Four Decades of Mizar

@article{Grabowski2015FourDO,
  title={Four Decades of Mizar},
  author={Adam Grabowski and Artur Kornilowicz and Adam Naumowicz},
  journal={Journal of Automated Reasoning},
  year={2015},
  volume={55},
  pages={191-198}
}
This special issue is dedicated to works related to Mizar, the theorem proving project started by Andrzej Trybulec in the 1970s, and other automated proof checking systems used for formalizing mathematics. 
Enhancement of Mizar Texts with Transitivity Property of Predicates
A typical proof step in mathematical reasoning consists of two parts – a formula to be proven and a list of references used to justify the formula. In addition, computer proof-assistants can use
Implementation of the Composition-nominative Approach to Program Formalization in Mizar
TLDR
An ongoing work on implementation of the composition-nominative approach to program formalization in Mizar proof assistant based on the first-order logic and axiomatic set theory is described.
Formal verification of the correctness of chosen algorithms in Mizar
TLDR
This paper defines operations of sequential composition of several programs, formalize and show soundness of new inference rules which can be used to prove partial correctness of programs involving these operations in the context of partial preand postconditions.
Natural Addition of Ordinals
TLDR
The uniqueness of the Cantor normal form of ordinals is proven and then used to formalize the natural sum of Ordinals.
Introducing Euclidean relations to Mizar
TLDR
The proposed enhancement of the Mizar proof checker based on enabling special processing of Euclidean predicates can be registered during their definition, and so the verification of some proof steps related to these relations can be automated to avoid explicit referencing.
Formalization of the algebra of nominative data in mizar
TLDR
A formalization of the notion of a nominative data with simple names and complex values in the Mizar proof assistant can be useful for formalizing semantics of programs that operate in real time environment and/or process complex data structures and for reasoning about the behavior of such programs.
About Graph Complements
TLDR
Different variants of the complement graph in the Mizar system are formalized, based on the formalization of graphs in [6], to form a single graph called a complement graph.
About Graph Unions and Intersections
TLDR
The union and intersection of a set of graphs are formalized in the Mizar system based on the formalization of graphs in [7].
Mizar Set Comprehension in Isabelle Framework
  • Karol Pak
  • Computer Science, Mathematics
    FedCSIS
  • 2018
TLDR
The progress in the development of the Isabelle/Mizar project whose main goal is independent cross-verification of the MML in Isabelle is presented and an infrastructure that provides a more elegant and recursive approach to construct and to provide the main property of set comprehension operators is proposed.
Building a Framework of Rough Inclusion Functions by Means of Computerized Proof Assistant
The paper describes some of the issues concerning the development of automated formal framework for the reasoning about rough inclusion functions, starting with the classical one, and generalizations
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References

SHOWING 1-10 OF 49 REFERENCES
Mizar in a Nutshell
This paper is intended to be a practical reference manual for basic Mizar terminology which may be helpful to get started using the system. The paper describes most important aspects of the Mizar
Interfacing external CA systems for Gröbner bases computation in Mizar proof checking
TLDR
A case study aimed at selecting a prospective CA system to be used by the Mizar proof-checking system for performing computations of Gröbner bases in Mizar’s module responsible for equality calculus.
A Brief Overview of Mizar
TLDR
An outline of the key features of the Mizar language, the ideas and theory behind the system, its main applications, and current development is given.
On the Integrity of a Repository of Formalized Mathematics
TLDR
The issues of maintaining integrity of an electronic repository of formal mathematics, based on the experience with MML, are discussed.
IZAR : the first 30 years
We present the story of the MIZAR project with focus on the years until 1989. A lot about MIZAR after 1989 is available at the web3.
A Declarative Language for the Coq Proof Assistant
TLDR
A new proof language for the Coq proof assistant that uses the declarative style, providing a simple, natural and robust alternative to the existing Ltac tactic language.
On Equivalents of Well-foundedness - An experiment in Mizar
Four statements equivalent to well-foundedness (well-founded induction, existence of recursively deened functions, uniqueness of recursively deened functions , and absence of descending !-chains)
Formal Mathematics for Mathematicians
TLDR
The issue features a collection of articles by practitioners of formalizing proofs who share a deep interest in making computerized mathematics widely available.
A Mizar Mode for HOL
TLDR
The addition of another style of HOL proofs, inspired by Mizar, is described, which is believed to combine the secure extensibility and interactivity of HOL with Mizar’s readability and lack of logical prescriptiveness.
A Comparison of Mizar and Isar
TLDR
A list of differences between Mizar and Isar is presented, highlighting the strengths of both systems from the perspective of end-users, and some key differences of the internal mechanisms of structured proof processing in either system are pointed out.
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