The Role of the Mizar Mathematical Library for Interactive Proof Development in Mizar
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Abstract
The Mizar system is one of the pioneering systems aimed at supporting mathematical proof development on a computer that have laid the groundwork for and eventually have evolved into modern interactive proof assistants. We claim that an important milestone in the development of these systems was the creation of organized libraries accumulating all previously available formalized knowledge in such a way that new works could effectively reuse all previously collected notions. In the case of Mizar, the turning point of its development was the decision to start building the Mizar Mathematical Library as a centrallymanaged knowledge base maintained together with the formalization language and the verification system. In this paper we show the process of forming this library, the evolution of its design principles, and also present some data showing its current use with the modern version of the Mizar proof checker, but also as a rich corpus of semantically linked mathematical data in various areas including webbased and natural language proof presentation, maths education, and machine learning based automated theorem proving.
Keywords
Proof assistant Repository Mizar Mathematical Library1 Introduction
Around 1970s, the advances in computer technology and its popularization together with the proliferation of more userfriendly programming languages allowed the mathematical community to initiate several seminal projects like de Bruijn’s Automath [20], Milner’s LCF [58] or Glushkov’s Evidence Algorithm [54]. The Mizar project [9] started in 1973 under the leadership of Andrzej Trybulec, first at the Płock Scientific Society and since 1976 at the University of Białystok (formerly the University of Warsaw, Białystok Branch), Poland. From the very beginning Trybulec postulated a language and a computer system for recording mathematical papers in such a way that [35, 56]: (a) the papers could be stored in a computer and later, at least partially, translated into natural languages, (b) the papers would be formal and concise, (c) it would form a basis for construction of an automated information system for mathematics, (d) it would facilitate detection of errors, verification of references, elimination of repeated theorems, etc., (e) it would open a way to machine assisted education of the art of proving theorems, (f) it would enable automated generation of input into typesetting systems. The initial ideas are still valid and with time and a growing support from more researchers involved in the project the current development can be geared towards more ambitious goals, offering more intelligent proof checking methods and better support for the users. A crucial factor that helped to establish Mizar’s position among leading proof assistants, stand the test of time and consequently be developed in the direction of achieving these goals, was the realization of the fact, that largescale formalizations require developing methods of efficient accumulating, maintaining and reuse of previously generated mathematical content. This approach is today taken by developers of dedicated formal libraries e.g. the Isabelle based Archive of Formal Proofs [14], as well as all large formalization projects, both in mathematics (like the Hales’s Flyspeck project [36], G. Gonthier’s formalization of the Feit–Thompson theorem [25]) as well as in formal computer science (e.g. NASA PVS Library [18], seL4: Formal Verification of an OperatingSystem Kernel [47], etc.).
2 The Beginnings of Mizar Mathematical Library
From the beginning of the Mizar project, the encoding of mathematical proofs was conducted in a dedicated formal language – the Mizar language. The formalization scripts were stored in plain text files, called articles, processed independently and with little connection to one another. The 1981 version of Mizar2 introduced the environment part of an article, at that time containing statements that were used as axioms and checked only syntactically, i.e., without requiring their justification. Later versions (Mizar3 and Mizar4 from the period 1982–1988) divided the processing of Mizar files into multiple passes with filebased communication, such as scanning, parsing, type and naturaldeduction analysis, and justification checking. The use of special vocabulary files for symbols together with infix, prefix, postfix notation and their combinations resulted in greater closeness to mathematical texts.
