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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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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...
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...
Reasoning about Redundant Patterns
, 1997
"... The extensional version of the simply typed calculus with product types enriched with layered, wildcard, and product patterns is studied. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. Two different confluent reduction systems, ..."
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Cited by 1 (1 self)
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The extensional version of the simply typed calculus with product types enriched with layered, wildcard, and product patterns is studied. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. Two different confluent reduction systems, called lwp \Gamma! and lw \Gamma! respectively, are obtained by turning the extensional axioms as expansion rules, and then adding some restrictions to these expansions to avoid reduction loops. It is shown that only layered and wildcard patterns are redundant in lw \Gamma!, while product patterns are unnecessary in lwp \Gamma!. Confluence of both reduction systems is proven by the composition of modular properties of the systems' extensional and nonextensional parts. Recursion is also added to both systems by keeping the modularity of the confluence property. 1 Introduction Patternmatching in function definitions is one of the most popular features of functional languages...
Goaldirected and Relative Dependency Pairs for Proving the Termination of Narrowing ⋆
"... Abstract. In this work, we first consider a goaloriented extension of the dependency pair framework for proving termination w.r.t. a given set of initial terms. Then, we introduce a new result for proving relative termination in terms of a dependency pair problem. Both contributions put together al ..."
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Cited by 1 (1 self)
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Abstract. In this work, we first consider a goaloriented extension of the dependency pair framework for proving termination w.r.t. a given set of initial terms. Then, we introduce a new result for proving relative termination in terms of a dependency pair problem. Both contributions put together allow us to define a simple and powerful approach to analyzing the termination of narrowing, an extension of rewriting that replaces matching with unification in order to deal with logic variables. Our approach could also be useful in other contexts where considering termination w.r.t. a given set of terms is also natural (e.g., proving the termination of functional programs). 1
A Complete Characterization of Termination of. . .
, 1994
"... We completely characterize termination of onerule string rewriting systems of the form 0 p 1 q ! 1 r 0 s for every choice of positive integers p, q, r, and s. For the simply terminating cases, we give a sharp estimate of the complexity of derivation lengths. 1 Introduction A term rewriting ..."
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We completely characterize termination of onerule string rewriting systems of the form 0 p 1 q ! 1 r 0 s for every choice of positive integers p, q, r, and s. For the simply terminating cases, we give a sharp estimate of the complexity of derivation lengths. 1 Introduction A term rewriting system R terminates, if every Rderivation t 1 !R t 2 !R \Delta \Delta \Delta is finite. Much of the success of term rewriting is due to the availability of powerful termination criteria. String rewriting is a special case of term rewriting where function symbols are of arity 1, and may be taken as characters. Termination of term rewriting systems is known to be undecidable, even for the special case of string rewriting systems [12], and even for leftlinear, onerule term rewriting systems [5]. The question whether termination is decidable for onerule string rewriting systems is still open. In this paper we give a decision procedure for a nontrivial subclass of onerule string rewriting ...
Rewriting with Extensional Polymorphic λcalculus
, 1996
"... We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over propositi ..."
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We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over proposition and the constant True. This result is an important step towards a new theory of reduction based on expansion rules, and gives a natural interpretation to the notion of second order jlong normal forms used in higher order resolution and unification, that are here just the normal forms of our reduction system.