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

Vezzosi, Andrea; Mörtberg, Anders; Abel, Andreas

doi:10.1017/s0956796821000034

Cited by 29 publications

(25 citation statements)

References 29 publications

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“…3 Following this discovery, Cohen et al (2018) ("CCHM") dropped the regularity condition from the De Morgan model and corrected it to obtain a constructive model of univalent type theory and a De Morgan cubical type theory containing univalent universes closed under , , path, natural number, circle, and propositional truncation types for which Huber (2019) proved canonicity. That type theory was implemented in the cubicaltt prototype proof assistant 4 and later integrated into Cubical Agda (Vezzosi et al, 2019). Two machine-checked formalizations of this model have been developed.…”

Section: The De Morgan (Cchm) Modelmentioning

confidence: 99%

Syntax and models of Cartesian cubical type theory

Angiuli

Brunerie

Coquand

et al. 2021

Math. Struct. Comp. Sci.

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We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, suspension, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgmental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of generating trivial cofibrations and a different set of generating cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model of this type theory in Cartesian cubical sets. We give a mechanized proof, using Agda as the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, suspension types, and the universe itself are Kan in this model, and that the universe is univalent. An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category, as used in the CCHM model, and bicubical sets, as used in directed type theory.

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Section: The De Morgan (Cchm) Modelmentioning

confidence: 99%

Syntax and models of Cartesian cubical type theory

Angiuli

Brunerie

Coquand

et al. 2021

Math. Struct. Comp. Sci.

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“…For the reader who wants more hands-on experience with the ideas presented in these notes we recommend trying out Cubical Agda (Vezzosi et al, 2019(Vezzosi et al, , 2021. A tutorial to Cubical Agda with many exercises can be found at https://github.com/HoTT/EPIT-2020/tree/main/ 04-cubical-type-theory.…”

Section: Conclusion and Further Readingmentioning

confidence: 99%

“…This, or some variation of it, is essentially the underlying type theoretic setup in the various cubical systems that have been implemented in recent years. These include cubical (2013), cubicaltt (2015), yacctt (2018), RedPRL (2016) (Angiuli et al, 2018a), redtt (2018), cooltt (2020), mlang (2019), and Cubical Agda (Vezzosi et al, 2019). All of these systems build on different cubical type theories and have different standard cubical models (Angiuli et al 2021a(Angiuli et al , 2018bBezem et al 2014Bezem et al , 2019Cavallo and Harper 2019;Cohen et al 2018;Coquand et al 2018); however, the ideas underlying them are very similar, and one of the goals of this paper is to give sufficient background to understand and work with the various systems.…”

Section: Introductionmentioning

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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“…Finally, in Cubical Type Theory (Cohen et al, 2018), function extensionality is provable because of the presence of the interval primitive and thus has computational content. Cubical type theory has recently been implemented as a special version of Agda (Vezzosi et al, 2021). Another (similar) version of homotopy type theory is implemented in the theorem prover Arend (JetBrains Research, 2021).…”

Section: Related Workmentioning

confidence: 99%

Extensional equality preservation and verified generic programming

Botta¹,

Brede²,

Jansson³

et al. 2021

J. Funct. Prog.

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In verified generic programming, one cannot exploit the structure of concrete data types but has to rely on well chosen sets of specifications or abstract data types (ADTs). Functors and monads are at the core of many applications of functional programming. This raises the question of what useful ADTs for verified functors and monads could look like. The functorial map of many important monads preserves extensional equality. For instance, if $$f,g \, : \, A \, \to \, B$$ are extensionally equal, that is, $$\forall x \in A$$ , $$f \, x = g \, x$$ , then $$map \, f \, : \, List \, A \to List \, B$$ and $$map \, g$$ are also extensionally equal. This suggests that preservation of extensional equality could be a useful principle in verified generic programming. We explore this possibility with a minimalist approach: we deal with (the lack of) extensional equality in Martin-Löf’s intensional type theories without extending the theories or using full-fledged setoids. Perhaps surprisingly, this minimal approach turns out to be extremely useful. It allows one to derive simple generic proofs of monadic laws but also verified, generic results in dynamical systems and control theory. In turn, these results avoid tedious code duplication and ad-hoc proofs. Thus, our work is a contribution toward pragmatic, verified generic programming.

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

Cited by 29 publications

References 29 publications

Syntax and models of Cartesian cubical type theory

Syntax and models of Cartesian cubical type theory

Cubical methods in homotopy type theory and univalent foundations

Extensional equality preservation and verified generic programming

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