Unifying Cubical Models of Univalent Type Theory

Cavallo, Evan; Mörtberg, Anders; Swan, Andrew

doi:10.4230/lipics.csl.2020.14

Cited by 5 publications

(6 citation statements)

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“…the base category. As such, it can in principle represent many different type theory extensions like guarded recursion [10,20], parametricity [1,2,4,33,30], univalence [6,12,15], nominal [34] and directed [36] type theory, etc., as well as combinations of these [11,41].…”

Section: Multimode Type Theorymentioning

confidence: 99%

“…The first function takes two syntactic modes and either provides a proof that these modes are syntactically equal or results in a type error if they are not. 12 The second function is the long awaited specification of the modality equivalence relation ≃ m , not as a predicate but as a sound decision procedure. As such, it tests whether two syntactic modalities of the same domain and codomain are equivalent, and if so, it produces a proof that the modalities' interpretations as DRAs are equivalent.…”

Section: Ctx : Ctxexprmentioning

confidence: 99%

“…Over the past years many authors have proposed new type systems that extend a theory like MLTT or CIC with new primitives which allow users to prove more theorems or write more programs. Examples of such extensions include guarded recursion [10,20], parametricity [1,2,4,33,30,11], univalence [6,12,15], directed type theory [36,41] and nominal reasoning [34].…”

Section: Introductionmentioning

confidence: 99%

“…The ℓ in Set ℓ is a universe level. This means that TCM can act on any Agda type, not just types in one particular universe 12. We check syntactic equality, as the semantic layer does not account for equivalence of modes, i.e.…”

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confidence: 99%

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Sikkel: Multimode Simple Type Theory as an Agda Library

Ceulemans

Nuyts

Devriese

2022

Electron. Proc. Theor. Comput. Sci.

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Many variants of type theory extend a basic theory with additional primitives or properties like univalence, guarded recursion or parametricity, to enable constructions or proofs that would be harder or impossible to do in the original theory. However, implementing such extended type theories (either from scratch or by modifying an existing implementation) is a big hurdle for their wider adoption. In this paper we present Sikkel, a library in the dependently typed programming language Agda that allows users to program in extended type theories. It uses a deeply embedded language that can be easily extended with additional type and term constructors, thus supporting a wide variety of type theories. Moreover, Sikkel has a type checker that is sound by construction in the sense that all well-typed programs are automatically translated to their semantics in a shallow embedding based on presheaf models. Additionally, our model supports combining different base categories by using modalities to transport definitions between them. This enables in particular a general approach for extracting definitions to the meta-level, so that we can use the extended type theories to define regular Agda functions and prove properties of them. In this paper, we demonstrate Sikkel theories with guarded recursion and parametricity, but other extensions can be easily plugged in. For now, Sikkel supports only simple type theories but its model already anticipates the future addition of dependent types and a universe.

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Section: Multimode Type Theorymentioning

confidence: 99%

Section: Ctx : Ctxexprmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Sikkel: Multimode Simple Type Theory as an Agda Library

Ceulemans

Nuyts

Devriese

2022

Electron. Proc. Theor. Comput. Sci.

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“…As all constructions in this section must be stable, it makes sense to work internally to Ô. This style has been previously used in normalization proofs [Coquand 2019] and also in the metatheory of cubical type theories [Cavallo et al 2020;Licata et al 2018;Orton and Pitts 2016].…”

Section: Working In ômentioning

confidence: 99%

Staged compilation with two-level type theory

Kovács

2022

Proc. ACM Program. Lang.

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The aim of staged compilation is to enable metaprogramming in a way such that we have guarantees about the well-formedness of code output, and we can also mix together object-level and meta-level code in a concise and convenient manner. In this work, we observe that two-level type theory (2LTT), a system originally devised for the purpose of developing synthetic homotopy theory, also serves as a system for staged compilation with dependent types. 2LTT has numerous good properties for this use case: it has a concise specification, well-behaved model theory, and it supports a wide range of language features both at the object and the meta level. First, we give an overview of 2LTT's features and applications in staging. Then, we present a staging algorithm and prove its correctness. Our algorithm is "staging-by-evaluation", analogously to the technique of normalization-by-evaluation, in that staging is given by the evaluation of 2LTT syntax in a semantic domain. The staging algorithm together with its correctness constitutes a proof of strong conservativity of 2LLT over the object theory. To our knowledge, this is the first description of staged compilation which supports full dependent types and unrestricted staging for types.

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Finitary Type Theories With and Without Contexts

Haselwarter,

Bauer

2023

J Autom Reasoning

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We give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin–Löf type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We prove several general meta-theorems about finitary type theories: weakening, admissibility of substitution and instantiation of metavariables, derivability of presuppositions, uniqueness of typing, and inversion principles. We then give a second formulation of finitary type theories in which there are no explicit contexts. Instead, free variables are explicitly annotated with their types. We provide translations between finitary type theories with and without contexts, thereby showing that they have the same expressive power. The context-free type theory is implemented in the nucleus of the Andromeda 2 proof assistant.

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Unifying Cubical Models of Univalent Type Theory

Cited by 5 publications

References 0 publications

Sikkel: Multimode Simple Type Theory as an Agda Library

Sikkel: Multimode Simple Type Theory as an Agda Library

Staged compilation with two-level type theory

Finitary Type Theories With and Without Contexts

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