Formal proof systems like Lean provide a unique training ground for LLMs. Unlike natural language reasoning, a proof's correctness can be programmatically verified. This creates a strong reward signal for training long-horizon planning and coherence, skills that can generalize to other tasks.

Languages like Lean allow mathematical proofs to be automatically verified. This provides a perfect, binary reward signal (correct/incorrect) for a reinforcement learning agent. It transforms the abstract art of mathematics into a well-defined environment, much like a game of Go, that an AI can be trained to master.

A team successfully formalized and simplified a complex proof from Fields Medalist Peter Scholze using Lean. The tool's interactive feedback guided them, allowing them to refactor and improve the mathematics akin to how a programmer refactors code, even without a complete intuitive grasp of the subject.

The act of creating a formal proof for a piece of software forces a level of rigor that surpasses even implementing it from scratch. This newfound confidence and clarity allows engineers to pursue aggressive optimizations without the fear of introducing subtle bugs, which they would otherwise avoid due to uncertainty.

A project demonstrated an AI successfully translating the zlib C library into the Lean language and proving its core compression/decompression property is correct. This shows AI's capability to handle real-world, complex codebases for formal verification, a task previously deemed infeasible.

A major hurdle for formal methods is the effort required to write proofs. Generative AI is becoming capable of producing proofs in formal languages like Lean, which can then be automatically verified by a machine. This could make verified software development scalable for the first time.

Because Lean is written in Lean, it's highly extensible. Users can write metaprograms in the same file as their proofs to build custom automation, visualizations, or entire domain-specific languages (DSLs). One mathematician even created a point-and-click, English-like proof interface this way without modifying Lean's core.

Instead of writing code and then tests, developers could define precise mathematical properties. AI would then synthesize both the program and a formal proof that the program meets those specifications, ensuring correctness by design and flipping the current code-first paradigm.

Beyond its use in formal mathematics for proof verification, Lean is a fully-featured, Turing-complete functional programming language. This dual nature allows developers to write standard code, like an autograd engine, and mathematical proofs within the same powerful system.