Three equivalent ordinal notation systems in cubical Agda

Forsberg, Fredrik Nordvall; Xu, Chuangjie; Ghani, Neil

doi:10.1145/3372885.3373835

Cited by 7 publications

(6 citation statements)

References 17 publications

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“…In the work on ordinals below ε 0 [19,39,53,58], one derives/implements transfinite induction directly from the selected representation of ordinals. Berger [6] instead extracts a program from Gentzen's proof [38] of transfinite induction up to ε 0 .…”

Section: Propertymentioning

confidence: 99%

“…for some natural number n and ordinals β i . If α < ε 0 , then β i < α, and we can represent α as a finite binary tree (with a condition) as follows [17,19,39,53]. Let T be the type of unlabeled binary trees, i.e.…”

Section: Three Constructions Of Types Of Ordinalsmentioning

confidence: 99%

“…each element of our type Cnf of Cantor normal forms denotes an actual ordinal. There are several reasonable ways in which such a type can be defined that can be shown to be equivalent to each other [53]; the construction that we choose defines Cnf as a subset of the type of binary trees.…”

Section: Introductionmentioning

confidence: 99%

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Type-theoretic approaches to ordinals

Kraus

Forsberg

2023

Theoretical Computer Science

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Section: Propertymentioning

confidence: 99%

Section: Three Constructions Of Types Of Ordinalsmentioning

confidence: 99%

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Type-theoretic approaches to ordinals

Kraus

Forsberg

2023

Theoretical Computer Science

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“…Since the conference version (Vezzosi et al, 2019) of this article was published, some interesting formalizations have been performed using Cubical Agda . Forsberg et al (2020) implemented three equivalent ordinal notations systems and transported programs and proofs between them. Altenkirch & Scoccola (2020) considered a higher inductive version of the integers which differs from the one in Section 2.4.1.…”

Section: Related Workmentioning

confidence: 99%

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021

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Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

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“…A tutorial to Cubical Agda with many exercises can be found at https://github.com/HoTT/EPIT-2020/tree/main/ 04-cubical-type-theory. There are also quite a few papers reporting on formalization projects using Cubical Agda, including a cubical version of the SIP (Angiuli et al, 2021b), synthetic homotopy theory (Mörtberg and Pujet, 2020), proof theory and ordinal notations (Forsberg et al, 2020), and a formalization of π-calculus (Veltri and Vezzosi, 2020).…”

Section: Conclusion and Further Readingmentioning

confidence: 99%

Cubical methods in homotopy type theory and univalent foundations

Mörtberg

2021

Math. Struct. Comp. Sci.

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Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was to give constructive meaning to Voevodsky’s univalence axiom, but they have since then led to a range of new results. Among the achievements of these methods is the design of new type theories and proof assistants with native support for notions from HoTT/UF, syntactic and semantic consistency results for HoTT/UF, as well as a variety of independence results and establishing that the univalence axiom does not increase the proof theoretic strength of type theory. This paper is based on lecture notes that were written for the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University. The goal of these lectures was to give an introduction to cubical methods and provide sufficient background in order to make the current research in this very active area of HoTT/UF more accessible to newcomers. The focus of these notes is hence on both the syntactic and semantic aspects of these methods, in particular on cubical type theory and the various cubical set categories that give meaning to these theories.

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Three equivalent ordinal notation systems in cubical Agda

Cited by 7 publications

References 17 publications

Type-theoretic approaches to ordinals

Type-theoretic approaches to ordinals

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

Cubical methods in homotopy type theory and univalent foundations

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