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Designing Mathematical Libraries Based on Requirements for Theorems

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Abstract

Theorems can be considered independent of abstract domains; a theorem rather depends on a set of properties necessary to prove the theorem correct. Following this observation theorems can be formulated and proven more generally thereby improving reuse of mathematical theorems. We discuss how this view influences the design of mathematical libraries and illustrate our approach with examples written in the Mizar language. We also argue that this approach allows for both stating requirements of generic algorithms and checking whether particular instantiations of generic algorithms are semantically correct.

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References

  1. J. Backer, P. Rudnicki and C. Schwarzweller, Ring ideals, Formalized Mathematics 9 (2000) 565–583; available by anonymous ftp from http://mizar.uwb.edu.pl/JFM/Vol12/ideal_1.html.

    Google Scholar 

  2. T. Becker and V. Weispfenning, Gröbner Bases – A Computational Approach to Commutative Algebra (Springer, Berlin, 1993).

    Google Scholar 

  3. B. Buchberger, T. Jebelean, F. Kriftner, M. Marin, E. Tomuta and D. Vasaru, A survey on the Theorema Project, in: Proceedings of ISSAC'97 (International Symposium on Symbolic and Algebraic Computation), ed. W. Küchlin (ACM Press, 1997) pp. 384–391.

  4. W.M. Farmer, J.D. Guttman and F.J. Thayer, Little theories, in: Automated Deduction – CADE-11, ed. D. Kapur, Lecture Notes in Computer Science, Vol. 607 (Springer, Berlin, 1992) pp. 567–581.

    Google Scholar 

  5. W.M. Farmer, J.D. Guttman and F.J. Thayer, IMPS: An Interactive Mathematical Proof System, Journal of Automated Reasoning 11 (1993) 213–248.

    Google Scholar 

  6. E. Kusak, W. Leonczuk and M. Muzalewski, Abelian groups, fields and vector spaces, Formalized Mathematics 1 (1990) 335–342; available by anonymous ftp from http://mizar.uwb.edu.pl/JFM/Vol1/vectsp_1.html.

    Google Scholar 

  7. W. McCune, Solution of the Robbins Problem, Journal of Automated Reasoning 19 (1997) 263–276.

    Google Scholar 

  8. D. Musser, The Tecton Concept Description Language (1998); available by anonymous ftp from http://www.cs.rpi.edu/~musser/gp/tecton.

  9. D. Musser, G. Derge and A. Saini, STL Tutorial and Reference Guide, 2nd edn., C++ Programming with the Standard Template Library (Addison-Wesley, Reading, MA, 2001).

    Google Scholar 

  10. P. Rudnicki and A. Trybulec, On equivalents of well-foundedness. An experiment in Mizar, Journal of Automated Reasoning 23 (1999) 197–234.

    Google Scholar 

  11. P. Rudnicki and A. Trybulec, Multivariate polynomials with arbitrary number of variables, Formalized Mathematics (1999), to appear; available by anonymous ftp from http://mizar.uwb.edu.pl/JFM/Vol11/polynom1.html.

  12. P. Rudnicki and A. Trybulec, Mathematical knowledge management in Mizar, in: Proceedings of the First International Workshop on Mathematical Knowledge Management (MKM2001), Linz, Austria (2001); available by anonymous ftp from http://www.risc.uni-linz.ac.at/institute/conferences/MKM2001/Proceedings/.

  13. P. Rudnicki, C. Schwarzweller and A. Trybulec, Commutative algebra in the Mizar system, Journal of Symbolic Computation 32 (2001) 143–169.

    Google Scholar 

  14. S. Schupp and R. Loos, SuchThat – generic programming works, in: Generic Programming – International Seminar on Generic Programming, eds. M. Jazayeri, R. Loos and D. Musser, Lecture Notes in Computer Science, Vol. 1766 (Springer, Berlin, 1998) pp. 133–145.

    Google Scholar 

  15. C. Schwarzweller, The ring of integers, Euclidean rings and modulo integers, Formalized Mathematics 8 (1999) 17–22; available by anonymous ftp from http://mizar.uwb.edu.pl/JFM/Vol11/int_3.html.

    Google Scholar 

  16. J.R. Shoenfield, Mathematical Logic (Addison-Wesley, Reading, MA, 1967).

    Google Scholar 

  17. I. Sommerville, Software Engineering, 4th edn. (Addison-Wesley, Reading, MA, 1992).

    Google Scholar 

  18. B. Stroustrup, The C++ Programming Language, 3rd edn. (Addison-Wesley, Reading, MA, 1997).

    Google Scholar 

  19. S. Wolfram, The Mathematica Book, 4th edn. (Addison-Wesley, Reading, MA, 1999).

    Google Scholar 

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  1. Wilhelm-Schickard-Institute for Computer Science, University of Tübingen, Sand 13, D-72076, Tübingen, Germany

    Christoph Schwarzweller

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  1. Christoph Schwarzweller
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Schwarzweller, C. Designing Mathematical Libraries Based on Requirements for Theorems. Annals of Mathematics and Artificial Intelligence 38, 193–209 (2003). https://doi.org/10.1023/A:1022924032739

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