The proper class of Conway’s surreal numbers forms a rich totally ordered algebraically closed field with many arithmetic and algebraic properties close to those of real numbers, the ordinals, and infinitesimal numbers. In this paper, we formalize the construction of Conway’s numbers in Mizar using two approaches and propose a bridge between them, aiming to combine their advantages for efficient formalization. By replacing transfinite induction-recursion with transfinite induction, we streamline their construction. Additionally, we introduce a method to merge proofs from both approaches using global choice, facilitating formal proof. We demonstrate that surreal numbers form a field, including the square root, and that they encompass subsets such as reals, ordinals, and powers of ω. We combined Conway’s work with Ehrlich’s generalization to formally prove Conway’s Normal Form, paving the way for many formal developments in surreal number theory.

The Isabelle Higher-order Tarski-Grothendieck object logic includes in its foundations both higher-order logic and set theory, which allows importing the libraries of Isabelle/HOL and Isabelle/Mizar. The two libraries, however, define all the basic concepts independently, which means that the results in the two are disconnected. In this paper, we align significant parts of these two libraries, by defining isomorphisms between their concepts, including the real numbers and algebraic structures. The isomorphisms allow us to transport theorems between the foundations and use the results from the libraries simultaneously.

We formally introduce a foundation for computer verified proofs based on higher-order Tarski-Grothendieck set theory. We show that this theory has a model if a 2-inaccessible cardinal exists. This assumption is the same as the one needed for a model of plain Tarski-Grothendieck set theory. The foundation allows the co-existence of proofs based on two major competing foundations for formal proofs: higher-order logic and TG set theory. We align two co-existing Isabelle libraries, Isabelle/HOL and Isabelle/Mizar, in a single foundation in the Isabelle logical framework. We do this by defining isomorphisms between the basic concepts, including integers, functions, lists, and algebraic structures that preserve the important operations. With this we can transfer theorems proved in higher-order logic to TG set theory and vice versa. We practically show this by formally transferring Lagrange’s four-square theorem, Fermat 3-4, and other theorems between the foundations in the Isabelle framework.

Declarative proof styles of different proof assistants include a number of incompatible features. In this paper we discuss and classify the differences between them and propose efficient algorithms for declarative proof outline translation. We demonstrate the practicality of our algorithms by automatically translating the proof outlines in 200 articles from the Mizar Mathematical Library to the Isabelle/Isar proof style. This generates the corresponding theories with 15301 proof outlines accepted by the Isabelle proof checker. The goal of our translation is to produce a declarative proof in the target system that is both accepted and short and therefore readable. For this three kinds of adaptations are required. First, the proof structure often needs to be rebuilt to capture the extensions of the natural deduction rules supported by the systems. Second, the references to previous items and their labels need to be matched and aligned. Finally, adaptations in the annotations of individual proof step may be necessary.

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 express Mizar types in a uniform way, provide a common type intersection operation, allow reasoning about type inhabitation, and develop a type inference mechanism. We provide Mizar-like definition mechanisms which require the same proof obligations and provide same derived properties. Structures and set comprehension operators can be defined as definitional extensions. Re-formalized proofs from various parts of the Mizar Library show the practical usability of the specified foundations.

The Mizar system provides two representations of the proofs present in its library.The syntactic representation preserves the human-friendly rich Mizar language, where the meaning of structures and expressions is still influenced by their context.The semantic one, on the other hand, explicitly reflects the meaning of all elements present in the proof scripts, however many features of the Mizar language are eliminated.In this article, we overcome the limitations of both representations of proofs, by proposing a method combining them.We show that we can simultaneously maintain the richness of the language and provide access to the derived proof information.We discuss how such combined information closer corresponds to that present in other proof assistant languages, for example that of Isabelle/Isar.

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 re-use 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 centrally-managed 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 web-based and natural language proof presentation, maths education, and machine learning based automated theorem proving.

We present a case study on a formalization of a textbook theorem that is listed as #39 at Freek Wiedijk's list of "Top 100 mathematical theorems".We focus on the formalization of the theorem that Pell's equation x 2-Dy 2 = 1 has infinitely many solutions in positive integers for a given non square natural number D. We present also a formalization of the theorem that based on the least fundamental solution of the equation we can simply calculate algebraically each remaining solution.

The Mizar Mathematical Library is one of the largest collections of machine understandable formal proofs encompassing many areas of today mathematics including results from algebra, analysis, topology, and lattice theory.The Mizar system has so far been the only tool able to completely process, certify, and make use of these developments.In this paper, we present the progress in the development of an independent certification mechanism of Mizar proofs based on the Isabelle logical framework.The approach allows rechecking the Mizar formal proofs based on a more succinct and more precisely specified formal infrastructure.Additionally, it necessitates a full formal specification of the mechanisms that ensure the correctness of the defined objects, in particular, the proofs that such mechanisms are correct.The development already covers an important part of the Mizar library foundations.We improve the mechanism for defining Mizar structures and show that it permits simpler validation of proof developments involving such objects.To demonstrate this, we perform a complete translation of the Mizar net of basic algebraic structures including their attributes and certify all the corresponding proofs in Isabelle.

In this paper we explore the possibility of emulating the Mizar environment as close as possible inside the Isabelle logical framework. We introduce adaptations to the Isabelle/FOL object logic that correspond to the logic of Mizar, as well as Isar inner syntax notations that correspond to these of the Mizar language. We show 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 show how they interact with the basic constructs of the Tarski-Grothendieck set theory. We discuss Mizar definitions and provide simple abbreviations that allow the introduction of Mizar predicates, functions, attributes and modes using the Isabelle/Pure language elements for introducing definitions and theorems. We finally consider the definite and indefinite description operators in Mizar and their use to introduce definitions by “means” and “equals”. We demonstrate the usability of the environment on a sample Mizar-style formalization, with cluster inferences and “by” steps performed manually.

Proof development in proof assistants such as HOL, Coq, Mizar, etc. is an activity where authors usually produce proofs by typing out proof scripts or system tactics. Quite frequently, however, authors also have to read existing proof scripts, either to imitate smart proof pieces, or to refactor fragments of reasoning to make some theorem stronger, more easily applicable and so on. Therefore, it is important to develop techniques to improve legibility of proofs, since it directly affects productivity of script writers. To analyze the legibility of natural deduction proofs, we investigate proof graphs that represent the flow of information in given reasoning. Our analysis of the information flow leads to methods of improving proof readability based on Behaghel’s First Law, which states that in legible text relevant pieces of information must occur close to each other. The presented method maximizes the number of close connections between premises and steps that use these steps as justification. In this paper we show that our optimization method is NP-hard.

The existing examples of natural deduction proofs, either declarative or procedural, indicate that often the legibility of proof scripts is of secondary importance to the authors. As soon as the computer accepts the proof script, many authors do not work on improving the parts that could be shortened and do not avoid repetitions of technical sub-deductions, which often could be replaced by a single lemma. This article presents selected properties of reasoning passages that may be used to determine if a reasoning passage can be extracted from a proof script, transformed into a lemma and replaced by a reference to the newly created lemma. Additionally, we present methods for improving the legibility of the reasoning that remains after the extraction of the lemmas.