Formalization of the Fundamental Group in Untyped Set Theory Using Auto2

  • Bohua ZhanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10499)


We present a new framework for formalizing mathematics in untyped set theory using auto2. Using this framework, we formalize in Isabelle/FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.



The author would like to thank the anonymous referees for their comments. This research is completed while the author is supported by NSF Award No. 1400713.


  1. 1.
    Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formalized Reason. 9(1), 101–148 (2016)MathSciNetGoogle Scholar
  2. 2.
    Bourbaki, N.: Theory of Sets. Springer, Heidelberg (2000)Google Scholar
  3. 3.
    Brunerie, G.: On the homotopy groups of spheres in homotopy type theory. Ph.D. thesis.
  4. 4.
    Grabowski, A., Kornilowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formaliz. Reason. Spec. Issue: User Tutor. I 3(2), 153–245 (2010)Google Scholar
  5. 5.
  6. 6.
    Kaliszyk, C., Pak, K., Urban, J.: Towards a Mizar environment for Isabelle: foundations and language. In: Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs (CPP 2016), New York, pp. 58–65 (2016)Google Scholar
  7. 7.
    Kornilowicz, A., Shidama, Y., Grabowski, A.: The fundamental group. Formalized Math. 12(3), 261–268 (2004)Google Scholar
  8. 8.
    Kuncar, O.: Reconstruction of the Mizar type system in the HOL light system. In: Pavlu, J., Safrankova, J. (eds.) WDS Proceedings of Contributed Papers: Part I - Mathematics and Computer Sciences, pp. 7–12. Matfyzpress (2010)Google Scholar
  9. 9.
    Lee, G., Rudnici, P.: Alternative aggregates in Mizar. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds.) Calculemus/MKM 2007. LNCS (LNAI), vol. 4573, pp. 327–341. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73086-6_26 CrossRefGoogle Scholar
  10. 10.
    Mahboubi, A., Tassi, E.: Canonical structures for the working Coq user. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 19–34. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39634-2_5 CrossRefGoogle Scholar
  11. 11.
    Megill, N.D.: Metamath: a computer language for pure mathematics.
  12. 12.
    Munkres, J.R.: Topology. Prentice Hall, Upper Saddle River (2000)Google Scholar
  13. 13.
    Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Automated Reason. 11(3), 353–389 (1993)Google Scholar
  14. 14.
    Paulson, L.C.: Set theory for verification: II. Induction and recursion. J. Automated Reason. 15(2), 167–215 (1995)Google Scholar
  15. 15.
    Trybulec, A.: Some features of the Mizar language. In: ESPRIT Workshop (1993)Google Scholar
  16. 16.
    Wiedijk, F.: Mizar’s soft type system. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 383–399. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74591-4_28 CrossRefGoogle Scholar
  17. 17.
    Zhan, B.: AUTO2, a saturation-based heuristic prover for higher-order logic. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 441–456. Springer, Cham (2016). doi: 10.1007/978-3-319-43144-4_27 CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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