Formalized Haar Measure

We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library \textsf{mathlib}, and discuss the construction of product measures and the proof of Fubini's theorem for the Bochner integral.Comment: 16 pages (excluding references

A Formalization of Forcing and the Unprovability of the Continuum Hypothesis

We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space 2^{omega_2 x omega} and formally verify the failure of the continuum hypothesis in the resulting model

TP53 Mutations in Serum Circulating Cell-Free Tumor DNA As Longitudinal Biomarker for High-Grade Serous Ovarian Cancer

The aim of this study was to determine an optimal workflow to detect TP53 mutations in baseline and longitudinal serum cell free DNA (cfDNA) from high-grade serous ovarian carcinomas (HGSOC) patients and to define whether TP53 mutations are suitable as biomarker for disease. TP53 was investigated in tissue and archived serum from 20 HGSOC patients by a next-generation sequencing (NGS) workflow alone or combined with digital PCR (dPCR). AmpliSeq™-focused NGS panels and customized dPCR assays were used for tissue DNA and longitudinal cfDNAs, and Oncomine NGS panel with molecular barcoding was used for baseline cfDNAs. TP53 missense mutations were observed in 17 tissue specimens and in baseline cfDNA for 4/8 patients by AmpliSeq, 6/9 patients by Oncomine, and 4/6 patients by dPCR. Mutations in cfDNA were detected in 4/6 patients with residual disease and 3/4 patients with disease progression within six months, compared to 5/11 patients with no residual disease and 6/13 patients with progression after six months. Finally, mutations were detected at progression in 5/6 patients, but not during chemotherapy. NGS with molecular barcoding and dPCR were most optimal workflows to detect TP53 mutations in baseline and longitudinal serum cfDNA, respectively. TP53 mutations were undetectable in cfDNA during treatment but re-appeared at disease progression, illustrating its promise as a biomarker for disease monitoring

Supplementary material for the CPP 2023 paper Formalising the h-Principle and Sphere Eversion

The supplementary material for the CPP 2023 paper Formalising the h-Principle and Sphere Eversion contains the formalisation of the result. It is a frozen version of the repository https://github.com/leanprover-community/sphere-eversion at commit 9cd599b74a419209e4204829efcd50008fdd1c2b Only cpp2023.zip is required to compile the project, by extracting the files and following the instructions in the README. These instructions will download mathlib (https://github.com/leanprover-community/mathlib) at commit cf9386b56953fb40904843af98b7a80757bbe7f9. For convenience, this version of mathlib has been provided as a separate compressed file mathlib.zip. Instead of following the step `leanproject get-mathlib-cache` in the README, one can extract that in the same folder

Maintaining a library of formal mathematics

The Lean mathematical library mathlib is developed by a community of users with very different backgrounds and levels of experience. To lower the barrier of entry for contributors and to lessen the burden of reviewing contributions, we have developed a number of tools for the library which check proof developments for subtle mistakes in the code and generate documentation suited for our varied audience

The Lean Theorem Prover (system description)

<p>Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.</p