In 1986, Mizar4 was ported to the IBM PC platform running MSDOS and later became PCMizar in 1988. In the years 1987–1991 the Mizar system and language played an important role in a Polish state research grant programme of the Polish Ministry of Science and Higher Education RPBP III.24 “Logical systems and algorithms for computerized checking of proof correctness”.
Trying to maintain all these works naturally led to issues concerning the reusability of formalizations. In the RPBP III.24 report from 1988 E. Woronowicz wrote:Each student had to solve several tasks from a given group of tasks. Initially, it was assumed that the students can reuse in their environment only the theorems and facts previously proved by others. But soon it turned out that this restriction cannot be applied to the simple facts from set theory (very often used in proving topological statements) obvious for students, while at the same time sometimes demanding tedious proofs. Therefore, each task contained in the description of its environment statements of theorems from set theory that were needed to solve a given task.
For each task group a special archive file was created, which was subsequently updated by adding new theorems together with their correct proofs. The care over the file archive was entrusted to one of the students, whose task was to take care of, among other things, the proper order of statements being added, as well as the integrity of operations and relations introduced by the students.^{3}
The first solution to address the reusability issue and avoid duplication of work was the implementation of a librarian utility which helped to include all the notions from previously processed files into a new article without the necessity to copy the whole environment by incorporating the contents of multiple files generated by several passes of the proof checker. This approach, however, turned out to be inefficient for bigger formalizations, which subsequently gave rise to developing methods of exporting only selected items from an article supported by corresponding changes in the Mizar language to facilitate both exporting and importing required notions. Later these concepts evolved into current forms used for importing theorems, definitions and other items from various articles. The functions of the librarian utility were superseded by extractor and accommodator, for extracting semantic information from an article into a database and including this information into a new formalization, respectively. From that time comes the conventional distinction between the library of Mizar articles (as a collection of userwritten files in the Mizar language) and the database consisting of the exported semantic information stored in dedicated machinereadable and optimized data formats. It was intended to avoid declaring adhoc axiomatics required for proving facts in each article, to help establish “a minimal axiomatization” for a particular article, to reduce the risk of introducing contradictory axioms and repetitions of already proved facts. January 1st, 1989 symbolically marks the date when the current Mizar library—Mizar Mathematical Library (MML) was born as the implementation of that proposal. The (still ongoing) process of building one common framework for verifying various branches of mathematics gave rise to a number of subsequent fundamental questions, e.g.:In previous versions of the Mizar system, including Mizar4, the text processed by the Mizar processor consists of two main parts:Such a text structure, mainly the fact that the user declares the environment, results in his full responsibility for the theory being developed (in the environment there may be contradictory items). Also the preparation time of Mizar articles is lengthened by the fact that when trying to correct errors e.g. in the proof of a statement, the processor processes the correct text of the environment each time. This is not a good solution neither in terms of methodology, nor technology.^{4}
declaration of the environment with definitions of basic concepts (modes, constants, functions) and statements of theorems playing the role of axioms for the developed theory,
the proper text, which consists of statements of new assertions and their proofs.

