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Mizar system
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| Mizar | |
|---|---|
 | |
Screenshot  Mizar MathWiki screenshot | |
| Paradigm | Declarative |
| Designed by | Andrzej Trybulec |
| First appeared | 1973 |
| Typing discipline | Weak, static |
| Filename extensions | .miz .voc |
| Website | www.mizar.org |
| Influenced by | |
| Automath | |
| Influenced | |
| OMDoc, HOL Light and Rocq mizar modes | |
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in the proof of new theorems.[1] The system is maintained and developed by the Mizar Project, formerly under the direction of its founder Andrzej Trybulec.
In 2009 the Mizar Mathematical Library was the largest coherent body of strictly formalized mathematics in existence.[2]
History
[edit]The Mizar Project was started around 1973 by Andrzej Trybulec as an attempt to reconstruct mathematical vernacular so it can be checked by a computer.[3] Its current goal, apart from the continual development of the Mizar System, is the collaborative creation of a large library of formally verified proofs, covering most of the core of modern mathematics. This is in line with the influential QED manifesto.[4]
Currently the project is developed and maintained by research groups at Białystok University, Poland, the University of Alberta, Canada, and Shinshu University, Japan. While the Mizar proof checker remains proprietary,[5] the Mizar Mathematical Library—the sizable body of formalized mathematics that it verified—is licensed open-source.[6]
Papers related to the Mizar system regularly appear in the peer-reviewed journals of the mathematic formalization academic community. These include Studies in Logic, Grammar and Rhetoric, Intelligent Computer Mathematics, Interactive Theorem Proving, Journal of Automated Reasoning and the Journal of Formalized Reasoning.
Mizar language
[edit]The distinctive feature of the Mizar language is its readability. As is common in mathematical text, it relies on classical logic and a declarative style.[7] Mizar articles are written in ordinary ASCII, but the language was designed to be close enough to the mathematical vernacular that most mathematicians could read and understand Mizar articles without special training.[1] Yet, the language enables the increased level of formality necessary for automated proof checking.
For a proof to be admitted, all steps have to be justified either by elementary logical arguments or by citing previously verified proofs.[8] This results in a higher level of rigor and detail than is customary in mathematical textbooks and publications. Thus, a typical Mizar article is about four times as long as an equivalent paper written in ordinary style.[9]
Formalization is relatively labor-intensive, but not impossibly difficult. Once one is versed in the system, it takes about one week of full-time work to have a textbook page formally verified. This suggests that its benefits are now within the reach of applied fields such as probability theory and economics.[2]
Mizar Mathematical Library
[edit]The Mizar Mathematical Library (MML) includes all theorems to which authors can refer in newly written articles. Once approved by the proof checker they are further evaluated in a process of peer-review for appropriate contribution and style. If accepted they are published in the associated Journal of Formalized Mathematics[10] and added to the MML.
Breadth
[edit]As of July 2012, the MML included 1150 articles written by 241 authors.[11] In aggregate, these contain more than 10,000 formal definitions of mathematical objects and about 52,000 theorems proved on these objects. More than 180 named mathematical facts have been given formal codification in this manner.[12] Some examples are the Hahn–Banach theorem, Kőnig's lemma, the Brouwer fixed point theorem, Gödel's completeness theorem, and the Jordan curve theorem.
This breadth of coverage has led some[13] to suggest Mizar as one of the leading approximations to the QED utopia of encoding all core mathematics in computer verifiable form.
Availability
[edit]All MML articles are available in PDF form as the papers of the Journal of Formalized Mathematics.[10] The full text of the MML is distributed with the Mizar checker and can be freely downloaded from the Mizar website. In an ongoing recent project[14] the library was also made available in an experimental wiki form[15] that only admits edits when they are approved by the Mizar checker.[16]
The MML Query website[11] implements a powerful search engine for the contents of the MML. Among other abilities, it can retrieve all MML theorems proved about any particular type or operator.[17][18]
Logical structure
[edit]The MML is built on the axioms of the Tarski–Grothendieck set theory. Even though semantically all objects are sets, the language allows one to define and use syntactical weak types. For example, a set may be declared to be of type Nat only when its internal structure conforms with a particular list of requirements. In turn, this list serves as the definition of the natural numbers and the set of all the sets that conform to this list is denoted as NAT.[19] This implementation of types seeks to reflect the way most mathematicians formally think of symbols[20] and so streamline codification.
