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Formalized Mathematics

(a computer assisted approach)

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Volume 24, Issue 2 (Jun 2016)

Issues

On Multiset Ordering

Grzegorz Bancerek
  • Association of Mizar Users, Białystok, Poland
Published Online: 2016-12-08 | DOI: https://doi.org/10.1515/forma-2016-0008

Summary

Formalization of a part of [11]. Unfortunately, not all is possible to be formalized. Namely, in the paper there is a mistake in the proof of Lemma 3. It states that there exists xM1 such that M1(x) > N1(x) and (∀yN1)xy. It should be M1(x) ⩾ N1(x). Nevertheless we do not know whether xN1 or not and cannot prove the contradiction. In the article we referred to [8], [9] and [10].

Keywords: ordering; Dershowitz-Manna ordering

MSC 2010: 06F05; 03B35

References

  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

  • [2] Grzegorz Bancerek. Reduction relations. Formalized Mathematics, 5(4):469–478, 1996.

  • [3] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

  • [5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

  • [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.

  • [7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

  • [8] Nachum Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science, 17(3):279–301, 1982. doi:10.1016/0304-3975(82)90026-3. [Crossref]

  • [9] Nachum Dershowitz and Zohar Manna. Proving termination with multiset orderings. Communications of the ACM, 22(8):465–476, 1979. doi:10.1145/359138.359142. [Crossref]

  • [10] Gerard Huet and Derek C. Oppen. Equations and rewrite rules: A survey. Technical report, Stanford, CA, USA, 1980.

  • [11] Jean-Pierre Jouannaud and Pierre Lescanne. On multiset ordering. Information Processing Letters, 15(2):57–63, 1982. doi:10.1016/0020-0190(82)90107-7. [Crossref]

  • [12] Robert Milewski. Natural numbers. Formalized Mathematics, 7(1):19–22, 1998.

  • [13] Eliza Niewiadomska and Adam Grabowski. Introduction to formal preference spaces. Formalized Mathematics, 21(3):223–233, 2013. doi:10.2478/forma-2013-0024. [Crossref]

  • [14] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.

  • [15] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.

  • [16] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.

About the article

Received: 2015-12-31

Published Online: 2016-12-08

Published in Print: 2016-06-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.1515/forma-2016-0008. Export Citation

© 2016 Grzegorz Bancerek, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

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