Methods of Lemma Extraction in Natural Deduction Proofs

Abstract

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.

References

  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)

    MATH  Google Scholar 

  3. 3.

    Jorgen, B., Gregory, G.: Digraphs: Theory, Algorithms and Applications. Springer. ISBN 1-85233-268-9 (2000)

  4. 4.

    Kornilowicz, A.: Tentative experiments with ellipsis in Mizar. In: AISC/MKM/Calculemus, pp. 453–457 (2012)

  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)

    MathSciNet  Google 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)

  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 0860-150X

    Google 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)

  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)

  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)

  12. 12.

    Urban, J.: Xml-izing Mizar: Making semantic processing and presentation of MML easy. In: Proceedings of MKM 2005, pp. 346–360 (2005)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Karol Pąk.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Pąk, K. Methods of Lemma Extraction in Natural Deduction Proofs. J Autom Reasoning 50, 217–228 (2013). https://doi.org/10.1007/s10817-012-9267-0

Download citation

Keywords

  • Lemma
  • Extraction
  • Natural
  • Deduction
  • Legibility