Lean

Gold definitionUpdated Apr 2, 2026

Definition

Lean is a formal proof assistant and programming language designed for the mechanical verification and formalization of mathematics. It enables the construction of rigorous, machine-checkable proofs and is increasingly integrated with AI agents for automated theorem proving and large-scale mathematical formalization.

At a glance

Executive summary

Lean is a specialized software system that helps mathematicians and computer scientists write and verify mathematical proofs with absolute certainty using a computer. It's like a highly rigorous spell-checker for math, ensuring every step of a proof is logically sound, and is increasingly used by AI to automate this process for large bodies of work.

TL;DR

Lean is a computer program that lets you write and check mathematical proofs to make sure they are perfectly correct, often with the help of AI.

Key points

Provides a type theory and functional programming language for constructing machine-checkable mathematical proofs.
Ensures absolute logical correctness of mathematical proofs, reducing human error and enabling mechanical verification.
Used by researchers in formal methods, mathematicians, computer scientists, and AI agents for automated theorem proving.
Unlike traditional informal proofs, Lean proofs are formally verified by a computer, eliminating ambiguity and potential human oversight.
Growing integration with AI agents for autonomous theorem proving and large-scale autoformalization of mathematical texts.

Use cases

Formal verification of critical software components, such as cryptographic algorithms or operating system kernels, to ensure their correctness.
Formalization of advanced mathematical theories (e.g., category theory, algebraic geometry) to build a robust, machine-verifiable knowledge base.
Automated theorem proving where AI agents use Lean to autonomously discover and verify proofs for complex mathematical problems.
Autoformalization of entire mathematical textbooks or research papers into a formal, machine-verifiable Lean library for enhanced learning and research.

Also known as

Lean theorem prover, Lean proof assistant, Lean language