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Abstract Induction as Deduction Modulo
"... 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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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 paper, we propose such an approach based on the general notion of deduction modulo. We extend slightly
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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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
Deduction modulo theory
"... 1.1 Weaker vs. stronger systems Contemporary proof theory goes into several directions at the same time. One of them aims at analysing proofs, propositions, connectives, etc., that is at decomposing them into more atomic objects. This often leads to design systems ..."
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1.1 Weaker vs. stronger systems Contemporary proof theory goes into several directions at the same time. One of them aims at analysing proofs, propositions, connectives, etc., that is at decomposing them into more atomic objects. This often leads to design systems
the Tortoise using Deduction Modulo⋆
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
On constructive cut admissibility in deduction modulo
 In Thorsten Altenkirch and Conor McBride, editors, TYPES for proofs and programs
"... Abstract. Deduction modulo is a theoretical framework which allows the introduction of computational steps in deductive systems. This approach is well suited to automated theorem proving. We describe a proofsearch method based upon tableaux for Gentzen’s intuitionistic LJ extended with rewrite ru ..."
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Cited by 6 (4 self)
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Abstract. Deduction modulo is a theoretical framework which allows the introduction of computational steps in deductive systems. This approach is well suited to automated theorem proving. We describe a proofsearch method based upon tableaux for Gentzen’s intuitionistic LJ extended with rewrite
Orthogonality and Boolean Algebras for Deduction Modulo
"... Abstract. Originating from automated theorem proving, deduction modulo removes computational arguments from proofs by interleaving rewriting with the deduction process. From a prooftheoretic point of view, deduction modulo defines a generic notion of cut that applies to any firstorder theory pre ..."
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Cited by 2 (0 self)
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Abstract. Originating from automated theorem proving, deduction modulo removes computational arguments from proofs by interleaving rewriting with the deduction process. From a prooftheoretic point of view, deduction modulo defines a generic notion of cut that applies to any firstorder theory
Embedding deduction modulo into a prover
 CSL. Lecture Notes in Computer Science
, 2010
"... Abstract. Deduction modulo consists in presenting a theory through rewrite rules to support automatic and interactive proof search. It induces proof search methods based on narrowing, such as the polarized resolution modulo. We show how to combine this method with more traditional ordering restric ..."
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Cited by 6 (1 self)
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Abstract. Deduction modulo consists in presenting a theory through rewrite rules to support automatic and interactive proof search. It induces proof search methods based on narrowing, such as the polarized resolution modulo. We show how to combine this method with more traditional ordering
Positive Deduction modulo Regular Theories
 in Proc. CSL '95, LNCS 1092
, 1995
"... . We propose a new technique for dealing with an equational theory E in the clausal framework. This consists of the definition of two inference rules called contextual superposition and extended superposition, and of an algorithm for computing the only needed applications of these last inference ..."
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Cited by 8 (1 self)
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the refutational completeness of their superpositionbased systems for any regular theory. We also combine a collection of strategies that decrease the number of possible deductions, without loss of completeness: the superposition strategy, the positive ordering strategy, and a simplification strategy
Unbounded prooflength speedup in deduction modulo
 CSL 2007, VOLUME 4646 OF LNCS
, 2007
"... In 1973, Parikh proved a speedup theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for h ..."
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Cited by 7 (3 self)
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for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We prove that i+1th order arithmetic can be linearly simulated into ith order
On the Complexity of Deduction Modulo Leaf Permutative Equations
"... Abstract. In the context of equational reasoning, it was proposed by J. Avenhaus and D. Plaisted, in [1], to deal with leaf permutative equations in a uniform, specialized way. The simplicity of these equations, and the useless variations that they produce are good incentives to lift theorem proving ..."
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proving to socalled stratified terms, in order to perform deduction modulo such equations. This requires specialized algorithms for standard problems involved in automated deduction. In order to analyse the computational complexity of these problems, we focus on the group theoretic properties
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