Interpreting j-conversion as an expansion rule in the simply-typed -calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where fi-contraction, as the local counit, and j-expansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fij-normal forms) can be recovered by "cutting" the loops.

) Delia Kesner CNRS and LRI, B at 490, Universit e Paris-Sud - 91405 Orsay Cedex, France. e-mail:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and non-extensional #-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 well-known 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 non-extensional versions. 1 Introduction The #-calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #-reduction. The...

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...

. Although the use of expansionary j-rewrite has become increasingly common in recent years, one area where j-contractions have until now remained the only possibility is in the more powerful type theories of the -cube. This paper rectifies this situation by applying j-expansions 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 fij-normal form and deduce the decidability of fij-equality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed -calculus requires j-conversion, 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 fi-reduction, 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 j-contractions beh...

This paper studies confluence of extensional and non-extensional -calculi with explicit substitutions, where extensionality is interpreted by j-expansion. 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 well-known 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 fi-reduction. The main mechanism used to perform fi-reduction 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...

. We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of j-reduction. 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...

An extension of the simply-typed lambda-calculus 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 lambda-calculus 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 Church-Rosser and strong normalization properties. Finally it is a conservative extension of the simply-typed lambda-calculus.

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...

The use of expansionary j-rewrite rules in various typed -calculi has become increasingly common in recent years as their advantages over contractive j-rewrite rules have become apparent. Not only does one obtain the decidability of fij-equality, but rewrite relations based on expansions give a natural interpretation of long fij-normal 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 fi-reductions and j-expansions satisfying certain restrictions. Further, we characterise the second order long fij-normal forms as precisely the normal forms of the restricted rewrite relation. These results are an important step towards showing that j-expansions are compatible with the m...

. 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...