Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic

@inproceedings{Kaliszyk2017PresentationAM,
  title={Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic},
  author={C. Kaliszyk and Karol Pak},
  booktitle={CICM},
  year={2017}
}
One of the crucial factors enabling an efficient use of a logical framework is the convenience of entering, manipulating, and presenting object logic constants, statements, and proofs. In this paper, we discuss various elements of the Mizar language and the possible ways how these can be represented in the Isabelle framework in order to allow a suitable way of working in typed set theory. We explain the interpretation of various components declared in each Mizar article environment and create… 
Semantics of Mizar as an Isabelle Object Logic
We formally define the foundations of the Mizar system as an object logic in the Isabelle logical framework. For this, we propose adequate mechanisms to represent the various components of Mizar. We
Isabelle Import Infrastructure for the Mizar Mathematical Library
TLDR
An infrastructure that allows importing an initial part of the Mizar Mathematical Library into the Isabelle/Mizar object logic is presented and it is shown that the imported 100 articles give rise to a usable Isabelle environment.
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.
Progress in the independent certification of mizar mathematical library in isabelle
  • C. Kaliszyk, Karol Pak
  • Computer Science, Mathematics
    2017 Federated Conference on Computer Science and Information Systems (FedCSIS)
  • 2017
TLDR
This paper improves the mechanism for defining Mizar structures and shows that it permits simpler validation of proof developments involving such objects and performs a complete translation of the Mizar net of basic algebraic structures including their attributes and certify all the corresponding proofs in Isabelle.
Isabelle Formalization of Set Theoretic Structures and Set Comprehensions
TLDR
This paper reformalize a number of Mizar definitions and theorems related to structures and set comprehensions, including both mathematical and programming language examples: groups, machines and properties of computer memory states.
A Tale of Two Set Theories
TLDR
It is shown how certain higher-order terms and propositions in Egal have equivalent first-order presentations and it is proved Tarski's Axiom A (an axiom in Mizar) in EGal and a Grothendieck Universe operator (a primitive with axioms in E Gal) is constructed.
Abstracts of the talks
s of the talks March 25 – 30, 2018, Aussois, France

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