Mizar Proof Checker
[edit]Distributions of the Mizar Proof Checker for all major operating systems are freely available for download at the Mizar Project website. Use of the proof checker is free for all non-commercial purposes. It is written in Free Pascal and the source code is available on GitHub.[21]
See also
[edit]References
[edit]- ^ a b Naumowicz, Adam; Kornilowicz, Artur (2009). "A Brief Overview of Mizar". In Berghofer, Stefan; Nipkow, Tobias; Urban, Christian; Wenzel, Makarius (eds.). Theorem Proving in Higher Order Logics, 22nd International Conference, TPHOLs 2009, Munich, Germany, August 17–20, 2009. Proceedings. Lecture Notes in Computer Science. Vol. 5674. Springer. pp. 67–72. doi:10.1007/978-3-642-03359-9_5. ISBN 978-3-642-03358-2.
- ^ a b Wiedijk, Freek (2009). "Formalizing Arrow's theorem". Sādhanā. 34 (1): 193–220. doi:10.1007/s12046-009-0005-1. hdl:2066/75428.
- ^ Matuszewski, Roman; Piotr Rudnicki (2005). "Mizar: the first 30 years" (PDF). Mechanized Mathematics and Its Applications. 4.
- ^ Wiedijk, Freek. "Mizar". Retrieved 24 July 2018.
- ^ Mailing list discussion Archived 2011-10-09 at the Wayback Machine referring to the close-sourcing of Mizar.
- ^ Mailing list announcement referring to the open-sourcing of MML.
- ^ Geuvers, H. (2009). "Proof assistants: History, ideas and future". Sādhanā. 34 (1): 3–25. doi:10.1007/s12046-009-0001-5. hdl:2066/75958.
- ^ Wiedijk, Freek (2008). "Formal Proof--Getting Started" (PDF). Notices of the AMS. 55 (11): 1408–1414.
- ^ Wiedijk, Freek. "The "de Bruijn factor"". Retrieved 24 July 2018.
- ^ a b Journal of Formalized Mathematics
- ^ a b The MML Query search engine
- ^ "A list of named theorems in the MML". Retrieved 22 July 2012.
- ^ Wiedijk, Freek (2007). "The QED Manifesto Revisited" (PDF). From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar and Rhetoric. 10 (23).
- ^ The MathWiki project homepage
- ^ The MML in wiki form
- ^ Alama, Jesse; Brink, Kasper; Mamane, Lionel; Urban, Josef (2011). "Large Formal Wikis: Issues and Solutions". In Davenport, James H.; Farmer, William M.; Urban, Josef; Rabe, Florian (eds.). Intelligent Computer Mathematics – 18th Symposium, Calculemus 2011, and 10th International Conference, MKM 2011, Bertinoro, Italy, July 18–23, 2011. Proceedings. Lecture Notes in Computer Science. Vol. 6824. Springer. pp. 133–148. arXiv:1107.3209. doi:10.1007/978-3-642-22673-1_10. ISBN 978-3-642-22672-4.
- ^ An example of an MML query, yielding all theorems proved on the exponent operator, by the number of times they are cited in subsequent theorems.
- ^ Another example of an MML query, yielding all theorems proved on sigma fields.
- ^ Grabowski, Adam; Artur Kornilowicz; Adam Naumowicz (2010). "Mizar in a Nutshell". Journal of Formalized Reasoning. 3 (2): 152–245.
- ^ Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press. ISBN 9780521631075. Archived from the original on 2015-06-23. Retrieved 2012-07-24.
