Abstract
Integrating decision procedures in proof assistants in a safe way is a major challenge. In this paper, we describe how, starting from Hilbert’s Nullstellensatz theorem, we combine a modified version of Buchberger’s algorithm and some reflexive techniques to get an effective procedure that automatically produces formal proofs of theorems in geometry. The method is implemented in the Coq system but, since our specialised version of Buchberger’s algorithm outputs explicit proof certificates, it could be easily adapted to other proof assistants.
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Allen, S.F., Constable, R.L., Howe, D.J., Aitken, W.E.: The Semantics of Reflected Proof. In: LICS, pp. 95–105. IEEE Computer Society, Los Alamitos (1990)
Buchberger, B.: Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Journal of Symbolic Computation 41(3-4) (2006)
Chaieb, A., Wenzel, M.: Context aware calculation and deduction. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds.) MKM/CALCULEMUS 2007. LNCS (LNAI), vol. 4573, pp. 27–39. Springer, Heidelberg (2007)
Char, B.W., Fee, G.J., Geddes, K.O., Gonnet, G.H., Monagan, M.B.: A Tutorial Introduction to MAPLE. Journal of Symbolic Computation 2(2), 179–200 (1986)
Chou, S.-C.: Mechanical geometry theorem proving. Kluwer Academic Publishers, Dordrecht (1987)
Chou, S.-C., Gao, X.-S.: Ritt-Wu’s Decomposition Algorithm and Geometry Theorem Proving. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 207–220. Springer, Heidelberg (1990)
Eisenbud, D.: Commutative Algebra: with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer, Heidelberg (1999)
Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (f4). Journal of Pure and Applied Algebra 139(1/3), 61–88 (1999)
Giusti, M., Heintz, J., Morais, J.E., Morgenstern, J., Pardo, L.M.: Straight-line programs in geometric elimination theory. Journal of Pure and Applied Algebra 124(1/3), 101–146 (1998)
Grayson, D.R., Stillman, M.E.: Macaulay2, http://www.math.uiuc.edu/Macaulay2/
Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: International Conference on Functional Programming 2002, pp. 235–246. ACM Press, New York (2002)
Grégoire, B., Mahboubi, A.: Proving equalities in a commutative ring done right in coq. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 98–113. Springer, Heidelberg (2005)
Grégoire, B., Théry, L., Werner, B.: A computational approach to pocklington certificates in type theory. In: Hagiya, M. (ed.) FLOPS 2006. LNCS, vol. 3945, pp. 97–113. Springer, Heidelberg (2006)
Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)
Harrison, J.: Complex quantifier elimination in HOL. In: TPHOLs 2001: Supplemental Proceedings. Division of Informatics, pp. 159–174. University of Edinburgh (2001), published as Informatics Report Series EDI-INF-RR-0046
Harrison, J.: Automating elementary number-theoretic proofs using gröbner bases. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 51–66. Springer, Heidelberg (2007)
Harrison, J.: Verifying nonlinear real formulas via sums of squares. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 102–118. Springer, Heidelberg (2007)
Harrison, J.: Handbook of Practical Logic and Automated Reasoning. Cambridge University Press, Cambridge (2009)
Hohenwarter, M., Preiner, J.: Dynamic Mathematics with GeoGebra. Journal of Online Mathematics 7, ID 1448 (March 2007)
Kapur, D.: Geometry theorem proving using Hilbert’s Nullstellensatz. In: SYMSAC 1986: Proceedings of the Fifth ACM Symposium on Symbolic and Algebraic Computation, pp. 202–208. ACM, New York (1986)
Kapur, D.: A refutational approach to geometry theorem proving. Artificial Intelligence 37(1-3), 61–93 (1988)
Kapur, D.: Automated Geometric Reasoning: Dixon Resultants, Gröbner Bases, and Characteristic Sets. In: Wang, D. (ed.) ADG 1996. LNCS, vol. 1360, pp. 1–36. Springer, Heidelberg (1998)
Kreisel, G., Krivine, J.L.: Elements of Mathematical Logic (Model Theory). Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1967)
Paulson, L.C.: Isabelle: A generic theorem prover. Journal of Automated Reasoning 828 (1994)
Pottier, L.: Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics. In: Proceedings of the LPAR Workshops: Knowledge Exchange: Automated Provers and Proof Assistants, and The 7th International Workshop on the Implementation of Logics. CEUR Workshop Proceedings, vol. (418) (2008)
Robu, J.: Geometry Theorem Proving in the Frame of the Theorema Project. Tech. Rep. 02-23, RISC Report Series, University of Linz, Austria, phD Thesis (September 2002)
Théry, L.: A Machine-Checked Implementation of Buchberger’s Algorithm. Journal of Automated Reasoning 26(2) (2001)
Wiedijk, F.: Formalizing 100 Theorems, http://www.cs.ru.nl/~freek/100
Wu, W.-T.: On the Decision Problem and the Mechanization of Theorem-Proving in Elementary Geometry. In: Automated Theorem Proving - After 25 Years, pp. 213–234. American Mathematical Society, Providence (1984)
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Grégoire, B., Pottier, L., Théry, L. (2011). Proof Certificates for Algebra and Their Application to Automatic Geometry Theorem Proving. In: Sturm, T., Zengler, C. (eds) Automated Deduction in Geometry. ADG 2008. Lecture Notes in Computer Science(), vol 6301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21046-4_3
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DOI: https://doi.org/10.1007/978-3-642-21046-4_3
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