Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Christoph Schwarzweller**- University of T\"ubingen

- We continue the Mizar formalization of Gr\"{o}bner bases following [6]. In this article we introduce S-polynomials and standard representations and show how these notions can be used to characterize Gr\"{o}bner bases.

- Preliminaries
- S-Polynomials
- Standard Representations

- [1]
Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller.
Ring ideals.
*Journal of Formalized Mathematics*, 12, 2000. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek.
Reduction relations.
*Journal of Formalized Mathematics*, 7, 1995. - [5]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [6] Thomas Becker and Volker Weispfenning. \em Gr\"obner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.
- [7]
Jozef Bialas.
Group and field definitions.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
*Journal of Formalized Mathematics*, 3, 1991. - [11]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [12]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Gilbert Lee and Piotr Rudnicki.
On ordering of bags.
*Journal of Formalized Mathematics*, 14, 2002. - [14]
Michal Muzalewski.
Construction of rings and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Michal Muzalewski and Wojciech Skaba.
From loops to abelian multiplicative groups with zero.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
*Journal of Formalized Mathematics*, 5, 1993. - [17]
Piotr Rudnicki and Andrzej Trybulec.
Multivariate polynomials with arbitrary number of variables.
*Journal of Formalized Mathematics*, 11, 1999. - [18]
Christoph Schwarzweller.
The binomial theorem for algebraic structures.
*Journal of Formalized Mathematics*, 12, 2000. - [19]
Christoph Schwarzweller.
More on multivariate polynomials: Monomials and constant polynomials.
*Journal of Formalized Mathematics*, 13, 2001. - [20]
Christoph Schwarzweller.
Polynomial reduction.
*Journal of Formalized Mathematics*, 14, 2002. - [21]
Christoph Schwarzweller.
Term orders.
*Journal of Formalized Mathematics*, 14, 2002. - [22]
Christoph Schwarzweller.
Characterization and existence of Gr\"obner bases.
*Journal of Formalized Mathematics*, 15, 2003. - [23]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [24]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [25]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [26]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Wojciech A. Trybulec.
Partially ordered sets.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [29]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [30]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [31]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [32]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [33]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989. - [34]
Edmund Woronowicz and Anna Zalewska.
Properties of binary relations.
*Journal of Formalized Mathematics*, 1, 1989. - [35]
Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura.
On replace function and swap function for finite sequences.
*Journal of Formalized Mathematics*, 12, 2000.

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