The Marriage of Univalence and Parametricity

TL;DR

This paper introduces univalent parametricity, a novel concept combining univalence and parametricity to enable seamless, automatic transport of programs and proofs across equivalent types in proof assistants like Coq.

Contribution

It develops the theory of univalent parametricity and implements a lightweight Coq framework for transparent transfer of definitions and theorems between equivalent types.

Findings

Enables switching between different representations of data types easily.

Supports transport of both basic and complex proofs across type equivalences.

Facilitates more efficient program extraction and proof reuse.

Abstract

Reasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, equality is appallingly syntactic and, as a result, exploiting equivalences is cumbersome at best. Parametricity and univalence are two major concepts that have been explored to transport programs and proofs across type equivalences, but they fall short of achieving seamless, automatic transport. This work first clarifies the limitations of these two concepts in isolation, and then devises a fruitful marriage between both. The resulting concept, univalent parametricity, is an heterogeneous extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences. In addition to the theory of univalent parametricity, we present a lightweight framework implemented in Coq that allows the user to transparently transfer definitions and theorems for a type to an equivalent one, as if they were equal. For instance, this makes it possible to conveniently switch between an easy-to-reason-about representation and a computationally-efficient representation, as soon as they are proven equivalent. The combination of parametricity and univalence supports transport \`a la carte: basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences between functions over these types, in order to be able to lift more programs and proofs, as well as to yield more efficient terms. We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq.