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Documents authored by Paulson, Lawrence C.

Document
Invited Talk
DOI: 10.4230/LIPIcs.ITP.2026.1
Nitro Isolation Engine: Formally Verifying a Production Hypervisor (Invited Talk)

Authors: Hanno Becker, Nathan Chong, Robert Dockins, Jim Grundy, Jason Hu, Ike Mulder, Dominic P. Mulligan, Paul Mure, Bryan Parno, Lawrence C. Paulson, and Konrad Slind

Published in: LIPIcs, Volume 382, 17th International Conference on Interactive Theorem Proving (ITP 2026)

Abstract
Cloud computing relies on hypervisors to enforce isolation between co-tenanted virtual machines. Hypervisors are therefore critical security infrastructure, and assurance of their correctness is paramount. Traditional engineering techniques - code review, testing, fuzzing - provide strong assurance but cannot exhaustively verify that isolation holds across all possible execution paths. Formal verification extends and complements these approaches by establishing mathematical guarantees about system behaviour. This talk presents our experience applying interactive theorem proving to verify a production hypervisor component: the Nitro Isolation Engine. This is a trusted, minimalist computing base written in Rust, enforcing isolation between virtual machines on AWS Graviton5 EC2 instances. Designed for verification from inception, we have specified the intended behaviour of this component and verified correctness in the Isabelle/HOL interactive theorem prover, producing approximately 330,000 lines of machine-checked models and proofs, and establishing three key classes of property: 1) Functional correctness: The system behaves as specified for all operations including virtual machine creation, memory mapping, and abort handling. Our total verification approach additionally establishes memory-safety, termination, and absence of runtime errors. 2) Confidentiality: A noninterference-style property demonstrates that guest virtual machine state remains hidden from an expansive definition of observer monitoring system actions, formalised as indistinguishability preservation up to permitted declassification flows. 3) Integrity: Guest virtual machine private state is unaffected by operations on distinct virtual machines. Currently, our proof coverage extends to verification of the core virtual machine-management hypercalls, guest power management, various utility hypercalls, and a subset of data, instruction, and asynchronous abort handling, and will continue to expand to cover more functionality including PCI device management and virtual GIC (Generic Interrupt Controller) handling. The talk will discuss the verification approach, key proof techniques, and challenges in applying formal methods to production systems. Note that this work builds on decades of academic research across interactive theorem proving, formal specification, separation logic and its automation, and programming language semantics.
Cite as

Hanno Becker, Nathan Chong, Robert Dockins, Jim Grundy, Jason Hu, Ike Mulder, Dominic P. Mulligan, Paul Mure, Bryan Parno, Lawrence C. Paulson, and Konrad Slind. Nitro Isolation Engine: Formally Verifying a Production Hypervisor (Invited Talk). In 17th International Conference on Interactive Theorem Proving (ITP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 382, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{becker_et_al:LIPIcs.ITP.2026.1,
  author =	{Becker, Hanno and Chong, Nathan and Dockins, Robert and Grundy, Jim and Hu, Jason and Mulder, Ike and Mulligan, Dominic P. and Mure, Paul and Parno, Bryan and Paulson, Lawrence C. and Slind, Konrad},
  title =	{{Nitro Isolation Engine: Formally Verifying a Production Hypervisor}},
  booktitle =	{17th International Conference on Interactive Theorem Proving (ITP 2026)},
  pages =	{1:1--1:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-436-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{382},
  editor =	{Komendantskaya, Ekaterina and Nipkow, Tobias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2026.1},
  URN =		{urn:nbn:de:0030-drops-269757},
  doi =		{10.4230/LIPIcs.ITP.2026.1},
  annote =	{Keywords: Isabelle/HOL, Rust, verification, separation logic, hypervisors}
}
Document
DOI: 10.4230/LIPIcs.ITP.2026.23
From Weierstraß to Dedekind via Jacobi: Formalising Foundations of Modular Forms

Authors: Manuel Eberl, Wenda Li, and Lawrence C. Paulson

Published in: LIPIcs, Volume 382, 17th International Conference on Interactive Theorem Proving (ITP 2026)

