Lean proof assistant

Extracted Attributes

• Paradigm

  Strict purely functional, dependently typed

• Co-designer

  Soonho Kong

• Official Website

  lean-lang.org

• Primary Designer

  Leonardo de Moura

• Software License

  Apache 2.0

• Foundational Theory

  Calculus of Inductive Constructions

• Initial Release Year

  2013

• Original Institution

  Microsoft Research

• Primary Applications

  Formal mathematics, software verification, and AI-assisted proof generation

• Implementation Language

  Lean, C++

• Supporting Organization

  Lean Focused Research Organization (Lean FRO)

Timeline

• Leonardo de Moura and Soonho Kong initiate the Lean project at Microsoft Research. (Source: https://leodemoura.github.io/files/CAV2024.pdf)

  2013-01-01

• Mathematicians complete the full formalization of Peter Scholze's Liquid Tensor Experiment challenge in Lean. (Source: https://leodemoura.github.io/files/CAV2024.pdf)

  2022-07-01

• Leonardo de Moura presents 'Lean 4: Bridging Formal Mathematics and Software Verification' at the 36th International Conference on Computer Aided Verification (CAV 2024). (Source: https://leodemoura.github.io/files/CAV2024.pdf)

  2024-07-25

Lean (proof assistant)

Lean is a proof assistant and a functional programming language. It is based on the calculus of constructions with inductive types (specifically, the Calculus of Inductive Constructions), the foundational type theory developed with the Coq theorem prover, which was renamed to Rocq in 2024. It is a free and open-source software project hosted on GitHub. Development is currently supported by the nonprofit Lean Focused Research Organization (FRO).

Web Search Results

• Lean (proof assistant) - Wikipedia

  Lean is a proof assistant and a functional programming language. It is based on the calculus of constructions with inductive types (specifically, the Calculus of Inductive Constructions), the foundational type theory developed with the Coq theorem prover, which was renamed to Rocq in 2024. It is a free and open-source software project hosted on GitHub. Development is currently supported by the nonprofit Lean Focused Research Organization (FRO). ## History [edit&action=edit&section=1 "Edit section: History")] Lean was developed primarily by Brazilian computer scientist Leonardo de Moura while employed by Microsoft Research and now Amazon Web Services and has had significant contributions from other coauthors and collaborators during its history. [...] Proof assistant and programming language | Lean | | Paradigm | Strict purely functional, dependently typed | | Family | Proof assistant | | Designed by | Leonardo de Moura | | Developer | Lean FRO | | First appeared | 2013; 13 years ago (2013) | | Stable release | 4.34.1 Edit this on Wikidata / 24 September 2026; 2 days ago (24 September 2026) | | | | Typing discipline | static, strong, inferred | | Implementation language | Lean, C++ | | Platform | x86-64, AArch64 | | OS | Cross-platform: Linux, macOS, Windows | | License | Apache 2.0 | | Website | lean-lang.org | | Influenced by | | ML "ML (programming language)"), Rocq (formerly named Coq), Haskell | [...] Jump to content Wikipedia The Free Encyclopedia Search ## Contents 1 History 2 Overview + 2.1 Libraries + 2.2 Editor integration 3 Examples (Lean 4)) 4 Usage + 4.1 Mathematics - 4.1.1 Noteworthy formalizations + 4.2 Physics - 4.2.1 Noteworthy formalizations + 4.3 Artificial intelligence 5 See also 6 References 7 External links # Lean (proof assistant) Čeština "Lean (programovací jazyk) – Czech") Deutsch "Lean (Beweisassistent) – German") Español فارسی "لین (دستیار اثبات) – Persian") Français "Lean (assistant de preuve) – French") 日本語 "Lean (証明アシスタント) – Japanese") 한국어 Nederlands "Lean (bewijsassistent) – Dutch") Português "Lean (linguagem de programação) – Portuguese") Русский Українська Tiếng Việt "Lean (trợ lý chứng minh) – Vietnamese") 文言 中文

• A Short Introduction to Formalising Mathematics in Lean

  • What is a proof assistant? A proof assistant is a programming language together with a corre-sponding interpreter/compiler that allow us to formalise mathematical objects and facts; this includes definitions, theorems, proofs, and ex-amples. The main task of a proof assistant is not to find proofs, but to check proofs for correctness. Proof assistants can thus provide certifi-cates for correctness (based on the assumption that the proof assistant in itself is correct). • Why do we need proof assistants? [...] ProofLab: A Short Introduction to Formalising Mathematics in Lean Seminar on Simplicial Topology Wintersemester 2021/22 Clara L¨ oh Version of October 27, 2021 clara.loeh@mathematik.uni-regensburg.de Fakult¨ at f¨ ur Mathematik, Universit¨ at Regensburg, 93040 Regensburg Contents 1 The Lean proof assistant 1 1.1 Proof assistants 2 1.2 Foundations 3 1.3 Proofs 4 1.4 First examples 5 1.4.1 Pen-and-paper 6 1.4.2 Lean 8 1.E Exercises 17 2 Further examples in Lean 19 2.1 Example: Induction 20 2.1.1 Pen-and-Paper 20 2.1.2 Lean 21 2.2 Example: Sums 23 2.3 Example: Commutators 24 2.3.1 Pen-and-Paper 25 2.3.2 Lean 26 2.E Exercises 28 A Source code A.1 Bibliography C.1 Index C.3 1 The Lean proof assistant Proof assistants allow us to formalise mathematical statements and to verify formalised [...] • Why Lean? The proof assistant Lean is an active and dynamically developing project; the Lean community already created many mathematical li-braries and Lean is used in several ambitious formalisation projects in Mathematics. Moreover, Lean offers a convenient web interface that allows to experiment with Lean without a proper installation. 1.2 Foundations The formalisation of Mathematics consists of the following components: • a universe of objects, • a language of logic, • a concept of proof, • and usually a meta language (in which all of this is formulated). The fact that these different levels interact with each other in various ways makes it challenging to give a complete and rigorous treatment of foundations of Mathematics.

