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30
Pure Type Systems with Explicit Substitution
, 2000
"... We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is ..."
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We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is established. A more complicated extension is proposed for which subject reduction does hold in general.
Oostrom, Uniform normalisation beyond orthogonality
 Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001
, 2001
"... Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal ..."
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Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal nonerasing term rewrite systems, e.g. the λIcalculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and nonerasingness to the nonlinear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first and secondorder term rewrite systems as well as to a λcalculus with explicit substitutions. 1
Absolute Explicit Unification
 in International Conference on Rewriting Techniques and Applications (RTA'2000
, 2000
"... . This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higherorder unification, and type inference for PTS using purely firstorder machinery. A novel feature of our system is that it combines substituti ..."
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Cited by 4 (0 self)
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. This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higherorder unification, and type inference for PTS using purely firstorder machinery. A novel feature of our system is that it combines substitutions and variable declarations. This allows as a sideeffect to type check letbindings. Our treatment of metavariables is also explicit, such that instantiations of metavariables is internalized in the calculus. This produces a confluent calculus with distinguished holes and explicit substitutions that is insensitive to ffconversion, and allows directly embedding the system into rewriting logic. 1 Introduction Explicit substitutions provide a convenient framework for encoding higherorder typed calculus using firstorder machinery. In particular, this allows to integrate higherorder unification with firstorder provers, rewriting logic, and to delay evaluation and resolve scoping...
A Theory of Explicit Substitutions with Safe and Full Composition
 Logical Methods in Computer Science
"... Vol. 5 (3:1) 2009, pp. 1–29 ..."
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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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.
Unification via the ...Style of Explicit Substitutions
, 2001
"... A unication method based on the se style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the sty ..."
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Cited by 2 (2 self)
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A unication method based on the se style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the style of explicit substitution. Correctness and completeness properties of the proposed seunication method are shown and its advantages, inherited from the qualities of the se calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambdacalculi. 1
The Soundness of Explicit Substitution with Nameless Variables
, 1995
"... We show the soundness of a λcalculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first fl ..."
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We show the soundness of a λcalculus B where de Bruijn indices are used, substitution is explicit, and reduction is stepwise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and stepwise reduction and the first clear account of exactly when ffreduction is needed.
Strong Normalisation of CutElimination that Simulates βReduction
"... This paper is concerned with strong normalisation of cutelimination for a standard intuitionistic sequent calculus. The cutelimination procedure is based on a rewrite system for proofterms with cutpermutation rules allowing the simulation of βreduction. Strong normalisation of the typed terms i ..."
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This paper is concerned with strong normalisation of cutelimination for a standard intuitionistic sequent calculus. The cutelimination procedure is based on a rewrite system for proofterms with cutpermutation rules allowing the simulation of βreduction. Strong normalisation of the typed terms is inferred from that of the simplytyped λcalculus, using the notions of safe and minimal reductions as well as a simulation in NederpeltKlop’s λIcalculus. It is also shown that the typefree terms enjoy the preservation of strong normalisation (PSN) property with respect to βreduction in an isomorphic image of the typefree λcalculus.
The NASA STI Program Office provides
, 2000
"... Since its founding, NASA has been dedicated to the advancement of aeronautics and space ..."
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Since its founding, NASA has been dedicated to the advancement of aeronautics and space