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Confluence properties of Weak and Strong Calculi of Explicit Substitutions
 JOURNAL OF THE ACM
, 1996
"... Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence prope ..."
Abstract

Cited by 120 (7 self)
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Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oecalculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oecalculus, named the Envcalculus in [23], called here the confluent oecalculus.
Conditional Linearization
"... A nonleftlinear term rewriting system lacking the ChurchRosser property can sometimes be shown to satisfy the unique normal form property by shifting attention to an associated conditional term rewriting system that is leftlinear. We call this the method of conditional linearization. In the presen ..."
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Cited by 1 (0 self)
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A nonleftlinear term rewriting system lacking the ChurchRosser property can sometimes be shown to satisfy the unique normal form property by shifting attention to an associated conditional term rewriting system that is leftlinear. We call this the method of conditional linearization. In the present paper the method is described in a general setting and some applications are discussed. In particular we present a simple proof of the unique normal form property for Combinatory Logic extended with 'Parallel Conditional', that is, with constants C, T and F (conditional, true, false) and extra reduction rules CTxy x, CFxy y and Czxx x. A special feature of this application is that it involves the use of negative conditions. Contents Introduction 1. Four nonleftlinar, nonconfluent TRSs 2. Conditional Term Rewriting Systems 3. Application of CTRSs to prove uniqueness of normal forms 4. The case of Combinatory Logic plus Parallel Conditional 5. Chew's theorem 6. Remarks and further ques...
Vrije Universiteit Amsterdam BARENDREGT’S LEMMA
"... Abstract. Barendregt’s Lemma in its original form is a statement on Combinatory Logic that holds also for the lambda calculus and gives important insight into the syntactic interplay between substitution and reduction. Its origin lies in undefinablity proofs, but there are other applications as well ..."
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Abstract. Barendregt’s Lemma in its original form is a statement on Combinatory Logic that holds also for the lambda calculus and gives important insight into the syntactic interplay between substitution and reduction. Its origin lies in undefinablity proofs, but there are other applications as well. It is connected to the socalled Square Brackets Lemma, introduced by van Daalen in proofs of strong normalization of typed lambda calculi and of the Hyland–Wadsworth labelled lambda calculus. In the generalization of the latter result to higherorder rewriting systems, finite family developments, van Oostrom introduced the property “Invert”, which is also related. In this short note we state the lemma, try to put it in perspective, and discuss the mentioned connections. We also present a yet unpublished alternative proof of SN of the Hyland–Wadsworth labelled lambda calculus, using a computability argument. Dedicated to Henk Barendregt, in celebration of his 60th anniversary