38 Search Results for "de Moura, Leonardo"


Volume

LIPIcs, Volume 237

13th International Conference on Interactive Theorem Proving (ITP 2022)

ITP 2022, August 7-10, 2022, Haifa, Israel

Editors: June Andronick and Leonardo de Moura

Document
An Extensible User Interface for Lean 4

Authors: Wojciech Nawrocki, Edward W. Ayers, and Gabriel Ebner

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


Abstract
Contemporary proof assistants rely on complex automation and process libraries with millions of lines of code. At these scales, understanding the emergent interactions between components can be a serious challenge. One way of managing complexity, long established in informal practice, is through varying external representations. For instance, algebraic notation facilitates term-based reasoning whereas geometric diagrams invoke spatial intuition. Objects viewed one way become much simpler than when viewed differently. In contrast, modern general-purpose ITP systems usually only support limited, textual representations. Treating this as a problem of human-computer interaction, we aim to demonstrate that presentations - UI elements that store references to the objects they are displaying - are a fruitful way of thinking about ITP interface design. They allow us to make headway on two fronts - introspection of prover internals and support for diagrammatic reasoning. To this end we have built an extensible user interface for the Lean 4 prover with an associated ProofWidgets 4 library of presentation-based UI components. We demonstrate the system with several examples including type information popups, structured traces, contextual suggestions, a display for algebraic reasoning, and visualizations of red-black trees. Our interface is already part of the core Lean distribution.

Cite as

Wojciech Nawrocki, Edward W. Ayers, and Gabriel Ebner. An Extensible User Interface for Lean 4. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{nawrocki_et_al:LIPIcs.ITP.2023.24,
  author =	{Nawrocki, Wojciech and Ayers, Edward W. and Ebner, Gabriel},
  title =	{{An Extensible User Interface for Lean 4}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{24:1--24:20},
  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.2023.24},
  URN =		{urn:nbn:de:0030-drops-183991},
  doi =		{10.4230/LIPIcs.ITP.2023.24},
  annote =	{Keywords: user interfaces, human-computer interaction, Lean}
}
Document
Complete Volume
LIPIcs, Volume 237, ITP 2022, Complete Volume

Authors: June Andronick and Leonardo de Moura

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


Abstract
LIPIcs, Volume 237, ITP 2022, Complete Volume

Cite as

13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 1-602, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@Proceedings{andronick_et_al:LIPIcs.ITP.2022,
  title =	{{LIPIcs, Volume 237, ITP 2022, Complete Volume}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{1--602},
  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},
  URN =		{urn:nbn:de:0030-drops-167080},
  doi =		{10.4230/LIPIcs.ITP.2022},
  annote =	{Keywords: LIPIcs, Volume 237, ITP 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: June Andronick and Leonardo de Moura

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


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

Cite as

13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{andronick_et_al:LIPIcs.ITP.2022.0,
  author =	{Andronick, June and de Moura, Leonardo},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{0:i--0:x},
  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.2022.0},
  URN =		{urn:nbn:de:0030-drops-167097},
  doi =		{10.4230/LIPIcs.ITP.2022.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Modelling and Verifying Properties of Biological Neural Networks (Invited Talk)

Authors: Amy Felty

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


Abstract
In this talk, I present a formal model of biological neural networks and discuss the use of model checking and interactive theorem proving to verify some of their properties. Having a formal model can increase our understanding of the behavior and properties of such networks, as well as provide insight into their response to external factors such as disease, medicine, and environmental changes. We focus on neuronal micro-networks, considering properties of single neurons as well as properties of slightly larger ones called archetypes, which represent specific computational functions. Archetypes, in turn, represent the building blocks of larger more complicated neuronal circuits. I first present work by colleagues on a model checking approach, and then present our joint work on a newer theorem proving approach. Using interactive theorem proving allows us to generalize the kinds of properties that we can prove. This work is joint with Abdorrahim Bahrami and Elisabetta De Maria.

Cite as

Amy Felty. Modelling and Verifying Properties of Biological Neural Networks (Invited Talk). In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{felty:LIPIcs.ITP.2022.1,
  author =	{Felty, Amy},
  title =	{{Modelling and Verifying Properties of Biological Neural Networks}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{1:1--1:2},
  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.2022.1},
  URN =		{urn:nbn:de:0030-drops-167100},
  doi =		{10.4230/LIPIcs.ITP.2022.1},
  annote =	{Keywords: Neuronal networks, Model checking, Theorem proving, Coq}
}
Document
Invited Talk
User Interface Design in the HolPy Theorem Prover (Invited Talk)

Authors: Bohua Zhan

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


Abstract
HolPy is a new interactive theorem prover implemented in Python. It is designed to achieve a small trusted-code-base with externally checkable proofs, writing proof automation using a Python API, and permit a wide variety of user interfaces for different application scenarios. In this talk, I will focus on the design of user interfaces in HolPy. While most interactive theorem provers today use text-based user interfaces, there have been several existing work aiming to build point-and-click interfaces where the user perform actions by clicking on part of the goal or choosing from a menu. In our work, we incorporate into the design extensive proof automation and heuristic suggestion mechanisms, allowing construction of proofs on a large scale using this method. We demonstrate the approach in two common scenarios: general-purpose theorem proving and symbolic computation in mathematics.