- ^ Mizar source code
External links
[edit]Mizar system
View on Grokipediafrom Grokipedia
History and Development
Origins
The Mizar system originated in 1973 when Andrzej Trybulec, a Polish mathematician, initiated the project at the Płock Scientific Society in Poland.[2][5][6] The name "Mizar" was adopted that year, recycled from an earlier unrelated initiative, and the effort began as an exploration into computer-assisted mathematical formalization.[5] The primary motivation was to develop a tool that would enable mathematicians to record their work in a formalized manner suitable for computer processing, including storage, translation into symbolic representations, error detection, and support for education in automated theorem proving.[2][6] Trybulec aimed to bridge the gap between natural mathematical language and rigorous verification, drawing from the traditions of the Polish school of mathematics, which emphasized precise and structured proofs.[5] Early experiments in 1974–1975 focused on creating a proof-checker for propositional logic, marking the system's initial technical foundation.[5][6] In 1976, the project relocated to the Białystok Branch of the University of Warsaw (now the University of Białystok), where it gained institutional support and began expanding under Trybulec's leadership.[2][6] This period saw the introduction of key features, such as the Mizar QC version in 1977, which incorporated quantifiers to handle more expressive logical statements.[5][6] By the late 1970s, a small group of collaborators, including Czesław Byliński and Piotr Rudnicki, contributed to refining the system's ability to formalize mainstream mathematics while preserving readability.[5][6]Key Milestones
The Mizar system originated in the early 1970s under the leadership of Andrzej Trybulec at the Płock Scientific Society in Poland, with the project's ideology first presented on November 14, 1973, at Warsaw University, emphasizing a formal language close to mathematical vernacular for computer-checked proofs.[7][5] Initial experiments began in 1974 on the ODRA-1204 computer using Algol 1204 to implement a proof checker for propositional logic, marking the system's foundational technical efforts.[7] By 1975, the first version, Mizar-PC, was implemented to process propositional calculus, supported by financing from the Płock Scientific Society and presented internationally in November at a conference in Ilmenau, East Germany.[7][6] In 1976, the project relocated to the Białystok Branch of Warsaw University (now the University of Białystok), where Czesław Byliński emerged as a primary developer.[6] The following year, 1977, saw the release of Mizar-QC in Pascal on CDC 6000 systems, extending capabilities to quantifier calculus and enabling formalization of theorems like the Chinese Remainder Theorem.[7][5] Advancements continued with Mizar-MS in 1978, introducing multi-sorted types, and Mizar-FC in 1979, adding functional notation to enhance expressiveness.[7][6] The 1980s brought significant expansions: Mizar-2 was released in July and September 1981, ported across machines like ICL 1900, IBM, and UNIX, solidifying the language's portability.[7][5] In 1982, Mizar-MSE was developed as a multi-sorted extension with equality, demonstrated at the International Congress of Mathematicians in Warsaw, highlighting its educational potential.[7][6] Mizar-4 followed in 1986, with PC ports by late 1986, allowing preliminary axioms and broader distribution to dozens of users.[7][5] By 1988, the final Mizar language design enabled cross-references among articles, and PC-Mizar was introduced for personal computers.[7][5] A pivotal milestone occurred on January 1, 1989, with the establishment of the Mizar Mathematical Library (MML), beginning with Trybulec's formalization of Tarski-Grothendieck Set Theory and growing to about 20 articles by 1988, reaching 279 by June 1992.[7][6][5] In 2000, Josef Urban developed a Mizar mode for Emacs, improving editorial workflows.[6] The 2002 launch of the Encyclopedia of Mathematics in Mizar introduced monographic articles, expanding the library's scope.[6] Recent integrations include the 2023 MizAR 60 system, an AI-assisted tool achieving automated proofs for about 60% of Mizar theorems as part of the project's 50th anniversary celebrations.[8] By 2025, the system remains actively maintained, with version 8.1.15 supporting ongoing library growth and theorem proving advancements.[1]Current Status
