Abstract
This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete Archimedean real closed field. This construction hence implements an interface of real closed field. Instances of such an interface immediately enjoy quantifier elimination thanks to a previous work. This work also intends to be a basis for the construction of complex algebraic numbers and to be a reference implementation for the certification of numerous algorithms relying on algebraic numbers in computer algebra.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barthe, G., Capretta, V., Pons, O.: Setoids in type theory. J. of Functional Programming 13(2), 261–293 (2003); Special Issue on Logical Frameworks and Metalanguages
Bostan, A.: Algorithmique efficace pour des opérations de base en Calcul formel. Ph.D. thesis, École polytechnique (2003), http://algo.inria.fr/bostan/these/These.pdf
Cohen, C.: Types quotients en COQ. In: Hermann (ed.) Actes des 21éme Journées Francophones des Langages Applicatifs (JFLA 2010), INRIA, Vieux-Port La Ciotat, France (January 2010), http://jfla.inria.fr/2010/actes/PDF/cyrilcohen.pdf
Cohen, C., Coquand, T.: A constructive version of Laplace’s proof on the existence of complex roots, http://hal.inria.fr/inria-00592284/PDF/laplace.pdf (unpublished)
Cohen, C., Mahboubi, A.: Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination. Logical Methods in Computer Science 8(1-02), 1–40 (2012), http://hal.inria.fr/inria-00593738
Delahaye, D.: A Tactic Language for the System COQ. In: Parigot, M., Voronkov, A. (eds.) LPAR 2000. LNCS (LNAI), vol. 1955, pp. 85–95. Springer, Heidelberg (2000)
Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L.: Packaging Mathematical Structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 327–342. Springer, Heidelberg (2009)
Geuvers, H., Niqui, M.: Constructive Reals in COQ: Axioms and Categoricity. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 79–95. Springer, Heidelberg (2002), http://dl.acm.org/citation.cfm?id=646540.696040
Krebbers, R., Spitters, B.: Computer Certified Efficient Exact Reals in COQ. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) MKM 2011 and Calculemus 2011. LNCS, vol. 6824, pp. 90–106. Springer, Heidelberg (2011)
Lang, S.: Algebra. Graduate texts in mathematics. Springer (2002)
Mines, R., Richman, F., Ruitenburg, W.: A course in constructive algebra. Universitext (1979); Springer-Verlag (1988)
O’Connor, R.: Certified Exact Transcendental Real Number Computation in COQ. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 246–261. Springer, Heidelberg (2008), http://dx.doi.org/10.1007/978-3-540-71067-7_21
Project, T.M.C.: SSReflect extension and libraries, http://www.msr-inria.inria.fr/Projects/math-components/index_html
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cohen, C. (2012). Construction of Real Algebraic Numbers in Coq . In: Beringer, L., Felty, A. (eds) Interactive Theorem Proving. ITP 2012. Lecture Notes in Computer Science, vol 7406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32347-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-32347-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32346-1
Online ISBN: 978-3-642-32347-8
eBook Packages: Computer ScienceComputer Science (R0)