Cite as

Bohua Zhan. User Interface Design in the HolPy Theorem Prover (Invited Talk). In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{zhan:LIPIcs.ITP.2022.2,
  author =	{Zhan, Bohua},
  title =	{{User Interface Design in the HolPy Theorem Prover}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{2:1--2:1},
  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.2022.2},
  URN =		{urn:nbn:de:0030-drops-167117},
  doi =		{10.4230/LIPIcs.ITP.2022.2},
  annote =	{Keywords: Proof assistants, User interface, Proof automation}
}
Document
Candle: A Verified Implementation of HOL Light

Authors: Oskar Abrahamsson, Magnus O. Myreen, Ramana Kumar, and Thomas Sewell

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


Abstract
This paper presents a fully verified interactive theorem prover for higher-order logic, more specifically: a fully verified clone of HOL Light. Our verification proof of this new system results in an end-to-end correctness theorem that guarantees the soundness of the entire system down to the machine code that executes at runtime. Our theorem states that every exported fact produced by this machine-code program is valid in higher-order logic. Our implementation consists of a read-eval-print loop (REPL) that executes the CakeML compiler internally. Throughout this work, we have strived to make the REPL of the new system provide a user experience as close to HOL Light’s as possible. To this end, we have, e.g., made the new system parse the same variant of OCaml syntax as HOL Light. All of the work described in this paper has been carried out in the HOL4 theorem prover.

Cite as

Oskar Abrahamsson, Magnus O. Myreen, Ramana Kumar, and Thomas Sewell. Candle: A Verified Implementation of HOL Light. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{abrahamsson_et_al:LIPIcs.ITP.2022.3,
  author =	{Abrahamsson, Oskar and Myreen, Magnus O. and Kumar, Ramana and Sewell, Thomas},
  title =	{{Candle: A Verified Implementation of HOL Light}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{3:1--3:17},
  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.2022.3},
  URN =		{urn:nbn:de:0030-drops-167126},
  doi =		{10.4230/LIPIcs.ITP.2022.3},
  annote =	{Keywords: Prover soundness, Higher-order logic, Interactive theorem proving}
}
Document
Use and Abuse of Instance Parameters in the Lean Mathematical Library

Authors: Anne Baanen

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


Abstract
The Lean mathematical library mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean’s mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the current Lean 3 version of mathlib, focussing on complications arising at scale and how the mathlib community deals with them.

Cite as

Anne Baanen. Use and Abuse of Instance Parameters in the Lean Mathematical Library. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{baanen:LIPIcs.ITP.2022.4,
  author =	{Baanen, Anne},
  title =	{{Use and Abuse of Instance Parameters in the Lean Mathematical Library}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{4:1--4:20},
  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.2022.4},
  URN =		{urn:nbn:de:0030-drops-167131},
  doi =		{10.4230/LIPIcs.ITP.2022.4},
  annote =	{Keywords: formalization of mathematics, dependent type theory, typeclasses, algebraic hierarchy, Lean prover}
}
Document
A Complete, Mechanically-Verified Proof of the Banach-Tarski Theorem in ACL2(R)

Authors: Jagadish Bapanapally and Ruben Gamboa

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


Abstract
This paper presents a formal proof of the Banach-Tarski theorem in ACL2(r). The Banach-Tarski theorem states that a unit ball can be partitioned into a finite number of pieces that can be rotated to form two identical copies of the ball. We have formalized 3D rotations and generated a free group of 3D rotations of rank 2. In prior work, the non-denumerability of the reals was proved in ACL2 (r), and a version of the Axiom of Choice that can consistently select a representative element from an equivalence class was introduced in ACL2 version 3.1. Using the free group of rotations, and this prior work, we show that the unit sphere can be decomposed into two sets, each equivalent to the original sphere. Then we show that the unit ball except for the origin can be decomposed into two sets each equivalent to the original ball by mapping the points of the unit ball to the points on the sphere. Finally, we handle the origin by rotating the unit ball around an axis such that the origin falls inside the sphere. Seemingly paradoxically, the construction results in two copies of the unit ball.

