definitions, higher inductive types, cubical Agda, programming, Electronic computers. Computer science, Computational Theory and Mathematics
    Subjects:Science > Mathematics > Electronic computers. Computer science
    Department:Faculty of Science > Computer and Information Sciences
    Depositing user: Pure Administrator
    Date deposited:27 Jan 2020 13:32
    Last modified:26 Jun 2022 02:25
    URI:https://strathprints.strath.ac.uk/id/eprint/71215
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