As of May 2025, the Mizar system remains under active development by the Mizar Project at the University of Białystok, Poland, with the latest release being version 8.1.15 of the core system and version 5.94.1493 of the Mizar Mathematical Library (MML).[1] This update reflects ongoing maintenance to support proof checking and library expansion, ensuring compatibility with modern computing environments while preserving the system's foundational reliance on natural deduction within Tarski-Grothendieck set theory.[1] The MML, serving as a centralized repository of formalized mathematics, continues to expand through peer-reviewed contributions published in the Journal of Formalized Mathematics. In 2025, Volume 33 (Issue 1) incorporated nine new articles, covering topics such as surreal numbers, their dyadic and real extensions, square roots in surreal arithmetic, and Conway's normal form for surreal representations.[9] These additions build on the library's established scope, which spans over 1,500 articles as of May 2025, encompassing definitions, theorems, and schemes across pure and applied mathematics.[1] The review process remains rigorous, with 21 articles in various stages of submission, correction, or evaluation as of late 2025.[9] Community engagement persists through workshops, conferences, and tool integrations. For instance, a 2023 reimplementation of core Mizar components in Rust (mizar-rs) has advanced efforts to enhance performance and extensibility, facilitating exports to other proof assistants and knowledge graph constructions like MMLKG for semantic querying of mathematical content.[10] In 2025, formalizations such as Conway's normal form demonstrate the system's applicability to contemporary mathematical constructs, underscoring its role in verifiable computing and automated reasoning.[11] The project emphasizes open access, with the full MML distributed alongside the system for unrestricted use in research and education.[1]Mizar Language
Design Principles
The Mizar language is designed to bridge the gap between informal mathematical writing and formal verification, prioritizing readability while ensuring machine-checkable rigor. It derives directly from the mathematical vernacular, employing natural language-like syntax with English keywords such as "for," "assume," and "thus" to mimic the structure of traditional mathematical papers. This approach allows mathematicians to author proofs that resemble their informal counterparts, facilitating adoption by non-computer scientists. The language supports declarative forward reasoning in a Jaśkowski-style natural deduction framework, where proofs are written as sequences of justified steps rather than low-level tactics, promoting clarity and maintainability over interactive scripting.[12][13] At its core, Mizar adheres to simple logical foundations based on classical first-order logic augmented with type information for all terms and Tarski-Grothendieck set theory as the underlying axiomatic system. This minimal axiomatization incorporates ZFC-like axioms (e.g., replacement and power set) plus Tarski's Axiom A for infinitary sets, avoiding overly complex primitives to reduce redundancy and potential contradictions in formalizations. The type system enforces non-empty types and correctness conditions for definitions, such as existence and uniqueness for functors and predicates, ensuring semantic consistency without burdening users with explicit type annotations in most cases. Free second-order variables enable higher-order schemes like mathematical induction, extending expressiveness while remaining verifiable in a first-order checker.[2][12][13] Verification principles emphasize efficiency and reusability through a non-interactive, batch-oriented proof checker that processes articles in seconds, supporting a "write-compile-correct" workflow. This design enables random-access authoring and large-scale library maintenance, with definitions, theorems, and proofs stored in a centralized database for modular reuse across formalizations. By concentrating on human-oriented structures like article environments and proof skeletons, Mizar fosters long-term readability and collaboration, distinguishing it from tactic-heavy systems.[14][2]Syntax and Features
The Mizar language employs a declarative style closely resembling mathematical vernacular, enabling mathematicians to write proofs that are both human-readable and machine-verifiable. It is based on classical first-order logic with equality, interpreted over Tarski-Grothendieck set theory, and features a soft type system that enforces non-emptiness of types while allowing type widening for flexibility.[15][16] Mizar articles are structured as plain ASCII text files, limited to 80 characters per line, beginning with an environment declaration followed by abegin section containing the main text. The environment is specified via directives such as environ with vocabularies for importing symbols, notations for operator definitions, and constructors for type constructors, which reference external library files to build the local context. The text proper consists of items like reservations, definitions, theorems, schemes, and registrations, processed in multiple verification passes including parsing, semantic analysis, and proof checking.[17][12]
Key syntactic constructs include declarations for modes (types), functors (functions), predicates, and attributes. For instance, modes define carrier types like mode Nat is Element of NAT;, while functors specify operations such as func succ(n) -> Nat means ... ;. Definitions are enclosed in blocks, e.g.,
definition
let n be Nat;
func [factorial](/page/Factorial)(n) -> Nat means
... [definiens with correctness conditions];
end;
definition
let n be Nat;
func [factorial](/page/Factorial)(n) -> Nat means
... [definiens with correctness conditions];
end;
