On Higher Inductive Types in Cubical Type Theory

Coquand, Thierry; Huber, Simon; Mörtberg, Anders

doi:10.1145/3209108.3209197

Cited by 60 publications

(97 citation statements)

References 23 publications

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“…The other two theories, on the other hand, were each presented with examples of what we call CITs. Coquand et al have expanded on their initial offering with further examples (such as pushouts and two approaches to the torus), sketched a schema, and proven consistency of these with a semantics in cubical sets [Coquand et al 2018]. Our definitions of the values (including the use of fhcom), operational semantics, and rules for a non-indexed HIT specialize (roughly speaking) to the definitions given for these examples.…”

Section: Eliminationmentioning

confidence: 99%

Higher inductive types in cubical computational type theory

Cavallo

Harper

2019

Proc. ACM Program. Lang.

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Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory does not specify the computational behavior of HITs. Computational interpretations have now been provided for univalence and specific HITs by way of cubical type theories, which use a judgmental infrastructure of dimension variables. We extend the cartesian cubical computational type theory introduced by Angiuli et al. with a schema for indexed cubical inductive types (CITs), an adaptation of higher inductive types to the cubical setting. In doing so, we isolate the canonical values of a cubical inductive type and prove a canonicity theorem with respect to these values.

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Section: Eliminationmentioning

confidence: 99%

Higher inductive types in cubical computational type theory

Cavallo

Harper

2019

Proc. ACM Program. Lang.

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“…Semantics for different subclasses of HITs are given by [8,24,25,26,27]. Cubical type theories were shown to support some HITs in a computational way [28,29].…”

Section: Related Workmentioning

confidence: 99%

Large and Infinitary Quotient Inductive-Inductive Types

Kovács

Kaposi

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QI-ITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QI-ITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.

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“…In the paper we define higher inductive types as initial algebras for a certain signature (Section 1.3). In the formalisation, we use higher inductive types as implemented in cubical Agda, which are based on [CHM18]. Although it is natural to assume that the Agda higher inductive type should be initial in the sense of Section 1.3, proving this fact is actually one of the main difficulties in the formalisation (Proposition 2.6).…”

Section: Formalization In Cubical Agdamentioning

confidence: 99%

“…For the formalisation we use cubical Agda [AMV19] which is based on the cubical type theory of [CCHM18]. The development of HIITs in Agda is based on [CHM18].…”

Section: Introductionmentioning

confidence: 99%

The Integers as a Higher Inductive Type

Altenkirch

Scoccola

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

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On Higher Inductive Types in Cubical Type Theory

Cited by 60 publications

References 23 publications

Higher inductive types in cubical computational type theory

Higher inductive types in cubical computational type theory

Large and Infinitary Quotient Inductive-Inductive Types

The Integers as a Higher Inductive Type

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