Three equivalent ordinal notation systems in cubical Agda

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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.
Original languageEnglish
Title of host publicationCPP 2020 : Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs
EditorsJasmin Blanchette, Catalin Hritcu
Place of PublicationNew York
Number of pages14
Publication statusPublished - 24 Jan 2020


  • theory of computation
  • proof theory
  • type theory
  • ordinal notation
  • cantor normal form
  • inductive-inductive definitions
  • higher inductive types
  • cubical Agda
  • programming


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