The exp-log normal form of types

TL;DR

This paper introduces the exp-log normal form of types, a systematic way to decide type isomorphism and simplify the beta-eta equality problem in lambda calculus with sums, leveraging high-school identities and type information.

Contribution

It presents the exp-log normal form of types, enabling a heuristic for type isomorphism and reducing the beta-eta equality problem while maintaining completeness, with an implementation in Coq.

Findings

Type isomorphism can be decided using the exp-log normal form.

The normal form reduces the beta-eta equality problem while preserving correctness.

An implemented converter in Coq facilitates practical application of the normal form.

Abstract

Lambda calculi with algebraic data types lie at the core of functional programming languages and proof assistants, but conceal at least two fundamental theoretical problems already in the presence of the simplest non-trivial data type, the sum type. First, we do not know of an explicit and implemented algorithm for deciding the beta-eta-equality of terms---and this in spite of the first decidability results proven two decades ago. Second, it is not clear how to decide when two types are essentially the same, i.e. isomorphic, in spite of the meta-theoretic results on decidability of the isomorphism. In this paper, we present the exp-log normal form of types---derived from the representation of exponential polynomials via the unary exponential and logarithmic functions---that any type built from arrows, products, and sums, can be isomorphically mapped to. The type normal form can be used as a simple heuristic for deciding type isomorphism, thanks to the fact that it is a systematic application of the high-school identities. We then show that the type normal form allows to reduce the standard beta-eta equational theory of the lambda calculus to a specialized version of itself, while preserving the completeness of equality on terms. We end by describing an alternative representation of normal terms of the lambda calculus with sums, together with a Coq-implemented converter into/from our new term calculus. The difference with the only other previously implemented heuristic for deciding interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we exploit the type information of terms substantially and this often allows us to obtain a canonical representation of terms without performing sophisticated term analyses.