BibTeX
@MISC{Ghani95βη-equalityfor,
author = {Neil Ghani},
title = {βη-equality for coproducts},
year = {1995}
}
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Abstract
The use of expansionary η-rewrite rules in various typed λ-calculi has become increasingly common in recent years as their advantages over contractive η-rewrite rules have become apparent. Not only does one obtain simultaneously a decision procedure for βη-equality and a rational reconstruction of the long βη-normal forms, but expansions retain key properties such as strong normalisation and confluence when combined with algebraic rewrite systems, are supported by a categorical theory of reduction and generalise more easily to other type constructors. This paper considers a type constructor for which a decision procedure for βη-equality has been sought for a long time, namely the coproduct. Categorical models of reduction are used to derive a new η-rewrite rule for the coproduct which turns out to be substantially more complex than that for the exponent or product. Not only is there a facility for expanding terms of sum type analogous to that for the product and exponential, but also the ability to permute the order in which different subterms of sum type occur. These different aspects of η-conversion for the sum type are 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 reducts. Finally decidability, confluence and quasi-normal forms for the full relation are derived by embedding the whole relation into this generalised commuting conversion relation.
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