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Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 112 (18 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
Deduction versus Computation: The Case of Induction
"... Abstract. The fundamental difference and the essential complementarity between computation and deduction are central in computer algebra, automated deduction, proof assistants and in frameworks making them cooperating.In this work we show that the fundamental proof method of induction can be underst ..."
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Cited by 1 (0 self)
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Abstract. The fundamental difference and the essential complementarity between computation and deduction are central in computer algebra, automated deduction, proof assistants and in frameworks making them cooperating.In this work we show that the fundamental proof method of induction can be understood and implemented as either computation or deduction. Inductive proofs can be built either explicitly by making use of an induction principle or implicitly by using the socalled induction by rewriting and inductionless induction methods.When mechanizing proof construction, explicit induction is used in proof assistants and implicit induction is used in rewrite based automated theorem provers.The two approaches are clearly complementary but up to now there was no framework able to encompass and to understand uniformly the two methods.In this work, we propose such an approach based on the general notion of deduction modulo.We extend slightly the original version of the deduction modulo
Sequent Calculus Viewed Modulo
 University of Birmingham
, 2000
"... The firstorder sequent calculus is generally considered as containing no computation but only pure deduction. But this is not completely true if we look at it carefully, using a deduction modulo framework. The origins of the computational part are first implicit behaviours of the calculus, then wel ..."
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Cited by 4 (0 self)
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The firstorder sequent calculus is generally considered as containing no computation but only pure deduction. But this is not completely true if we look at it carefully, using a deduction modulo framework. The origins of the computational part are first implicit behaviours of the calculus, then well known consequences that we do not want to prove any more. We end up with a calculus fully in the spirit of deduction modulo [DHK98].
Proof Search and Proof Check for Equational and Inductive Theorems
 Conference on Automated Deduction  CADE19
, 2003
"... Abstract. This paper presents ongoing researches on theoretical and practical issues of combining rewriting based automated theorem proving and userguided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof a ..."
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Abstract. This paper presents ongoing researches on theoretical and practical issues of combining rewriting based automated theorem proving and userguided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN rewriting based system, focusing first on equational and then on inductive proofs. Different concepts, especially rewriting calculus and deduction modulo, contribute to define and to relate proof search, proof representation and proof check.
ProjectTeam PROTHEO Constraints, Mechanized Deduction and Proofs of Software Properties
"... d ' ctivity ..."
Induction as Deduction Modulo
, 2001
"... Inductive proofs can be built either explicitly by making use of an induction principle or implicitly by using the socalled induction by rewriting and inductionless induction methods. When mechanizing proof construction, explicit induction is used in proof assistants and implicit induction is used ..."
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Inductive proofs can be built either explicitly by making use of an induction principle or implicitly by using the socalled induction by rewriting and inductionless induction methods. When mechanizing proof construction, explicit induction is used in proof assistants and implicit induction is used in automated theorem provers. The two approaches are clearly complementary but up to now there was no framework able to encompass and to understand uniformly the two methods. In this paper, we propose such an approach based on the general notion of deduction modulo. We extend slightly the original version of the deduction modulo framework and we provide modularity properties for it. We show how this applies to a uniform understanding of the so called induction by rewriting and inductionless induction methods and how this relates directly to the general use of an induction principle. 1 Introduction Proof by induction is a fundamental proof method in mathematics. Since the emergence of comput...
ProjectTeam PROTHEO Constraints, Mechanized Deduction and Proofs of Software Properties
"... c t i v it y e p o r t 2007 Table of contents ..."
Programming Logics Group
"... This report 1 covers the seminar no. 01101 on Deduction, held at Dagstuhl, Germany during March 4–March 9, 2001. This seminar was organized by U. Furbach (Koblenz, Germany), H. Ganzinger (Saarbrücken, Germany) and D. Kapur (Albuquerque, USA). It brought together about 50 researchers from various cou ..."
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This report 1 covers the seminar no. 01101 on Deduction, held at Dagstuhl, Germany during March 4–March 9, 2001. This seminar was organized by U. Furbach (Koblenz, Germany), H. Ganzinger (Saarbrücken, Germany) and D. Kapur (Albuquerque, USA). It brought together about 50 researchers from various countries. Dagstuhl, a place being developed exclusively for research activities in Computer Science, provides an excellent atmosphere for researchers to meet and exchange ideas. During this seminar we had 40 talks and a discussion session on an ’open source software repository’.
Table des matires
"... Introduction 2 1.1 Prsentation du stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Notations et dnitions prliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Le DAWG 3 2.1 Dnitions et proprits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..."
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Introduction 2 1.1 Prsentation du stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Notations et dnitions prliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Le DAWG 3 2.1 Dnitions et proprits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Exemple, remarques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Algorithme de grappe 5 3.1 Mise jour du DAWG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Ajout d'une lettre un mot cl . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 Dchargement d'un mot du DAWG . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 L'algorithme original . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Le DAWG tendu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.2 L'algorithme de pattern matching . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Extensions 8 4.1 Trous bor
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