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lambdacalculi with explicit substitutions and composition which preserve beta strong normalization (Extended Abstract)
, 1996
"... ) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de ParisSud, 91405 Orsay Cede ..."
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Cited by 27 (3 self)
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) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de ParisSud, 91405 Orsay Cedex, France, fkesner,puelg@lri.fr. Abstract. We study preservation of fistrong normalization by d and dn , two confluent calculi with explicit substitutions defined in [10]; the particularity of these calculi is that both have a composition operator for substitutions. We develop an abstract simulation technique allowing to reduce preservation of fistrong normalization of one calculus to that of another one, and apply said technique to reduce preservation of fistrong normalization of d and dn to that of f , another calculus having no composition operator. Then, preservation of fistrong normalization of f is shown using the same technique as in [2]. As a consequence, d and dn become the fir...
Strong Normalization of Explicit Substitutions via Cut Elimination in Proof Nets
, 1997
"... In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #xcalculus [30, 29] is strongly normalizing, as ..."
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Cited by 22 (4 self)
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In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #xcalculus [30, 29] is strongly normalizing, as well as of all the calculi isomorphic to it such as # # [24], # s [19], # d [21], and # f [11]. In order to achieve this result, we introduce a new notion of reduction in Proof Nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it correspond to more identifications of proofs in Linear Logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.
Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus
, 1996
"... Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe calculus of explicit s ..."
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Cited by 20 (4 self)
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Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe calculus of explicit substitutions, proposed by Abadi, Cardelli, Curien andL evy, is a firstorder rewriting system that implements substitution and renaming mechanism of calculus. However, oe does not preserve strong normalisation of calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation. 1. Explicit substitutions The calculus is a higherorder theor...
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...
ProofTerm Synthesis on Dependenttype Systems via Explicit Substitutions
, 1999
"... Typed #terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the CurryHoward isomorphism relates proof trees with typed #terms. The proofsasterms principle can be used to check a proof by type checking the #term extracted from the compl ..."
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Cited by 8 (1 self)
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Typed #terms are used as a compact and linear representation of proofs in intuitionistic logic. This is possible since the CurryHoward isomorphism relates proof trees with typed #terms. The proofsasterms principle can be used to check a proof by type checking the #term extracted from the complete proof tree. However, proof trees and typed #terms are built differently. Usually, an auxiliary representation of unfinished proofs is needed, where type checking is possible only on complete proofs. In this paper we present a proof synthesis method for dependenttype systems where typed open terms are built incrementally at the same time as proofs are done. This way, every construction step, not just the last one, may be type checked. The method is based on a suitable calculus where substitutions as well as metavariables are firstclass objects.
The theory of calculi with explicit substitutions revisited
 CSL 2007
, 2007
"... Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we fi ..."
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Cited by 6 (1 self)
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Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambdacalculus which enjoys fundamental properties such as simulation of onestep betareduction, confluence on metaterms, preservation of betastrong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proofnets.
A weak calculus with explicit operators for pattern matching and substitution
 In Tison [Tis02
, 2002
"... Abstract. In this paper we propose a Weak Lambda Calculus called λPw having explicit operators for Pattern Matching and Substitution. This formalism is able to specify functions defined by cases via pattern matching constructors as done by most modern functional programming languages such as OCAML. ..."
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Cited by 5 (0 self)
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Abstract. In this paper we propose a Weak Lambda Calculus called λPw having explicit operators for Pattern Matching and Substitution. This formalism is able to specify functions defined by cases via pattern matching constructors as done by most modern functional programming languages such as OCAML. We show the main property enjoyed by λPw, namely subject reduction, confluence and strong normalization. 1
Dependent Types and Explicit Substitutions
, 1999
"... We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization. ..."
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Cited by 3 (0 self)
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We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
Two equivalent calculi of explicit substitution with confluence on metaterms and preservation of strong normalization (one with names and one firstorder) (Extended Abstract)
 In Proceedings of the 1st Int. Workshop on Explicit Substitutions: Theory and Applications to Programs and Proofs
, 1998
"... We propose a solution to the standing open problem of finding a calculus of explicit substitution with the following four properties: 1. simulates onestep βreduction, 2. is confluent on metaterms (also known as "open terms"), 3. has a strongly normalizing substitution subcalculus, and 4. prese ..."
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Cited by 3 (1 self)
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We propose a solution to the standing open problem of finding a calculus of explicit substitution with the following four properties: 1. simulates onestep βreduction, 2. is confluent on metaterms (also known as "open terms"), 3. has a strongly normalizing substitution subcalculus, and 4. preserves βstrong normalization. Our solution, λxci, is based on insights gained by studying the critical pair between two metaterms that makes calculi without substitution composition nonconfluent (on metaterms). The insight is closely tied to the fact that this critical pair is essentially an explicit representation of the "substitution lemma" of λcalculus, and the missing link in the solution is to express finiteness of all reductions starting from any reachable development of the source term. We give an encoding of the system as a first order system using de Bruijn's explicit variable indexing idea, and show that it enjoys the same properties by an easy equivalence.