should various axiomatizations be allowed or not (and if only one, then which one should be chosen);

how to build the knowledge database and what information should be stored in it;

how to organize, manage and ensure the integrity of the database.
3 Axiomatics
Technically, the initial axiomatization was introduced in two special axiomatic files: HIDDEN and TARSKI. The former contained a selection of primitive notions built into the checker e.g. the types Any and set, equality and membership relations, but also types Element of, Subset of, and the powerset operation bool. The latter presented only the statements of TG axioms. The idea was to keep the proof checking system independent from any particular set theory, and so the type Any was introduced to represent any arbitrary object, not necessarily being a proper set. This, in principle, enabled developing other libraries with different axiomatizations. However, the library developed in terms of TG contained an additional axiom that the type Any is also of type set. Since the interest in developing alternative libraries was rather small, later the type Any was completely removed from the axiomatic files in favor of using the type set alone. Interestingly, in Mizar Ver. 8.1.01 from 2012, to filter out some futile definitional expansions automatically generated by a newly implemented mechanism in the Mizar checker [52], the most general root type was restored into the HIDDEN file, but under a new name: object.For every set N there exists a system M of sets which satisfies the following conditions:
 (i)
\(N\in M\);
 (ii)
if \(X\in M\) and \(Y \subseteq X\), then \(Y\in M\);
 (iii)
if \(X\in M\) and Z is the system of all subsets of X, then \(Z \in M\);
 (iv)
if \(X \subseteq M\) and X and M do not have the same potency, then \(X\in M.\)
The current form of the TARSKI file representing the set theory axioms is the result of library reorganization performed in connection with exploring finegrained dependencies in the library [6] that took place in 2013. The original TARSKI file was split into TARSKI_0, TARSKI_A and TARSKI. The TARSKI_0 file contains only ZF axioms, TARSKI_A represents the Tarski’s axiom alone, and the TARSKI file contains more userfriendly formulations of the axioms proved as consequences of the raw axioms from the other two files, e.g. using a defined notion of an ordered set rather than the axiom that such a pair exists.
It should also be noted that the original axiomatics was extended by a selection of properties of some notions commonly used in various formalizations that were built into the system to improve its usability. The extra axiomatic file AXIOMS contained the definitional axioms of the following concepts: element, subset, Cartesian product, domain (non empty set), subdomain (non empty subset of a domain), set domain (domain consisting of sets), as well as the axioms of strong arithmetic of real numbers [76]. The process of restructuring this part of the library had two major steps: first the properties of settheoretic notions were consequently proved as consequences of basic axioms and moved to ordinary Mizar articles (i.e. ZFMISC_1, SUBSET_1), while the full stepbystep construction of real numbers was consequently defined in the ARYTM* series of articles in the years 1995–1998. The current form of the arithmetic in the MML was shaped around the year 2003 in the course of developing a series of encyclopedic articles XCMPLX* and XREAL* extracted from the library in order to simplify the browsing for selected useful properties of real, complex, and extended real numbers. In consequence, the sets of real and complex numbers were defined together with proving all their usual properties without resolving to any extra axioms, so the original article AXIOMS was removed.
On the other hand, the Mizar system was enhanced by introducing special automation of selected commonly used notions, as so called requirements [59]. In particular this concerned the arithmetic of complex numbers, so that the users could decide whether they want the arithmetic facts to be obvious for the Mizar checker [63, 68]. At the same time, the automatically obvious facts found their justification in corresponding ordinary Mizar articles to enable their crossverification and allow switching off the automation in special contexts, e.g. for educational purposes or for experimenting with axiomatic systems [6].
The separation and consequent use of the minimal axiomatization was an important landmark and since then has been the main approach taken in developing the current MML. This allows to shift the focus in extending the proof checking mechanisms from implementing hardcoded system procedures towards language extensions like registrations [62], properties [49, 60], or reductions [51].
4 MML Information Storage
The number of constructors and notations
Kind  Number of constructors  Number of notations 

Attribute  2890  3237 
Functor  8873  9259 
Mode  518  1348 
Predicate  1137  1304 
Selector  187  187 
Structure  166  166 
Total  13,937  15,667 
The number of symbols
Symbol  Number 

Functor  4825 
Attribute  1933 
Mode  935 
Predicate  746 
Selector  175 
Structure  168 
Left bracket  35 
Right bracket  35 
Total  8852 
Top 10 symbols wrt the number of notations
Symbol  Notations  Formats  Constructors 

.  219  8  219 
\(*\)  197  6  191 
\(+\)  150  6  148 
−  141  4  133 
Element  65  2  60 
@  59  7  59 
”  54  3  45 
(#)  52  4  51 
  52  3  50 
\({<}\)*  47  8  47 
Top 10 symbols wrt the number of formats
Symbol  Notations  Formats  Constructors 

\(\{\)  38  10  38 
\(\}\)  38  10  38 
.  233  8  233 
\({<}\)*  49  8  49 
*\({>}\)  49  8  49 
@  65  7  65 
\(*\)  207  6  201 
\(+\)  155  6  153 
.:  42  6  40 
.]  13  6  13 
Properties of predicates, functors and modes
Property  Occurrences  Articles 

Predicates  
Reflexivity  141  95 
Irreflexivity  11  10 
Symmetry  125  86 
Asymmetry  7  7 
Connectedness  4  4 
Total  288  202 
Functors  
Involutiveness  38  32 
Projectivity  21  18 
Commutativity  161  89 
Idempotence  20  13 
Total  240  152 
Modes  
Sethood  8  8 
The number of registrations
Registration  Number 

Conditional  2579 
Existential  2871 
Functorial  8213 
Term identification  152 
Term reduction  229 
Total  14,044 
The number of definitions, theorems and schemes
Item  Number 

