Three equivalent ordinal notation systems in cubical Agda
Fredrik Nordvall Forsberg, Chuangjie Xu, Neil Ghani · Strathprints: The University of Strathclyde institutional repository (University of Strathclyde) · 2020
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.