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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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. 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...
A Generic Normalisation Proof for Pure Type Systems
, 1996
"... We prove the strong normalisation for any PTS, provided the existence of a certainset A * (s) for every sort s of the system. The properties verified by the A * (s)'s depend of the axiom and rules of the type system. 1 Introduction 1.1 Brief History This work is an attempt to deal with ..."
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We prove the strong normalisation for any PTS, provided the existence of a certainset A * (s) for every sort s of the system. The properties verified by the A * (s)'s depend of the axiom and rules of the type system. 1 Introduction 1.1 Brief History This work is an attempt to deal with the structure of complex Type Theories. Historically, once Girard had transposed the BuraliForti paradox to type theory, MartinLof replied by suppressing the guilty Type : Type rule and remediated to the resulting loss of expressiveness by introducing a new concept of stratified universes [10]. Today this notion can be found, in different forms and variants, in most Type Theories, especially the ones with foundational ambitions. For example, it appears in the theories used in actually implemented proofcheckers (NuPRL, Coq, Lego. . . ). The main idea is that all types are no longer equal. Each one inhabits a certain universe (MartinLof) or sort (Pure Type Systems). In general, universes are emb...
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...