Definition  12,114 
Theorem  59,076 
Scheme  858 
Total  72,048 
Importing all the aforementioned information from the database to the environment of new articles is done with the help of a set of designated directives in the Mizar language [34]. A first group of directives enables formulating and disambiguating formal Mizar texts considering overloaded notations i.e. vocabularies, constructors, notations, and partially also registrations and requirements (because of the overloading, the order of used notations is important [66]). Another group of directives concerns encoding proofs, i.e. specifying their skeletons (definitions) and conducting reasoning steps to justify a given goal (theorems and schemes). The rest of import directives influence the way in which the Mizar proof checker uses its built in automation procedures that offer shorter justifications of selected proof steps. This includes the directives requirements, registrations (also importing term identifications and reductions), expansions, and equalities (that automatically provide references to the definitions of used atomic formulas and terms, respectively) [26].
Various independently processed import directives in the Mizar language allow controlling the amount of imported information and accessing only the necessary items from the database. Thanks to this approach, processing information from the database should be similarly timeconsuming, despite the level of complexity of the formalization environment.
5 Organization and Management
The development of the MML has been driven by several main factors. Initially, most formalization attempts were targeted at covering background knowledge in chosen fields of mathematics. When the library was advanced enough, it was also possible to start works directed at formalizing individual theorems with nontrivial proofs. Moreover, several projects aimed to prove whole papers and monographs were carried out.
Main fields developed during the first 3 years of the MML
#  MSC  MSC category  NOA 

1  03  Mathematical logic and foundations  70 
2  51  Geometry  27 
3  26  Real functions  23 
4  54  General topology  16 
5  15  Linear and multilinear algebra; matrix theory  13 
6  06  Order, lattices, ordered algebraic structures  11 
7  11  Number theory  9 
8  14  Algebraic geometry  9 
9  18  Category theory, homological algebra  9 
10  20  Group theory and generalizations  7 
11  12  Field theory and polynomials  6 
12  16  Associative rings and algebras  6 
13  05  Combinatorics  5 
14  40  Sequences, series, summability  4 
15  46  Functional analysis  4 
16  57  Manifolds and cell complexes  4 
17  60  Probability theory and stochastic processes  3 
18  28  Measure and integration  2 
19  33  Special functions  2 
20  13  Commutative algebra  1 
21  32  Several complex variables and analytic spaces  1 
22  68  Computer science  1 
23  81  Quantum theory  1 
Removed  24  
Total  258 
It should be noted that the formalizations from these initial years were mostly done independently with no actual connection between formalized theories (e.g. the hierarchies of functions and relations were disjoint, various geometries were constructed in different formal frameworks, groups, rings, fields, and vector spaces were all distinct, without any intrinsic hierarchy). Collecting as many articles as possible was the main priority then. It was the development of the library that eventually allowed to identify common works and emphasize on the quality of the formalizations and the organization of the collected library.^{9}
Main fields developed in the current MML
#  MSC  MSC category  NOA 