Cite as

Jagadish Bapanapally and Ruben Gamboa. A Complete, Mechanically-Verified Proof of the Banach-Tarski Theorem in ACL2(R). In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bapanapally_et_al:LIPIcs.ITP.2022.5,
  author =	{Bapanapally, Jagadish and Gamboa, Ruben},
  title =	{{A Complete, Mechanically-Verified Proof of the Banach-Tarski Theorem in ACL2(R)}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{5:1--5:15},
  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.2022.5},
  URN =		{urn:nbn:de:0030-drops-167142},
  doi =		{10.4230/LIPIcs.ITP.2022.5},
  annote =	{Keywords: ACL2(r), Banach-Tarski, Rotations}
}
Document
Dandelion: Certified Approximations of Elementary Functions

Authors: Heiko Becker, Mohit Tekriwal, Eva Darulova, Anastasia Volkova, and Jean-Baptiste Jeannin

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


Abstract
Elementary function operations such as sin and exp cannot in general be computed exactly on today’s digital computers, and thus have to be approximated. The standard approximations in library functions typically provide only a limited set of precisions, and are too inefficient for many applications. Polynomial approximations that are customized to a limited input domain and output accuracy can provide superior performance. In fact, the Remez algorithm computes the best possible approximation for a given polynomial degree, but has so far not been formally verified. This paper presents Dandelion, an automated certificate checker for polynomial approximations of elementary functions computed with Remez-like algorithms that is fully verified in the HOL4 theorem prover. Dandelion checks whether the difference between a polynomial approximation and its target reference elementary function remains below a given error bound for all inputs in a given constraint. By extracting a verified binary with the CakeML compiler, Dandelion can validate certificates within a reasonable time, fully automating previous manually verified approximations.

Cite as

Heiko Becker, Mohit Tekriwal, Eva Darulova, Anastasia Volkova, and Jean-Baptiste Jeannin. Dandelion: Certified Approximations of Elementary Functions. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{becker_et_al:LIPIcs.ITP.2022.6,
  author =	{Becker, Heiko and Tekriwal, Mohit and Darulova, Eva and Volkova, Anastasia and Jeannin, Jean-Baptiste},
  title =	{{Dandelion: Certified Approximations of Elementary Functions}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{6:1--6:19},
  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.2022.6},
  URN =		{urn:nbn:de:0030-drops-167155},
  doi =		{10.4230/LIPIcs.ITP.2022.6},
  annote =	{Keywords: elementary functions, approximation, certificate checking}
}
Document
The Zoo of Lambda-Calculus Reduction Strategies, And Coq

Authors: Małgorzata Biernacka, Witold Charatonik, and Tomasz Drab

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


Abstract
We present a generic framework for the specification and reasoning about reduction strategies in the lambda calculus, representable as sets of term decompositions. It is provided as a Coq formalization that features a novel format of phased strategies. It facilitates concise description and algebraic reasoning about properties of reduction strategies. The formalization accommodates many well-known strategies, both weak and strong, such as call by name, call by value, head reduction, normal order, full β-reduction, etc. We illustrate the use of the framework as a tool to inspect and categorize the "zoo" of existing strategies, as well as to discover and study new ones with particular properties.

Cite as

Małgorzata Biernacka, Witold Charatonik, and Tomasz Drab. The Zoo of Lambda-Calculus Reduction Strategies, And Coq. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{biernacka_et_al:LIPIcs.ITP.2022.7,
  author =	{Biernacka, Ma{\l}gorzata and Charatonik, Witold and Drab, Tomasz},
  title =	{{The Zoo of Lambda-Calculus Reduction Strategies, And Coq}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{7:1--7:19},
  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.2022.7},
  URN =		{urn:nbn:de:0030-drops-167165},
  doi =		{10.4230/LIPIcs.ITP.2022.7},
  annote =	{Keywords: Lambda calculus, Reduction strategies, Coq}
}
Document
Seventeen Provers Under the Hammer

Authors: Martin Desharnais, Petar Vukmirović, Jasmin Blanchette, and Makarius Wenzel

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


Abstract
One of the main success stories of automatic theorem provers has been their integration into proof assistants. Such integrations, or "hammers," increase proof automation and hence user productivity. In this paper, we use Isabelle/HOL’s Sledgehammer tool to find out how useful modern provers are at proving formulas in higher-order logic. Our evaluation follows in the steps of Böhme and Nipkow’s Judgment Day study from 2010, but instead of three provers we use 17, including SMT solvers and higher-order provers. Our work offers an alternative yardstick for comparing modern provers, next to the benchmarks and competitions emerging from the TPTP World and SMT-LIB.

