# Parametric Cubical Type Theory

@article{Cavallo2019ParametricCT, title={Parametric Cubical Type Theory}, author={Evan Cavallo and Robert Harper}, journal={ArXiv}, year={2019}, volume={abs/1901.00489} }

We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing out the similarities and distinctions between the two along the way. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic types, including functions between higher inductive types, and we…

## 10 Citations

### Internal Parametricity for Cubical Type Theory

- MathematicsCSL
- 2020

A computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives is defined, which supports both univalence and its relational equivalent, which is called relativity.

### Higher Inductive Types and Parametricity in Cubical Type Theory

- Mathematics
- 2019

Cubical type theory is a novel extension of dependent type theory with a form of equality called a path. Path equality enjoys extensionality properties missing from traditional treatments of…

### Freedom for Proofs!

- Mathematics
- 2021

Representation independence allows programmers to give different implementations for an abstract interface. Reynolds’ characterization of representation independence for System F uses parametricity,…

### A Constructive Model of Directed Univalence in Bicubical Sets

- MathematicsLICS
- 2020

This paper gives a constructive model of a directed type theory with directed univalence in bicubical, rather than bisimplicial, sets, and formalizes much of this model using Agda as the internal language of a 1-topos, following Orton and Pitts.

### Gradualizing the Calculus of Inductive Constructions

- Computer ScienceACM Trans. Program. Lang. Syst.
- 2022

A crucial trade-off is observed between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys, which informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.

### The Marriage of Univalence and Parametricity

- Computer ScienceJ. ACM
- 2021

This work first clarifies the limitations of these two concepts when considered in isolation and then devises a fruitful marriage between both, which is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences.

### Cubical Categories for Higher-Dimensional Parametricity Extended Version

- Mathematics, Computer Science
- 2019

A generalization of cubical sets is introduced, which is called cubical categories, and used to develop a framework for higher-dimensional parametricity, all the way up to and including infinity, which has the crucial property that if a model is p-parametric according to this definition, then it is l- parametric for every l < p.

### Leibniz equality is isomorphic to Martin-Löf identity, parametrically

- Mathematics, PhilosophyJournal of Functional Programming
- 2020

It is shown that the two definitions of equality are isomorphic: any proof of Leibniz equality can be converted to one of Martin-Löf identity and vice versa, and each conversion followed by the other is the identity.

### Computational Semantics of Cartesian Cubical Type Theory

- Geology
- 2021

Many students complete PhDs in functional programming each year. As a service to the community, twice per year the Journal of Functional Programming publishes the abstracts from PhD dissertations…

### Principles of Program Verification for Arbitrary Monadic Effects. (Principes de la Vérification de Programmes à Effets Monadiques Arbitraires)

- Computer Science
- 2019

The goal of this thesis is to devise a principled semantic framework for verifying programs with arbitrary monadic effects in a generic way with respect to rich specifications, for properties such as program equivalence.

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