2000
DOI: 10.1007/3-540-46432-8_5
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Proof Nets and Explicit Substitutions

Abstract: Abstract. We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λ l -calculus with de Bruijn indices (a calculus with full composition defined in [8]) u… Show more

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Cited by 19 publications
(19 citation statements)
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“…The push equivalence 9 has already been studied in the literature on proof nets and explicit substitutions [8,9]. The pull reduction may seem somewhat complicated, however it is a generalization of the reduction pulling out weakenings from boxes [9].…”
Section: Proposition 4 (Stability Of Correctness) If π Is Switching mentioning
confidence: 99%
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“…The push equivalence 9 has already been studied in the literature on proof nets and explicit substitutions [8,9]. The pull reduction may seem somewhat complicated, however it is a generalization of the reduction pulling out weakenings from boxes [9].…”
Section: Proposition 4 (Stability Of Correctness) If π Is Switching mentioning
confidence: 99%
“…equivalences on nets. We thus generalize the equivalences and reductions of [9], providing as a byproduct the first proof of confluence 1 for such LL proof nets with equivalences in the completely pure case, as previous works concentrated on normalization in the typed one. Finally, we are able to introduce one more equivalence potentially giving the right to always consider boxes without sums inside (the bang sum equivalence).…”
Section: Introductionmentioning
confidence: 99%
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“…It is a leading principle behind, among others, explicit substitution calculi [1,18,8,9,15,2], term calculi with strategies or higher-order transformations [14,3], and sharing graphs in the style of Lamping [17,4,21]. The atomic lambda-calculus represents a novel category in this range.…”
Section: Introductionmentioning
confidence: 99%
“…But the λ ws -calculus has a complicated syntax and its named version [13] is even less intelligible. However, the strong normalisation proof for λ ws given in [13] reveals a natural semantics for composition of ES via Linear Logic's proof-nets [19], suggesting that weakening (explicit erasure) and contraction (explicit duplication) can be added to the calculus without losing strong normalisation.…”
Section: Calculusmentioning
confidence: 99%