Formalizing a discrete model of the continuum in Coq from a discrete geometry perspective

  • Nicolas Magaud
  • Agathe Chollet
  • Laurent Fuchs


This work presents a formalization of the discrete model of the continuum introduced by Harthong (1989), the Harthong-Reeb line. This model was at the origin of important developments in the Discrete Geometry field (Reveillès and Richard, Ann. Math. Artif. Intell. Math. Inform. 16(14), 89–152 (1996)). The formalization is based on the work presented in Chollet et al. (2012, 2009) where it was shown that the Harthong-Reeb line satisfies the axioms for constructive real numbers introduced by Bridges (1999). Laugwitz-Schmieden numbers (Laugwitz 1983) are then introduced and their limitations with respect to being a model of the Harthong-Reeb line is investigated (Chollet et al., Theor. Comput. Sci. 466, 2–19 (2012)). In this paper, we transpose all these definitions and properties into a formal description using the Coq proof assistant. We also show that Laugwitz-Schmieden numbers can be used to actually compute continuous functions. We hope that this work could improve techniques for both implementing numeric computations and reasoning about them in geometric systems.


Coq Formal proofs Discrete geometry Non-standard arithmetic Arithmetisation scheme Exact computations 

Mathematics Subject Classification (2010)

03F55 03H15 03F60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development, Coq’Art: The Calculus of Inductive Constructions. Springer, Berlin Heidelberg New York (2004)CrossRefGoogle Scholar
  2. 2.
    Bridges, D., Palmgren, E.: Constructive Mathematics. The Stanford Encyclopedia of Philosophy. available from (2013)
  3. 3.
    Bridges, D., Reeves, S.: Constructive mathematics, in theory and programming practice. Technical Report CDMTCS-068, Centre for Discrete Mathematics and Theorical Computer Science (1997)Google Scholar
  4. 4.
    Bridges, D.S.: Constructive mathematics: A foundation for computable analysis. Theor. Comput. Sci. 219 (1–2), 95–109 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chlipala, A.: Certified programming with dependent types: A pragmatic introduction to the coq proof assistant. MIT Press, Cambridge (2013)Google Scholar
  6. 6.
    Chollet, A.: Formalismes non classiques pour le traitement informatique de la topologie et de la géométrie discrète. PhD thesis, Université de la Rochelle (2010)Google Scholar
  7. 7.
    Chollet, A., Wallet, G., Fuchs, L., Andres, E., Largeteau-Skapin, G.: Foundational aspects of multiscale digitization. Theor. Comput. Sci. 466, 2–19 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chollet, A., Wallet, G., Fuchs, L., Largeteau-Skapin, G., Andres, E.: Insight in discrete geometry and computational content of a discrete model of the continuum. Pattern Recog. 42, 2220–2228 (2009)CrossRefGoogle Scholar
  9. 9.
    Chollet, A., Wallet, G., Fuchs, L., Largeteau-Skapin, G., Andres, E.: Ω-Arithmetization: A Discrete Multi-resolution Representation of Real Functions. In: Wiederhold, P., Barneva, P.R. (eds.) Combinatorial Image Analysis: 13th International Workshop, IWCIA of Lecture Notes in Computer Science (LNCS), vol. 5852, pp 316–329, Mexico (2009)Google Scholar
  10. 10.
    Coq development team: The Coq Proof Assistant Reference Manual, Version 8.2. LogiCal Project (2008)Google Scholar
  11. 11.
    Diener, F., Reeb, G.: Analyse non standard. Hermann, Paris (1989)Google Scholar
  12. 12.
    Diener, M.: Application du calcul de Harthong-Reeb aux routines graphiques. In: Salanskis, J.-M., Sinaceurs, H. (eds.) Le Labyrinthe du Continu, pp 424–435. Springer, Berlin Heidelberg New York (1992)Google Scholar
  13. 13.
