Towards a Cubical Type Theory without an Interval

Following the cubical set model of type theory which validates the univalence axiom, cubical type theories have been developed that interpret the identity type using an interval pretype. These theories start from a geometric view of equality. A proof of equality is encoded as a term in a context extended by the interval pretype. Our goal is to develop a cubical theory where the identity type is defined recursively over the type structure, and the geometry arises from these definitions. In this theory, cubes are present explicitly, e.g., a line is a telescope with 3 elements: two endpoints and the connecting equality. This is in line with Bernardy and Moulin\u27s earlier work on internal parametricity. In this paper we present a naive syntax for internal parametricity and by replacing the parametric interpretation of the universe, we extend it to univalence. However, we do not know how to compute in this theory. As a second step, we present a version of the theory for parametricity with named dimensions which has an operational semantics. Extending this syntax to univalence is left as further work

Towards a cubical type theory without an interval

Following the cubical set model of type theory which validates the univalence axiom, cubical type theories have been developed that interpret the identity type using an interval pretype. These theories start from a geometric view of equality. A proof of equality is encoded as a term in a context extended by the interval pretype. Our goal is to develop a cubical theory where the identity type is defined recursively over the type structure, and the geometry arises from these definitions. In this theory, cubes are present explicitly, e.g. a line is a telescope with 3 elements: two endpoints and the connecting equality. This is in line with Bernardy and Moulin's earlier work on internal parametricity. In this paper we present a naive syntax for internal parametricity and by replacing the parametric interpretation of the universe, we extend it to univalence. However, we don't know how to compute in this theory. As a second step, we present a version of the theory for parametricity with named dimensions which has an operational semantics. Extending this syntax to univalence is left as further work

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, and we give an account of the identity extension lemma for internal parametricity