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- Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama
- Formalized Mathematics
- 2014

In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, there exists a submodule with any given rank that satisfies the above condition. In the… (More)

- Kazuhisa Nakasho, Yuichi Futa, Yasunari Shidama
- Formalized Mathematics
- 2014

In this article, we formalize topological properties of real nor-med spaces. In the first few parts, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. In the middle of the article, we discuss linear functions between real normed speces. Several kinds of subspaces induced… (More)

- Noboru Endou, Kazuhisa Nakasho, Yasunari Shidama
- Formalized Mathematics
- 2015

In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets… (More)

- Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama
- Formalized Mathematics
- 2014

In this article, we formalize a torsion Z-module and a torsion-free Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lovász) base… (More)

- Kazuhisa Nakasho, Yasunari Shidama
- CICM
- 2015

The purpose of this project is to collect symbol information in the Mizar Mathematical Library and manipulate it into practical and organized documentation. Inspired by the MathWiki project and API reference systems for computer programs, we developed a documentation generator focusing on symbols for the HTML-ized Mizar library. The system has several… (More)

- Kazuhisa Nakasho, Noboru Endou
- Formalized Mathematics
- 2015

In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the… (More)

- Kazuhisa Nakasho, Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama
- Formalized Mathematics
- 2015

In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of of commutative… (More)

- Takuya Sasaki, Cairos Cuadra, Hirokazu Madokoro, Kazuhisa Nakasho, Nobuhiro Shimoi
- 2016 16th International Conference on Control…
- 2016

At present, approximately 60% of world's population lives in masonry structures. Some of these structures are found in developing countries experiencing frequent earthquakes. Strong earthquakes tend to collapse these brittle, "non-engineered" structures, causing great harm to humans when these are residential structures. The purpose of this… (More)

- Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama
- Formalized Mathematics
- 2016

In this article, we formalize a torsion Z-module and a torsion-free Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lovász) base… (More)