- Three equivalent ordinal notation systems in Cubical Agda -

       Fredrik Nordvall Forsberg  and  Chuangjie Xu

We present three ordinal notation systems representing ordinals below
ε₀ in type theory, using recent type-theoretical innovations such as
mutual inductive-inductive definitions and higher inductive types.  As
case studies, we show how basic 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

The source files are available at


All the files are tested with

• Agda development version 2.6.1
  (commit: 4af7b92663d2eb1dd94383b140eb7acfeb1b1eb0)

• Cubical Agda library
  (commit: d7e345d3bcaefbc066d057487fca9677de7e29c7)


{-# OPTIONS --cubical --safe #-}

module index where

-- Table of contents --

-- §2 Cubical Agda
import Preliminaries

-- §3 Notation systems for ordinals below ε₀

-- §3.1 The subset approach
import SigmaOrd

-- §3.2 The mutual approach
import MutualOrd

-- MutualOrd can also be defined without nested and
-- inductive-recursive definitions
import MutualOrdWithoutNestednessAndRecursion

-- §3.3 The higher inductive approach
import HITOrd

-- §3.4 Equivalences between the three approaches
import Equivalences

-- §4 Ordinal arithmetic
import Arithmetic

-- §5 Transfinite induction
import TransfiniteInduction