Abstract
Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LK-proof with only atomic cuts.
In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cut-elimination method. The system CERES already proved efficient in handling very large proofs.
Supported by the Austrian Science Fund (project no. P17995-N12).
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References
Aigner, M., Ziegler, G.M.: Proofs from THE BOOK. Springer, Heidelberg (1998)
Andrews, P.B.: Resolution in Type Theory. Journal of Symbolic Logic 36, 414–432 (1971)
Leitsch, A., Baaz, M., Hetzl, S., Richter, C., Spohr, H.: Cut-Elimination: Experiments with CERES. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 481–495. Springer, Heidelberg (2005)
Baaz, M., Leitsch, A.: On skolemization and proof complexity. Fundamenta Informaticae 20(4), 353–379 (1994)
Baaz, M., Leitsch, A.: Cut normal forms and proof complexity. Annals of Pure and Applied Logic 97, 127–177 (1999)
Baaz, M., Leitsch, A.: Cut-Elimination and Redundancy-Elimination by Resolution. Journal of Symbolic Computation 29, 149–176 (2000)
Baaz, M., Leitsch, A.: Towards a Clausal Analysis of Cut-Elimination. Journal of Symbolic Computation 41, 381–410 (2006)
Eder, E.: Relative complexities of first-order calculi. Vieweg (1992)
Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 405–431 (1935)
Girard, J.Y.: Proof Theory and Logical Complexity. In: Studies in Proof Theory, Bibliopolis, Napoli (1987)
Luckhardt, H.: Herbrand-Analysen zweier Beweise des Satzes von Roth: polynomiale Anzahlschranken. The Journal of Symbolic Logic 54, 234–263 (1989)
Nieuwenhuis, R., Rubio, A.: Paramodulation-based Theorem Proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 371–443. Elsevier, Amsterdam (2001)
Polya, G.: Mathematics and plausible reasoning: Induction and Analogy in Mathematics, vol. I. Princeton University Press, Princeton (1954)
Polya, G.: Mathematics and plausible reasoning: Patterns of Plausible Inference, vol. II. Princeton University Press, Princeton (1954)
Takeuti, G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987)
Urban, C.: Classical Logic and Computation Ph.D. Thesis, University of Cambridge Computer Laboratory (2000)
Degtyarev, A., Voronkov, A.: Equality Reasoning in Sequent-Based Calculi. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I ch. 10, pp. 611–706. Elsevier Science, Amsterdam (2001)
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Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H. (2006). Proof Transformation by CERES. In: Borwein, J.M., Farmer, W.M. (eds) Mathematical Knowledge Management. MKM 2006. Lecture Notes in Computer Science(), vol 4108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11812289_8
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