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Regaining Cut Admissibility in Deduction Modulo using Abstract Completion
, 2009
"... Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to p ..."
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Cited by 9 (2 self)
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Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cutfree proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the KnuthBendix completion in a nontrivial way, using the framework of abstract canonical systems. These
ProjectTeam Pareo Formal Islands: Foundations and Applications
"... c t i v it y e p o r t 2009 Table of contents ..."
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Consistency Implies Cut Admissibility
 PSATTT'11: INTERNATIONAL WORKSHOP ON PROOFSEARCH IN AXIOMATIC THEORIES AND TYPE THEORIES (2011)
, 2011
"... For any finite and consistent firstorder theory, we can find a presentation as a rewriting system that enjoys cut admissibility. ..."
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For any finite and consistent firstorder theory, we can find a presentation as a rewriting system that enjoys cut admissibility.
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