Volume 31 (2023): Issue 1 (September 2023)

The article concerns about formalizing multivariable formal power series and polynomials [3] in one variable in terms of “bag” (as described in detail in [9]), the same notion as multiset over a finite set, in the Mizar system [1], [2]. Polynomial rings and ring of formal power series, both in one variable, have been formalized in [6], [5] respectively, and elements of these rings are represented by infinite sequences of scalars. On the other hand, formalization of a multivariate polynomial requires extra techniques of using “bag” to represent monomials of variables, and polynomials are formalized as a function from bags of variables to the scalar ring. This means the way of construction of the rings are different between single variable and multi variables case (which implies some tedious constructions, e.g. in the case of ten variables in [8], or generally in the problem of prime representing polynomial [7]). Introducing bag-based construction to one variable polynomial ring provides straight way to apply mathematical induction to polynomial rings with respect to the number of variables. Another consequence from the article, a polynomial ring is a subring of an algebra [4] over the same scalar ring, namely a corresponding formal power series. A sketch of actual formalization of the article is consists of the following four steps:

1. translation between Bags 1 (the set of all bags of a singleton) and N;

2. formalization of a bag-based formal power series in multivariable case over a commutative ring denoted by Formal-Series(n, R);

3. formalization of a polynomial ring in one variable by restricting one variable case denoted by Polynom-Ring(1, R). A formal proof of the fact that polynomial rings are a subring of Formal-Series(n, R), that is R-Algebra, is included as well;

4. formalization of a ring isomorphism to the existing polynomial ring in one variable given by sequence: Polynom-Ring(1, R) →˜ Polynom-Ring .

This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals.

Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.

In this paper problems 48, 80, 87, 89, and 124 from [7] are formalized, using the Mizar formalism [1], [2], [4]. The work is natural continuation of [5] and [3] as suggested in [6].

In this article sets of certain subgraphs of a graph are formalized in the Mizar system [7], [1], based on the formalization of graphs in [11] briefly sketched in [12]. The main result is the spanning subgraph theorem.

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].

Since isosceles triangular and trapezoidal membership functions [4] are easy to manage, they were applied to various fuzzy approximate reasoning [10], [13], [14]. The centroids of isosceles triangular and trapezoidal membership functions are mentioned in this article [16], [9] and formalized in [11] and [12]. Some propositions of the composition mapping (f + · g, or f +* g using Mizar formalism, where f, g are a ne mappings), are proved following [3], [15]. Then different notations for the same isosceles triangular and trapezoidal membership function are formalized.

We proved the agreement of the same function expressed with different parameters and formalized those centroids with parameters. In addition, various properties of membership functions on intervals where the endpoints of the domain are fixed and on general intervals are formalized in Mizar [1], [2]. Our formal development contains also some numerical results which can be potentially useful to encode either fuzzy numbers [7], or even fuzzy implications [5], [6] and extends the possibility of building hybrid rough-fuzzy approach in the future [8].

A classical algebraic geometry is study of zero points of system of multivariate polynomials [3], [7] and those zero points would be corresponding to points, lines, curves, surfaces in an affine space. In this article we give some basic definition of the area of affine algebraic geometry such as algebraic set, ideal of set of points, and those properties according to [4] in the Mizar system[5], [2].

We treat an affine space as the n-fold Cartesian product kⁿ as the same manner appeared in [4]. Points in this space are identified as n-tuples of elements from the set k. The formalization of points, which are n-tuples of numbers, is described in terms of a mapping from n to k, where the domain n corresponds to the set n = {0, 1, . . ., n − 1}, and the target domain k is the same as the scalar ring or field of polynomials. The same approach has been applied when evaluating multivariate polynomials using n-tuples of numbers [10].

This formalization aims at providing basic notions of the field which enable to formalize geometric objects such as algebraic curves which is used e.g. in coding theory [11] as well as further formalization of the fields [8] in the Mizar system, including the theory of polynomials [6].

In this article regular graphs, both directed and undirected, are formalized in the Mizar system [7], [2], based on the formalization of graphs as described in [10]. The handshaking lemma is also proven.

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.

This is a “quality of life” article concerning product groups, using the Mizar system [2], [4]. Like a Sonata, this article consists of three movements.

The first act, the slowest of the three, builds the infrastructure necessary for the rest of the article. We prove group homomorphisms map arbitrary finite products to arbitrary finite products, introduce a notion of “group yielding” families, as well as families of homomorphisms. We close the first act with defining the inclusion morphism of a subgroup into its parent group, and the projection morphism of a product group onto one of its factors.

The second act introduces the universal property of products and its consequences as found in, e.g., Kurosh [7]. Specifically, for the product of an arbitrary family of groups, we prove the center of a product group is the product of centers. More exciting, we prove for a product of a finite family groups, the commutator subgroup of the product is the product of commutator subgroups, but this is because in general: the direct sum of commutator subgroups is the subgroup of the commutator subgroup of the product group, and the commutator subgroup of the product is a subgroup of the product of derived subgroups. We conclude this act by proving a few theorems concerning the image and kernel of morphisms between product groups, as found in Hungerford [5], as well as quotients of product groups.

The third act introduces the notion of an internal direct product. Isaacs [6] points out (paraphrasing with Mizar terminology) that the internal direct product is a predicate but the external direct product is a [Mizar] functor. To our delight, we find the bulk of the “recognition theorem” (as stated by Dummit and Foote [3], Aschbacher [1], and Robinson [11]) are already formalized in the heroic work of Nakasho, Okazaki, Yamazaki, and Shimada [9], [8]. We generalize the notion of an internal product to a set of subgroups, proving it is equivalent to the internal product of a family of subgroups [10].

In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E. We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F[T] = {p(a₁, . . . a_n) | p ∈ F[X], a_i ∈ T} and F(T) = F[T] for finite algebraic T ⊆ E. We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).

In this paper, we introduce indefinite integrals [8] (antiderivatives) and proof integration by substitution in the Mizar system [2], [3]. In our previous article [15], we have introduced an indefinite-like integral, but it is inadequate because it must be an integral over the whole set of real numbers and in some sense it causes some duplication in the Mizar Mathematical Library [13]. For this reason, to define the antiderivative for a function, we use the derivative of an arbitrary interval as defined recently in [7]. Furthermore, antiderivatives are also used to modify the integration by substitution and integration by parts.

In the first section, we summarize the basic theorems on continuity and derivativity (for interesting survey of formalizations of real analysis in another proof-assistants like ACL2 [12], Isabelle/HOL [11], Coq [4], see [5]). In the second section, we generalize some theorems that were noticed during the formalization process. In the last section, we define the antiderivatives and formalize the integration by substitution and the integration by parts. We referred to [1] and [6] in our development.