Canonicity and homotopy canonicity for cubical type theory

Coquand, Thierry; Huber, Simon; Sattler, Christian

doi:10.46298/lmcs-18(1:28)2022

Cited by 3 publications

(4 citation statements)

References 26 publications

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“…The idea of dependently typed equational logical frameworks goes back to Cartmell [29] (for theories without binding), and was further developed by Martin-Löf for theories with binding of arbitrary order [97]. Because we work only with typed terms up to judgmental equality, we may use semantic methods such as Artin gluing to succinctly prove syntactic results as in several recent works [6,36,37,71,124,126,127].…”

Section: Algebraicmentioning

confidence: 99%

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Sterling

Harper

2021

J. ACM

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The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction , computational effects , and type abstraction . We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure . Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof- relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.” Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types , by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

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Section: Algebraicmentioning

confidence: 99%

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Sterling

Harper

2021

J. ACM

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“…The past several years have witnessed an explosion in semantic computability techniques for establishing syntactic metatheorems [4,20,22,26,38,51,55,56,61]. What makes semantic computability different from "free-hand" computability is that it is expressed as a gluing model, parameterized in the generic model of the type theory; hence one is always working with typed terms up to judgmental equality.…”

Section: Semantic and Proof-relevant Computabilitymentioning

confidence: 99%

“…Uemura [61] proved a general gluing theorem for certain dependent type theories in the language of Shulman's type theoretic fibration categories; Kaposi et al [38] proved a similar result in the language of categories with families. Coquand et al [22] employed gluing to prove a homotopy canonicity result for a version of cubical type theory that omits certain computation rules, and Kapulkin and Sattler [39] used gluing to prove homotopy canonicity for homotopy type theory (as famously conjectured by Voevodsky). Sterling et al [56] adapted Coquand's gluing argument to prove the first non-operational strict canonicity result for a cubical type theory.…”

Section: )mentioning

confidence: 99%

Normalization for Cubical Type Theory

Sterling,

Angiuli

2021

Preprint

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We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major open problem in the syntactic metatheory of cubical type theory. Our normalization result is reduction-free, in the sense of yielding an injective function from equivalence classes of terms in context to a tractable language of β/η-normal forms. As corollaries we obtain both decidability of judgmental equality as well as the injectivity of type constructors in judgmentally consistent contexts.

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“…Coquand et al interpreted several HITs in the cubical sets model [17,24]. Note that one can constructively prove univalence in the cubical sets model [23] and that cubical type theory satisfies homotopy canonicity [25]. Furthermore, cubical type theory has been implemented in Agda with support for higher inductive types [62].…”

Section: Related Workmentioning

confidence: 99%

Constructing Higher Inductive Types as Groupoid Quotients

Weide

2020

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

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In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. We finish by constructing a biinitial object in the bicategory of algebras in groupoids. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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Canonicity and homotopy canonicity for cubical type theory

Cited by 3 publications

References 26 publications

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Normalization for Cubical Type Theory

Constructing Higher Inductive Types as Groupoid Quotients

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