• [PDF] Lean 4: Bridging Formal Mathematics and Software Verification

  of crucial intermediary lemma, with only minor corrections July 2022: Completion of the full challenge in Lean “The Lean Proof Assistant was really that: an assistant in navigating through the thick jungle that this proof is. Really, one key problem I had when I was trying to find this proof was that I was essentially unable to keep all the objects in my RAM, and I think the same problem occurs when trying to read the proof.” Peter Scholze 18 Abstract Formalities Johan Commelin's talk: Abstraction boundaries in Mathematics. [...] Conclusion ● Lean is an efficient programming language and proof assistant ● Machine checkable proofs eliminate the trust bottleneck ● Lean enables decentralized collaboration ● Lean is very extensible: ○ Users extend Lean using Lean itself without fear of introducing unsoundness ● The Mathlib community is changing how math is done ● Lean proofs are maintainable, stable, and transparent ● The FRO model has been instrumental in supporting Lean It is not just about proving but also understanding complex objects and proofs, getting new insights, and navigating through the “thick jungles” that are beyond our cognitive abilities. [...] Lean 4: Bridging Formal Mathematics and Software Verification 36th International Conference on Computer Aided Verification July 25th 2024 Leo de Moura Senior Principal Applied Scientist – AWS Chief Architect – Lean FRO What is ? An efficient pure functional programming language and a proof assistant. Lean is implemented in Lean, and is very extensible. Main applications: ● Formal Mathematics ● Software/Hardware verification ● AI for Mathematics and code synthesis ● Education Small trusted kernel, and many external type/proof checkers. Soonho Kong and I started the project at Microsoft Research in 2013. What is ? Lean is based on dependent type theory What is ? Lean is based on dependent type theory is and IDE for formal methods Lean is a development environment for formal methods.

• Lean Programming Language

  The Proof in the Code by Kevin Hartnett FROM Quanta Books | available now Lean is an open-source programming language and proof assistant that enables correct, maintainable, and formally verified code InstallLearn Powerful automation Mathematics `-- 'Grind' efficiently manages complex pattern matching and -- case analysis beyond standard tactics. example (x : Nat) : 0 < match x with | 0 => 1 | n + 1 => x + n := by grind -- Automatically solves systems of linear inequalities. example (x y : Int) : 27 ≤ 11 x + 13 y → 11 x + 13 y ≤ 45 → - 10 ≤ 7 x - 9 y → 7 x - 9 y > 4 := by grind` [...] ## Tudor Achim Co-Founder and CEO, Harmonic " > Lean is the core verification technology behind Cedar, the open-source authorization language that powers cloud services like Amazon Verified Permissions and AWS Verified Access. Our team rigorously formalizes and verifies core components of Cedar using Lean’s proof assistant, and we leverage Lean’s lightning-fast runtime to continuously test our production Rust code against the Lean formalization. Lean’s efficiency, extensive libraries, and vibrant community enable us to develop and maintain Cedar at scale, while ensuring the key correctness and security properties that our users depend on. ## Emina Torlak Senior Principal Applied Scientist, AWS SUPPORTERS # Sponsors and Partners [...] MATHEMATICS ## Fermat's Last Theorem The Fermat's Last Theorem project is formalizing one of mathematics' most complex proofs in Lean, demonstrating the proof assistant's capacity to handle frontier research mathematics. This ambitious work is creating formalized definitions for advanced number-theoretic objects, enabling mathematicians to build machine-verified proofs of cutting-edge results while establishing Lean as a powerful platform for mathematical collaboration. Read More Pierre de Fermat painting by Rolland Lefebvre PROGRESS # Growth and Impact Tracking Lean's reach across math, research, and industry AUGUST 2026

• How does Lean work

  Interactive Proof Assistants allow you to define a set of objects and then prove statements about the objects using logic. This could be proving that hardware or software meets a specification, or proving mathematical theorems. The allure here is that the system guarantees soundness for us: if our theorem gets that blue checkmark, then we know it is true (as it's written at least, Lean can't preclude incorrectly defined goal statements). Automated theorem proving tools help with finding these proofs. # Lean at 10,000 feet Here's my understanding of how the different bits of the lean language map to formal verification: