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22
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 92 (35 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.
Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection
 IN CSN95: COMPUTER SCIENCE IN THE NETHERLANDS
, 1995
"... In this paper we introduce and study a new lambdacalculus with explicit substitution, lambdaxgc, which has two distinguishing features: first, it retains the use of traditional variable names, specifying terms modulo renaming; this simplifies the reduction system. Second, it includes reduction rul ..."
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Cited by 87 (7 self)
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In this paper we introduce and study a new lambdacalculus with explicit substitution, lambdaxgc, which has two distinguishing features: first, it retains the use of traditional variable names, specifying terms modulo renaming; this simplifies the reduction system. Second, it includes reduction rules for explicit garbage collection; this simplifies several proofs. We show that lambdaxgc is a conservative extension which preserves strong normalisation (PSN) of the untyped lambdacalculus. The result is obtained in a modular way by first proving it for garbagefree reduction and then extending to `reductions in garbage'. This provides insight into the counterexample to PSN for lambdasigma of Melliès (1995); we exploit the abstract nature of lambdaxgc to show how PSN is in conflict with any reasonable substitution composition rule (except for trivial composition rules of which we mention one). Key words: lambda calculus, explicit substitution, strong normalisation, garbage collection.
Unification via Explicit Substitutions: The Case of HigherOrder Patterns
 PROCEEDINGS OF JICSLP'96
, 1998
"... In [6] we have proposed a general higherorder unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higherorder patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient ..."
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Cited by 58 (15 self)
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In [6] we have proposed a general higherorder unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higherorder patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an efficient implementation of the abstract algorithm and its generalization to constraint simplification, which has yielded good experimental results at the core of a higherorder constraint logic programming language.
Implementation of HigherOrder Unification Based on Calculus of Explicit Substitution
, 1995
"... . In this paper, we present several improvements of an algorithm for a higherorder unification based on the calculus of explicit substitutions. The main difference between our algorithm and the already known version is, that we try to postpone normalisation of oeterms as long as possible, i.e. unt ..."
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Cited by 20 (1 self)
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. In this paper, we present several improvements of an algorithm for a higherorder unification based on the calculus of explicit substitutions. The main difference between our algorithm and the already known version is, that we try to postpone normalisation of oeterms as long as possible, i.e. until some information of these oeterms is necessary for the next step of the unification algorithm. 1 Introduction In this paper, we describe an improved version of a higherorder unification algorithm, which was presented in [DHK95]. The main idea of this algorithm is based on a calculus of explicit substitutions in a simply typed theory (for definitions and details, see [ACCL90]), which integrates substitutions in the framework of the firstorder formalism. In this calculus, substitutions are treated as the firstorder objects, i.e. all basic operations over substitutions, like an application, a composition and a concatenation are defined in the firstorder theory (their semantic is descri...
A Calculus of Substitutions for IncompleteProof Representation in Type Theory
, 1997
"... : In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In ..."
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Cited by 18 (1 self)
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: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambdacalculus with new operators. First, we consider typed metavariables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing metavariables. Unfortunately, the theory of explicit substitution calculi with typed metavariables is more complex than that of lambdacalculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...
Calculi of Generalised βReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1998
"... Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substit ..."
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Cited by 16 (8 self)
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Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the  calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs, and show that welltyped terms are strongly normalizing and that other properties,...
Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus
, 1995
"... Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, f ..."
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Cited by 15 (0 self)
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Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, for instance, to represent incomplete proofs in type based proof systems. The oe calculus of explicit substitutions proposed by Abadi, Cardelli, Curien and L'evy gives an elegant way to deal with management of variable names and substitutions 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 an we present the full proofs of its main properties. This is, as far as we know, the...
Bridging de Bruijn indices and variable names in explicit substitutions calculi
 Logic Journal of the Interest Group of Pure and Applied Logic (IGPL
, 1996
"... Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ffconversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renew ..."
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Cited by 11 (7 self)
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Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ffconversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. Formal systems of explicit substitutions using variable names is a new area however and we believe, it should not develop without being welltied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names. In our aim to do so, we provide the tcalculus: a calculus `a la de Bruijn which can be translated into a calculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for t. Moreover, we show several prope...
Proof Representation in Type Theory: State of the Art
, 1996
"... In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to repres ..."
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Cited by 6 (0 self)
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In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called placesholders. We present a theoretical approach to placeholders in type theory. In this approach placeholders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities.
HigherOrder Equational Unification via Explicit Substitutions
 in Proceedings of the tenth UNIF Workshop
, 1996
"... . We show how to reduce the unification problem modulo fij conversion and a firstorder equational theory E, into a firstorder unification problem in a union of two nondisjoint equational theories including E and a calculus of explicit substitutions. A rulebased unification procedure in thi ..."
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Cited by 5 (3 self)
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. We show how to reduce the unification problem modulo fij conversion and a firstorder equational theory E, into a firstorder unification problem in a union of two nondisjoint equational theories including E and a calculus of explicit substitutions. A rulebased unification procedure in this combined theory is described and may be viewed as an extension of the one initially designed by G. Dowek, T. Hardin and C. Kirchner for performing unification of simply typed terms in a firstorder setting via the oecalculus of explicit substitutions. Additional rules are used to deal with the interaction between E and oe. 1 Introduction Unification modulo an equational theory plays an important role in automated deduction and in logic programming systems. For example, Prolog[NM88] is based on higherorder unification, ie. unification modulo the fijconversion. In order to design more expressive higherorder logic programming systems enhanced with a firstorder equational theory E,...