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We present three ordinal notation systems representing ordinals below$\varepsilon_0$ in type theory, using recent type-theoretical innovations suchas mutual inductive-inductive definitions and higher inductive types. We showhow ordinal arithmetic can be developed for these systems, and how they admit atransfinite induction principle. We prove that all three notation systems areequivalent, so that we can transport results between them using the univalenceprinciple. All our constructions have been implemented in cubical Agda.

Three Equivalent Ordinal Notation Systems in Cubical Agda