Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic

  title={Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic},
  author={C. Kaliszyk and Karol Pak},
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
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
  • 2018
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
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
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
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


Towards a Mizar environment for Isabelle: foundations and language
It is shown how Isabelle types can be used to differentiate between the syntactic categories of the Mizar language, such as sets and Mizar types including modes and attributes, and how they interact with the basic constructs of the Tarski-Grothendieck set theory.
Set theory for verification: I. From foundations to functions
The paper describes the derivation of rules for descriptions, relations, and functions and discusses interactive proofs of Cantor's Theorem, the Composition of Homomorphisms challenge, and Ramsey’s Theorem.
The Twelf Proof Assistant
This work designs special purpose logical frameworks for capturing reoccurring concepts for special domains, such as, for example, variable renaming, substitution application, and resource management for programming language theory.
Isabelle: The Next 700 Theorem Provers
A thorough history of Isabelle is given, beginning with its origins in the LCF system, and an account of how logics are represented is presented, illustrated using classical logic.
On Rewriting Rules in Mizar
This paper presents some tentative experiments in using a special case of rewriting rules in Mizar rewriting a term as its subterm based on another Mizar mechanism called functor identification.
The Isabelle Framework
Isabelle, which is available from , is a generic framework for interactive theorem proving. The Isabelle/Puremeta-logic allows the formalization of the syntax and inference
Type Inference for ZFH
The type inference algorithm for ZFH, a descendant of Agerholm's and Gordon's HOL-ST but does not allow the use of type variables nor the definition of new types, is described and proved.
On the structure of Mizar types
  • G. Bancerek
  • Computer Science, Mathematics
    Electron. Notes Theor. Comput. Sci.
  • 2003
A logical framework combining model and proof theory
  • Florian Rabe
  • Computer Science, Philosophy
    Mathematical Structures in Computer Science
  • 2013
This framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs.
ATP-based Cross-Verification of Mizar Proofs: Method, Systems, and First Experiments
A translation of Mizar natural-deduction proofs to the TPTP format used for recording derivations from first-order automated theorem proving systems, and verification of the resulting TPTP style derivations indicate that cross-verification of the whole MML is feasible.