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)
AI-greement's Lean formal verification not only verified the safety of the new protocol but also revealed several previously unreported vulnerabilities in existing consensus protocols.
The AI-greement framework integrates three key components: a concept-oriented protocol knowledge base, a modular implementation skeleton, and a Lean-based formal verification framework.
The core work of mathematicians intervening in AI mathematical results includes translating natural-language proofs into formal languages (such as Lean and Coq) and checking the logical validity of each step of reasoning.
OpenAI announced on October 6 a result called the "quasi-Riemann proof," initially covering 377 papers and later expanded to 722; the first version of the proof, dated September 30, shows all nontrivial zeros lie between 1/8 and 7/8 and was formally verified in Lean; the second version, dated October 5, proves the weaker 1/12 to 11/12 bound in 49 pages.
Maestro is a Lean prover framework whose core is a coordinator with a full proof-tree view that can respond based on failure causes with actions such as supplementary guidance, lemma correction, or withdrawing a subtree and replanning.
In Lean, mathematical propositions are expressed as types, proofs are terms of that type, and the kernel independently checks the correctness of proofs.
In Lean, tensors can be represented as indexed finite-dimensional array types, with shape constraints statically checked at compile time via dependent types.
A work titled "Provably Correct Tensor Puzzles" built a Jax-to-Lean translator for the formal verification of Python machine learning code.
Lean's standard mathematical library is mainly based on exact real and rational numbers and does not directly support floating-point reasoning.
Lean was initiated by Microsoft Research and has since developed the vast mathematics library mathlib; Coq is widely used for program correctness proofs.
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%VerifiedLean is an interactive theorem prover developed by Leonardo de Moura at Microsoft Research, based on dependent type theory.
80%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%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%VerifiedThe core principle of formal verification tools is type theory or higher-order logic, which transforms mathematical correctness into a mechanically checkable computational problem.
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%VerifiedLarge language models are demonstrating increasingly strong capabilities in mathematical reasoning, theorem proving, and formal verification.
75%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%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%