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)
Dozens of AI agents took only 24 days to convert a 15,000-page finite simple group classification proof into about 43 million lines of Lean code, and the scale of AI-written mathematical code is growing at a rate of doubling every 37 days.
In Lean, `sorry` is a placeholder indicating a proof has been skipped; a sorry-free proof means every line of reasoning has been fully machine-checked with no unproven gaps in between.
When giving a talk in the Princeton mathematics department, Alex judged that AI was at least three years away from formalizing Fermat's Last Theorem, but two days after the talk (Friday) a 13-million-line, sorry-free Lean formal proof generated by Anthropic's AI system appeared.
Lean's mathlib library continues to accumulate formalized theorems, and the four color theorem, the Kepler conjecture and others have all been partially or fully formalized.
Formalizing CFSG in Lean means first building a vast foundational library covering core results in group theory, representation theory, groups of Lie type, and other areas.
Proofs that compile in Lean have all passed strict logical checking by the kernel, thereby providing an extremely high degree of correctness assurance.
Lean formalization can serve as a validator for technical building blocks in mathematical research, and reviewers can trust these parts to a limited extent, thereby focusing their energy on the more creative core arguments.
Directions proposed by the guest include crowdsourced verification, limited trust in Lean formalization, and more flexible version control and annotation mechanisms, as supplements to rather than subversions of peer review.
Representative formal proof tools include Lean, Coq, and Isabelle, and their accompanying mathematical libraries (such as Lean's mathlib) have accumulated a vast number of formalized theorems.
The success criterion for open rungs is providing an exact certificate or a Lean-checkable proof, along with numerical evidence and known dead ends.
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%VerifiedTest cases provide AI with a machine-verifiable objective function, converting vague natural language requirements into precise pass/fail signals.
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
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