<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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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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