Three equivalent ordinal notation systems in cubical Agda

Fredrik Nordvall Forsberg; Chuangjie Xu; Neil Ghani

doi:10.1145/3372885.3373835

Three equivalent ordinal notation systems in cubical Agda

Fredrik Nordvall Forsberg, Chuangjie Xu, Neil Ghani

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

3 Citations (Scopus)

28 Downloads (Pure)

Abstract

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.

Original language	English
Title of host publication	CPP 2020 : Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs
Editors	Jasmin Blanchette, Catalin Hritcu
Place of Publication	New York
Pages	172–185
Number of pages	14
DOIs	https://doi.org/10.1145/3372885.3373835
Publication status	Published - 24 Jan 2020

Keywords

theory of computation
proof theory
type theory
ordinal notation
cantor normal form
inductive-inductive definitions
higher inductive types
cubical Agda
programming

Access to Document

10.1145/3372885.3373835

Forsberg-etal-SIGPLAN-2020-Three-equivalent-ordinal-notation-systems-in-cubical-agdaAccepted author manuscript, 840 KB

Fredrik Nordvall Forsberg
- fredrik.nordvall-forsberg@strath.ac.uk
- Computer And Information Sciences - Lecturer B
- SICSA
Person: Academic, Research Only

1 Finished

Homotopy Type Theory: Programming and Verification
Ghani, N. & McBride, C.
EPSRC (Engineering and Physical Sciences Research Council)
1/04/15 → 30/09/19
Project: Research

Cite this

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title = "Three equivalent ordinal notation systems in cubical Agda",

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Three equivalent ordinal notation systems in cubical Agda. / Nordvall Forsberg, Fredrik; Xu, Chuangjie; Ghani, Neil.

CPP 2020 : Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs. ed. / Jasmin Blanchette; Catalin Hritcu. New York, 2020. p. 172–185.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

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AB - 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.

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KW - proof theory

KW - type theory

KW - ordinal notation

KW - cantor normal form

KW - inductive-inductive definitions

KW - higher inductive types

KW - cubical Agda

KW - programming

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DO - 10.1145/3372885.3373835

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Three equivalent ordinal notation systems in cubical Agda

Abstract

Keywords

Access to Document

Fredrik Nordvall Forsberg

Homotopy Type Theory: Programming and Verification

Cite this