1  03  Mathematical logic and foundations  161 
2  06  Order, lattices, ordered algebraic structures  110 
3  26  Real functions  101 
4  54  General topology  100 
5  68  Computer science  97 
6  14  Algebraic geometry  84 
7  11  Number theory  77 
8  46  Functional analysis  70 
9  15  Linear and multilinear algebra; matrix theory  62 
10  08  General algebraic systems  49 
11  57  Manifolds and cell complexes  42 
12  05  Combinatorics  39 
13  51  Geometry  38 
14  18  Category theory, homological algebra  33 
15  20  Group theory and generalizations  32 
16  28  Measure and integration  31 
17  94  Information and communication, circuits  26 
18  13  Commutative algebra  16 
19  60  Probability theory and stochastic processes  16 
20  12  Field theory and polynomials  15 
21  16  Associative rings and algebras  15 
22  65  Numerical analysis  14 
23  33  Special functions  13 
24  40  Sequences, series, summability  10 
25  30  Functions of a complex variable  9 
26  55  Algebraic topology  7 
27  52  Convex and discrete geometry  5 
28  47  Operator theory  4 
29  32  Several complex variables and analytic spaces  3 
30  91  Game theory, economics, social and behaviour sciences  3 
31  41  Approximation and expansions  2 
32  58  Global analysis, analysis on manifolds  2 
33  22  Topological groups  1 
34  39  Difference and functional equations  1 
35  81  Quantum theory  1 
36  92  Biology and other natural sciences  1 
Total  1290 
The largest area covered in the MML (in terms of the number of articles) are mathematical logic and foundations (MSC #03)—the repository contains the model of the Mizar language itself (the construction of the firstorder language, propositional tautologies and satisfiability, Gödel completeness theorem), but also the formalization of various systems of nonclassical propositional logic, including linear temporal time or Grzegorczyk’s logics. Alongside with the building blocks of Tarski–Grothendieck set theory (descriptive and combinatorial set theory and large cardinals), other systems based on nonstandard membership relation, such as fuzzy and rough sets [33], are also widely represented. The theory of ordered algebraic structures (MSC #06) follows closely the seminal handbook of G. Grätzer (for lattices and orders) and [23] (for domains). MSC #26 devoted to real functions is one of the most fundamental parts of the MML in terms of knowledge reuse, extending in a straightforward way classical set theory—essentially it covers the standard undergraduate course in mathematical analysis. We can point out that general algebraic systems [31] seem to be underrepresented—summing up the number of files from underlying AMS MSC sections (08, 12, 13, 15, 16, 20), we obtain 186 articles formalizing the classical S. Lang’s course of algebra.
Among the formalizations of particular theorems that stimulated the development of the library we can name an early article formalizing a fundamental theorem of functional analysis—the Hahn–Banach theorem [70] submitted to the MML in 1993. Formalizing the fundamental theorem of algebra [57] completed in 2000 served as an example of a parallel development in several different proof checking systems including HOL Light (2001) and Coq (2002). A collaborative effort to formalize an elementary proof of the Jordan curve theorem (JCT) [50] is notable for developing a vast number of facts concerning the twodimensional real space and properties of special sequences.
Along with the appearance of “The Hundred Greatest Theorems” list published by Paul and Jack Abad [1] and later expanded by Freek Wiedijk^{10} the library gained another important source from which the theorems to be formalized have been selected by various Mizar authors. At the moment of writing this paper, the Mizar system, with a total of 65 verified theorems from the list, is placed at the third position among all systems involved.
That project stimulated an extensive development of lattice theory, both in algebraic and topological sense, together with relevant category theory results (compare the position of lattice theory in the list in Tables 8 and 9). On the other hand, a formalization of this size, carried out by an international collaborating group of developers, greatly influenced the development of the Mizar system [13]. The system had to efficiently cope with importing multiple theories, the users had to learn how to routinely work with and share a local database, some mechanisms were developed to deal with various representations of related objects in different formalisms (e.g. lattices as algebraic structures and as ordered sets) [28, 32]. Moreover, these developments created the possibility of formalizing selected papers from current research frontier in that field. For example, the welldeveloped theory of ordered sets served as a basis for translating a paper on betterquasiordering countable seriesparallel orders (cf. [82] and [74]).This is also the place to report on an activity of the Mizar project group located primarily at the University of Bialystok, Poland, the University of Alberta, Edmonton, Canada, and the Shinshu University, Nagano, Japan. It is the aim of the Mizar project to codify mathematical knowledge in a database. The codification means the formalization of concepts and proofs mechanically checked for logical correctness. The Mizar language is a formal language derived from the mathematical vernacular. The principal idea was to design a language that is readable by mathematicians, and simultaneously, is sufficiently rigorous to enable processing and verifying by computer software.
Our monograph A Compendium of Continuous Lattices was chosen by the Mizar group for testing their system. Since 1995, the Compendium has been translated piece by piece into the language Mizar. As of August 2002, sixteen authors have worked on this specific project; they have produced fiftyseven Mizar articles.
Top 10 authors
Author  Country  Number of submissions 

