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14
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.
Confluence Properties of Extensional and NonExtensional lambdaCalculi with Explicit Substitutions (Extended Abstract)
 in Proceedings of the Seventh International Conference on Rewriting Techniques and Applications
, 1996
"... ) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For ..."
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Cited by 22 (5 self)
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) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many wellknown calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and nonextensional versions. 1 Introduction The #calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #reduction. The...
Simulating Expansions Without Expansions
, 1993
"... We add extensional equalities for the functional and product types to the typed calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing ..."
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Cited by 17 (9 self)
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We add extensional equalities for the functional and product types to the typed calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus, that stays confluent when allowing unbounded recursion. For that, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence directly form our simulation...
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...
Confluence of Extensional and NonExtensional λcalculi with Explicit Substitutions
 Theoretical Computer Science
"... This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. O ..."
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Cited by 12 (2 self)
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This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many wellknown calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, calculi, explicit substitutions, confluence, extensionality. 1 Introduction The calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fireduction. The main mechanism used to perform fireduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the calculus `a la de Bruijn by using natur...
Expanding Extensional Polymorphism
, 1995
"... . We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of jreduction. Our proof technique, based on a simple abstract lemma and a labelled calculus, can also be successfully used to simplify the proofs of confluence and nor ..."
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Cited by 11 (2 self)
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. We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of jreduction. Our proof technique, based on a simple abstract lemma and a labelled calculus, can also be successfully used to simplify the proofs of confluence and normalization for first order calculi, and can be applied to various extensions of the calculus presented here. 1 Introduction The typed lambda calculus provides a convenient framework for studying functional programming and offers a natural formalism to deal with proofs in intuitionistic logic. It comes traditionally equipped with the fi equality (x:M)N = M [N=x] as fundamental computational mechanism, and with the j (extensional) equality x:Mx = M as a tool for reasoning about programs. This basic calculus can then be extended by adding further types, like products, unit and second order types, each coming with its own computational mechanism and/or its extensional equalities. To reason abou...
On the Power of Simple Diagrams
, 1996
"... . In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by ..."
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Cited by 9 (3 self)
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. In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systematically confluence and/or normalization (or decidability of equality) in a simple way. Many result are new, but we also discuss systems for which our technique allows to provide a much simpler proof than what can be found in the literature. 1 Introduction During a recent investigation of confluence and normalization properties of polymorphic lambda calculus with an expansive version of the # rule, we came across a nice lemma that gives a simple but quite powerful sufficient condition to check the Church Rosser property for a compound rewriting system...
A Modal Lambda Calculus with Iteration and Case Constructs
 TYPES FOR PROOFS AND PROGRAMS: INTERNATIONAL WORKSHOP, TYPES ’98, KLOSTER IRSEE
, 1997
"... An extension of the simplytyped lambdacalculus allowing iteration and case reasoning over terms defined by means of higher order abstract syntax has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operato ..."
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Cited by 8 (1 self)
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An extension of the simplytyped lambdacalculus allowing iteration and case reasoning over terms defined by means of higher order abstract syntax has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operator ` ' of modal logic IS4. Here we give a new presentation of their system, with reduction rules, instead of evaluation judgments, that compute the canonical forms of terms. Our presentation is based on a modal lambdacalculus that is better from the user's point of view, is more concise and we do not impose a particular strategy of reduction during the computation. Our system enjoys the decidability of typability, soundness of typed reduction with respect to typing rules, the ChurchRosser and strong normalization properties. Finally it is a conservative extension of the simplytyped lambdacalculus.
Combining Algebraic Rewriting, Extensional Lambda Calculi, and Fixpoints
"... It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for #, or recursion together with the usual contract ..."
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Cited by 8 (3 self)
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It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for #, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and strong normalization are modular properties for the combination of algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for # and surjective pairing. We also show how to preserve confluence in a modular way when adding fixpoints to di#erent rewriting systems. This result is also obtained by a simple translation technique allowing to simulate bounded recursion. 1 Introduction Confluence and strong normalization for the combination of lambda calculus and algebraic rewriting systems have been the object of many studies [BT88, JO91, BTG94, HM90], where the modularity of these properties is s...
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...