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- Karol Pa̧k
- 2007

In this paper we define Stirling numbers of the second kind by cardinality of certain functional classes so that S(n, k) = {f where f is function of n, k : f is onto increasing} After that we show basic properties of this number in order to prove recursive dependence of Stirling number of the second kind. Consecutive theorems are introduced to prove formula

- Grzegorz Bancerek, Czeslaw Bylinski, +5 authors Josef Urban
- CICM
- 2015

- Karol Pa̧k
- 2007

In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement sgn(p) = Y 1≤i<j≤n sgn(p(j) − p(i)), where p is any fixed permutation of a set with n elements. I prove that the sign of a product of two permutations… (More)

- Karol Pak
- Journal of Automated Reasoning
- 2012

The existing examples of natural deduction proofs, either declarative or procedural, indicate that often the legibility of proof scripts is of secondary importance to the authors. As soon as the computer accepts the proof script, many authors do not work on improving the parts that could be shortened and do not avoid repetitions of technical sub-deductions,… (More)

- Karol Pa̧k
- 2007

In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the… (More)

- Karol Pak
- Formalized Mathematics
- 2009

We continue Mizar formalization of general topology according to the book [16] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that… (More)

- Karol Pak
- Formalized Mathematics
- 2009

We present the concept and basic properties of the Menger-Urysohn small inductive dimension of topological spaces according to the books [7]. Namely, the paper includes the formalization of main theorems from Sections 1.1 and 1.2. The terminology and notation used here are introduced in the following articles: For simplicity, we adopt the following rules: T… (More)

- Karol Pa̧k
- 2007

In this paper we define a discrete subset family of a topological space and basis sigma locally finite and sigma discrete. First, we prove an auxiliary fact for discrete family and sigma locally finite and sigma discrete basis. We also show the necessary condition for the Nagata Smirnov theorem: every metrizable space is T3 and has a sigma locally finite… (More)

- Karol Pak
- Logical Methods in Computer Science
- 2014

In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable.… (More)

- Karol Pak
- Formalized Mathematics
- 2010

In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set… (More)