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

Search

Volume 28 (2020): Issue 4 (December 2020)

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

Search

3 Articles
Open Access

Functional Sequence in Norm Space

Hiroshi Yamazaki
Hiroshi Yamazaki
Published Online: 21 May 2021
Page range: 263 - 268

Abstract

Summary

In this article, we formalize in Mizar [1], [2] functional sequences and basic operations on functional sequences in norm space based on [5]. In the first section, we define functional sequence in norm space. In the second section, we define pointwise convergence and prove some related theorems. In the last section we define uniform convergence and limit of functional sequence.

Keywords

  • pointwise convergence
  • functional sequence
  • formalized mathematics

MSC

  • 46A19
  • 46A32
  • 68V20
Open Access

General Theory and Tools for Proving Algorithms in Nominative Data Systems

Adrian Jaszczak
Adrian Jaszczak
Published Online: 21 May 2021
Page range: 269 - 278

Abstract

Summary

In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future.

This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16].

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 postconditions [17], [19], [7], [5].

Keywords

  • nominative data
  • program verification
  • inference rules

MSC

  • 68Q60
  • 03B70
  • 68V20
Open Access

Partial Correctness of an Algorithm Computing Lucas Sequences

Adrian Jaszczak
Adrian Jaszczak
Published Online: 21 May 2021
Page range: 279 - 288

Abstract

Summary

In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing n-th element of Lucas sequence [23], [20] with given P and Q coefficients as well as two first elements (x and y). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1].

i := 0

s := x

b := y

c := x

while (i <> n)

c := s

s := b

ps := p*s

qc := q*c

b := ps − qc

i := i + j

return s

This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [17]. 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 [18], [21], [7], [5].

Keywords

  • nominative data
  • program verification
  • Lucas sequences

MSC

  • 68Q60
  • 03B70
  • 68V20
3 Articles
Open Access

Functional Sequence in Norm Space

Hiroshi Yamazaki
Hiroshi Yamazaki
Published Online: 21 May 2021
Page range: 263 - 268

Abstract

Summary

In this article, we formalize in Mizar [1], [2] functional sequences and basic operations on functional sequences in norm space based on [5]. In the first section, we define functional sequence in norm space. In the second section, we define pointwise convergence and prove some related theorems. In the last section we define uniform convergence and limit of functional sequence.

Keywords

  • pointwise convergence
  • functional sequence
  • formalized mathematics

MSC

  • 46A19
  • 46A32
  • 68V20
Open Access

General Theory and Tools for Proving Algorithms in Nominative Data Systems

Adrian Jaszczak
Adrian Jaszczak
Published Online: 21 May 2021
Page range: 269 - 278

Abstract

Summary

In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future.

This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16].

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 postconditions [17], [19], [7], [5].

Keywords

  • nominative data
  • program verification
  • inference rules

MSC

  • 68Q60
  • 03B70
  • 68V20
Open Access

Partial Correctness of an Algorithm Computing Lucas Sequences

Adrian Jaszczak
Adrian Jaszczak
Published Online: 21 May 2021
Page range: 279 - 288

Abstract

Summary

In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing n-th element of Lucas sequence [23], [20] with given P and Q coefficients as well as two first elements (x and y). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1].

i := 0

s := x

b := y

c := x

while (i <> n)

c := s

s := b

ps := p*s

qc := q*c

b := ps − qc

i := i + j

return s

This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [17]. 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 [18], [21], [7], [5].

Keywords

  • nominative data
  • program verification
  • Lucas sequences

MSC

  • 68Q60
  • 03B70
  • 68V20