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Documents authored by Kaliszyk, Cezary

Document
DOI: 10.4230/LIPIcs.ITP.2024.29
Conway Normal Form: Bridging Approaches for Comprehensive Formalization of Surreal Numbers

Authors: Karol Pąk and Cezary Kaliszyk

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
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.
Cite as

Karol Pąk and Cezary Kaliszyk. Conway Normal Form: Bridging Approaches for Comprehensive Formalization of Surreal Numbers. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{pak_et_al:LIPIcs.ITP.2024.29,
  author =	{P\k{a}k, Karol and Kaliszyk, Cezary},
  title =	{{Conway Normal Form: Bridging Approaches for Comprehensive Formalization of Surreal Numbers}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.29},
  URN =		{urn:nbn:de:0030-drops-207573},
  doi =		{10.4230/LIPIcs.ITP.2024.29},
  annote =	{Keywords: Surreal numbers, Conway normal form, Mizar}
}
Document
DOI: 10.4230/LIPIcs.ITP.2023.19
MizAR 60 for Mizar 50

Authors: Jan Jakubův, Karel Chvalovský, Zarathustra Goertzel, Cezary Kaliszyk, Mirek Olšák, Bartosz Piotrowski, Stephan Schulz, Martin Suda, and Josef Urban

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract
As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60% of the Mizar theorems in the hammer setting. We also automatically prove 75% of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically.
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Jan Jakubův, Karel Chvalovský, Zarathustra Goertzel, Cezary Kaliszyk, Mirek Olšák, Bartosz Piotrowski, Stephan Schulz, Martin Suda, and Josef Urban. MizAR 60 for Mizar 50. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jakubuv_et_al:LIPIcs.ITP.2023.19,
  author =	{Jakub\r{u}v, Jan and Chvalovsk\'{y}, Karel and Goertzel, Zarathustra and Kaliszyk, Cezary and Ol\v{s}\'{a}k, Mirek and Piotrowski, Bartosz and Schulz, Stephan and Suda, Martin and Urban, Josef},
  title =	{{MizAR 60 for Mizar 50}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{19:1--19:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.19},
  URN =		{urn:nbn:de:0030-drops-183942},
  doi =		{10.4230/LIPIcs.ITP.2023.19},
  annote =	{Keywords: Mizar, ENIGMA, Automated Reasoning, Machine Learning}
}
Document
DOI: 10.4230/OASIcs.FMBC.2022.4
Proofgold: Blockchain for Formal Methods

Authors: Chad E. Brown, Cezary Kaliszyk, Thibault Gauthier, and Josef Urban

Published in: OASIcs, Volume 105, 4th International Workshop on Formal Methods for Blockchains (FMBC 2022)

Abstract
Proofgold is a peer to peer cryptocurrency making use of formal logic. Users can publish theories and then develop a theory by publishing documents with definitions, conjectures and proofs. The blockchain records the theories and their state of development (e.g., which theorems have been proven and when). Two of the main theories are a form of classical set theory (for formalizing mathematics) and an intuitionistic theory of higher-order abstract syntax (for reasoning about syntax with binders). We have also significantly modified the open source Proofgold Core client software to create a faster, more stable and more efficient client, Proofgold Lava. Two important changes are the cryptography code and the database code, and we discuss these improvements. We also discuss how the Proofgold network can be used to support large formalization efforts.
Cite as

Chad E. Brown, Cezary Kaliszyk, Thibault Gauthier, and Josef Urban. Proofgold: Blockchain for Formal Methods. In 4th International Workshop on Formal Methods for Blockchains (FMBC 2022). Open Access Series in Informatics (OASIcs), Volume 105, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brown_et_al:OASIcs.FMBC.2022.4,
  author =	{Brown, Chad E. and Kaliszyk, Cezary and Gauthier, Thibault and Urban, Josef},
  title =	{{Proofgold: Blockchain for Formal Methods}},
  booktitle =	{4th International Workshop on Formal Methods for Blockchains (FMBC 2022)},
  pages =	{4:1--4:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-250-1},
  ISSN =	{2190-6807},
  year =	{2022},
  volume =	{105},
  editor =	{Dargaye, Zaynah and Schneidewind, Clara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.FMBC.2022.4},
  URN =		{urn:nbn:de:0030-drops-171851},
  doi =		{10.4230/OASIcs.FMBC.2022.4},
  annote =	{Keywords: Formal logic, Blockchain, Proofgold}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.16
The Isabelle ENIGMA

Authors: Zarathustra A. Goertzel, Jan Jakubův, Cezary Kaliszyk, Miroslav Olšák, Jelle Piepenbrock, and Josef Urban

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
We significantly improve the performance of the E automated theorem prover on the Isabelle Sledgehammer problems by combining learning and theorem proving in several ways. In particular, we develop targeted versions of the ENIGMA guidance for the Isabelle problems, targeted versions of neural premise selection, and targeted strategies for E. The methods are trained in several iterations over hundreds of thousands untyped and typed first-order problems extracted from Isabelle. Our final best single-strategy ENIGMA and premise selection system improves the best previous version of E by 25.3% in 15 seconds, outperforming also all other previous ATP and SMT systems.
Cite as

