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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.
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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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
Term Indexing for the LEOII Prover
"... We present a new term indexing approach which shall support efficient automated theorem proving in classical higher order logic. Key features of our indexing method are a shared representation of terms, the use of partial syntax trees to speedup logical computations and indexing of subterm occurrenc ..."
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We present a new term indexing approach which shall support efficient automated theorem proving in classical higher order logic. Key features of our indexing method are a shared representation of terms, the use of partial syntax trees to speedup logical computations and indexing of subterm occurrences. For the implementation of explicit substitutions, additional support is offered by indexing of bound variable occurrences. A preliminary evaluation of our approach shows some encouraging first results. 1
A Practical Analysis of Explicit Substitutions Calculi
 Proceedings of 11 th International Workshop on Unification UNIF'97
, 1997
"... Explicit substitutions calculi provide a method to compute over expressions by using firstorder term rewriting systems. Many calculi have been proposed that differ by their signature, their number of rules and their theoretical properties. Our aim is to compare these systems from an operational ..."
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Explicit substitutions calculi provide a method to compute over expressions by using firstorder term rewriting systems. Many calculi have been proposed that differ by their signature, their number of rules and their theoretical properties. Our aim is to compare these systems from an operational point of view. For a selected class of such calculi, we have developed prototypes written in ELAN. In this paper, we briefly present our first experimental results. 1
Operated by Universities Space Research Association
"... CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus 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 ..."
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CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus 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.
SUBSTITUTIONS
"... Abstract. Typed λterms 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 λterms. The proofsasterms principle can be used to check a proof by type checking the λterm extracted from ..."
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Abstract. Typed λterms 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 λterms. The proofsasterms principle can be used to check a proof by type checking the λterm extracted from the complete proof tree. However, proof trees and typed λterms 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.
Comparing Calculi of Explicit . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 67 (2002)
, 2002
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substi ..."
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The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substitutions are treated in this paper: the and the s e which have proved useful for solving practical problems like higher order uni cation, and the suspension calculus related to the implementation of the language Prolog. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. Additionally, we prove that and s e as well as and the suspension calculus are non comparable while s e is more adequate than the suspension calculus.
Bridging de Bruijn Indices and Variable Names in Explicit Substitutions Calculi
"... Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding αconversion 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 renewe ..."
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Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding αconversion 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. We believe that the study of these systems 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 and to do so, we provide the λtcalculus: a λcalculus à la de Bruijn which can be translated into a λcalculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for λt. Moreover, we show several properties of λt, including confluence on closed terms and efficiency to simulate βreduction. Furthermore, λt is a good example of a calculus written in the λsstyle (cf. [19]) that possesses the updating mechanism of the calculi à la λσ (cf. [1, 7, 26]).
The NASA STI Program Office provides
, 1999
"... 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