DeepAI

# Internal Parametricity for Cubical Type Theory

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, observe how cubical equality regularizes parametric type theory, and examine the similarities and discrepancies between cubical and parametric type theory, which are closely related. We abstract a formal interface to the computational interpretation and show that this also has a presheaf model, and we explore cohesive type theory as a means of connecting parametric and non-parametric theories.

• 6 publications
• 16 publications
01/02/2019

### Parametric Cubical Type Theory

We exhibit a computational type theory which combines the higher-dimensi...
05/17/2018

### Presheaf Models of Relational Modalities in Dependent Type Theory

This report is an extension of 'A Model of Parametric Dependent Type The...
12/06/2022

### Type Theories with Universe Level Judgements

The aim of this paper is to refine and extend Voevodsky's draft "A unive...
09/11/2020

### Internalizing Representation Independence with Univalence

In their usual form, representation independence metatheorems provide an...
04/30/2018

### A General Framework for Relational Parametricity

Reynolds' original theory of relational parametricity was intended to ca...
09/02/2022

### A Reasonably Gradual Type Theory

Gradualizing the Calculus of Inductive Constructions (CIC) involves deal...
10/11/2021

### Free Commutative Monoids in Homotopy Type Theory

We develop a constructive theory of finite multisets, defining them as f...