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Lean
Paradigm	Functional programming, Imperative programming
Family	Proof assistant
Developer	Lean FRO
First appeared	2013; 11 years ago
Stable release	4.11.0 / 1 September 2024; 3 months ago
Preview release	4.12.0-rc1 / 2 September 2024; 3 months ago
Typing discipline	Static, strong, inferred
Implementation language	Lean, C++
OS	Cross-platform
License	Apache License 2.0
Website	lean-lang.org
Influenced by
	ML; Coq; Haskell

Lean is a proof assistant and a functional programming language.^[1] It is based on the calculus of constructions with inductive types. It is an open-source project hosted on GitHub. It was developed primarily by 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. Development is currently supported by the non-profit Lean Focused Research Organization (FRO).

History

Lean was launched by Leonardo de Moura at Microsoft Research in 2013.^[2] The initial versions of the language, later known as Lean 1 and 2, were experimental and contained features such as support for homotopy type theory – based foundations that were later dropped.

Lean 3 (first released Jan 20, 2017) was the first moderately stable version of Lean. It was implemented primarily in C++ with some features written in Lean itself. After version 3.4.2 Lean 3 was officially end-of-lifed while development of Lean 4 began. In this interim period members of the Lean community developed and released unofficial versions up to 3.51.1.

In 2021, Lean 4 was released, which was a reimplementation of the Lean theorem prover capable of producing C code which is then compiled, enabling the development of efficient domain-specific automation.^[3] Lean 4 also contains a macro system and improved type class synthesis and memory management procedures over the previous version. Another benefit compared to Lean 3 is the ability to avoid touching C++ code in order to modify the frontend and other key parts of the core system, as they are now all implemented in Lean and available to the end user to be overridden as needed.^{[citation needed]}

Lean 4 is not backwards-compatible with Lean 3.^[4]

In 2023, the Lean FRO was formed, with the goals of improving the language's scalability and usability, and implementing proof automation.^[5]

Overview

Libraries

The official lean package includes a standard library batteries, which implements common data structures that may be used for both mathematical research and more conventional software development.^[6]

In 2017, a community-maintained project to develop a Lean library mathlib began, with the goal to digitize as much of pure mathematics as possible in one large cohesive library, up to research level mathematics.^[7]^[8] As of September 2024, mathlib had formalised over 165,000 theorems and 85,000 definitions in Lean.^[9]

Editors integration

Lean integrates with:^[10]

Interfacing is done via a client-extension and Language Server Protocol server.

It has native support for Unicode symbols, which can be typed using LaTeX-like sequences, such as "\times" for "×". Lean can also be compiled to JavaScript and accessed in a web browser and has extensive support for meta-programming.

Examples (Lean 4)

The natural numbers can be defined as an inductive type. This definition is based on the Peano axioms and states that every natural number is either zero or the successor of some other natural number.

inductive Nat : Type
| zero : Nat
| succ : Nat → Nat

Addition of natural numbers can be defined recursively, using pattern matching.

def Nat.add : Nat → Nat → Nat
| n, Nat.zero   => n                      -- n + 0 = n  
| n, Nat.succ m => Nat.succ (Nat.add n m) -- n + succ(m) = succ(n + m)

This is a simple proof of $(P\wedge Q)\implies (Q\wedge P)$ for two propositions $P$ and $Q$ (where $\wedge$ is the conjunction and $\implies$ the implication) in Lean using tactic mode:

theorem and_swap (p q : Prop) : p ∧ q → q ∧ p := by
  intro h            -- assume p ∧ q with proof h, the goal is q ∧ p
  apply And.intro    -- the goal is split into two subgoals, one is q and the other is p
  · exact h.right    -- the first subgoal is exactly the right part of h : p ∧ q
  · exact h.left     -- the second subgoal is exactly the left part of h : p ∧ q

This same proof in term mode:

theorem and_swap (p q : Prop) : p ∧ q → q ∧ p :=
    fun ⟨hp, hq⟩ => ⟨hq, hp⟩

Usage

Mathematics

Lean has received attention from mathematicians such as Thomas Hales,^[11] Kevin Buzzard,^[12] and Heather Macbeth.^[13] Hales is using it for his project, Formal Abstracts.^[14] Buzzard uses it for the Xena project.^[15] One of the Xena Project's goals is to rewrite every theorem and proof in the undergraduate math curriculum of Imperial College London in Lean. Macbeth is using Lean to teach students the fundamentals of mathematical proof with instant feedback.^[16]

