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        <title>Formalized Mathematics Feed</title>
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        <item>
            <title><![CDATA[General Linear Group over Arbitrary Rings]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2026-0003</link>
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            <pubDate>Fri, 16 Oct 2026 00:00:00 GMT</pubDate>
            <description><![CDATA[We formalize the general linear group over an arbitrary (unital, associative) ring R as the group of units for the matrix ring over R. We loosely follow Elements of Mathematics by Bourbaki.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[The Fundamental Theorem of Galois Theory]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2026-0002</link>
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            <pubDate>Sat, 13 Jun 2026 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article we prove the Fundamental Theorem of Galois Theory using the Mizar formalism.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[The Galois Connection between IntermediateFields(E,F) and Subgroups of Aut(E,F)]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2026-0001</link>
            <guid isPermaLink="false">/article/10.2478/forma-2026-0001</guid>
            <pubDate>Sat, 13 Jun 2026 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article we establish the Galois connection between the intermediate fields of an extension E over F and the subgroups of the automorphism group Aut(E, F ). We show that if E is a finite Galois extension of F , then this connection induces a bijection between all intermediate fields of E and F and all subgroups of Aut(E, F ).]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Application of Complex Classes to Number Theory]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0016</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[The paper illustrates how the formal framework proposed earlier for complex numbers can be applied to integer and natural numbers, facilitating proofs by means of clusters’ registrations for Mizar adjectives. The contents of the article are closely related to the series of Mizar articles “Elementary Number Theory Problems” (with MML identifiers NUMBER*), but it is not solving any of “250 Problems in Elementary Number Theory” by Wacław Sierpiński book.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Measure for Product Space of Real Normed Spaces]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0010</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[This paper deals with Cartesian product spaces of real linear spaces and Cartesian product spaces of real normed spaces. In principle, both are attributed to the direct product of real linear spaces, but the direct product of normed spaces is needed to deal with the calculus of n-dimensional spaces. We also prove that the Lebesgue measure on tuple-type sets introduced in [6] is σ-finite.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Elementary Number Theory Problems. Part XVII]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0012</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[This paper furthers the formalization of problems from Wacław Sierpiński book “250 Problems in Elementary Number Theory” in the Mizar system. The selected twelve items are 37, 101, 115, 117, 145, 157, 159, 161–163, 165, and 169.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Elementary Number Theory Problems. Part XVIII]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0017</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this paper another seven problems from Wacław Sierpiński’s book “250 Problems in Elementary Number Theory” are formalized, using the Mizar formalism, namely: 53, 61, 81, 90, 100, 156, and 167.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Triangular Fuzzy Set Composed of Two Intersecting Affine Maps]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0011</link>
            <guid isPermaLink="false">/article/10.2478/forma-2025-0011</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article, various properties of triangular membership functions are formally proven, including the relationship between a triangular membership function composed of two straight lines and a MAX function, as well as a triangular membership function defined by the horizontal coordinates of the triangle’s vertices. Furthermore, we formalize defuzzified value of a triangular membership function and the integration of two connected functions.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Integrability of Multivariable Continuous Functions]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0015</link>
            <guid isPermaLink="false">/article/10.2478/forma-2025-0015</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article, we prove the integrability of continuous functions on n-dimensional real normed spaces, using the Mizar formalism. Generalizing selected theorems from the Mizar Mathematical Library, we prove the integrability of continuous real n-variable functions and then, using the correspondence between product-type and tuple-type spaces, we demonstrate the integrability of continuous functions on the desired multidimensional spaces.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Introduction to Galois Theory]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0014</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[This article continues a series devoted to the formalization of the Fundamental Theorem of Galois Theory using the Mizar proof assistant. We define groups of automorphisms and fixed fields and establish their fundamental properties. We also introduce an encoding of conjugates for groups of automorphisms and Galois extensions, and present the classical example demonstrating that the field of complex numbers is a Galois extension of the field of real numbers.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Formalization of Separable Version of Banach–Alaoglu Theorem]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0018</link>
            <guid isPermaLink="false">/article/10.2478/forma-2025-0018</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article, we first formalize the weak sequential compactness in dual normed spaces; then we prove the separable version of Banach– Alaoglu theorem.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Higher Order Partial Differentiable Functions]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0009</link>
            <guid isPermaLink="false">/article/10.2478/forma-2025-0009</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[This article formalizes higher-order partial differentiable functions in Mizar: it introduces the concept of partial differentiation for functions between real normed spaces and develops the theory of partial derivatives of arbitrary order. Key results include properties of partial derivatives, their relationship with total derivatives, and criteria for the continuity of higher-order partial derivatives.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[The Lattice of Intermediate Fields and Other Preliminaries to Galois Theory]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0013</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[This article initiates a series devoted to the formalization of the Fundamental Theorem of Galois Theory. It presents several preliminaries required for the formal development of Galois theory. In particular, as a main result, we define the lattice of intermediate fields of an extension E/F; we also treat sets of functions, groups, and intermediate fields in order to provide the necessary cluster registrations that enable effective Mizar automation.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Characterization of Finite Galois Extensions]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0019</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article we prove the well-known characterization of finite Galois extensions: a finite extension E of F is a Galois extension of F i E is both normal and separable i E is the splitting field of a separable polynomial p ∈ F [X]. We also prove some applications of the characterization, so for example that F (a1, . . ., an) is a separable extension of F if and only if all the ai are separable, or that every finite separable extension of F is contained in a Galois extension of F .]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Elementary Number Theory Problems. Part XIX]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0020</link>
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            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this paper, we present formal solutions to twelve problems selected from Wacław Sierpiński’s book 250 Problems in Elementary Number Theory. The selected problems are: 108, 112–114, 118–119, 127, 129, 130, and 132–134 formalized in the Mizar system.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Formalization of Wallis Infinite Product Formula for π and the Wallis Integral]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0007</link>
            <guid isPermaLink="false">/article/10.2478/forma-2025-0007</guid>
            <pubDate>Tue, 23 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article, we develop the proof of the so-called Wallis formula using the Mizar formalism. The purpose of this formalization is to complete the proof of Stirling’s formula using elementary techniques of calculus; however, the formal encoding of the Wallis-type integral is also included.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[A Formal Proof of Stirling’s Formula]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0008</link>
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            <pubDate>Tue, 23 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[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:



limn→∞n!nn+12e-n=2π
\mathop {\lim }\limits_{n \to \infty } {{n!} \over {{n^{n + {1 \over 2}}}{e^{ - n}}}} = \sqrt {2\pi }


using the Mizar formalism.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Extensions of Languages in Polish Notation]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0005</link>
            <guid isPermaLink="false">/article/10.2478/forma-2025-0005</guid>
            <pubDate>Wed, 17 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[This article prepares a collection of basic tools for adding new operators to a language in Polish notation, a parenthesis-free logical system where operators precede their operands. A need for this arose while attempting to formalize some extensions of Roman Suszko’s basic non-Fregean logic SCI, namely WB and WH.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Semidirect Products of Groups]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0006</link>
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            <pubDate>Wed, 17 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[We formalize the semidirect product of groups in Mizar, following §10 of Aschbacher’s Finite Group Theory [2]. We also prove the universal property for semidirect products as found in Bourbaki [5, III §2.10] Proposition 27. In an appendix, we define the dihedral group of the regular n-gon and the infinite dihedral group.]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Free Product of Groups]]></title>
            <link>https://reference-global.com/article/10.2478/forma-2025-0004</link>
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            <pubDate>Mon, 08 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[In this article the free product of groups is formalized in the Mizar system.]]></description>
            <category>ARTICLE</category>
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