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The Virtues of Etaexpansion
, 1993
"... Interpreting jconversion as an expansion rule in the simplytyped calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where ficontraction, as the local counit, and jexpansion, as the local unit, are li ..."
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Cited by 36 (4 self)
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Interpreting jconversion as an expansion rule in the simplytyped calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where ficontraction, as the local counit, and jexpansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fijnormal forms) can be recovered by "cutting" the loops.
Adjoint Rewriting
, 1995
"... This thesis concerns rewriting in the typed calculus. Traditional categorical models of typed calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these s ..."
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Cited by 25 (11 self)
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This thesis concerns rewriting in the typed calculus. Traditional categorical models of typed calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these structures providing an equational theory for terms. One then seeks a rewrite relation which, by transforming terms into canonical forms, provides a decision procedure for this equational theory. Unfortunately the rewrite relations which have been proposed, apart from for the most simple of calculi, either generate the full equational theory but contain no decision procedure, or contain a decision procedure but only for a subtheory of that required. Our proposal is to unify the semantics and reduction theory of the typed calculus by generalising the notion of model from categorical structures based on term equality to categorical structures based on term reduction. This is accomplished via...
EtaExpansions in Dependent Type Theory  The Calculus of Constructions
 Proceedings of the Third International Conference on Typed Lambda Calculus and Applications (TLCA'97
, 1997
"... . Although the use of expansionary jrewrite has become increasingly common in recent years, one area where jcontractions have until now remained the only possibility is in the more powerful type theories of the cube. This paper rectifies this situation by applying jexpansions to the Calculus of ..."
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Cited by 13 (0 self)
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. Although the use of expansionary jrewrite has become increasingly common in recent years, one area where jcontractions have until now remained the only possibility is in the more powerful type theories of the cube. This paper rectifies this situation by applying jexpansions to the Calculus of Constructions  we discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long fijnormal form and deduce the decidability of fijequality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed calculus requires jconversion, whose interpretation as a rewrite rule has traditionally been as a contraction x : T:fx ) f where x 6 2 FV(t). When combined with the usual fireduction, the resulting rewrite relation is strongly normalising and confluent, and thus reduction to normal form provides a decision procedure for the associated equational theory. However jcontractions beh...
Equality Between Functionals in the Presence of Coproducts
 Information and Computation
, 1995
"... We consider the lambdacalculus obtained from the simplytyped calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any settheoretic model with an infinite base type. 1 Introduction The mod ..."
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Cited by 9 (1 self)
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We consider the lambdacalculus obtained from the simplytyped calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any settheoretic model with an infinite base type. 1 Introduction The model theory of the simplytyped lambda calculus, ! , has been well developed in the last two decades. For the most part, techniques and results generalize readily to the calculus when product types are added. Indeed, a categorical treatment goes more smoothly in the presence of products. But very little is known about the model theory of the simplytyped lambda calculus with coproducts for two chief reasons. First, techniques in the model theory of ! often rely heavily on the strong syntactic properties of the calculus; many of these properties fail in the presence of coproducts. Second, the natural generalizations of several key theorems in the model theory of ! fail in the setting wi...
Eta Expansions in System F
 LIENSDMI, Ecole Normale Superieure
, 1996
"... The use of expansionary jrewrite rules in various typed calculi has become increasingly common in recent years as their advantages over contractive jrewrite rules have become apparent. Not only does one obtain the decidability of fijequality, but rewrite relations based on expansions give a natu ..."
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Cited by 6 (0 self)
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The use of expansionary jrewrite rules in various typed calculi has become increasingly common in recent years as their advantages over contractive jrewrite rules have become apparent. Not only does one obtain the decidability of fijequality, but rewrite relations based on expansions give a natural interpretation of long fijnormal forms, generalise more easily to other type constructors, retain key properties when combined with other rewrite relations, and are supported by a categorical theory of reduction. This paper extends the initial results concerning the simply typed calculus to System F, that is, we prove strong normalisation and confluence for a rewrite relation consisting of traditional fireductions and jexpansions satisfying certain restrictions. Further, we characterise the second order long fijnormal forms as precisely the normal forms of the restricted rewrite relation. These results are an important step towards showing that jexpansions are compatible with the m...
betaetaEquality for Coproducts
 In Typed calculus and Applications, number 902 in Lecture Notes in Computer Science
, 1995
"... . Recently several researchers have investigated fijequality for the simply typed calculus with exponentials, products and unit types. In these works, jconversion was interpreted as an expansion with syntactic restrictions imposed to prevent the expansion of introduction terms or terms which for ..."
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. Recently several researchers have investigated fijequality for the simply typed calculus with exponentials, products and unit types. In these works, jconversion 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 fijnormal forms. Thus reduction to normal form provides a decision procedure for fijequality. This paper extends these methods to sum types. Although this extension was originally thought to be straight forward, the proposed jrule 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 e...
EtaExpansions III  F omega
, 1996
"... The use of expansionary jrewrite rules in various typed calculi has become increasingly common in recent years as their advantages over contractive jrewrite rules have become apparent. Not only does one obtain simultaneously a decision procedure for fij equality and a procedure for the calculat ..."
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The use of expansionary jrewrite rules in various typed calculi has become increasingly common in recent years as their advantages over contractive jrewrite rules have become apparent. Not only does one obtain simultaneously a decision procedure for fij equality and a procedure for the calculation of the long fijnormal form of a term, but rewrite relations using expansions retain key properties when combined with first order rewrite systems, generalise more easily to other type constructors and are supported by a categorical theory of reduction. However, until now jcontractions have remained the only possibility in the more powerful type systems of the cube. In this paper we begin to rectify this situation by extending the techniques previously developed to a higher order polymorphic calculus called F ! , where reduction no longer occurs only at the level of terms but also at the level of types. 1 Introduction Extensional equality for terms of the simply typed calculus req...