# The Yoneda Reduction of Polymorphic Types (Extended Version)

## TL;DR

This paper investigates Yoneda-inspired type isomorphisms in System F, enabling quantifier elimination through a process called Yoneda reduction, with applications in counting inhabitants and program equivalence.

## Contribution

It introduces Yoneda reduction as a novel type rewriting technique that eliminates quantifiers in polymorphic types, extending the understanding of type isomorphisms in System F.

## Key findings

- Yoneda reduction can eliminate quantifiers from polymorphic types.
- Conditions for full quantifier elimination are established.
- Applications include counting inhabitants and checking program equivalence.

## Abstract

In this paper we explore a family of type isomorphisms in System F whose validity corresponds, semantically, to some form of the Yoneda isomorphism from category theory. These isomorphisms hold under theories of equivalence stronger than beta-eta-equivalence, like those induced by parametricity and dinaturality. We show that the Yoneda type isomorphisms yield a rewriting over types, that we call Yoneda reduction, which can be used to eliminate quantifiers from a polymorphic type, replacing them with a combination of monomorphic type constructors. We establish some sufficient conditions under which quantifiers can be fully eliminated from a polymorphic type, and we show some application of these conditions to count the inhabitants of a type and to compute program equivalence in some fragments of System F.