Journal of Automated Reasoning

, Volume 50, Issue 2, pp 217–228 | Cite as

Methods of Lemma Extraction in Natural Deduction Proofs

  • Karol PąkEmail author
Open Access


The existing examples of natural deduction proofs, either declarative or procedural, indicate that often the legibility of proof scripts is of secondary importance to the authors. As soon as the computer accepts the proof script, many authors do not work on improving the parts that could be shortened and do not avoid repetitions of technical sub-deductions, which often could be replaced by a single lemma. This article presents selected properties of reasoning passages that may be used to determine if a reasoning passage can be extracted from a proof script, transformed into a lemma and replaced by a reference to the newly created lemma. Additionally, we present methods for improving the legibility of the reasoning that remains after the extraction of the lemmas.


Lemma Extraction Natural Deduction Legibility 


  1. 1.
    Bonarska, E.: An Introduction to PC Mizar. Mizar Users Group. Fondation Philippe le Hodey, Brussels (1990)Google Scholar
  2. 2.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formal. Reason. 3(2), 153–245 (2010)zbMATHGoogle Scholar
  3. 3.
    Jorgen, B., Gregory, G.: Digraphs: Theory, Algorithms and Applications. Springer. ISBN 1-85233-268-9 (2000)Google Scholar
  4. 4.
    Kornilowicz, A.: Tentative experiments with ellipsis in Mizar. In: AISC/MKM/Calculemus, pp. 453–457 (2012)Google Scholar
  5. 5.
    Matuszewski, R., Rudnicki, P.: Mizar: the first 30 years. Mech. Math. Its Appl. 4(1), 3–24 (2005)Google Scholar
  6. 6.
    Milewski, R.: New auxiliary software for MML database management. Mech. Math. Its Appl. 5(2), 1–10 (2006)MathSciNetGoogle Scholar
  7. 7.
    Milewski, R.: Algorithms analyzing formal deduction support systems/Algorytmy analizy systemu wspomagania edukcji formalnej (in Polish). PhD thesis, Faculty of Computer Science, Białystok University of Technology (2008)Google Scholar
  8. 8.
    Pąk. K.: The algorithms for improving and reorganizing natural deduction proofs. Stud. Log. Gramm. Rheto. 22(35), 95–112 (2010). ISBN 978-83-7431-273-8, ISSN 0860-150XGoogle Scholar
  9. 9.
    Rahul, S.P., Necula, G.C.: Proof optimization using lemma extraction. UCB/CSD-01-1143, Computer Science Division (EECS), University of California (2001)Google Scholar
  10. 10.
    Rudnicki, P.: An overview of the Mizar project. In: Proceedings of the 1992 Workshop on Types for Proofs and Programs, Chalmers University of Technology. Bastad, pp. 311–332 (1992)Google Scholar
  11. 11.
    Rudnicki, P., Trybulec, A.: On the integrity of a repository of formalized mathematics. In: Proceedings of MKM 2003, Lecture Notes in Computer Science, vol. 2594 (2003)Google Scholar
  12. 12.
    Urban, J.: Xml-izing Mizar: Making semantic processing and presentation of MML easy. In: Proceedings of MKM 2005, pp. 346–360 (2005)Google Scholar

Copyright information

© The Author(s) 2012

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BiałystokBiałystokPoland

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