Canonicity for Cubical Type Theory

Huber, Simon

doi:10.1007/s10817-018-9469-1

Cited by 22 publications

(33 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also extended cubical type theory with some higher inductive types. While [14] only proves canonicity for cubical type theory extended with the circle and propositional truncation, it should be straightforward to extend this result to the higher inductive types presented in this paper using the obvious operational semantics obtained by orienting the judgmental equalities given here. It also remains to prove normalization and decidability of type-checking for cubical type theory and in particular also for our extension with higher inductive types.…”

Section: Conclusion and Related Workmentioning

confidence: 85%

On Higher Inductive Types in Cubical Type Theory

Coquand

Huber

Mörtberg

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Self Cite

View full text Add to dashboard Cite

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions, truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.

show abstract

Section: Conclusion and Related Workmentioning

confidence: 85%

On Higher Inductive Types in Cubical Type Theory

Coquand

Huber

Mörtberg

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…This hence provides semantic consistency proofs for the cubical type theory that Cubical Agda is based on. A syntactic consistency proof, using Tait's computability method, for this cubical type theory was given in Huber [2016] by defining an operational semantics and proving that any term of natural numbers type computes to a numeral.…”

Section: Metatheory Of Cubical Type Theory and Cubical Agdamentioning

confidence: 99%

“…Interesting further directions would be to study meta-theoretical properties of cubical type theory, including a proof of decidability of type-checking and a complete correctness proof of the conversion checking algorithm with respect to a declarative specification of equality. We believe this can be done by extending the canonicity proof of Huber [2016] using ideas from Abel et al [2017].…”

Section: Future Workmentioning

confidence: 99%

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Vezzosi

Mörtberg

Abel

2019

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

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. 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 higher inductive types. These new primitives make function and propositional extensionality as well as quotient types directly definable with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. This extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.or invariance up to equivalence, in the sense that equivalent types will share the same structures and properties. The fact that equality is proof relevant in dependent type theory is the key to enabling this; the data of an equality proof can store the equivalence and transporting along this equality should then apply the function underlying the equivalence. In particular, this allows programs and properties to be transported between equivalent types, hereby increasing modularity and decreasing code duplication. A concrete example are the equivalent representations of natural numbers in unary and binary format. In a univalent system it is possible develop theory about natural numbers using the unary representation, but compute using the binary representation, and as the two representations are equivalent they share the same properties.The principle of univalence is the major new addition in Homotopy Type Theory and Univalent Foundations (HoTT/UF) [Univalent Foundations Program 2013]. However, these new type theoretic foundations add univalence as an axiom which disrupts the good constructive properties of type theory. In particular, if we transport addition on binary numbers to the unary representation we will not be able to compute with it as the system would not know how to reduce the univalence axiom. Cubical type theory [Cohen et al. 2018] addresses this problem by introducing a novel representation of equality proofs and thereby providing computational content to univalence. This makes it possible to constructively transport programs and properties between equivalent types. This representation of equality proofs has many other useful consequences, in particular function and propositional extensionality (pointwise equal functions and logically equivalent propositions are equal), and the equivalence between bisi...

show abstract

Section: Arxiv:161109263v2 [Cslo] 6 Oct 2017mentioning

confidence: 99%

“…CTT has sound denotational semantics in (fibrations in) cubical sets, a presheaf category that is used to model homotopy types. CTT enjoys canonicity for terms of natural number type [18] and is conjectured to have decidability of type-checking. Moreover, a type-checker has been implemented 1 .In Section 3 of this paper we propose guarded cubical type theory (GCTT), a combination of the two type theories 2 which supports non-trivial proofs about guarded recursive types via path equality, while retaining the potential for good syntactic properties such as canonicity for base types and decidable type-checking.…”

mentioning

confidence: 99%

Guarded Cubical Type Theory

et al. 2018

View full text Add to dashboard Cite

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory (GCTT), provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation. We show that CTT can be given semantics in presheaves on C × D, where C is the cube category, and D is any small category with an initial object. We then show that the category of presheaves on C × ω provides semantics for GCTT. arXiv:1611.09263v2 [cs.LO] 6 Oct 2017Clouston et al. [11] developed guarded recursive types in a simply-typed setting, following earlier work [27,3,1], with semantics in the presheaf category ω known as the topos of trees, and also presented a logic for reasoning about programs with guarded recursion. For large examples, such as models of program logics, we would like to be able to formalise such reasoning. A major approach to formalisation is via dependent types, used for example in the proof assistants Coq [24] and Agda [28]. Bizjak et al. [10], following earlier work [6,26], introduced guarded dependent type theory (GDTT), integrating the type-former into a dependently typed calculus, and supporting the definition of guarded recursive types as fixed-points of functions on universes, and guarded recursive operations on these types.We wish to formalise non-trivial theorems about equality between guarded recursive constructions, but such arguments often cannot be accommodated within intensional Martin-Löf type theory. For example, we may need to be able to reason about the extensions of streams in order to prove the equality of different stream functions. Hence GDTT includes an equality reflection rule, which is well known to make type checking undecidable. This problem is close to well-known problems with functional extensionality [16, Section 3.1.3], and indeed this analogy can be developed. Just as functional extensionality involves mapping terms of type (x : A) → Id B (f x) (gx) to proofs of Id (A → B) f g, extensionality for guarded recursion requires an extensionality principle for later types, namely the ability to m...

show abstract

Canonicity for Cubical Type Theory

Cited by 22 publications

References 11 publications

On Higher Inductive Types in Cubical Type Theory

On Higher Inductive Types in Cubical Type Theory

Cubical agda: a dependently typed programming language with univalence and higher inductive types

Guarded Cubical Type Theory

Contact Info

Product

Resources

About