Zarathustra A. Goertzel, Jan Jakubův, Cezary Kaliszyk, Miroslav Olšák, Jelle Piepenbrock, and Josef Urban. The Isabelle ENIGMA. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goertzel_et_al:LIPIcs.ITP.2022.16,
  author =	{Goertzel, Zarathustra A. and Jakub\r{u}v, Jan and Kaliszyk, Cezary and Ol\v{s}\'{a}k, Miroslav and Piepenbrock, Jelle and Urban, Josef},
  title =	{{The Isabelle ENIGMA}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.16},
  URN =		{urn:nbn:de:0030-drops-167253},
  doi =		{10.4230/LIPIcs.ITP.2022.16},
  annote =	{Keywords: E Prover, ENIGMA, Premise Selection, Isabelle/Sledgehammer}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.26
Formalizing a Diophantine Representation of the Set of Prime Numbers

Authors: Karol Pąk and Cezary Kaliszyk

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert’s 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the existence of diophantine equations with a reduced number of variables, as well as the explicit construction of polynomials that represent specific sets, in particular the set of primes. In this work, we formalize these constructions in the Mizar system. We focus on the set of prime numbers and its explicit representation using 10 variables. It is the smallest representation known today. For this, we show that the exponential function is diophantine, together with the same properties for the binomial coefficient and factorial. This formalization is the next step in the research on formal approaches to diophantine sets following the DPRM theorem.
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Karol Pąk and Cezary Kaliszyk. Formalizing a Diophantine Representation of the Set of Prime Numbers. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 26:1-26:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pak_et_al:LIPIcs.ITP.2022.26,
  author =	{P\k{a}k, Karol and Kaliszyk, Cezary},
  title =	{{Formalizing a Diophantine Representation of the Set of Prime Numbers}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{26:1--26:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.26},
  URN =		{urn:nbn:de:0030-drops-167350},
  doi =		{10.4230/LIPIcs.ITP.2022.26},
  annote =	{Keywords: DPRM theorem, Polynomial reduction, prime numbers}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.ITP.2021
LIPIcs, Volume 193, ITP 2021, Complete Volume

Authors: Liron Cohen and Cezary Kaliszyk

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
LIPIcs, Volume 193, ITP 2021, Complete Volume
Cite as

12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 1-560, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{cohen_et_al:LIPIcs.ITP.2021,
  title =	{{LIPIcs, Volume 193, ITP 2021, Complete Volume}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{1--560},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021},
  URN =		{urn:nbn:de:0030-drops-138943},
  doi =		{10.4230/LIPIcs.ITP.2021},
  annote =	{Keywords: LIPIcs, Volume 193, ITP 2021, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.ITP.2021.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Liron Cohen and Cezary Kaliszyk

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 0:i-0:viii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohen_et_al:LIPIcs.ITP.2021.0,
  author =	{Cohen, Liron and Kaliszyk, Cezary},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{0:i--0:viii},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.0},
  URN =		{urn:nbn:de:0030-drops-138955},
  doi =		{10.4230/LIPIcs.ITP.2021.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.ITP.2019.9
Higher-Order Tarski Grothendieck as a Foundation for Formal Proof

Authors: Chad E. Brown, Cezary Kaliszyk, and Karol Pąk

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract
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.
Cite as

Chad E. Brown, Cezary Kaliszyk, and Karol Pąk. Higher-Order Tarski Grothendieck as a Foundation for Formal Proof. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brown_et_al:LIPIcs.ITP.2019.9,
  author =	{Brown, Chad E. and Kaliszyk, Cezary and P\k{a}k, Karol},
  title =	{{Higher-Order Tarski Grothendieck as a Foundation for Formal Proof}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.9},
  URN =		{urn:nbn:de:0030-drops-110643},
  doi =		{10.4230/LIPIcs.ITP.2019.9},
  annote =	{Keywords: model, higher-order, Tarski Grothendieck, proof foundation}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.ITP.2019.35
Declarative Proof Translation (Short Paper)

Authors: Cezary Kaliszyk and Karol Pąk

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract
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.
Cite as

Cezary Kaliszyk and Karol Pąk. Declarative Proof Translation (Short Paper). In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 35:1-35:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kaliszyk_et_al:LIPIcs.ITP.2019.35,
  author =	{Kaliszyk, Cezary and P\k{a}k, Karol},
  title =	{{Declarative Proof Translation}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{35:1--35:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.35},
  URN =		{urn:nbn:de:0030-drops-110903},
  doi =		{10.4230/LIPIcs.ITP.2019.35},
  annote =	{Keywords: Declarative Proof, Translation, Isabelle/Isar, Mizar}
}
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