    Fleuriot, J.: Exploring the foundations of discrete analytical geometry in Isabelle/HOL. In: Schreck, P., Richter-Gebert, J., Narboux, J. (eds.) Proceedings of Automated Deduction in Geometry 2010 of LNAI, vol. 6877. Springer, Berlin Heidelberg New York (2011)Google Scholar
  14. 14.
    Fleuriot, J.D., Paulson, L.C.: A combination of nonstandard analysis and geometry theorem proving, with application to newton’s principia. In: Kirchner, C., Kirchner, H. (eds.) CADE of Lecture Notes in Computer Science, vol. 1421, pp 3–16. Springer, Berlin Heidelberg New York (1998)Google Scholar
  15. 15.
    Geuvers, H., Niqui, M., Spitters, B., Wiedijk, F.: Constructive analysis, types and exact real numbers. Math. Struct. Comput. Sci. 17 (1), 3–36 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Harthong, J.: Une théorie du continu. In: Barreau, H., Harthong, J. (eds.) La mathématiques non standard, pp 307–329, Paris (1989). Éditions du CNRSGoogle Scholar
  17. 17.
    Huth, M., Ryan, M., 2nd ed: Logic in Computer Science. Cambridge University Press (2004)Google Scholar
  18. 18.
    Kaye, R.: Models of Peano Arithmetic. Oxford Science Publications (1991)Google Scholar
  19. 19.
    Krebbers, R., Spitters, B.: Type classes for efficient exact real arithmetic in Coq. Log. Meth. Comput. Sci. 9(1) (2011)Google Scholar
  20. 20.
    Laugwitz, D.: Ω-calculus as a generalization of field extension an alternative approach to nonstandard analysis. In: Hurd, A. (ed.) Nonstandard Analysis - Recent developments, volume 983 of Lecture Notes in Mathematics, pp 120–133. Springer (1983)Google Scholar
  21. 21.
    Laugwitz, D.: Leibniz’ principle and Ω-calculus. In: Salanskis, J., Sinacoeur, H. (eds.) Le Labyrinthe du Continu, pp 144–155. Springer, France (1992)Google Scholar
  22. 22.
    Laugwitz, D., Schmieden, C.: Eine erweiterung der infinitesimalrechnung. Mathematische Zeitschrift 89, 1–39 (1958)MathSciNetGoogle Scholar
  23. 23.
    Lutz, R.: La force modélisatrice des théories infinitésimales faibles. In: Salanskis, J.-M., Sinaceur, H. (eds.) Le Labyrinthe du Continu, pp 414–423. Springer-Verlag (1992)Google Scholar
  24. 24.
    Magaud, N., Chollet, A., Fuchs, L.: Formalizing a Discrete Model of the Continuum in Coq from a Discrete Geometry Perspective. In: ADG’2010 (2010). Accepted for presentation at the conferenceGoogle Scholar
  25. 25.
    Martin-Löf, P.: Intuitionnistic Type Theory. Bibliopolis, Napoli (1984)Google Scholar
  26. 26.
    Martin-Löf, P.: Mathematics of infinity. In: COLOG-88 Computer Logic, Lecture Notes in Computer Science, pp 146–197. Springer-Verlag , Berlin (1990)Google Scholar
  27. 27.
    Moerdijk, I.: A model for intuitionistic non-standard arithmetic. Ann. Pure Appl. Logic 73(1), 37–51 (1995)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Nelson, E.: Internal set theory: A new approach to nonstandard analysis. Bull. Am. Math. Soc. 83(6), 1165–1198 (1977)CrossRefGoogle Scholar
  29. 29.
    Nelson, E.: Radically Elementary Theory. Annals of Mathematics Studies. Princeton University Press (1987)Google Scholar
  30. 30.
    O’Connor, R.: Certified exact transcendental real number computation in Coq. In: Mohamed, O.A., Muñoz, C.A., Tahar, S. (eds.) TPHOLs, volume 5170 of Lecture Notes in Computer Science, pp 246–261. Springer (2008)Google Scholar
  31. 31.
    Reveillès, J.-P., Richard, D.: Back and forth between continuous and discrete for the working computer scientist. Ann. Math. Artif. Intell., Math. Inform. 16(1–4), 89–152 (1996)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Icube UMR 7357 CNRS - Université de StrasbourgIllkirch CedexFrance
  2. 2.Laboratoire MIA - Université de La RochelleLa Rochelle CedexFrance
  3. 3.Laboratoire XLIM-SIC UMR 6172 CNRS - Université de PoitiersFuturoscope Chasseneuil CedexFrance

Personalised recommendations