Flattening subtyping by eta expansion

TL;DR

This paper explores how eta-expansion can replace deep subtyping in type systems, simplifying reasoning and design by transforming complex subtyping relations into more manageable forms.

Contribution

It provides a comprehensive tutorial, historical context, and formal proof for using eta-expansion to avoid deep subtyping in type system design.

Findings

Eta-expansion can effectively replace deep subtyping in type systems.

The paper offers a complete proof of the eta-expansion technique.

Discussion on how eta-expansion simplifies type system reasoning.

Abstract

To design type systems that use subtyping, we have to make tradeoffs. Deep subtyping is more expressive than shallow subtyping, because deep subtyping compares the entire structure of types. However, shallow subtyping is easier to reason about. By eta-expanding source programs, we can get the effect of deep subtyping with less of its complexity. An early paper on filter models (Barendregt et al. 1983) examined two similar intersection type systems. The first included a subsumption rule that used a rich subtyping relation, including multiple rules for the top type and a distributivity rule. Their second type system dropped the subsumption rule, but added a rule that allowed a term to be eta-expanded before typing it. This rule in their second type system compensated for the lack of subsumption: where their first type system used subtyping to manipulate intersections deep inside types, their second type system used introduction and elimination rules directly on the subterms created by eta-expansion. Viewed as a computation, their proof of completeness for the second (shallow) system performs eta-expansion. Thus, we can regard their proof as inventing the application of eta-expansion to avoid deep subtyping. This paper serves as a tutorial on using eta-expansion to obviate deep subtyping, puts the invention of the technique by Barendregt et al. (1983) into context, gives a complete proof of the relevant lemma, and discusses how the technique can be used in type system design.