- Journal Details
- Format
- Journal
- eISSN
- 1898-9934
- ISSN
- 1426-2630
- First Published
- 09 Jun 2008
- Publication timeframe
- 4 times per year
- Languages
- English

#### Search

#### Abstract

This is the first part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field

In this first part we show that an irreducible polynomial

Because

Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields _{2} we construct for every field ^{′} of _{n}, and we have _{n} ∩ _{n}[

In the fourth part we finally define field extensions:

#### Keywords

- roots of polynomials
- field extensions
- Kronecker’s construction

#### MSC 2010

- 12E05
- 12F05
- 68T99
- 03B35

Open Access

#### Isomorphisms from the Space of Multilinear Operators

Page range: 101 - 106

#### Abstract

In this article, using the Mizar system [5], [2], the isomorphisms from the space of multilinear operators are discussed. In the first chapter, two isomorphisms are formalized. The former isomorphism shows the correspondence between the space of multilinear operators and the space of bilinear operators.

The latter shows the correspondence between the space of multilinear operators and the space of the composition of linear operators. In the last chapter, the above isomorphisms are extended to isometric mappings between the normed spaces. We referred to [6], [11], [9], [3], [10] in this formalization.

#### Keywords

- linear operators
- bilinear operators
- multilinear operators
- isomorphism of linear operator spaces

#### MSC 2010

- 46-00
- 47A07
- 47A30
- 68T99
- 03B35

#### Abstract

In this article, using the Mizar system [2], [1], we discuss invertible operators on Banach spaces. In the first chapter, we formalized the theorem that denotes any operators that are close enough to an invertible operator are also invertible by using the property of Neumann series.

In the second chapter, we formalized the continuity of an isomorphism that maps an invertible operator on Banach spaces to its inverse. These results are used in the proof of the implicit function theorem. We referred to [3], [10], [6], [7] in this formalization.

#### Keywords

- Banach space
- invertible operator
- Neumann series
- isomorphism of linear operator spaces

#### MSC 2010

- 47A05
- 47J07
- 68T99
- 03B35

#### Abstract

In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here.

In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

#### Keywords

- implicit function theorem
- Lipschitz continuity
- differentiability
- implicit function

#### MSC 2010

- 26B10
- 47A05
- 47J07
- 53A07
- 03B35

#### Abstract

This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field

In the first part we show that an irreducible polynomial

Because

Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields _{2} we construct for every field _{n}, and we have _{n} ∩ _{n}[

In the fourth part we finally define field extensions:

#### Keywords

- roots of polynomials
- field extensions
- Kronecker’s construction

#### MSC 2010

- 12E05
- 12F05
- 68T99
- 03B35

#### Abstract

In [3] the existence of the Cantor normal form of ordinals was proven in the Mizar system [6]. In this article its uniqueness is proven and then used to formalize the natural sum of ordinals.

#### Keywords

- ordinal numbers
- Cantor normal form
- Hessenberg sum
- natural sum

#### MSC 2010

- 03E10
- 68T99
- 03B35

#### Abstract

The previous articles [5] and [6] introduced formalizations of the step-by-step operations we use to construct finite graphs by hand. That implicitly showed that any finite graph can be constructed from the trivial edgeless graph _{1} by applying a finite sequence of these basic operations. In this article that claim is proven explicitly with Mizar[4].

#### Keywords

- supergraph
- graph operations
- construction of finite graphs

#### MSC 2010

- 05C76
- 68T99
- 03B35

Open Access

#### Partial Correctness of a Factorial Algorithm

Page range: 181 - 187

#### Abstract

In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm:

computing the factorial of given natural number n, where variables

This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[14],[10],[11],[12]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [13],[15],[7],[5].

#### Keywords

- factorial
- nominative data
- program verification

#### MSC 2010

- 68Q60
- 68T37
- 03B70
- 03B35

Open Access

#### Partial Correctness of a Power Algorithm

Page range: 189 - 195

#### Abstract

This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[15],[11],[12],[13]. In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm:

computing the natural n power of given complex number b, where variables

The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [14],[16],[7],[5].

#### Keywords

- power
- nominative data
- program verification

#### MSC 2010

- 68Q60
- 68T37
- 03B70
- 03B35

#### Abstract

The article is the next in a series aiming to formalize the MDPR-theorem using the Mizar proof assistant [3], [6], [4]. We analyze four equations from the Diophantine standpoint that are crucial in the bounded quantifier theorem, that is used in one of the approaches to solve the problem.

Based on our previous work [1], we prove that the value of a given binomial coefficient and factorial can be determined by its arguments in a Diophantine way. Then we prove that two products

where

The formalization follows [10], Z. Adamowicz, P. Zbierski [2] as well as M. Davis [5].

#### Keywords

- Hilbert’s 10th problem
- Diophantine relations

#### MSC 2010

- 11D45
- 68T99
- 03B35

Open Access

#### Formalization of the MRDP Theorem in the Mizar System

Page range: 209 - 221

#### Abstract

This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem.

In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form ^{2}^{2} − 1)^{2} = 1 [8] and its solutions considered as two sequences _{i}(^{z} is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9].

In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that

The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

#### Keywords

- Hilbert’s 10th problem
- Diophantine relations

#### MSC 2010

- 11D45
- 68T99
- 03B35