theorem [compact statement];, often justified by prior results using by or expanded via proofs.[17]
Proofs adopt a block-structured natural deduction style, inspired by Jaśkowski and Fitch, where reasoning is organized into nested blocks for assumptions and conclusions. Common constructs include assume [proposition]; for hypotheses, let [variable] be [type]; for universal quantification, and thus [statement]; or hence [statement]; to discharge goals. Justifications reference prior theorems or schemes with by [label]:[number];, supporting concise "by" proofs or detailed expansions. An example proof fragment for reflexivity might appear as:
proof
let a be Element of X;
thus a = a by XBOOLE:3;
end;
proof
let a be Element of X;
thus a = a by XBOOLE:3;
end;
scheme [name] {parameters}: [conclusion] provided [premise];, which generalize over predicates without explicit higher-order syntax. The language also supports registrations for clustering properties (e.g., cluster non empty for set;) and correctness conditions like existence and uniqueness in definitions, preventing ill-formed expressions. Terms use prefix notation for functions (e.g., add(x,y)) and infix for relations, with Fraenkel notation for comprehensions like {x where x in S: P(x)}. These elements prioritize conceptual clarity over exhaustive formalism, facilitating the formalization of over 70,000 theorems in the Mizar Mathematical Library as of 2025.[17][12][18]
Mizar Mathematical Library
Scope and Content
The Mizar Mathematical Library (MML) is a comprehensive repository of formalized mathematical knowledge, consisting of articles written in the Mizar language that have been mechanically verified by the Mizar proof checker. It serves as a foundational resource for users of the Mizar system, enabling the reuse of established theorems, definitions, and proofs in new developments. The library is built upon Tarski-Grothendieck set theory and emphasizes declarative, natural-language-like formalizations to bridge informal mathematics with rigorous verification.[2] The scope of the MML encompasses a broad spectrum of mathematical disciplines, classified according to the Mathematics Subject Classification (MSC). As of 2023, it includes over 1,400 articles covering areas such as mathematical logic and foundations (MSC 03), order theory and lattices (MSC 06), real analysis (MSC 26), general topology (MSC 54), algebra (MSC 13–20), geometry (MSC 51–53), and computer science (MSC 68). The largest category by number of articles is mathematical logic and foundations, reflecting the library's emphasis on foundational aspects. Other notable areas include number theory, functional analysis, and measure theory, with the content progressing from elementary concepts to advanced results.[3][19][2] In terms of content, the MML contains a diverse array of mathematical objects, including definitions, theorems, schemes (generalized theorems), and registrations (type inclusions). As of 2023, it featured over 65,000 theorems and more than 13,000 definitions, along with thousands of supporting lemmas and corollaries, all interconnected through a dependency graph that ensures reusability. Articles are organized thematically, starting with foundational texts like those on set theory and progressing to specialized topics; for instance, the library includes formalizations of classical results such as the Hahn-Banach theorem in functional analysis and the Fundamental Theorem of Algebra in algebra. Representative examples also encompass the Jordan Curve Theorem in topology and a full compendium of continuous lattices in order theory. As of May 2025, the library comprises 1,493 articles in version 5.94, continuing to grow through community contributions.[3][1][2] The content prioritizes conceptual depth over exhaustive coverage, focusing on verifiable proofs that support further formalization. Quantitative growth has been steady, with the library expanding from 258 articles in its first three years (covering primarily logic, geometry, and real functions) to its current scale, demonstrating its role as one of the largest mechanically verified mathematical corpora. This structure facilitates applications in automated theorem proving and mathematical software development.[19][2]| MSC Code | Category | Approximate Articles (2018) |
|---|---|---|
| 03 | Mathematical logic and foundations | 161 |
| 06 | Order, lattices, ordered structures | 110 |
| 26 | Real functions | 101 |
| 54 | General topology | 100 |
| 68 | Computer science | 97 |
Organization and Structure
