Lean 4 Explained: The Theorem Prover Powering AI Math

Lean 4 is a pure functional programming language and proof assistant, originating from Microsoft Research and now developed primarily by the non-profit Lean FRO. Its defining feature is an expressive type system based on dependent types, which lets programmers "prove" that an algorithm returns what it was intended to return. If you write a proof and it compiles, you can be confident the proof is correct — that is the core mechanism. Lean also offers assistance features such as partial automation of proofs and automatic detection of overly strong assumptions. Thanks to the community-maintained mathlib4 library, a substantial portion of undergraduate-level mathematics has already been formalized in Lean, and the browser-based "Lean 4 Web" lets anyone try it without installing anything. Since early 2026, Lean 4 has rapidly become the underlying infrastructure for AI-driven mathematical research.

Large language models can produce plausible-sounding but factually wrong answers — hallucinations. In mathematical proof, where a single logical gap invalidates everything, that weakness is fatal. This is where Lean 4 comes in: it acts as a mechanical referee that checks whether an AI-generated proof candidate actually holds. If a Lean proof compiles, its correctness is mathematically guaranteed. In other words, AI can generate large numbers of proof candidates while Lean rigorously verifies them — a clear division of labor. Combining the AI's exploratory power with Lean's rigorous verification is what has made previously intractable proofs newly reachable, and that shift is the essence of what has been happening throughout 2026.

Since the start of 2026, a series of results combining AI with Lean have been reported. Here is a summary of the major ones.

AlphaProof traces back to an AlphaZero-style reinforcement learning agent that achieved silver-medal-equivalent performance at the 2024 IMO. AlphaProof Nexus, its successor, is notable for extending that lineage into the uncharted territory of open Erdős problems.

The easiest way to experience Lean 4 firsthand is to open Lean 4 Web in a browser — no installation required. You can write simple function definitions or proofs and see the results immediately. Lean's type system is based on dependent types, meaning the type itself can depend on a value. For example, you can express "a list of exactly three elements," embedding length information directly into the type, and the compiler statically checks that constraint. Applied to mathematics, a proposition itself is expressed as a type, and if you can actually construct a value of that type — a proof term — the code compiles and the proposition's truth is guaranteed.

- Lean 4 Web — the official browser-based environment, no setup required
- mathlib4 — a community-maintained library formalizing much of undergraduate mathematics
- lean-ja — a Japanese-language community site for learning Lean (see below)

Most Lean 4 resources are in English, but lean-ja is a community site offering Japanese-language learning material, covering everything from Lean's basic concepts to practical usage explained in Japanese. It is a useful entry point for anyone hesitant about English-only documentation.

Lean is a specialized tool for mathematical proof, but the underlying idea — AI generates at scale, and a machine rigorously verifies — is not confined to mathematics. How to verify AI-generated code safely is a growing concern in everyday software development too. The approach described in OpenAI Codex Security's verification phase, where AI output is mechanically checked, shares the same underlying logic as Lean. Rather than trusting AI generation unconditionally, designing a solid verification layer is likely to become a common theme across AI adoption going forward.

Lean 4 is a proof assistant built on dependent types, where the compiler mechanically guarantees the correctness of a proof. Since 2026 began, a wave of results combining AI and Lean has emerged: AlphaProof Nexus's work on open Erdős problems, Gauss's formalization of the strong prime number theorem, and gold-medal-equivalent results at IMO 2025. The pattern of AI generating proof candidates at scale while Lean rigorously verifies them extends beyond mathematics into the broader question of how to verify AI-generated output. For those who want to learn in Japanese, lean-ja is a good place to start.