Lean
Lean is an open-source interactive theorem prover and functional programming language developed by Microsoft Research. It supports dependent type theory, allowing users to write mathematical definitions, theorems, and their formal proofs, with a built-in verification mechanism that automatically checks proof correctness. Lean is widely used in mathematical formalization, program verification, and logic research. Its companion mathematical library, Mathlib, contains a large collection of formalized mathematical results.
Core Facts
Lean is an interactive theorem prover that requires every step of a proof to hold within a strict formal logic framework.
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 based on dependent type theory and expresses proofs and programs in a unified manner.
Timeline (last 90 days)
For the arcsin subroutine in quantum simulation, the implementation written by Argon was formally verified with Lean, reducing space-time cost by more than 100 times compared with the published quantum CORDIC baseline.
Formal mathematics (such as the languages used by proof assistants like Lean and Coq) requires every step of reasoning to be strictly verifiable, so it is naturally suited as a source of reward signals for reinforcement learning; the core value of organizing formal data into reinforcement learning environments lies in the automatic scorer, which can greatly reduce training and evaluation costs.
Mechanized verification refers to the use of formal proof assistants such as Lean, Coq, and Isabelle to convert mathematical proofs into a formal language that can be automatically checked by a computer.
Every proof in SqueezeBench is checked by two independent Lean kernels, and the entire mathematical correctness review requires no human referee.
A record in SqueezeBench refers to a computable Lean function accompanied by a machine-verifiable proof.
Cross-validation by two independently implemented Lean kernels is an additional defensive measure; if both accept the same proof, the probability of a common error is extremely low.
On October 7, OpenAI released 722 mathematics manuscripts all at once, with the papers and some Lean proofs uploaded to GitHub.
Approximately 42% of the 722 manuscripts released by OpenAI were formalized in Lean, with each result consuming on average about three hours of ChatGPT Pro inference compute.
LeanCopilot is a tool that auto-completes proof steps in Lean/Coq environments.
Cross-validation by multiple independent checkers cannot solve the mis-translation problem, because the target proposition may be substituted.
40 more timeline events
All Facts (20)
Lean is an interactive theorem prover that requires every step of a proof to hold within a strict formal logic framework.
90%VerifiedFormal 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 based on dependent type theory and expresses proofs and programs in a unified manner.
90%VerifiedLean is an interactive theorem prover developed by Microsoft Research, with the latest version currently being Lean 4.
90%VerifiedLean was created in 2013 by Leonardo de Moura of Microsoft Research and has since evolved to Lean 4.
85%VerifiedFormal proof systems (such as Lean, Coq, and Isabelle) require every reasoning step to conform to a predefined axiomatic system and be mechanically verified by computer
80%VerifiedLean is an interactive theorem prover developed by Leonardo de Moura at Microsoft Research, based on dependent type theory.
80%VerifiedMathlib is a core mathematics library maintained by the Lean community; it currently contains approximately 2.5 million lines of code and includes a large number of formalized definitions and theorems spanning group theory, topology, and measure theory.
80%VerifiedLean requires that every step of derivation be written in a strict type-theoretic language, with logical completeness automatically verified by a computational kernel.
80%VerifiedDeepMind's AlphaProof and AlphaGeometry systems adopt a step-by-step verification approach.
75%VerifiedLean is both a programming language and an interactive theorem prover, in which users write mathematical propositions in a formal language and have them verified step by step by the machine.
75%VerifiedLean's mathematics library Mathlib has accumulated more than one million lines of machine-verified mathematical theorems.
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%VerifiedLean's core is based on Dependent Type Theory, realizing the Curry-Howard correspondence of 'types as propositions, programs as proofs'.
75%VerifiedOnly when a proof passes the rigorous type-checking of proof assistants such as Lean can its conclusions be trusted; combining AI output with formal verification tools is the key pathway to ensuring mathematical correctness.
70%VerifiedInteractive theorem provers such as Lean, Coq, and Isabelle allow proof steps to be encoded in machine-verifiable formal languages
70%VerifiedDeepMind's AlphaProof combined large language models with Lean 4 in 2024 and used reinforcement learning to self-play in a formal environment to generate and verify proof steps.
70%VerifiedLean is an interactive theorem prover developed by Microsoft Research that allows mathematicians to write proofs in a formal language and have their logical correctness verified by a computer.
65%VerifiedLean and Coq are interactive theorem provers (ITPs) that require users to translate mathematical proofs into machine-verifiable formal languages, where every step of reasoning must conform to strict type-theoretic rules.
65%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.
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