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Confluence Properties of Extensional and NonExtensional lambdaCalculi with Explicit Substitutions (Extended Abstract)
 in Proceedings of the Seventh International Conference on Rewriting Techniques and Applications
, 1996
"... ) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For ..."
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Cited by 22 (5 self)
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) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many wellknown calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and nonextensional versions. 1 Introduction The #calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #reduction. The...
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.
Functional BackEnds within the LambdaSigma Calculus
, 1996
"... We define a weak calculus, oe w , as a subsystem of the full calculus with explicit substitutions oe * . We claim that oe w could be the archetypal output language of functional compilers, just as the calculus is their universal input language. Furthermore, oe * could be the adequate theory to e ..."
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Cited by 19 (0 self)
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We define a weak calculus, oe w , as a subsystem of the full calculus with explicit substitutions oe * . We claim that oe w could be the archetypal output language of functional compilers, just as the calculus is their universal input language. Furthermore, oe * could be the adequate theory to establish the correctness of simplified functional compilers. Here, we illustrate these claims by proving the correctness of four simplified compilers and runtime systems modeled as abstract machines. The four machines we prove are the Krivine machine, the SECD, the FAM and the CAM. Thereby, we give the first formal proofs of Cardelli's FAM and of its compiler.
Intersection types for explicit substitutions
, 2003
"... We present a new system of intersection types for a compositionfree calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical inte ..."
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Cited by 17 (6 self)
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We present a new system of intersection types for a compositionfree calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical intersection types system, which characterized head normalization and weak normalization, but was not complete for strong normalization. An important role is played by the notion of available variable in a term, which is a generalization of the classical notion of free variable.
Pure Type Systems in Rewriting Logic
 In Proc. of LFM’99: Workshop on Logical Frameworks and MetaLanguages
, 1999
"... . The logical and operational aspects of rewriting logic as a logical framework are illustrated in detail by representing pure type systems as object logics. More precisely, we apply membership equational logic, the equational sublogic of rewriting logic, to specify pure type systems as they can be ..."
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Cited by 10 (2 self)
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. The logical and operational aspects of rewriting logic as a logical framework are illustrated in detail by representing pure type systems as object logics. More precisely, we apply membership equational logic, the equational sublogic of rewriting logic, to specify pure type systems as they can be found in the literature and also a new variant of pure type systems with explicit names that solves the problems with closure under conversion in a very satisfactory way. Furthermore, we use rewriting logic itself to give a formal operational description of type checking, that directly serves as an ecient type checking algorithm. The work reported here is part of a more ambitious project concerned with the development in Maude [7] of a proof assistant for OCC, the open calculus of constructions, an equational extension of the calculus of constructions. 1 Introduction This paper is a detailed case study on the ease and naturalness with which a family of higherorder formal systems, namely...
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.
Explicit Substitutions and Reducibility
 Journal of Logic and Computation
, 2001
"... . We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simplytyped callbyname ~calculus enriched with op ..."
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Cited by 7 (1 self)
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. We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simplytyped callbyname ~calculus enriched with operators of explicit unary substitutions. The ~calculus, dened by Curien & Herbelin, is a variant of calculus with a let operator that exhibits symmetries such as terms/contexts and callbyname /callbyvalue reduction. The ~calculus embeds various standard calculi (and Gentzen's style sequent calculi too) and as an application we derive the strong normalization of Parigot's simplytyped calculus with explicit substitution. Introduction Explicit substitution in calculus The traditional theory of calculus relies on reduction, that is the capture by a function of its argument followed by the process of substituting this argument to the places where it is used. The ...
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.
Explicit Substitutions and All That
, 2000
"... Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In ..."
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Cited by 3 (3 self)
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Explicit substitution calculi are extensions of the lambdacalculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution calculi: the lambdasigma and lambda_secalculi.
Addressed Term rewriting Systems: . . .
, 2006
"... We present a formalism called Addressed Term Rewriting Systems, which can be used to model implementations of theorem proving, symbolic computation, and programming languages, especially aspects of sharing, recursive computations and cyclic data structures. Addressed Term Rewriting Systems are there ..."
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Cited by 3 (1 self)
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We present a formalism called Addressed Term Rewriting Systems, which can be used to model implementations of theorem proving, symbolic computation, and programming languages, especially aspects of sharing, recursive computations and cyclic data structures. Addressed Term Rewriting Systems are therefore well suited for describing objectbased languages, and as an example we present a language called λObj a, incorporating both functional and objectbased features. As a case study in how reasoning about languages is supported in the ATRS formalism a type system for λObj a is defined and a type soundness result is proved.