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

Part of: ICFP 2019

Published online by Cambridge University Press:  06 April 2021

ANDREA VEZZOSI Open the ORCID record for ANDREA VEZZOSI [Opens in a new window] ,
ANDERS MÖRTBERG  and
ANDREAS ABEL
ANDREA VEZZOSI
Affiliation:
Department of Computer Science, IT University of Copenhagen, Copenhagen, Denmark (e-mail: avez@itu.dk)
ANDERS MÖRTBERG
Affiliation:
Department of Mathematics, Stockholm University, Stockholm, Sweden (e-mail: anders.mortberg@math.su.se)
ANDREAS ABEL
Affiliation:
Department of Computer Science and Engineering, Chalmers and Gothenburg University, Gothenburg, Sweden (e-mail: andreas.abel@gu.se)
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Abstract

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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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Type
Research Article
Information
Journal of Functional Programming , Volume 31 , 2021 , e8
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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