In 2021, a team of researchers used Lean to verify the correctness of a proof by Peter Scholze in the area of condensed mathematics. The project garnered attention for formalizing a result at the cutting edge of mathematical research.^[17] In 2023, Terence Tao used Lean to formalize a proof of the Polynomial Freiman-Ruzsa (PFR) conjecture, a result published by Tao and collaborators in the same year.^[18]

Artificial intelligence

In 2022, OpenAI and Meta AI independently created AI models to generate proofs of various high-school-level olympiad problems in Lean.^[19] Meta AI's model is available for public use with the Lean environment.^[20]

In 2023, Vlad Tenev and Tudor Achim co-founded startup Harmonic, which aims to reduce AI hallucinations by generating and checking Lean code.^[21]

In 2024, Google DeepMind created AlphaProof^[22] which proves mathematical statements in Lean at the level of a silver medalist at the International Mathematical Olympiad. This was the first AI system that achieved a medal-worthy performance on a math olympiad's problems.^[23]

References

^ Moura, Leonardo de; Ullrich, Sebastian (2021). "The Lean 4 Theorem Prover and Programming Language". In Platzer, André; Sutcliffe, Geoff (eds.). Automated Deduction – CADE 28. Lecture Notes in Computer Science. Vol. 12699. Cham: Springer International Publishing. pp. 625–635. doi:10.1007/978-3-030-79876-5_37. ISBN 978-3-030-79876-5.
^ "About". Lean Language. Retrieved 2024-03-13.
^ Moura, Leonardo de; Ullrich, Sebastian (2021). Platzer, Andr'e; Sutcliffe, Geoff (eds.). Automated Deduction -- CADE 28. Springer International Publishing. pp. 625–635. doi:10.1007/978-3-030-79876-5_37. ISBN 978-3-030-79876-5. S2CID 235800962. Retrieved 24 March 2023.
^ "Significant changes from Lean 3". Lean Manual. Retrieved 24 March 2023.
^ "Mission". Lean FRO. 2023-07-25. Retrieved 2024-03-14.
^ "batteries". GitHub. Retrieved 2024-09-22.
^ "Building the Mathematical Library of the Future". Quanta Magazine. October 2020. Archived from the original on 2 Oct 2020.
^ "Lean community". leanprover-community.github.io. Retrieved 2023-10-24.
^ "Mathlib statistics". leanprover-community.github.io. Retrieved 2024-09-22.
^ "Installing Lean 4 on Linux". leanprover-community.github.io. Retrieved 2023-10-24.
^ Hales, Thomas (September 18, 2018). "A Review of the Lean Theorem Prover". Jigger Wit. Archived from the original on 21 Nov 2020.
^ Buzzard, Kevin. "The Future of Mathematics?" (PDF). Retrieved 6 October 2020.
^ Macbeth, Heather. "The Mechanics of Proof". hrmacbeth.github.io. Archived from the original on 5 Jun 2024.
^ "Formal Abstracts". Github.
^ "What is the Xena project?". Xena. 8 May 2019.
^ Roberts, Siobhan (July 2, 2023). "A.I. Is Coming for Mathematics, Too". New York Times. Archived from the original on 1 May 2024.
^ Hartnett, Kevin (July 28, 2021). "Proof Assistant Makes Jump to Big-League Math". Quanta Magazine. Archived from the original on 2 Jan 2022.
^ Sloman, Leila (2023-12-06). "'A-Team' of Math Proves a Critical Link Between Addition and Sets". Quanta Magazine. Retrieved 2023-12-07.
^ "Solving (some) formal math olympiad problems". OpenAI. February 2, 2022. Retrieved March 13, 2024.
^ "Teaching AI advanced mathematical reasoning". Meta AI. November 3, 2022. Retrieved March 13, 2024.
^ Metz, Cade (23 September 2024). "Is Math the Path to Chatbots That Don't Make Stuff Up?". New York Times. Archived from the original on 24 Sep 2024.
^ "AI achieves silver-medal standard solving International Mathematical Olympiad problems". Google DeepMind. 2024-05-14. Retrieved 2024-07-25.
^ Roberts, Siobhan (July 25, 2024). "Move Over, Mathematicians, Here Comes AlphaProof". New York Times. Archived from the original on July 29, 2024.