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Forsberg, Fredrik Nordvall; Xu, Chuangjie; Ghani, Neil (2020): Three Equivalent Ordinal Notation Systems in Cubical Agda. In: CPP '20: Proceedings of the 9Th Acm Sigplan International Conference on Certified Programs and Proofs: pp. 172-185
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We present three ordinal notation systems representing ordinals below epsilon(0) 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.