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Categorical Completeness Results for the SimplyTyped LambdaCalculus
 Proceedings of TLCA '95, Springer LNCS 902
, 1995
"... . We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the catego ..."
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. We investigate, in a categorical setting, some completeness properties of betaeta conversion between closed terms of the simplytyped lambda calculus. A cartesianclosed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the category that distinguishes them. It is said to have a complete interpretation if there is some interpretation that equates only interconvertible terms. We give simple necessary and sufficient conditions on the category for each of the two forms of completeness to hold. The classic completeness results of, e.g., Friedman and Plotkin are immediate consequences. As another application, we derive a syntactic theorem of Statman characterizing betaeta conversion as a maximum consistent congruence relation satisfying a property known as typical ambiguity. 1 Introduction In 1970 Friedman proved that betaeta conversion is complete for deriving all equalities between the (simplytyped) lambdadefinable...
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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. 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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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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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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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...
A Confluent Reduction for the Extensional Typed LambdaCalculus With Pairs, Sums, Recursion and Terminal Object
, 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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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 simulatio...
Reasoning about Layered, Wildcard and Product Patterns
 ALP '94, volume 850 of LNCS
, 1994
"... We study the extensional version of the simply typed calculus with product types and fixpoints enriched with layered, wildcard and product patterns. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. We obtain a confluent reduction syst ..."
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We study the extensional version of the simply typed calculus with product types and fixpoints enriched with layered, wildcard and product patterns. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. We obtain a confluent reduction system by turning the extensional axioms as expansion rules, and then adding some restrictions to these expansions in order to avoid reduction loops. Confluence is proved by composition of modular properties of the extensional and nonextensional subsystems of the reduction calculus. 1 Introduction Patternmatching function definitions is one of the most popular features of functional languages, allowing to specify the behavior of functions by cases, according to the form of their arguments. Lefthand sides of function definitions are usually expressed using Layered, Wildcard and Product Patterns (LWPP), as for example the following Caml Light [ea93] program where the function cons new pair ta...
Bigstep Normalisation
 UNDER CONSIDERATION FOR PUBLICATION IN J. FUNCTIONAL PROGRAMMING
, 2007
"... Traditionally, decidability of conversion for typed λcalculi is established by showing that smallstep reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflec ..."
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Traditionally, decidability of conversion for typed λcalculi is established by showing that smallstep reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simplytyped λcalculus with explicit substitutions and βηequality, a system which is not strongly normalising. We also show how the construction can be extended to System T with the usual βrules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the ChurchRosser property of a term rewriting system.