Yasunari Shidama  Japan  153 
Yatsuka Nakamura  Japan  135 
Grzegorz Bancerek  Poland  124 
Andrzej Trybulec  Poland  123 
Artur Korniłowicz  Poland  102 
Noboru Endou  Japan  92 
Adam Grabowski  Poland  66 
Piotr Rudnicki  Canada  60 
Xiquan Liang  China  48 
Hiroyuki Okazaki  Japan  45 
The number of authors by countries
Country  Number of authors 

Poland  109 
Japan  62 
China  38 
Canada  10 
Germany  10 
Russia  4 
USA  4 
Austria  3 
Italy  2 
Netherlands  2 
Ukraine  2 
Belgium  1 
Czech Republic  1 
Denmark  1 
Finland  1 
Israel  1 
Myanmar  1 
Spain  1 
Total  253 
The interplay of formalizations developed by Mizar users from various groups is seen when analyzing the theorem dependence tree of Mizar articles based on the quantitative information transfer, i.e. the relation induced by the use of the by keyword for straightforward justification within proofs.
In this relation, an article A is an ancestor of an article B if B refers to theorems in A, that is A transfers information to B. The direct ancestor A of an article B is the article which transfers the largest quantity of information into B.
The amount of information that an article A transfers to an article B is calculated as the sum of information transferred by all theorems from A which are referred to in B.

a is the number of all references to T in B,

n is the number of all references to T in all articles at the time when B was the last article accepted into the MML,

N is the number of all references to theorems in the MML at the given time.
 18.
EUCLID11 “Morley’s Trisector Theorem” by Roland Coghetto
 17.
EUCLID10 “Some Facts about Trigonometry and Euclidean Geometry” by Roland Coghetto
 16.
EUCLID_6 “Heron’s Formula and Ptolemy’s Theorem” by Marco Riccardi
 15.
COMPLEX2 “Inner Products and Angles of Complex Numbers” by Wenpai Chang, Yatsuka Nakamura and Piotr Rudnicki
 14.
COMPLEX1 “The Complex Numbers” by Czesław Byliński
 13.
SQUARE_1 “Some Properties of Real Numbers. Operations: min, max, square, and square root” by Andrzej Trybulec and Czesław Byliński
 12.
XREAL_1 “Real Numbers—Basic Theorems” by Library Committee
 11.
XCMPLX_1 “Complex Numbers—Basic Theorems” by Library Committee
 10.
XCMPLX_0 “Complex Numbers—Basic Definitions” by Library Committee
 9.
ARYTM_0 “Introduction to Arithmetics” by Andrzej Trybulec
 8.
ARYTM_1 “Non Negative Real Numbers. Part II” by Andrzej Trybulec
 7.
ARYTM_2 “Non Negative Real Numbers. Part I” by Andrzej Trybulec
 6.
ARYTM_3 “Arithmetic of Non Negative Rational Numbers” by Grzegorz Bancerek
 5.
ORDINAL3 “Ordinal Arithmetics” by Grzegorz Bancerek
 4.
ORDINAL2 “Sequences of Ordinal Numbers. Beginnings of Ordinal Arithmetics” by Grzegorz Bancerek
 3.
ORDINAL1 “The Ordinal Numbers. Transfinite Induction and Defining by Transfinite Induction” by Grzegorz Bancerek
 2.
TARSKI “Tarski Grothendieck Set Theory” by Andrzej Trybulec
 1.
TARSKI_0 “Axioms of Tarski Grothendieck Set Theory” by Andrzej Trybulec

an authored revision—consists of small changes in some articles in the library when the author of a new submission notices a possible generalization of already existing theorems or definitions. For such a task, usually it is necessary to improve some older articles that depend on the change. As a rule, however, a rather small part of the library is affected.

an automatic revision—takes place frequently whenever either a new revision software is developed (e.g. software for checking equivalence of theorems, which enables removing one or two equivalent theorems), or the Mizar verifier is strengthened and existing revision programs can use it to simplify articles [64, 69] or utilize newly implemented language features [53].

prettyprinting—if changes touch only the parts which are not exported to the library; when newly designed mechanisms allow shorter proofs [71, 72, 73].

