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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...
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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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...
Normalisation by Evaluation for System F using Staged Outermost Reduction
, 2000
"... We give a strikingly simple presentation of Normalisation by Evaluation (NbE) for System F. It is formulated using straightforward type theory in the form of System F NbE , a purposedefined twolevel version of System F. One level of System F NbE contains copies of all the System F terms while ..."
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We give a strikingly simple presentation of Normalisation by Evaluation (NbE) for System F. It is formulated using straightforward type theory in the form of System F NbE , a purposedefined twolevel version of System F. One level of System F NbE contains copies of all the System F terms while the other level contains copies of all the long #(#)normal forms of System F. Overlap is allowed only at variable type. We derive the NbE algorithm as the canonical coercers of terms between the levels in System F NbE , upto outermostonly #normalisation (a.k.a. evaluation) in just the one level. In doing so, we prove the NbE algorithm correct: for any System F term, NbE computes its long #(#)normal form. 1 Introduction Normalisation by Evaluation was first studied in its own right by Berger and Schwichtenberg approximately a decade ago [5]. It is a method by which you can use the meaning function (a.k.a. evaluation functional) of a model of a language to normalise terms of the langua...