On Rewriting Rules in Mizar

Abstract

This paper presents some tentative experiments in using a special case of rewriting rules in Mizar (Mizar homepage: http://www.mizar.org/): rewriting a term as its subterm. A similar technique, but based on another Mizar mechanism called functor identification (Korniłowicz 2009) was used by Caminati, in his paper on basic first-order model theory in Mizar (Caminati, J Form Reason 3(1):49–77, 2010, Form Math 19(3):157–169, 2011). However for this purpose he was obligated to introduce some artificial functors. The mechanism presented in the present paper looks promising and fits the Mizar paradigm.

References

  1. 1.

    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)

  2. 2.

    Caminati, M.B.: Basic first-order model theory in Mizar. J. Form. Reason. 3(1), 49–77 (2010)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Caminati, M.B.: Preliminaries to classical first-order model theory. Form. Math. 19(3), 157–169 (2011)

    Google Scholar 

  4. 4.

    Davis, M.: Obvious logical inferences. In: Proceedings of the Seventh International Joint Conference on Artificial Intelligence, pp. 530–531 (1981)

  5. 5.

    Downey, P.J., Sethi, R., Tarjan, R.E.: Variations on the common subexpression problem. J. ACM 27, 758–771 (1980). doi:10.1145/322217.322228

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Form. Reason., Special Issue: User Tutorials I 3(2), 153–245 (2010)

    MATH  Google Scholar 

  7. 7.

    Korniłowicz, A.: How to define terms in Mizar effectively. In: Grabowski, A., Naumowicz, A. (eds.) Computer Reconstruction of the Body of Mathematics. Studies in Logic, Grammar and Rhetoric, vol. 18(31), pp. 67–77. University of Białystok (2009)

  8. 8.

    Naumowicz, A., Byliński, C.: Improving Mizar texts with properties and requirements. In: A. Asperti (ed.) MKM-2004. LNCS, vol. 3119, pp. 290–301. Springer, Berlin Heidelberg (2004)

    Google Scholar 

  9. 9.

    Naumowicz, A., Korniłowicz, A.: A brief overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Proc. 22nd International Conference, TPHOLs, Munich, Germany. LNCS, vol. 5674, pp. 67–72. Springer, Berlin Heidelberg (2009)

    Google Scholar 

  10. 10.

    Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. ACM 27, 356–364 (1980). doi:10.1145/322186.322198

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Nieuwenhuis, R., Oliveras, A.: Proof-producing congruence closure. In: Giesl, J. (ed.) 16th International Conference on Term Rewriting and Applications, RTA’05. Lecture Notes in Computer Science, vol. 3467, pp. 453–468. Springer (2005)

  12. 12.

    Rudnicki, P.: Obvious inferences. J. Autom. Reasoning 3(4), 383–393 (1987). doi:10.1007/BF00247436

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Rudnicki, P., Trybulec, A.: Mathematical knowledge management in Mizar. In: Proc. of MKM 2001 (2001)

  14. 14.

    Shostak, R.E.: An algorithm for reasoning about equality. Commun. ACM 21, 583–585 (1978). doi:10.1145/359545.359570

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Artur Korniłowicz.

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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.

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Korniłowicz, A. On Rewriting Rules in Mizar. J Autom Reasoning 50, 203–210 (2013). https://doi.org/10.1007/s10817-012-9261-6

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Keywords

  • Proof assistant
  • Natural deduction
  • Computer algebra system
  • Term rewriting
  • Mizar