a reorganization of the library—it involves changing the order of article processing used for (re)creating the Mizar database—the ordering is stored in a special file mml.lar distributed with each MML version.

concrete, which does not use the notion of structure (set theory, relations, functions, arithmetic and so on);

abstract, i.e. the article STRUCT_0 and its descendants, all of them directly or indirectly using Mizar structures.
Apart from that, it turned out to be convenient to also separate a part of the articles in which Random Access Turing Machines were modeled (named “SCM part”). Isolating these articles and placing them at the end of the mml.lar list enabled frequent revisions avoiding the need to constantly update the rest of the library.
A similar approach was taken when the Encyclopedia of Mathematics in Mizar, “EMM”, was formed as another distinctive part of the library in years 2002–2012. Its fourteen articles (with MML identifiers starting with a capital ‘X’) were extracted in order to simplify the browsing for selected, most commonly used notions and their properties (e.g. of real, complex, and extended real numbers, boolean properties of sets, families of subsets, ordered tuples, etc.).

keeping the repository as small as possible,

preserving a clear organization of the repository in order to attract authors,

establishing “elegant” mathematics, e.g. using short definitions (without unnecessary properties) or better proofs.
Another popular software, MML CVS—the usual concurrent version system for the MML was active for quite some time, but then was postponed, because the changes were too cryptic for the reader due to the lack of proper marking of items. Actually, one of the most general problems is that there are no absolute names for MML items and the changes are sometimes too massive to find out what really matters; hence the usefulness of the tools of this type is very limited.
then, quite naturally, we can call odd all integers which are not even. As long as the assumption of a variable i to be integer is not needed in the formulation of the above definition, it can be marked for removal by the library software (it can be a real number, at least). But then, the number \(\pi \) can be shown to be odd as it is not even (although it is not an integer, hence such a classification is void). Any automation of the process of dropping assumption about the types of used loci in the definition of attributes, however useful from a general point of view, could be dangerous.
As a rule, building an extensive encyclopedia of knowledge needs some investment; on the one hand, it can be considered by purely financial means as “information wants to be free, people want to be paid” [2], and it can be observed that projects like the aforementioned formalization of CCL or JCT always go in hand with a significant expansion of the library.
6 Other Applications of the MML
The size and the diversity of the MML makes it not only an indispensable standard library for Mizar users, but also an important resource for various projects related to processing mathematical knowledge.
Among the main activities based on the content of the MML, we can mention the development of representations of formal mathematics in human readable LaTeX form (e.g. the Formalized Mathematics journal [12]), XMLbased semanticallylinked web pages [85, 90], more semantic variants of the Mizar language (e.g. WSMizar [61]), or general semanticallyrich formats like OMDoc [37]. The library’s content available under an open source license [4] was also used as a test bed for developing formal wikis [3]. Moreover, it is worthwhile that the development of first MMLbased semanticallylinked presentations of mathematical content in the form of the online Journal of Formalized Mathematics ^{11} active in years 1995–2004 predated the advent of Wikipedia and other online maths journals and services popular today.
As already mentioned in Sect. 2, the development of the Mizar library since its beginnings has been connected with various educational projects involving earlier versions of Mizar. More recently, the organization and scientific content of several courses for mathematics and computer science students which used the modern Mizar versions 7 and 8 have been reported by Retel and Zalewska [75], Borak and Zalewska [16], as well as Naumowicz [65]. Over the last decade, a number of Mizar tutorials at conferences^{12} and at summer schools^{13} have been addressed to students and young researchers. The community of Mizar users has benefited from creating a number of introductory manuals^{14} and a technical reference manual [34]. Some directions in the development of the Mizar proof checking system have also been targeted at beginner users, e.g. the works on making the MML more easily accessible using dedicated auxiliary tools [66] and simplified environment building [67].
