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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 75 (14 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.
Inductive Data Type Systems
 THEORETICAL COMPUTER SCIENCE
, 1997
"... In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, w ..."
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Cited by 44 (10 self)
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In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.
From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
Embedding pure type systems in the lambdaPicalculus modulo
 TLCA
, 2007
"... The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all fu ..."
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Cited by 19 (5 self)
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The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.
Arithmetic as a theory modulo
 Proceedings of RTA’05
, 2005
"... Abstract. We present constructive arithmetic in Deduction modulo with rewrite rules only. In natural deduction and in sequent calculus, the cut elimination theorem and the analysis of the structure of cut free proofs is the key to many results about predicate logic with no axioms: analyticity and no ..."
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Cited by 18 (3 self)
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Abstract. We present constructive arithmetic in Deduction modulo with rewrite rules only. In natural deduction and in sequent calculus, the cut elimination theorem and the analysis of the structure of cut free proofs is the key to many results about predicate logic with no axioms: analyticity and nonprovability results, completeness results for proof search algorithms, decidability results for fragments, constructivity results for the intuitionistic case... Unfortunately, the properties of cut free proofs do not extend in the presence of axioms and the cut elimination theorem is not as powerful in this case as it is in pure logic. This motivates the extension of the notion of cut for various axiomatic theories such as arithmetic, Church’s simple type theory, set theory and others. In general, we can say that a new axiom will necessitate a specific extension of the notion of cut: there still is no notion of cut general enough to be applied to any axiomatic theory. Deduction modulo [2, 3] is one attempt, among others, towards this aim.
Semantic cut elimination in the intuitionistic sequent calculus
 Typed Lambda Calculi and Applications, number 3461 in Lectures
, 2005
"... Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to ..."
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Cited by 9 (5 self)
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Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.
CoLoR: a Coq library on wellfounded rewrite relations and its application to the automated verification of termination certificates
, 2010
"... ..."
WellFounded Recursive Relations
 Proc. 15th Int. Workshop Computer Science Logic, LNCS 2142:484497
, 2001
"... We give a short constructive proof of the fact that certain binary relations > are wellfounded, given a lifting a la FerreiraZantema and a wellfounded relation .. This construction generalizes several variants of the recursive path ordering on terms and of the KnuthBendix ordering. It also appl ..."
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Cited by 6 (1 self)
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We give a short constructive proof of the fact that certain binary relations > are wellfounded, given a lifting a la FerreiraZantema and a wellfounded relation .. This construction generalizes several variants of the recursive path ordering on terms and of the KnuthBendix ordering. It also applies to other domains, of graphs, of infinite terms, of word and tree automata notably. We then extend this construction further; the resulting family of wellfounded relations generalizes Jouannaud and Rubio's higherorder recursive path orderings. Keywords: Termination, wellfoundedness, path orderings, KnuthBendix orderings, calculus, higherorder path orderings, graphs, automata. 1
A completeness theorem for strong normalization in minimal deduction modulo
, 2009
"... Abstract. Deduction modulo is an extension of firstorder predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of ..."
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Cited by 6 (2 self)
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Abstract. Deduction modulo is an extension of firstorder predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. Dowek and Werner have given a semantic sufficient condition for a theory to have the strong normalization property: they have proved a ”soundness ” theorem of the form: if a theory has a model (of a particular form) then it has the strong normalization property. In this paper, we refine their notion of model in a way allowing not only to prove soundness, but also completeness: if a theory has the strong normalization property, then it has a model of this form. The key idea of our model construction is a refinement of Girard’s notion of reducibility candidates. By providing a sound and complete semantics for theories having the strong normalization property, this paper contributes to explore the idea
Truth value algebras and proof normalization
 In TYPES 2006
, 2006
"... Abstract. We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a Bvalued model for some non trivial truth values algebra B. A theory that has a Bvalued model for all truth values algebras B is said to be superconsi ..."
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Cited by 5 (3 self)
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Abstract. We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a Bvalued model for some non trivial truth values algebra B. A theory that has a Bvalued model for all truth values algebras B is said to be superconsistent. We prove that superconsistency is a modeltheoretic sufficient condition for strong normalization. 1