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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 order to see the pr ..."
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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...
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
A Leftlinear Variant of λσ
, 1997
"... In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcal ..."
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In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcalculus enjoys the same general characteristics as oe, i.e. a simple and finitary firstorder presentation, confluent on expressions with metavariables of terms and weakly normalizing on typed expressions. Moreover, L does not have the nonleftlinear surjective pairing rule of oe which raises technical problems in some frameworks.