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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 ..."
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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.
A lambdacalculus à la de Bruijn with explicit substitutions
, 1995
"... The aim of this paper is to present the scalculus which is a very simple calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of terms. We shall prove strong normalisation of the corresponding calculus of substitution by tra ..."
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Cited by 78 (26 self)
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The aim of this paper is to present the scalculus which is a very simple calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oecalculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the scalculus is obtained by the "interpretation method" ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the AEcalculus (cf. [BBLRD95]). The relation between s and AE is also studied.
Confluence of Extensional and NonExtensional λcalculi with Explicit Substitutions
 Theoretical Computer Science
"... This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. O ..."
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Cited by 12 (2 self)
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This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many wellknown calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, calculi, explicit substitutions, confluence, extensionality. 1 Introduction The calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fireduction. The main mechanism used to perform fireduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the calculus `a la de Bruijn by using natur...
Proving Correctness of the Translation from MiniML to the CAM with the Coq Proof Development System
 with the Coq Proof Development System. Research report RR2536, INRIA, Rocquencourt
, 1995
"... In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (MiniML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to ..."
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Cited by 4 (0 self)
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In this article we show how we proved correctness of the translation from a small applicative language with recursive definitions (MiniML) to the Categorical abstract machine (CAM) using the Coq system. Our aim was to mechanise the proof of J. Despeyroux [10]. Like her, we use natural semantics to axiomatise the semantics of our languages. The axiomatisations of inferences systems and of the languages is nicely performed by the mechanism of inductive definitions in the Coq system. Unfortunately both the source and the target semantics involve nested structures that cannot be formalised inductively. We have overcome this problem by making some slight modifications of both the source and target semantics and show how the changes in the source and target semantics are related. For the remaining tranlation we explain how we can use the Coq system to formalize nonterminating programs and incorrect programs, objects that are impossible to explain with only the formalism of natural semantic...
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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Cited by 3 (0 self)
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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.
TypeFree CurryHoward Isomorphisms (A ProofTheory Inspired Exposition of the Isomorphism between the Untyped Calculus with Variable Names and à la de Bruijn)
"... We give an alternative, prooftheory inspired proof of the wellknown result that the untyped calculus presented with variable names and `a la de Bruijn are isomorphic. The two presentations of the calculus come about from two isomorphic logic formalisations by observing that, for the logic in ..."
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We give an alternative, prooftheory inspired proof of the wellknown result that the untyped calculus presented with variable names and `a la de Bruijn are isomorphic. The two presentations of the calculus come about from two isomorphic logic formalisations by observing that, for the logic in question, the CurryHoward correspondence is formulaindependent. We identify the exchange rule as the the prooftheoretical difference between the two representations of the systems. 1 Introduction The CurryHoward correspondence relates formal inference systems of symbolic logic to typed like calculi. An inference system for formal, symbolic logic is said to be in Hilbertstyle if, 1 no logical rule (i.e., excluding cut, weakening, etc.) change the set of assumptions. Such systems are also referred to as combinatory logics, in that they typically consist of a set of tautologies (or combinators) which are combined by the, socalled, Modus Ponens rule: A ! B A (Modus Ponens) B For...
Preuve de correction de la compilation de MiniML en code CAM dans le système d'aide à la démonstration COQ
, 1995
"... Machine). Notre objectif a 'et'e de m'ecaniser une preuve pr'esent'ee dans l'article de J. Despeyroux [9] et 'ecrite `a l'aide du langage Typol. Nous utilisons des s'emantiques naturelles pour mod'eliser l"evaluation de nos langages. Nous ne sommes parvenus que partiellement `a m'ecaniser cette preu ..."
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Machine). Notre objectif a 'et'e de m'ecaniser une preuve pr'esent'ee dans l'article de J. Despeyroux [9] et 'ecrite `a l'aide du langage Typol. Nous utilisons des s'emantiques naturelles pour mod'eliser l"evaluation de nos langages. Nous ne sommes parvenus que partiellement `a m'ecaniser cette preuve de correction. En effet, les sp'ecifications naturelles des langages source et cible contiennent des termes rationnels difficiles `a axiomatiser dans l"etat actuel du syst`eme. Nous proposons un d'ecoupage de la preuve isolant cette difficult'e. (Abstract: pto) Samuel.Boutin@inria.fr Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11  T'el'ecopie : (33 1) 39 63 53 30 Proving Correctness of the Translation from MiniML to the CAM with the Coq Proof Development System Abstract: In this report we show how we proved correctness of the translation from a small applicative language with rec...