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Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
Explicit Substitutions for Control Operators
, 1997
"... . The \Deltacalculus is a calculus with a local operator closely related to normalisation procedures in classical logic and control operators in functional programming. We introduce \Deltaexp, an explicit substitution calculus for \Delta, show it preserves strong normalisation and that its simply ..."
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. The \Deltacalculus is a calculus with a local operator closely related to normalisation procedures in classical logic and control operators in functional programming. We introduce \Deltaexp, an explicit substitution calculus for \Delta, show it preserves strong normalisation and that its simply typed version is strongly normalising. Interestingly, \Deltaexp is the first example for which the decency method of showing preservation of strong normalisation (PSN) works whereas the structure preserving method which is based on the decency method does not. In particular, \Deltaexp is a very simple calculus yet is not structure preserving. This shows that the structure preserving notion intended to give a general description of calculi of explicit substitution that satisfy PSN, is restrictive. To our knowledge, \Deltaexp is the first calculus of explicit substitution that is not structure preserving. 5 1 Introduction Explicit substitutions were introduced in [1] as a bridge between cal...
Explicit Substitutions for the λΔcalculus
"... The λΔcalculus is a λcalculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisa ..."
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The λΔcalculus is a λcalculus with a controllike operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisation, which provides us with the first example  moreover a very natural one indeed  of explicit substitution calculus which is not structurepreserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of λΔexp is strongly normalising. In addition, we show that Plotkin's callbyname continuationpassing style translation may be extended to λΔexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions.
A Leftlinear Variant of λσ
, 1997
"... In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcal ..."
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In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcalculus enjoys the same general characteristics as oe, i.e. a simple and finitary firstorder presentation, confluent on expressions with metavariables of terms and weakly normalizing on typed expressions. Moreover, L does not have the nonleftlinear surjective pairing rule of oe which raises technical problems in some frameworks.
Dependent Types with Explicit Substitutions: A metatheoretical development
, 1997
"... We present a theory of dependent types with explicit substitutions. We follow a metatheoretical approach where open expressions expressions with metavariables are firstclass objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normal ..."
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We present a theory of dependent types with explicit substitutions. We follow a metatheoretical approach where open expressions expressions with metavariables are firstclass objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.