Advertisement

Construction of Real Algebraic Numbers in Coq

  • Cyril Cohen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7406)

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.

Keywords

Arithmetic Operation Cauchy Sequence Algebraic Number Proof Assistant Polynomial Evaluation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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 MetalanguagesMathSciNetzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    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
  4. 4.
    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version of Laplace’s proof on the existence of complex roots, http://hal.inria.fr/inria-00592284/PDF/laplace.pdf (unpublished)
  5. 5.
    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 MathSciNetGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    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 CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Lang, S.: Algebra. Graduate texts in mathematics. Springer (2002)Google Scholar
  11. 11.
    Mines, R., Richman, F., Ruitenburg, W.: A course in constructive algebra. Universitext (1979); Springer-Verlag (1988)Google Scholar
  12. 12.
    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 CrossRefGoogle Scholar
  13. 13.
    Project, T.M.C.: SSReflect extension and libraries, http://www.msr-inria.inria.fr/Projects/math-components/index_html

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Cyril Cohen
    • 1
    • 2
    • 3
  1. 1.INRIA Saclay–Île-de-FranceFrance
  2. 2.LIX École PolytechniqueFrance
  3. 3.Microsoft Research - INRIA Joint CentreFrance

Personalised recommendations