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A Calculus of Substitutions for Incomplete-Proof Representation in Type Theory

Abstract : In the framework of intuitionnistic logic and type theory, the concepts of «propositions» and «types» are identified. This principle is known as the Curry-Howard isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambda-terms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambda-calculus with new operators. First, we consider typed meta-variables 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 meta-variables. Unfortunately, the theory of explicit substitution calculi with typed meta-variables is more complex than that of lambda-calculus. 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 properties of confluence and strong normalization are not incompatible with explicit substitution calculi. This thesis also proposes a calculus with explicit substitutions and typed meta-variables for dependent type systems, in particular for the Calculus of Constructions, which allows incomplete proof-terms to be represented. For these type systems, we prove the main typing properties: Type Uniqueness, Subject Reduction, Weak Normalization, Confluence and Typing Decidability. Finally, we give an application of this formalism to proof synthesis. The proposed method merges a procedure for term enumeration with a technique of higher-order unification via explicit substitutions where unification variables are coded as meta-variables.
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https://hal.inria.fr/inria-00073380
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 12:40:04 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
Long-term archiving on: : Sunday, April 4, 2010 - 11:44:38 PM

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  • HAL Id : inria-00073380, version 1

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César Muñoz. A Calculus of Substitutions for Incomplete-Proof Representation in Type Theory. [Research Report] RR-3309, INRIA. 1997. ⟨inria-00073380⟩

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