Abstract
We present an Isabelle/HOL formalisation of the foundations of analytic number theory related to modular forms. We begin by refactoring and extending the existing library on elliptic functions, adding the theorem that every elliptic function can be written in terms of the Weierstraß elliptic function ℘ and the addition theorem for ℘, which links complex lattices to elliptic curves. Next, we develop an extensive library on the Jacobi theta functions, including well-known results such as the Jacobi triple product, the Pentagonal Number Theorem, and the Rogers-Ramanujan identities. Finally, we apply this library to the study of the Dedekind η function and "forbidden" Eisenstein series G₂. In all of this, we aim for short and clean proofs, building a library of reusable lemmas.
Cite as

Manuel Eberl, Wenda Li, and Lawrence C. Paulson. From Weierstraß to Dedekind via Jacobi: Formalising Foundations of Modular Forms. In 17th International Conference on Interactive Theorem Proving (ITP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 382, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{eberl_et_al:LIPIcs.ITP.2026.23,
  author =	{Eberl, Manuel and Li, Wenda and Paulson, Lawrence C.},
  title =	{{From Weierstra{\ss} to Dedekind via Jacobi: Formalising Foundations of Modular Forms}},
  booktitle =	{17th International Conference on Interactive Theorem Proving (ITP 2026)},
  pages =	{23:1--23:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-436-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{382},
  editor =	{Komendantskaya, Ekaterina and Nipkow, Tobias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2026.23},
  URN =		{urn:nbn:de:0030-drops-269976},
  doi =		{10.4230/LIPIcs.ITP.2026.23},
  annote =	{Keywords: Isabelle/HOL, elliptic functions, Eisenstein series, modular forms, theta functions, number theory, complex analysis, formalisation of mathematics}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.18
Formalising New Mathematics in Isabelle: Diagonal Ramsey

Authors: Lawrence C. Paulson

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
The formalisation of mathematics is becoming routine, but its value to research mathematicians remains unproven. There are few examples of using proof assistants to verify new work. This paper reports the formalisation - inspired by a Lean one by Bhavik Mehta - of a major new result [Marcelo Campos et al., 2023] about Ramsey numbers. One unexpected finding was a heavy role for computer algebra techniques.
Cite as

Lawrence C. Paulson. Formalising New Mathematics in Isabelle: Diagonal Ramsey. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{paulson:LIPIcs.ITP.2025.18,
  author =	{Paulson, Lawrence C.},
  title =	{{Formalising New Mathematics in Isabelle: Diagonal Ramsey}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.18},
  URN =		{urn:nbn:de:0030-drops-246163},
  doi =		{10.4230/LIPIcs.ITP.2025.18},
  annote =	{Keywords: Isabelle, formalisation of mathematics, Ramsey’s theorem, computer algebra}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.ITP.2024.40
Formalising Half of a Graduate Textbook on Number Theory (Short Paper)

Authors: Manuel Eberl, Anthony Bordg, Lawrence C. Paulson, and Wenda Li

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

Abstract
Apostol’s Modular Functions and Dirichlet Series in Number Theory [Tom M. Apostol, 1990] is a graduate text covering topics such as elliptic functions, modular functions, approximation theorems and general Dirichlet series. It relies on complex analysis, winding numbers, the Riemann ζ function and Laurent series. We have formalised several chapters and can comment on the sort of gaps found in pedagogical mathematics. Proofs are available from https://github.com/Wenda302/Number_Theory_ITP2024.
Cite as

Manuel Eberl, Anthony Bordg, Lawrence C. Paulson, and Wenda Li. Formalising Half of a Graduate Textbook on Number Theory (Short Paper). In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 40:1-40:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{eberl_et_al:LIPIcs.ITP.2024.40,
  author =	{Eberl, Manuel and Bordg, Anthony and Paulson, Lawrence C. and Li, Wenda},
  title =	{{Formalising Half of a Graduate Textbook on Number Theory}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{40:1--40:7},
  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.2024.40},
  URN =		{urn:nbn:de:0030-drops-207686},
  doi =		{10.4230/LIPIcs.ITP.2024.40},
  annote =	{Keywords: Isabelle/HOL, number theory, complex analysis, formalisation of mathematics}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.11
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.
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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)

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@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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