International audienceDedukti est un vérificateur de types pour le lambda-Pi -calcul modulo, un formalisme alliant types dépendants et réécriture qui permet d'exprimer et de vérifier les preuves de nombreux systèmes logiques. Nous proposons d'utiliser DEDUKTI comme un vérificateur de preuves universel en traduisant HOL, Coq et FoCaLize vers DEDUKTI

Saillard, Ronan

HAL-MINES ParisTech

Dedukti : a Universal Proof CheckerRonan SaillardTo cite this version:Ronan Saillard. Dedukti : a Universal Proof Checker. Foundation of Mathematics forComputer-Aided Formalization Workshop, Jan 2013, Padova, Italy. <hal-00833992>HAL Id: hal-00833992https://hal-mines-paristech.archives-ouvertes.fr/hal-00833992Submitted on 14 Jun 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Dedukti: a Universal Proof CheckerRonan SaillardMINES ParisTechronan.saillard@cri.ensmp.frContext The success of formal methods both as tools of practical importanceand as objects of intellectual curiosity, has spawned a bewildering variety ofsoftware systems to support them. While the field has developed to maturityin academia and has registered some important successes in the industry, thefull benefit of formal methods in an industrial setting remains largely untapped.We submit that a lack of standards and easy interoperability in the field is oneexplanation to this state of affairs.The λΠ-calculus modulo To address this issue we propose the λΠ-calculusmodulo as a universal proof language. This calculus, introduced by Cousineauand Dowek [5], is a dependent typed λ-calculus where the definitional equalityhas been generalized to an arbitrary congruence generated by rewrite rules. Ouropinion is that this formalism is well suited for encoding foreign logics and makethem cooperate. Indeed, by allowing the addition of user-defined rewrite rules,logics benefit from shallower encodings and do not lose their computationalcontent.To support this language we have developed a new type checker called Dedukti [3].A type-checker generator In order to provide an efficient and yet simpletype checker for the λΠ-calculus modulo we chose to implement it as a typechecker generator: a parser/front-end generates a specialized type checker in atarget language which is then run. This scheme has two key advantages:First it conveniently allows us to sidestep the problem of implementing alpha-equivalence and substitution by making use of Higher Order Abstract Syntax(HOAS).Second, it permits us to make use of normalization by evaluation (NbE)which is one particularly efficient way to implement normalization scheme thatalternates phases of weak reduction (i.e. evaluation) and reification phases (orreadback phases [6]). The idea is to use the application of the target languageto code the β-reduction or any other rewrite rules.A Just-in-Time compiler as our backend As we intend to be a universalproof checker we need to be versatile. In particular some logics allow proofs toembed computational content and some do not. This is a real challenge becausethe efficiency of the normalization strategy highly depends on the computationalcontent of the proof term. Indeed in the first case λ-terms should be compiledand in the latter λ-terms should be interpreted. The matter of choosing the rightstrategy is very difficult but it is already adressed by the Just-in-Time compilercommunity, so we chose to rely on the state-of-the-art LuaJIT [8] compiler, asour backend.A Bidirectional and Context-Free type checking We use a domain-freerepresentation of terms as introduced in [1], meaning that abstractions may notbe annotated by the type of the variable that they bind. This information is notnecessary if one adopts a bidirectional typing discipline, a technique going backto [4] which splits the usual type judgments into two forms: checking judgmentsand synthesis judgments. Hence, input terms are smaller at no cost to thecomplexity or size of the type checker.When using HOAS, abstractions are encoded as functions of the implemen-tation language. They are therefore opaque structures that cannot be directlyanalyzed. Functions are black boxes whose only interface is that they can beapplied. In HOAS, it is therefore necessary to draw the parameter/variabledistinction common in presentations of first-order logic — variables are alwaysbound and parameters are variables that are free. This distinction is madeexplicit by the use of context-free type checking [2].Features and Translators Dedukti features also dot-pattern a` la Agda [7]which permits to avoid type checking subterms well-typed by construction,opaque definitions and a basic notion of modules. We provide a translatorto the λΠ-calculus modulo for the Calculus of Inductive Construction as imple-mented by the Coq proof assistant, and a translator for the Higher Order Logicas implemented by HOL/Isabelle. We plan to develop translations for the logicof PVS and the set theory of Atelier B.References[1] Gilles Barthe and Morten Heine Sørensen. Domain-free pure type systems. In J.Funct. Program., pages 9–20. Springer, 1993.[2] Mathieu Boespflug. Conception d’un noyau de ve´rification de preuves pour le λΠ-calcul modulo. PhD thesis, Ecole polytechnique, January 2011.[3] Mathieu Boespflug, Quentin Carbonneaux, Olivier Hermant, and Ronan Saillard.Dedukti : https://www.rocq.inria.fr/deducteam/dedukti/index.html.[4] Thierry Coquand. An algorithm for type-checking dependent types. Sci. Comput.Program., 26(1-3):167–177, 1996.[5] Denis Cousineau and Gilles Dowek. Embedding pure type systems in the lambda-pi-calculus modulo. In Simona Ronchi Della Rocca, editor, TLCA, volume 4583 ofLNCS, pages 102–117. Springer, 2007.[6] B. Gre´goire and X. Leroy. A compiled implementation of strong reduction. InMitchell Wand and Simon L. Peyton Jones, editors, ICFP, pages 235–246. ACM,2002.[7] Ulf Norell. Dependently typed programming in agda. In Pieter W. M. Koop-man, Rinus Plasmeijer, and S. Doaitse Swierstra, editors, Advanced FunctionalProgramming, volume 5832 of LNCS, pages 230–266. Springer, 2008.[8] The LuaJIT Project. http://www.http://luajit.org/.

Dedukti : a Universal Proof Checker

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Dedukti : a Universal Proof Checker

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