<abstract xmlns="http://www.w3.org/1999/xhtml">

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.
</abstract>

Functional Sequence in Norm Space

<abstract xmlns="http://www.w3.org/1999/xhtml">

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].
</abstract>

General Theory and Tools for Proving Algorithms in Nominative Data Systems

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing <italic>n</italic>-th element of Lucas sequence [23], [20] with given <italic>P</italic> and <italic>Q</italic> coefficients as well as two first elements (<italic>x</italic> and <italic>y</italic>). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1].
<disp-quote><monospace>i := 0</monospace>
<monospace>s := x</monospace>
<monospace>b := y</monospace>
<monospace>c := x</monospace>
<monospace>while (i &lt;> n)</monospace>
 <monospace>c := s</monospace>
 <monospace>s := b</monospace>
 <monospace>ps := p*s</monospace>
 <monospace>qc := q*c</monospace>
 <monospace>b := ps − qc</monospace>
 <monospace>i := i + j</monospace>
<monospace>return s</monospace></disp-quote>
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].
</abstract>

Partial Correctness of an Algorithm Computing Lucas Sequences

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  Pioneering Journal in the field of formalized and mechanically checked mathematics. Excellent articles on latest research in formal methods and computer-checked mathematics. Full-length original research articles in English. Archival issues available at the journal's home page fm.mizar.org/.   To build a uniform and extensible computer-checked repository of mathematical knowledge.  Reconstruction of mathematics based on mechanization of mathematical reasoning.  To check a wide range of mathematical theorems using a computer in an automatic fashion.  General mathematics: especially topology, lattice theory, functional analysis.  Applications: computer-aided verification of software and hardware, computer-aided teaching of mathematics.   Why subscribe and read   Pioneering source of high quality research in formalized and mechanically checked mathematics.  The oldest journal in the field, a medium for the publication of research reporting pioneering findings of broad mathematical significance that are likely to advance the direction and thinking of the field.  Long-term experience in publishing original articles on mechanizing mathematics and various aspects of automated reasoning, since 1990.   Why submit   Fair and constructive peer review by international experts in the field (three referees are designated for the assessment of each manuscript) - double-blind review process.  New publishing technology: articles automatically translated from a formal language into English and automatically typeset, available in both forms - electronic and printed.  Articles abstracted by online indexing services and databases.   Rejection Rate   59% of submitted articles.  The rejection percentage is described in detail in the following paper, pp.239-240: dx.doi.org/10.1007/978-3-540-73086-6_20 Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. Back Issues fm.mizar.org/   Other Files  Addenda 2008Formalized Mathematics Bibliography File.Formalized Mathematics External Bibliography File.  "In years: 2016-2017, the English version of Formalized Mathematics was financed under agreements 548/P-DUN/2016 and 548/1/P-DUN/2016 with the funds from the Polish Minister of Science and Higher Education for the dissemination of science." 

Formalized Mathematics

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

Functional Sequence in Norm Space

General Theory and Tools for Proving Algorithms in Nominative Data Systems

Partial Correctness of an Algorithm Computing Lucas Sequences

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