The Mizar Mathematical Library (MML) is organized as an incremental collection of articles, each a self-contained text file written in the Mizar language and mechanically verified by the system's proof checker.[20] These articles, numbering 1,493 as of May 2025, form a database of formalized mathematical knowledge built cumulatively since January 1, 1989, with contributions from more than 250 authors across various countries.[21][20][1] The library's foundational articles establish the axioms of Tarski-Grothendieck set theory, based on Zermelo-Fraenkel set theory extended by an axiom for strongly inaccessible cardinals, serving as the logical basis for all subsequent formalizations.[20] Structurally, each article begins with an environ section that defines its context by importing essential components from prior library elements, including vocabularies for terminology, constructors for types and operations, notations, theorems, definitions, schemes, registrations, and cluster expansions.[22] This environment mechanism ensures modular dependencies, allowing new articles to build upon exported facts and concepts from earlier ones while maintaining a directed acyclic graph of inter-article relationships.[21] The library is divided into core sections: concrete mathematics covering foundational topics like sets, relations, functions, and orders; abstract mathematics employing algebraic structures and modes for higher-level concepts; and supplementary areas such as the SCM (for scheme clusters and modes) and Addenda (for corrections and extensions).[20] Processed articles are compiled into a semantic database in XML format, storing over 59,000 theorems, 12,000 definitions, 14,000 constructors, and 15,000 notations as of 2017, which supports querying and reuse.[20] For categorization, the MML employs a two-tiered classification system derived from the Encyclopedic Dictionary of Mathematics, with broad divisions (e.g., algebra, analysis, geometry) and finer items within them (e.g., groups under algebra).[21] Approximately 450 articles are additionally tagged using the Mathematics Subject Classification (MSC) for cross-referencing.[21] Algebraic structures within the library form a hierarchical network through multiple inheritance, starting from basic types like sets and relations, extending to specialized ones such as groups, topologies, and vector spaces, with dependencies visualized as layered graphs to highlight inheritance and import relations.[23][21] The overall development and maintenance are overseen by the Mizar Library Committee, which handles revisions including authorship checks, automatic updates, pretty-printing, and reorganizations to ensure consistency and growth.[20]Access and Usage
The Mizar Mathematical Library (MML) is freely available for download from the official Mizar project website, allowing users to obtain the complete collection of formalized mathematical articles alongside the Mizar system itself. The current version, MML 5.94.1493 (as of May 2025), can be accessed via HTTP at http://mizar.uwb.edu.pl/~softadm/current/ or FTP at ftp://mizar.uwb.edu.pl/pub/system/current/, typically bundled with the Mizar system release for platforms including Windows, Linux (i386 and ARM), macOS, Solaris, and FreeBSD. Upon installation, the library integrates directly into the Mizar environment, with files placed in a designated directory (e.g., /usr/local/share/mizar/mml on Linux), enabling seamless reference during proof development without additional configuration. Installation instructions are provided in platform-specific README files, such as readme.txt for Windows, which guide users through unpacking the archive and verifying the setup via the miz command-line tool.[24] Once installed, users access the MML through the Mizar system's built-in mechanisms, where articles are referenced in new proofs using directives like :: for environment imports, vocabularies for notation, and notations for symbols. The library's 1,493 articles as of May 2025, containing over 65,000 theorems and more than 13,000 definitions as of 2023 with ongoing growth, serve as a foundational knowledge base, with dependencies automatically resolved by the system's analyzer during proof checking. For exploratory use, the MML can be browsed offline via the provided mml.ini configuration file or online through the HTML-indexed directory at https://mizar.uwb.edu.pl/version/current/html/, which organizes content by abstract and concrete categories for navigation. This structure supports conceptual exploration, such as tracing definitions from basic set theory in early articles to advanced topology in later ones, without requiring full system installation.[25][22][3] Advanced access is facilitated by the MML Query tool at http://mmlquery.mizar.org, a web-based service that enables searching the library using Mizar-like syntax or filters for articles, theorems, schemes, and authors. Users can query for specific elements, such as theorems involving real numbers (e.g., via keywords like "Real" or scheme names), retrieving results with contextual excerpts and links to source articles. This tool, powered by the MMLQT transformation engine, supports both novice discovery—listing articles by lexical order or author—and expert tasks like verifying dependencies across the library's 278 contributors' works. Integration with authoring environments, such as the Emacs Mizar Mode, further streamlines usage by providing syntax highlighting and direct library lookups during editing. For hands-on learning, introductory tutorials, including slides and exercises from the 2016 CICM conference, demonstrate basic querying and article writing with MML references.[26][22]Proof Verification
Mizar Proof Checker
