Journal of Automated Reasoning

, Volume 62, Issue 2, pp 281–300 | Cite as

Formally Verified Approximations of Definite Integrals

  • Assia Mahboubi
  • Guillaume Melquiond
  • Thomas Sibut-PinoteEmail author


Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton–Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. Our approach also handles improper integrals, provided that a factor of the integrand belongs to a catalog of identified integrable functions. This work has been integrated to the CoqInterval library.


Formal proof Numeric computations Definite integrals Improper integrals Decision procedure Interval arithmetic Polynomial approximations Real analysis 



We would like to thank Érik Martin-Dorel for his improvements to the Coq framework for computing rigorous polynomial approximations and Philippe Dumas for stimulating discussions and suggestions.


  1. 1.
    Ahmed, Z.: Ahmed’s integral: the maiden solution. Math. Spectr. 48(1), 11–12 (2015)Google Scholar
  2. 2.
    Boespflug, M., Dénès, M., Grégoire, B.: Full reduction at full throttle. In: Jouannaud, J.P., Shao, Z. (eds.) Certified Programs and Proofs, LNCS, vol. 7086, pp. 362–377. Springer, Kenting (2011). CrossRefGoogle Scholar
  3. 3.
    Boldo, S., Lelay, C., Melquiond, G.: Coquelicot: a user-friendly library of real analysis for Coq. Math. Comput. Sci. 9(1), 41–62 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corliss, G.F., Rall, L.B.: Adaptive, self-validating numerical quadrature. SIAM J. Sci. Stat. Comput. 8(5), 831–847 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eaton, J.W., Bateman, D., Hauberg, S., Wehbring, R.: GNU Octave version 3.8.1 manual: a high-level interactive language for numerical computations (2014).
  6. 6.
    Hass, J., Schlafly, R.: Double bubbles minimize. Ann. Math. Second Ser. 151(2), 459–515 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Helfgott, H.A.: Major arcs for Goldbach’s problem (2014). arXiv:1305.2897
  8. 8.
    Immler, F.: Formally verified computation of enclosures of solutions of ordinary differential equations. In: Badger, J.M., Rozier, K.Y. (eds.) NASA Formal Methods (NFM), LNCS, vol. 8430, pp. 113–127. Springer, Kenting (2014). CrossRefGoogle Scholar
  9. 9.
    Mahboubi, A., Melquiond, G., Sibut-Pinote, T.: Formally verified approximations of definite integrals. In: Blanchette, J.C., Merz, S. (eds.) 7th Conference on Interactive Theorem Proving, LNCS, vol. 9807, pp. 274–289, Nancy (2016).
  10. 10.
    Makarov, E., Spitters, B.: The Picard algorithm for ordinary differential equations in Coq. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) 4th International Conference on Interactive Theorem Proving, LNCS, vol. 7998, pp. 463–468. Springer, Rennes (2013). CrossRefGoogle Scholar
  11. 11.
    Martin-Dorel, É., Melquiond, G.: Proving tight bounds on univariate expressions with elementary functions in Coq. J. Autom. Reason. (2015). zbMATHGoogle Scholar
  12. 12.
    Mayero, M.: Formalisation et automatisation de preuves en analyses réelle et numérique. Ph.D. Thesis, Université Paris VI (2001)Google Scholar
  13. 13.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009). CrossRefzbMATHGoogle Scholar
  14. 14.
    Nedialkov, N.S.: Interval tools for ODEs and DAEs. In: Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN) (2006).
  15. 15.
    O’Connor, R., Spitters, B.: A computer verified, monadic, functional implementation of the integral. Theoret. Comput. Sci. 411(37), 3386–3402 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010).
  17. 17.
    Tucker, W.: Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Inria, LS2NUniversité de NantesNantes Cedex 3France
  2. 2.InriaUniversité Paris-SaclayOrsay CedexFrance
  3. 3.École Polytechnique, InriaUniversité Paris-SaclayPalaiseauFrance

Personalised recommendations