Lean 4
Lean 4 is an open-source functional programming language and interactive theorem prover developed by Microsoft Research, representing the fourth generation of the Lean language. It integrates programming language capabilities with formal mathematical verification, supports a dependent type system, and can be used for constructing mathematical proofs, verifying program correctness, and developing general-purpose software. Lean 4 is widely used in the mathematical formalization community, including the Mathlib project, and features metaprogramming, a macro system, and high-performance compilation.
Core Facts
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
Timeline (last 90 days)
SemantEq judges equivalence by checking proof terms rather than tactic scripts, to avoid bias introduced by differences in tactic style.
SemantEq is a bidirectional equivalence relation for judging the equivalence of formal mathematical statements, aimed at solving the problem that existing logical equivalence tests are too permissive.
Lean 4 and its accompanying mathematics library Mathlib have become the de facto standard for formalized mathematics, and benchmarks such as arXivLean and PutnamBench are built on top of them.
LeanGraphSearch is a graph retrieval method for Lean 4 formal theorem proving that, by constructing a graph from Mathlib's forward dependencies, first performs semantic matching to locate anchor points, then uses a graph ranking algorithm to propagate relevance along dependency edges, and finally re-ranks the candidate set.
The OjasX repository contains 558 machine-verified Lean 4 theorems, 28 Rocq 9.2 proof files, and 840 automated PyTorch tests.
The authors constructed an end-to-end benchmark covering four formal languages, including Dafny contracts, Lean 4, Isabelle, and Rocq theorem statements, totaling 3,234 lines of paired data.
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.
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All Facts (20)
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.
90%VerifiedLean is an interactive theorem prover developed by Microsoft Research, with the latest version currently being Lean 4.
90%VerifiedLean 4 has an active mathematical library, Mathlib, containing more than 100,000 formalized theorems
85%VerifiedLean 4 has an active community and a continuously expanding mathematics library, Mathlib, and has been used by mathematicians to formalize modern mathematical theorems.
75%VerifiedLean 4 is based on dependent type theory at its core, and formally proving a theorem is equivalent to writing a type-correct program.
75%VerifiedTheorem provers such as Lean, Coq, and Isabelle require every reasoning step to be translated into strict logical notation and automatically checked for compliance.
75%VerifiedThe 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.
70%VerifiedLean 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.
65%VerifiedLean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
65%VerifiedAt 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
80%UnverifiedLean拥有名为Mathlib的庞大数学库，涵盖数论、代数、拓扑等众多分支
70%UnverifiedSemantEq judges equivalence by checking proof terms rather than tactic scripts, to avoid bias introduced by differences in tactic style.
50%UnverifiedSemantEq is a bidirectional equivalence relation for judging the equivalence of formal mathematical statements, aimed at solving the problem that existing logical equivalence tests are too permissive.
50%UnverifiedLean 4 and its accompanying mathematics library Mathlib have become the de facto standard for formalized mathematics, and benchmarks such as arXivLean and PutnamBench are built on top of them.
50%UnverifiedLeanGraphSearch is a graph retrieval method for Lean 4 formal theorem proving that, by constructing a graph from Mathlib's forward dependencies, first performs semantic matching to locate anchor points, then uses a graph ranking algorithm to propagate relevance along dependency edges, and finally re-ranks the candidate set.
50%UnverifiedThe OjasX repository contains 558 machine-verified Lean 4 theorems, 28 Rocq 9.2 proof files, and 840 automated PyTorch tests.
50%UnverifiedThe authors constructed an end-to-end benchmark covering four formal languages, including Dafny contracts, Lean 4, Isabelle, and Rocq theorem statements, totaling 3,234 lines of paired data.
50%UnverifiedAll numbered results in the paper, except for two clauses concerning the planner, are machine-verified in Lean 4, conditional on citing two published results.
50%UnverifiedLean 4 is also a general-purpose programming language.
50%UnverifiedLean 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.
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