The Mizar Proof Checker is a core component of the Mizar system, designed to verify the logical correctness of mathematical proofs written in the Mizar language. It ensures that proofs adhere to a formal deductive system based on Tarski-Grothendieck set theory and classical first-order logic, without attempting to discover proofs autonomously.[1][13] Developed initially by Andrzej Trybulec in 1973, the checker has evolved through versions like Mizar-2 (1981) to its current iteration (version 8.1.15 as of 2025), supporting interactive proof development while maintaining rigorous verification.[2][1] The verification process operates non-interactively on entire proof files (typically.miz extensions) using the mizf program, which performs multiple passes: scanning for syntax, parsing the structure, analyzing types and modes, and checking justifications against axioms and prior theorems. During justification checking, the system evaluates proof steps—such as assumptions, implications, and conclusions—ensuring each inference is valid and references to the Mizar Mathematical Library (MML) are accurate, while eliminating duplicates via a semantic database. Errors, such as invalid inferences or unresolved references, are flagged directly in the file with diagnostic messages (e.g., *4 for logical flaws), requiring manual correction before re-verification.[13][2][27]
Key features include integration with the MML, which contains over 1,500 articles (1,493 as of May 30, 2025) allowing the checker to import definitions, theorems, and automations like registrations for enhanced reusability.[1] The system exports verified items in XML format for semantic storage, using constructors to represent mathematical objects, and supports tools like the accommodator and extractor to resolve ambiguities in proof references. This batch-oriented approach, combined with minimal axiomatization, enables scalable verification of complex mathematics, as demonstrated in the formalization of J. B. Conway's A Compendium of Continuous Lattices.[2]
Verification Process
The verification process in the Mizar system involves the automated checking of formalized mathematical proofs written in the Mizar language, ensuring their logical correctness, syntactic validity, and semantic consistency against a foundational axiomatic system based on Tarski-Grothendieck set theory.[19] This process is declarative and non-tactical, meaning proofs are structured as sequences of natural deduction steps that the system verifies step-by-step without user-guided tactics, allowing mathematicians to write proofs in a style close to informal mathematical vernacular.[22] Each proof article, saved as a.miz file, is processed independently to confirm that all inferences follow from axioms or previously verified theorems in the Mizar Mathematical Library (MML).[27]
The core component is the Mizar proof checker, which performs multiple passes over the input file: initial scanning and parsing to validate lexical and grammatical structure; type and natural deduction analysis to infer types dynamically using a sup-semilattice type system and ensure mode correctness; and justification checking via modules like the Preparator, Prechecker, Equalizer, and Unifier.[22] The Equalizer module employs a dependent congruence-closure algorithm to identify and merge semantically equivalent terms, while the Unifier handles substitutions for proof steps.[22] Logical validity is assessed by transforming inferences into refutable forms in classical first-order logic and verifying all disjuncts using a built-in disprover.[22] The system draws on the MML's database of over 1,500 articles (1,493 as of May 30, 2025), which stores exported facts (theorems, definitions, and schemes) in an XML-based format for efficient reuse and dependency resolution.[19][1]
To initiate verification, users compile the article using the mizf command (or equivalent in the environment), which invokes the librarian to manage dependencies and the checker to process the file against specified requirements (e.g., importing vocabularies like BOOLE or SUBSET_1).[27] Successful verification registers new facts into the MML if the article passes all checks, including reductions and registrations that automate routine inferences.[19] Errors, such as invalid justifications or type mismatches, are reported with codes and line references, prompting iterative correction in a "write-compile-correct" loop, though feedback can be general rather than diagnostic.[27] This process ensures minimal axiomatization and reduces contradictions by enforcing strict semantic accuracy, with ongoing enhancements like AI-assisted automation verifying up to 40% of MML theorems automatically.[19]
Key features include soft-typing for flexible mode declarations and support for scheme-based generalizations, which the checker expands during verification to handle parametric proofs.[22] The system's independence from external provers in core checking maintains reliability, though extensions like ATP integration enable cross-verification for complex steps.[19] Overall, this rigorous pipeline has formalized vast swaths of mathematics, with the MML containing tens of thousands of theorems as of May 30, 2025, demonstrating the process's scalability for interactive proof development.[1]