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lean community lean community lean and its mathematical library the lean theorem prover is a proof assistant developed principally by leonardo de moura the lean mathematical library mathlib is a community driven effort to build a unified library of mathematics formalized in the lean proof assistant the library also contains definitions useful for programming this project is very active with many regular contributors and daily activity you can get a bird s eye view of what is in the mathlib library by reading the library overview and read about recent additions on our blog the design and community organization of mathlib are described in the 2020 article the lean mathematical library although the library has grown by more than an order of magnitude since that article appeared there is also a paper arxiv version about the challenges with mathlib s growth on a technical and social level you can also have a look at our repository statistics to see how the library grows and who contributes to it try it you can try lean in your web browser install it in an isolated folder or go for the full install lean is free open source software it works on linux windows and macos try the online version of lean installation instructions learn to lean you can learn by playing a game following tutorials or reading books learning resources theorem proving in lean 4 an introduction api documentation of mathlib meet the community lean has a very diverse and active community it gathers mostly on a zulip chat and on github you can get involved and join the fun meet us how to contribute papers involving lean what is a proof assistant a proof assistant is a piece of software that provides a language for defining objects specifying properties of these objects and proving that these specifications hold the system checks that these proofs are correct down to their logical foundation these tools are often used to verify the correctness of programs but they can also be used for abstract mathematics which is something of interest to the mathlib community in a formalization all definitions are precisely specified and all proofs are virtually guaranteed to be correct formalized with lean and mathlib liquid tensor experiment by commelin topaz et al liquid vector spaces are crucial objects in clausen and scholze s condensed mathematics framework blending number theory and functional analysis the main theorem about them guarantees they have suitable properties to achieve that goal learn more polynomial freiman ruzsa conjecture by tao et al this is a famous conjecture in additive combinatorics whose solution was announced in november 2023 and formalized in three weeks thanks to a large collaborative project learn more cedar specification by emina torlak et al cedar is a language for defining permissions as policies which describe who should have access to what it is also a specification for evaluating those policies the specification has been formalized and the implementation in lean is used to check the rust implementation by comparison learn more independence of the continuum hypothesis by han and van doorn the continuum hypothesis states that there is no cardinality between the smallest infinite cardinal and the cardinality of the continuum it was posed by cantor in 1878 and was the first problem on hilbert s list of twenty three unsolved problems in mathematics gödel and cohen proved this is independent i e neither provable nor disprovable from zfc learn more perfectoid spaces by buzzard commelin and massot perfectoid spaces are sophisticated objects in arithmetic geometry introduced by peter scholze in 2012 we formalised enough definitions and theorems in topology algebra and geometry to define perfectoid spaces in the lean theorem prover this experiment confirms that a proof assistant can handle complexity in that direction which is rather different from formalising a long proof about simple objects learn more cairo verification by avigad cairo is a blockchain based programming language various components of the language and its library have been formalized learn more fermat s last theorem for regular primes by best birkbeck brasca rodriguez van de velde yang altough this is far from the general case of fermat s last theorem the case of regular prime exponents is a nice test case for the foundations of algebraic number theory learn more the sphere eversion by massot nash van doorn smale s famous sphere eversion theorem ensures you can turn a sphere inside out in euclidean 3 space it is an instance of a much more general flexibility result due to gromov its formalization shows that proof assistants are not limited to algebraic parts of mathematics and can handle differential topology learn more data types as quotients of polynomial functors by avigad carneiro and hudon a broad class of data types including arbitrary nestings of inductive types coinductive types and quotients can be represented as quotients of polynomial functors this provides perspicuous ways of constructing them and reasoning about them in an interactive theorem prover learn more the cap set problem by dahmen hölzl and lewis in 2016 ellenberg and gijswijt established a new upper bound on the size of subsets of 𝔽 n_q with no three term arithmetic progression this problem has received much mathematical attention particularly in the case q 3 where it is commonly known as the cap set problem ellenberg and gijswijt s proof was published in the annals of mathematics and is noteworthy for its clever use of elementary methods learn more you can find many more in the list of papers and list of projects lean community community zulip chat github blog community information community guidelines teams papers about lean citing mathlib licence projects using lean teaching using lean events use lean online version no installation install lean more options documentation learning resources start here documentation overview api documentation declaration search loogle language reference tactic list glossary did you really prove it about mwes library overviews library overview undergraduate maths wiedijk s 100 theorems 1000 theorems theories natural numbers linear algebra sets and finite sets topology category theory contributing pull request lifecycle mathlib review and triage dashboard git guide for mathlib4 contributors naming conventions code style guideline documentation style commit conventions pull request review guide contribution statistics tags and branches