Lean 4

Formal verification tools such as Lean and Coq can be used to verify complete mathematical proofs generated by AI, thereby guaranteeing their correctness in a mathematical sense.

Lean is an interactive theorem prover developed by Microsoft Research, with the latest version currently being Lean 4.

Lean 4 has an active mathematical library, Mathlib, containing more than 100,000 formalized theorems

Lean 4 is based on dependent type theory at its core, and formally proving a theorem is equivalent to writing a type-correct program.

Theorem provers such as Lean, Coq, and Isabelle require every reasoning step to be translated into strict logical notation and automatically checked for compliance.

The key to judging whether news of an AI mathematical breakthrough is true or false lies in whether the chain of evidence is complete and reproducible; one should check three aspects: the original prompt, the complete proof, and the formal verification record.

Lean is an interactive theorem prover driven by dependent type theory, developed by Leonardo de Moura's team at Microsoft Research, with Lean 4 being the currently active version.

Lean是由微软研究院Leonardo de Moura开发的交互式定理证明器和函数式编程语言

At the end of 2023, Terence Tao and collaborators used Lean and LLM assistance to prove an important special case of the polynomial Freiman-Ruzsa conjecture

Lean 4 has an active community and a continuously expanding mathematics library, Mathlib, and has been used by mathematicians to formalize modern mathematical theorems.

All numbered results in the paper, except for two clauses concerning the planner, are machine-verified in Lean 4, conditional on citing two published results.

Lean 4 is also a general-purpose programming language.

Lean 4's kernel is solely responsible for type checking; all tactics, macros, and automation tools run outside the kernel, generating proof terms that are then independently verified by the kernel.

Lean 4 has gained widespread attention through the Mathlib project (a large-scale formalized mathematics library), which has formalized a large number of theorems in fields such as algebraic geometry and number theory.

The correctness of results containing a Lean 4 proof can be independently verified by anyone by running the Lean checker.

StochBench is a Lean 4 benchmark suite specifically targeting the field of stochastic processes.

The Lean 4 language and its mathematics library Mathlib have formalized over 150,000 mathematical theorems and definitions, with more than 400 contributors.

AlphaProof combines a large language model with the Lean 4 formalization system and is trained under a reinforcement learning framework to perform multi-step mathematical reasoning.

The Fermat's Last Theorem proof project was led by Tianyi Peng, an assistant professor at Columbia University; the AI autonomously completed more than 30,000 intermediate theorems, generated approximately 13 million lines of Lean 4 code, a volume five times that of the Mathlib core library.

Lean 4 represents a qualitative leap over Lean 3 in performance, metaprogramming capability, and user experience, and also possesses full general-purpose programming capability