Apart from various forms of presenting mathematics, the creation and development of the MML into a large corpus of formalized knowledge enabled experiments with mining the dependencies in formal mathematics [5], as well as training automated theorem provers [88]. The MML served as a basis for preparing largetheory problems to be solved by provers during the competitions of the CADE ATP System Competition [79] and provided a sufficiently big data for machine learning oriented ATP training [21, 40].
Among the most important projects related to the interplay of ATP and the MML we can mention here the J. Urban’s Mizar Proof Advisor, one of the first hammerstyle systems i.e. giving the authors of formal proofs a semiintelligent brute force tool that can take advantage of very large lemma libraries [15]. Thanks to these works Mizar gained ways of crossverification for its proof checking system by using external provers [89]. This technology was later adopted in the MizAR system which now provides an online automated reasoning service for Mizar users employing a large suite of AI/ATP methods trained over the Mizar library. Reportedly, the system is able to automatically prove 40% of the theorems from the whole MML [45].
The MML is of great value for various machine learning experiments not only for its size, but also for its inherent structure. The application of deep learning techniques [55] taking into account the semantic features of formalized statements significantly improves the premise selection procedure [38] which is at the heart of efficient ATP proof search. The semantic features preserved in the MML, in particular intermediate proof steps, can be used to select interesting lemmas for ATP proofs [42, 44]. Machine learning methods tested over the MML were also applied to define metrics between proofs [8, 41] as well as to align concepts across the libraries of different proof assistants [22]. There were also attempts to translate the MML contents to the formalisms of other proof systems [39].
Finally, it is worthwhile to mention most recent progress in parsing informal mathematics where machine learning methods refer to the correspondence of the formal MML articles and their informal English rendering automatically generated for the Formalized Mathematics journal [43].
7 Conclusion
We claim that the library is much more important [than] the system. A good system without a library is useless. A good library for a bad system still is very interesting (the system might be improved or the library might be ported to a different, better, system). So the library is what counts. [92]
Footnotes
 1.
 2.
 3.
 4.
 5.
Current MML Ver. 5.40.1289 is distributed with Mizar system Ver. 8.1.05.
 6.
The Library Committee of the Association of Mizar Users was officially established on November 11, 1989 with the main aim to collect Mizar articles and to organize them into a central repository (called Main Mizar Library at that time as multiple repositories were supposed to be created, being a direct successor of Central Archive of Mizar Texts coming with Mizar4).
 7.
A constructor is a unique number representing a given formal object with respect to its article’s environment.
 8.
In Mizar, a registration is an umbrella term for three kinds of automation techniques concerning the use of notions represented as adjectives, i.e. existential registrations, conditional registrations, as well as term adjectives registrations, also known as functorial registrations, cf. [34].
 9.
With time, the initial articles were thoroughly adjusted, 24 articles from that period were entirely removed from today’s library. Overall, the Library Committee decided on removing 52 original articles from the library.
 10.
 11.
 12.
E.g. at ISLA 2010 (http://ali.cmi.ac.in/isla2010/workshops/mizar/), CADE 2013 (http://mizar.uwb.edu.pl/CADE23tutorial/), CICM 2016 (http://mizar.org/cicm_tutorial/).
 13.
E.g. at the TYPES 2007 Summer School (http://typessummerschool07.cs.unibo.it/#mizar), ESSLLI 2012 (http://www.esslli2012.pl/index.php?id=177).
 14.
The most recent and uptodate is F. Wiedijk’s “Writing a Mizar article in nine easy steps” (https://www.cs.ru.nl/F.Wiedijk/mizar/mizman.pdf). Other manuals are available at http://mizar.uwb.edu.pl/project/bibliography.html.
Notes
Acknowledgements
A. Naumowicz’s work on this paper was partially supported by the project \(\hbox {N}^\circ \) 2017/01/X/ST6/00012 financed by the National Science Center, Poland. K. Pąk’s work was supported by the resources of the Polish National Science Center granted by decision \(\hbox {N}^\circ \) DEC2015/19/D/ST6/01473. The English version of Formalized Mathematics was financed under agreement 548/PDUN/2016 with the funds from the Polish Minister of Science and Higher Education for the dissemination of science. The processing and analysis of the Mizar library has been performed using the infrastructure of the University of Bialystok High Performance Computing Center (http://uco.uwb.edu.pl).
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