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Divisible ℤ-modules Cover
By: Yuichi Futa and  Yasunari Shidama  
Open Access
|Aug 2016

References

  1. [1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.
  2. [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
  3. [3] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.10.1007/978-3-319-20615-8
  4. [4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
  5. [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  6. [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  7. [7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  8. [8] Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.10.1007/978-3-658-00360-9
  9. [9] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.10.2478/v10037-012-0007-z
  10. [10] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205-214, 2012. doi:10.2478/v10037-012-0024-y.10.2478/v10037-012-0024-y
  11. [11] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free ℤ-module. Formalized Mathematics, 20(4):275-280, 2012. doi:10.2478/v10037-012-0033-x.10.2478/v10037-012-0033-x
  12. [12] Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free Z-module. Formalized Mathematics, 22(4):277-289, 2014. doi:10.2478/forma-2014-0028.10.2478/forma-2014-0028
  13. [13] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Torsion part of ℤ-module. Formalized Mathematics, 23(4):297-307, 2015. doi:10.1515/forma-2015-0024.10.1515/forma-2015-0024
  14. [14] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
  15. [15] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982.10.1007/BF01457454
  16. [16] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective. The International Series in Engineering and Computer Science, 2002.10.1007/978-1-4615-0897-7_8
  17. [17] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.
  18. [18] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
  19. [19] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
  20. [20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
DOI: https://doi.org/10.1515/forma-2016-0004 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630
Language: English
Page range: 37 - 47
Submitted on: Dec 30, 2015
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Published on: Aug 31, 2016
In partnership with: Paradigm Publishing Services
Publication frequency: 1 issue per year
Keywords:

© 2016 Yuichi Futa, Yasunari Shidama, published by University of Białystok
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