βη-Equality for coproducts

Abstract

Recently several researchers have investigated βη-equality for the simply typed λ-calculus with exponentials, products and unit types. In these works, η-conversion was interpreted as an expansion with syntactic restrictions imposed to prevent the expansion of introduction terms or terms which form the major premise of elimination rules. The resulting rewrite relation was shown confluent and strongly normalising to the long βη-normal forms. Thus reduction to normal form provides a decision procedure for βη-equality.

This paper extends these methods to sum types. Although this extension was originally thought to be straight forward, the proposed η-rule for the sum is substantially more complex than that for the exponent or product and contains features not present in the previous systems. Not only is there a facility for expanding terms of sum type analogous to that for product and exponential, but also the ability to permute the order in which different subterms of sum type are eliminated.

These different aspects of η-conversion for the sum type is reflected in our analysis. The rewrite relation is decomposed into two parts, a strongly normalising and confluent fragment resembling that found in the calculus without coproducts and a relation which generalises the ‘commuting conversions’ appearing in the literature. This second fragment is proved decidable by constructing for each term its (finite) set of quasi-normal forms and, by embedding the whole relation into this conversion relation, decidability, confluence and quasi-normal forms for the full relation are derived.