Induction as Deduction Modulo

Rapport interne.Inductive proofs can be built either explicitly by making use of an induction principle or implicitly by using the so-called 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 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 method and how this relates directly to the general use of an induction principle

Système de preuve modulo récurrence

Methods and systems for proof by induction are very different. The most general methods are difficult to automatize. Automated systems are sometimes difficult to justify.This thesis establishes at proof level a link between noetherian induction and induction bt rewriting, which will enable systems to cooperate in a skeptical mode in which the proof is verified thanks to the Curry-Howard isomorphism.The formalism of deduction modulo is extended to conditional congruences which are evaluated with respect to a context. Moreover,the induction ordering, which cannot be compatible with the congruence, is made protective, which means that it blocks the application of the congruence.Proof by induction by rewriting is seen as the result of the internalization of induction hypotheses in deduction modulo, which enables to explain some of the behavior of the induction by rewriting method.Les méthodes et systèmes de preuve par récurrence sont très diverses. Les méthodes les plus générales sont difficiles à automatiser. Les systèmes automatiques parfois difficiles à justifier.Cette thèse établit au niveau des preuves un lien entre récurrence noethérienne et récurrence par réécriture, ce qui permettra la coopération de systèmes dans un mode sceptique où la preuve est vérifiée grâce à l'isomorphisme de Curry-Howard. Le formalisme de la déduction modulo est étendu au traitement de congruences conditionnelles dont l'évaluation tient compte du contexte. De plus, l'ordre de récurrence qui ne peut pas être compatible avec la congruence, est rendu protecteur, c'est-à-dire qu'il bloque l'application de la congruence. La preuve par récurrence par réécriture est vue comme le résultat de l'internalisation en déduction modulo des hypothèses de récurrence, ce qui permet d'expliquer certains comportements de la méthode de récurrence par réécriture

Sequent Calculus Viewed Modulo

The first-order 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]

Système de preuve modulo récurrence

Methods and systems for proof by induction are very different. The most general methods are difficult to automatize. Automated systems are sometimes difficult to justify.This thesis establishes at proof level a link between noetherian induction and induction bt rewriting, which will enable systems to cooperate in a skeptical mode in which the proof is verified thanks to the Curry-Howard isomorphism.The formalism of deduction modulo is extended to conditional congruences which are evaluated with respect to a context. Moreover,the induction ordering, which cannot be compatible with the congruence, is made protective, which means that it blocks the application of the congruence.Proof by induction by rewriting is seen as the result of the internalization of induction hypotheses in deduction modulo, which enables to explain some of the behavior of the induction by rewriting method.Les méthodes et systèmes de preuve par récurrence sont très diverses. Les méthodes les plus générales sont difficiles à automatiser. Les systèmes automatiques parfois difficiles à justifier.Cette thèse établit au niveau des preuves un lien entre récurrence noethérienne et récurrence par réécriture, ce qui permettra la coopération de systèmes dans un mode sceptique où la preuve est vérifiée grâce à l'isomorphisme de Curry-Howard. Le formalisme de la déduction modulo est étendu au traitement de congruences conditionnelles dont l'évaluation tient compte du contexte. De plus, l'ordre de récurrence qui ne peut pas être compatible avec la congruence, est rendu protecteur, c'est-à-dire qu'il bloque l'application de la congruence. La preuve par récurrence par réécriture est vue comme le résultat de l'internalisation en déduction modulo des hypothèses de récurrence, ce qui permet d'expliquer certains comportements de la méthode de récurrence par réécriture

Système de preuve modulo récurrence

Les méthodes et systèmes de preuve par récurrence sont très diverses. Les méthodes les plus générales sont difficiles à automatiser. Les systèmes automatiques parfois difficiles à justifier.Cette thèse établit au niveau des preuves un lien entre récurrence noethérienne et récurrence par réécriture, ce qui permettra la coopération de systèmes dans un mode sceptique où la preuve est vérifiée grâce à l'isomorphisme de Curry-Howard. Le formalisme de la déduction modulo est étendu au traitement de congruences conditionnelles dont l'évaluation tient compte du contexte. De plus, l'ordre de récurrence qui ne peut pas être compatible avec la congruence, est rendu protecteur, c'est-à-dire qu'il bloque l'application de la congruence. La preuve par récurrence par réécriture est vue comme le résultat de l'internalisation en déduction modulo des hypothèses de récurrence, ce qui permet d'expliquer certains comportements de la méthode de récurrence par réécriture.Methods and systems for proof by induction are very different. The most general methods are difficult to automatize. Automated systems are sometimes difficult to justify.This thesis establishes at proof level a link between noetherian induction and induction bt rewriting, which will enable systems to cooperate in a skeptical mode in which the proof is verified thanks to the Curry-Howard isomorphism.The formalism of deduction modulo is extended to conditional congruences which are evaluated with respect to a context. Moreover,the induction ordering, which cannot be compatible with the congruence, is made protective, which means that it blocks the application of the congruence.Proof by induction by rewriting is seen as the result of the internalization of induction hypotheses in deduction modulo, which enables to explain some of the behavior of the induction by rewriting method.NANCY1-SCD Sciences & Techniques (545782101) / SudocSudocFranceF

Deduction versus Computation: the Case of Induction

Colloque sur invitation. internationale.International audienceThe 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 fondamental 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 so-called 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 framework and we provide modularity properties for it. We show how this applies to a uniform understanding of the so called induction by rewriting method and how this relates directly to the general use of an induction principle

Abstract Induction as Deduction Modulo

Inductive proofs can be built either explicitly by making use of an induction principle or implicitly by using the so-called 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 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 method and how this relates directly to the general use of an induction principle. Key words: Induction, rewriting, deduction modulo