Abstract
We present a theorem prover ArgoCLP based on coherent logic that can be used for generating both readable and formal (machine verifiable) proofs in various theories, primarily geometry. We applied the prover to various axiomatic systems and proved tens of theorems from standard university textbooks on geometry. The generated proofs can be used in different educational purposes and can contribute to the growing body of formalized mathematics. The system can be used, for instance, in showing that modifications of some axioms do not change the power of an axiom system. The system can also be used as an assistant for proving appropriately chosen subgoals of complex conjectures.
This work has been partly supported by the grant 174021 of the Ministry of Science of Serbia and by the SNF SCOPES grant IZ73Z0_127979/1.
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References
Avigad, J., Dean, E., Mumma, J.: A formal system for Euclid’s Elements. The Review of Symbolic Logic (2009)
Bezem, M., Coquand, T.: Automating coherent logic. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 246–260. Springer, Heidelberg (2005)
Bezem, M., Hendriks, D.: On the Mechanization of the Proof of Hessenberg’s Theorem in Coherent Logic. Journal of Automated Reasoning 40(1) (2008)
Borsuk, K., Szmielew, W.: Foundations of Geometry. Norht-Holland Publishing Company, Amsterdam (1960)
Buchberger, B.: An Algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal. PhD thesis. University of Innsbruck (1965)
Buchberger, B., et al.: Theorema: Towards computer-aided mathematical theory exploration. Journal of Applied Logic (2006)
Chou, S.-C.: Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, Dordrecht (1988)
Chou, S.-C., Gao, X.-S.: Automated reasoning in geometry. In: Handbook of Automated Reasoning. Elsevier (2001)
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine Proofs in Geometry. World Scientific, Singapore (1994)
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Automated production of traditional proofs for constructive geometry theorems. In: IEEE Symposium on Logic in Computer Science LICS. IEEE Computer Society Press (1993)
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Automated generation of readable proofs with geometric invariants, II. theorem proving with full-angles. Journal of Automated Reasoning 17 (1996)
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering. Journal Automated Reasoning 25(3) (2000)
Dehlinger, C., Dufourd, J.-F., Schreck, P.: Higher-order intuitionistic formalization and proofs in hilbert’s elementary geometry. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 306–323. Springer, Heidelberg (2001)
Duprat, J.: Une axiomatique de la géométrie plane en coq. In: Actes des JFLA 2008, INRIA (2008)
Fisher, J., Bezem, M.: Skolem Machines and Geometric Logic. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) ICTAC 2007. LNCS, vol. 4711, pp. 201–215. Springer, Heidelberg (2007)
Gelernter, H., Hanson, J.R., Loveland, D.W.: Empirical explorations of the geometry-theorem proving machine. In: Computers and Thought. MIT Press (1995)
Guilhot, F.: Formalisation en coq d’un cours de géométrie pour le lycée. Journées Francophones des Langages Applicatifs (2004)
Harrison, J.: Hol light: A Tutorial Introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)
Heath, T.L.: The Thirteen Books of Euclid’s Elements. Dover Publications, New-York (1956)
Hilbert, D.: Grundlagen der Geometrie, Leipzig (1899)
Janičić, P., Kordić, S.: EUCLID — the geometry theorem prover. FILOMAT 9(3) (1995)
Kahn, G.: Constructive geometry according to Jan von Plato. Coq contribution. Coq V5.10 (1995)
Kapur, D.: Using Gröbner bases to reason about geometry problems. Journal of Symbolic Computation 2(4) (1986)
Magaud, N., Narboux, J., Schreck, P.: Formalizing Projective Plane Geometry in Coq. In: Sturm, T., Zengler, C. (eds.) ADG 2008. LNCS, vol. 6301, pp. 141–162. Springer, Heidelberg (2011)
Magaud, N., Narboux, J., Schreck, P.: Formalizing Desargues’ theorem in Coq using ranks. In: ACM Symposium on Applied Computing. ACM (2009)
Meikle, L.I., Fleuriot, J.D.: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In: Basin, D., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 319–334. Springer, Heidelberg (2003)
Narboux, J.: Formalisation et automatisation du raisonnement géométrique en Coq. PhD thesis. Université Paris Sud (2006)
Narboux, J.: Mechanical theorem proving in tarski’s geometry. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 139–156. Springer, Heidelberg (2007)
Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)
von Plato, J.: The axioms of constructive geometry. Annals of Pure and Applied Logic 76 (1995)
von Plato, J.: Formalization of Hilbert’s geometry of incidence and parallelism. Synthese 110 (1997)
Polonsky, A.: Proofs, Types, and Lambda Calculus. PhD thesis. University of Bergen (2011)
Quaife, A.: Automated development of Tarski’s geometry. Journal of Automated Reasoning 5(1) (1989)
Schwabhuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie. Springer, Berlin (1983)
Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Journal of Automated Reasoning 43(4) (2009)
Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. Journal of ACM 22(2) (1975)
Tarski, A.: What is elementary geometry? In: The Axiomatic Method, with Special Reference to Geometry and Physics. North-Holland (1959)
The Coq development team. The Coq proof assistant reference manual, Version 8.2. TypiCal Project (2009)
Trybulec, A.: Mizar. In: Wiedijk, F. (ed.) The Seventeen Provers of the World. LNCS (LNAI), vol. 3600, pp. 20–23. Springer, Heidelberg (2006)
Wenzel, M.T.: Isar - a Generic Interpretative Approach to Readable Formal Proof Documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 167–183. Springer, Heidelberg (1999)
Wu, W.-T.: On the decision problem and the mechanization of theorem proving in elementary geometry. Scientia Sinica 21 (1978)
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Stojanović, S., Pavlović, V., Janičić, P. (2011). A Coherent Logic Based Geometry Theorem Prover Capable of Producing Formal and Readable Proofs. In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds) Automated Deduction in Geometry. ADG 2010. Lecture Notes in Computer Science(), vol 6877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25070-5_12
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