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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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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
RhoCalculi for Computation and Logic
"... The rhocalculi provide enlightening concepts for both computing and reasoning as well as their combination. They consist in the generalization of lambdacalculus to structures like terms, propositions or graphs and we will show how their interrelations with deduction provide powerful frameworks for ..."
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The rhocalculi provide enlightening concepts for both computing and reasoning as well as their combination. They consist in the generalization of lambdacalculus to structures like terms, propositions or graphs and we will show how their interrelations with deduction provide powerful frameworks for the next generation of proof assistants.