Cite as

Martin Desharnais, Petar Vukmirović, Jasmin Blanchette, and Makarius Wenzel. Seventeen Provers Under the Hammer. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{desharnais_et_al:LIPIcs.ITP.2022.8,
  author =	{Desharnais, Martin and Vukmirovi\'{c}, Petar and Blanchette, Jasmin and Wenzel, Makarius},
  title =	{{Seventeen Provers Under the Hammer}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{8:1--8:18},
  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.2022.8},
  URN =		{urn:nbn:de:0030-drops-167178},
  doi =		{10.4230/LIPIcs.ITP.2022.8},
  annote =	{Keywords: Automatic theorem proving, interactive theorem proving, proof assistants}
}
Document
Formalising Szemerédi’s Regularity Lemma in Lean

Authors: Yaël Dillies and Bhavik Mehta

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


Abstract
Szemerédi’s Regularity Lemma is a fundamental result in graph theory with extensive applications to combinatorics and number theory. In essence, it says that all graphs can be approximated by well-behaved unions of random bipartite graphs. We present a formalisation in the Lean theorem prover of a strong version of this lemma in which each part of the union must be approximately the same size. This stronger version has not been formalised previously in any theorem prover. Our proof closely follows the pen-and-paper method, allowing our formalisation to provide an explicit upper bound on the number of parts. An application of this lemma is also formalised, namely Roth’s theorem on arithmetic progressions in qualitative form via the triangle removal lemma.

Cite as

Yaël Dillies and Bhavik Mehta. Formalising Szemerédi’s Regularity Lemma in Lean. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{dillies_et_al:LIPIcs.ITP.2022.9,
  author =	{Dillies, Ya\"{e}l and Mehta, Bhavik},
  title =	{{Formalising Szemer\'{e}di’s Regularity Lemma in Lean}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{9:1--9:19},
  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.2022.9},
  URN =		{urn:nbn:de:0030-drops-167185},
  doi =		{10.4230/LIPIcs.ITP.2022.9},
  annote =	{Keywords: Lean, formalisation, formal proof, graph theory, combinatorics, additive combinatorics, Szemer\'{e}di’s Regularity Lemma, Roth’s Theorem}
}
Document
Formalized functional analysis with semilinear maps

Authors: Frédéric Dupuis, Robert Y. Lewis, and Heather Macbeth

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


Abstract
Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear and conjugate-linear maps. We implement this generalization in Lean’s mathlib library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fréchet-Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces. We also show that semilinear maps have applications beyond functional analysis by formalizing the one-dimensional case of a theorem of Dieudonné and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic.

Cite as

Frédéric Dupuis, Robert Y. Lewis, and Heather Macbeth. Formalized functional analysis with semilinear maps. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Copy BibTex To Clipboard

@InProceedings{dupuis_et_al:LIPIcs.ITP.2022.10,
  author =	{Dupuis, Fr\'{e}d\'{e}ric and Lewis, Robert Y. and Macbeth, Heather},
  title =	{{Formalized functional analysis with semilinear maps}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{10:1--10:19},
  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.2022.10},
  URN =		{urn:nbn:de:0030-drops-167191},
  doi =		{10.4230/LIPIcs.ITP.2022.10},
  annote =	{Keywords: Functional analysis, Lean, linear algebra, semilinear, Hilbert space}
}
Document
Formalising Fisher’s Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics

Authors: Chelsea Edmonds and Lawrence C. Paulson

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


Abstract
The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents formal linear algebraic techniques for proofs on incidence structures in Isabelle/HOL, and their application to the first formalisation of Fisher’s inequality. In addition to formalising incidence matrices and simple techniques for reasoning on linear algebraic representations, the formalisation focuses on the linear algebra bound and rank arguments. These techniques can easily be adapted for future formalisations in combinatorics, as we demonstrate through further application to proofs of variations on Fisher’s inequality.

Cite as

Chelsea Edmonds and Lawrence C. Paulson. Formalising Fisher’s Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Copy BibTex To Clipboard

@InProceedings{edmonds_et_al:LIPIcs.ITP.2022.11,
  author =	{Edmonds, Chelsea and Paulson, Lawrence C.},
  title =	{{Formalising Fisher’s Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{11:1--11:19},
  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.2022.11},
  URN =		{urn:nbn:de:0030-drops-167204},
  doi =		{10.4230/LIPIcs.ITP.2022.11},
  annote =	{Keywords: Isabelle/HOL, Mathematical Formalisation, Fisher’s Inequality, Linear Algebra, Formal Proof Techniques, Combinatorics}
}
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