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Tom: Piggybacking Rewriting on Java
, 2007
"... We present the Tom language that extends Java with the purpose of providing high level constructs inspired by the rewriting community. Tom furnishes a bridge between a general purpose language and higher level specifications that use rewriting. This approach was motivated by the promotion of rewriti ..."
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Cited by 34 (6 self)
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We present the Tom language that extends Java with the purpose of providing high level constructs inspired by the rewriting community. Tom furnishes a bridge between a general purpose language and higher level specifications that use rewriting. This approach was motivated by the promotion of rewriting techniques and their integration in large scale applications. Powerful matching capabilities along with a rich strategy language are among Tom’s strong points, making it easy to use and competitive with other rule based languages.
Revisiting cutelimination: One difficult proof is really a proof
 RTA 2008
, 2008
"... Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. ..."
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Cited by 5 (3 self)
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Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the first authors PhD. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.
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 3 (2 self)
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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 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 arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speedup between ith order arithmetic modulo this system and ith order arithmetic without modulo. All this allows us to prove that the speedup conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.
Normalization in supernatural deduction and in deduction modulo. Available at http://hal.inria.fr/ inria00141720
, 2007
"... Abstract. Deduction modulo and Supernatural deduction are two extentions of predicate logic with computation rules. Whereas the application of computation rules in deduction modulo is transparent, these rules are used to build nonlogical deduction rules in Supernatural deduction. In both cases, add ..."
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Cited by 3 (2 self)
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Abstract. Deduction modulo and Supernatural deduction are two extentions of predicate logic with computation rules. Whereas the application of computation rules in deduction modulo is transparent, these rules are used to build nonlogical deduction rules in Supernatural deduction. In both cases, adding computation rules may jeopardize proof normalization, but various conditions have been given in both cases, so that normalization is preserved. We prove in this paper that normalization in Supernatural deduction and in Deduction modulo are equivalent, i.e. the set of computation rules for which one system strongly normalizes is the same as the set of computation rules for which the other is. 1
Efficiently Simulating HigherOrder Arithmetic by a FirstOrder Theory Modulo
"... Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. Ho ..."
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Cited by 2 (0 self)
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Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. However, a theorem stated by Gödel and proved by Parikh expresses that proofs in secondorder arithmetic may be unboundedly shorter than proofs in firstorder arithmetic, even when considering only formulæ provable in firstorder arithmetic. We investigate how deduction modulo can be used to translate proofs of higherorder arithmetic into firstorder proofs without inflating their length. First we show how higher orders can be encoded through a quite simple (finite, terminating, confluent, leftlinear) rewrite system. Then, a proof in higherorder arithmetic can be linearly translated into a proof in firstorder arithmetic modulo this system. Second, in the continuation of a work of Dowek and Werner, we show how to express the whole higherorder arithmetic as a rewrite system. Then, proofs of higherorder arithmetic can be linearly translated into proofs in the empty theory modulo this rewrite system. These results show that the speedup between firstand secondorder arithmetic, and more generally between i th and i +1 storder arithmetic, can in fact be expressed as computation, and does not lie in the really deductive part of the proofs.
Superdeduction at Work
"... Abstract Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalization of a proof term language that ..."
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Abstract Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalization of a proof term language that models appropriately superdeduction. We finaly examplify on several examples, including equality and noetherian induction, the usefulness of this approach which is implemented in the lemuridæ system, written in TOM. 1
Proof Analysis with HLK, CERES and ProofTool: Current Status and Future Directions
, 2008
"... CERES, HLK and ProofTool form together a system for the computeraided analysis of mathematical proofs. This analysis is based on a proof transformation known as cutelimination, which corresponds to the elimination of lemmas in the corresponding informal proofs. Consequently, the resulting formal p ..."
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CERES, HLK and ProofTool form together a system for the computeraided analysis of mathematical proofs. This analysis is based on a proof transformation known as cutelimination, which corresponds to the elimination of lemmas in the corresponding informal proofs. Consequently, the resulting formal proof in atomiccut normal form corresponds to a direct, i.e. without lemmas, informal mathematical proof of the given theorem. In this paper, we firstly describe the current status of the whole system from the point of view of its usage. Subsequently, we discuss each component in more detail, briefly explaining the formal calculi (LK and LKDe) used, the intermediary language HandyLK, the CERES method of cutelimination by resolution and the extraction of Herbrand sequents. Three successful cases of application of the system to mathematical proofs are then summarized. And finally we discuss extensions of the system that are currently under development or that are planned for the shortterm future.
Superdeduction in λµ˜µ
, 2010
"... Abstract. Superdeduction is a method specially designed to ease the use of firstorder theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proofterm language and a cutelimination reduction already exis ..."
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Abstract. Superdeduction is a method specially designed to ease the use of firstorder theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proofterm language and a cutelimination reduction already exist for superdeduction, both based on Christian Urban’s work on classical sequent calculus. However Christian Urban’s calculus is not directly related to the CurryHoward correspondence contrarily to the λµ˜µcalculus which relates straightaway to the λcalculus. This short paper is my first step towards a further exploration of the computational content of superdeduction proofs, for I extend the λµ˜µcalculus in order to obtain a proofterm langage together with a cutelimination reduction for superdeduction. I also prove strong normalisation for this extension of the λµ˜µcalculus.
Axiom directed Focusing long version
, 2008
"... Abstract. Superdeduction and deduction modulo are methods specially designed to ease the use of firstorder theories in predicate logic. Superdeduction modulo, which combines both, enables the user to make a distinct use of computational and reasoning axioms. Although soundness is ensured, using sup ..."
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Abstract. Superdeduction and deduction modulo are methods specially designed to ease the use of firstorder theories in predicate logic. Superdeduction modulo, which combines both, enables the user to make a distinct use of computational and reasoning axioms. Although soundness is ensured, using superdeduction and deduction modulo to extend deduction with awkward theories can jeopardize essential properties of the extended system such as cutelimination or completeness w.r.t. predicate logic. Therefore one has to design criteria for theories which can safely be used through superdeduction and deduction modulo. In this paper we revisit the superdeduction paradigm by comparing it with the focusing approach. In particular we prove a focalization theorem for cutfree superdeduction modulo: we show that permutations of inference rules can transform any cutfree proof in deduction modulo into a cutfree proof in superdeduction modulo and conversely, provided that some hypotheses on the synchrony of reasoning axioms are verified. It implies that cutelimination for deduction modulo and for superdeduction modulo are equivalent. Since several criteria have already been proposed for theories that do not break cutelimination of the corresponding deduction modulo system, these criteria also imply cutelimination of the superdeduction modulo system, provided our synchrony hypotheses hold. Finally we design a tableaux method for superdeduction modulo which is sound and complete provided cutelimination holds. Key words: proof theory, superdeduction, focusing, deduction modulo 1