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 (

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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.


Nordvall Forsberg, Fredrik ORCID logoORCID:, Xu, Chuangjie and Ghani, Neil ORCID logoORCID:; Blanchette, Jasmin and Hritcu, Catalin