An Isabelle/HOL Formalisation of Green’s Theorem

  • Mohammad AbdulazizEmail author
  • Lawrence C. Paulson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9807)


We formalise a statement of Green’s theorem in Isabelle/HOL, which is its first formalisation to our knowledge. The theorem statement that we formalise is enough for most applications, especially in physics and engineering. An interesting aspect of our formalisation is that we neither formalise orientations nor region boundaries explicitly, with respect to the outwards-pointing normal vector. Instead we refer to equivalences between paths.


Elementary Region Line Integral Horizontal Edge Original Boundary Regular Region 
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.



This research was supported in part by an Australian National University - International Alliance of Research Universities Travel Grant and by an Australian National University, College of Engineering and Computer Science Dean’s Travel Grant Award. Also, the first author thanks Katlyn Quenzer for helpful discussions.


  1. 1.
    Federer, H.: Geometric Measure Theory. Springer, Heidelberg (2014)zbMATHGoogle Scholar
  2. 2.
    Green, G.: An essay on the application of mathematical analysis to the theories of electricity and magnetism (1828)Google Scholar
  3. 3.
    Harrison, J.: Formalizing basic complex analysis. In: From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar and Rhetoric, vol. 10(23), pp. 151–165 (2007)Google Scholar
  4. 4.
    Hölzl, J., Heller, A.: Three chapters of measure theory in Isabelle/HOL. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 135–151. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Hölzl, J., Immler, F., Huffman, B.: Type classes and filters for mathematical analysis in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 279–294. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Jurkat, W., Nonnenmacher, D.: The general form of Green’s theorem. Proc. Am. Math. Soc. 109(4), 1003–1009 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Michael, J.: An approximation to a rectifiable plane curve. J. Lond. Math. Soc. 1(1), 1–11 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: a proof assistant for higher-order logic, vol. 2283. Springer, Heidelberg (2002)Google Scholar
  9. 9.
    Protter, M.H.: Basic Elements of Real Analysis. Springer Science & Business Media, New York (2006)zbMATHGoogle Scholar
  10. 10.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish, Inc., University of Tokyo Press (1981)Google Scholar
  11. 11.
    Zorich, V.A., Cooke, R.: Mathematical Analysis II. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Canberra Research Laboratory, NICTACanberraAustralia
  2. 2.Australian National UniversityCanberraAustralia
  3. 3.Computer LaboratoryUniversity of CambridgeCambridgeEngland

Personalised recommendations