Summary In this article, we formalized the proof of the Stirling’s formula, which is considered an essential item in the field of statistics, as shown below: lim n → ∞ n ! n n + 1 2 e - n = 2 π \mathop {\lim }\limits_{n \to \infty } {{n!} \over {{n^{n + {1 \over 2}}}{e^{ - n}}}} = \sqrt {2\pi } using the Mizar formalism.

Summary This paper formalizes problems 38, 58, 160, 164, 168, 171, 188, 195, 196, and 198 from “250 Problems in Elementary Number Theory” by Wacław Sierpiński.

Summary In this article, using the Mizar system, we introduce some standard examples of vector spaces, e.g., the vector space of linear transformations between vector spaces. We formulate some conditions for the isomorphism of finite-dimensional vector spaces and prove that linear transformations are uniquely determined by their values with respect to the basis.

Summary This paper deals with the notions of U-small set, U-small category, and U-locally small category (U is non-empty Grothendieck universe). We reuse the first Mizar formalization of categories contained in CAT_* series of Mizar articles in order to show the expressive power of the Tarski-Grothendieck set theory (which is the base for the Mizar Mathematical Library) in this area. We encode parts of SGA 4 by Nicolas Bourbaki.

Summary In this article we continue our proofs on integrals of continuous functions of three variables in Mizar. In fact, we use similar techniques as in the case of two variables: we deal with projections of continuous function, the continuity of three variable functions in general, aiming at pure real-valued functions (not necessarily extended real-valued functions), concluding with integrability and iterated integrals of continuous functions of three variables.

Summary In this article, we formalize the Gram-Schmidt process in the Mizar system [2], [3] (compare another formalization using Isabelle/HOL proof assistant [1]). This process is one of the most famous methods for orthonormalizing a set of vectors. The method is named after Jørgen Pedersen Gram and Erhard Schmidt [4]. There are many applications of the Gram-Schmidt process in the field of computer science, e.g., error correcting codes or cryptology [8]. First, we prove some preliminary theorems about real unitary space. Next, we formalize the definition of the Gram-Schmidt process that finds orthonormal basis. We followed [5] in the formalization, continuing work developed in [7], [6].

Summary In this paper we present the Mizar formalization of the 36th problem from W. Sierpiński’s book “250 Problems in Elementary Number Theory” [10].

Abstract In this paper problems 25, 86, 88, 105, 111, 137–142, and 184–185 from [12] are formalized, using the Mizar formalism [3], [1], [4]. This is a continuation of the work from [5], [6], and [2] as suggested in [8]. The automatization of selected lemmas from [11] proven in this paper as proposed in [9] could be an interesting future work.

Summary In this article we formalize in Mizar [1], [2] the final step of our attempt to formally construct a prime representing polynomial with 10 variables proposed by Yuri Matiyasevich in [4]. The first part of the article includes many auxiliary lemmas related to multivariate polynomials. We start from the properties of monomials, among them their evaluation as well as the power function on polynomials to define the substitution for multivariate polynomials. For simplicity, we assume that a polynomial and substituted ones as i-th variable have the same number of variables. Then we study the number of variables that are used in given multivariate polynomials. By the used variable we mean a variable that is raised at least once to a non-zero power. We consider both adding unused variables and eliminating them. The second part of the paper deals with the construction of the polynomial proposed by Yuri Matiyasevich. First, we introduce a diophantine polynomial over 4 variables that has roots in integers if and only if indicated variable is the square of a natural number, and another two is the square of an odd natural number. We modify the polynomial by adding two variables in such a way that the root additionally requires the divisibility of these added variables. Then we modify again the polynomial by adding two variables to also guarantee the nonnegativity condition of one of these variables. Finally, we combine the prime diophantine representation proved in [7] with the obtained polynomial constructing a prime representing polynomial with 10 variables. This work has been partially presented in [8] with the obtained polynomial constructing a prime representing polynomial with 10 variables in Theorem (85).

Summary In this article, we formalize in Mizar [1], [2] the 3-fold product space of real normed spaces for usefulness in application fields such as engineering, although the formalization of the 2-fold product space of real normed spaces has been stored in the Mizar Mathematical Library [3]. First, we prove some theorems about the 3-variable function and 3-fold Cartesian product for preparation. Then we formalize the definition of 3-fold product space of real linear spaces. Finally, we formulate the definition of 3-fold product space of real normed spaces. We referred to [7] and [6] in the formalization.