Abstract
We present a new framework for formalizing mathematics in untyped set theory using auto2. We demonstrate that many difficulties with using set theory for formalization of mathematics can be addressed by improvements to automation, without sacrificing the inherent flexibility of the logic. As applications, we formalize in Isabelle/FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group of an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.
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References
Arthan, R.D.: Some Mathematical Case Studies in ProofPower-HOL. https://www.lemma-one.com/papers/51.pdf. Accessed 22 July 2018
Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formaliz. Reason. 9(1), 101–148 (2016)
Bourbaki, N.: Theory of Sets. Springer, Berlin (2000)
Brunerie, G.: On the homotopy groups of spheres in homotopy type theory. Ph.D. Thesis. https://arxiv.org/abs/1606.05916
Grabowski, A., Kornilowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formaliz. Reason. Spec. Issue: User Tutor. I 3(2), 153–245 (2010)
IsarMathLib: http://www.nongnu.org/isarmathlib/
Kaliszyk C., Pak, K., and 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), ACM, New York, NY, USA, pp. 58–65 (2016)
Kornilowicz, A., Shidama, Y., Grabowski, A.: The fundamental group. Formaliz. Math. 12(3), 261–268 (2004)
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)
Lee, G., Rudnicki, P.: Alternative Aggregates in Mizar. In: Kauers M., Kerber M., Miner R., Windsteiger W. (eds) Towards mechanized mathematical assistants. LNCS, vol 4573, Springer, Berlin (2007)
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, Springer, Berlin, pp. 19–34 (2013)
Megill, N.D.: Metamath: A computer language for pure mathematics. http://us.metamath.org/downloads/metamath.pdf. Accessed 22 July 2018
Moura, L., Bjourner, N.: Efficient E-matching for SMT Solvers. In: Automated Deduction, CADE-21, LNCS 4603, pp. 183–198 (2007)
Munkres, J.R.: Topology. Prentice Hall, Upper Saddle River (2000)
Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Autom. Reason. 11(3), 353–389 (1993)
Paulson, L.C.: Set theory for verification: II. Induction and recursion. J. Autom. Reason. 15(2), 167–215 (1995)
Trybulec, A.: Some Features of the Mizar Language. In: ESPRIT Workshop (1993)
Wiedijk, F.: Mizar’s Soft Type System. In: Schneider K., Brandt J. (eds.) Theorem Proving in Higher Order Logics, LNCS, vol 4732, Springer, Berlin (2007)
Zhan, B.: AUTO2: a saturation-based heuristic prover for higher-order logic. In: Blanchette, J.C., Merz, S. (eds.): ITP 2016, LNCS 9807, pp. 441–456 (2016)
Acknowledgements
The author would like to thank the anonymous reviewers for their helpful comments. Most of this work is completed while the author is at the Department of Mathematics at MIT, and supported by NSF Award No. 1400713. Part of the work is done at the Technical University of Munich, while the author is supported by DFG Grant NI 491/16-1.
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Zhan, B. Formalization of the Fundamental Group in Untyped Set Theory Using Auto2. J Autom Reasoning 63, 517–538 (2019). https://doi.org/10.1007/s10817-018-9478-0
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DOI: https://doi.org/10.1007/s10817-018-9478-0