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The
Lean theorem prover
is a proof assistant developed principally by Leonardo de Moura at Amazon. The Lean mathematical library—
mathlib
—is a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant.
Installation
Lean 4 using elan
elan
is a tool to manage installs of the Lean theorem prover. It facilitates maintaining several concurrent versions of Lean.
In addition, as Lean 4 is under heavy development, using it often requires installing nightly builds, which is made easy using elan.
elan
installs Lean in subdirectories of
$HOME/.elan/toolchains
, and does not require root access.
Install
the
elan-lean
AUR
package.
Note
The rest of these instructions explain how to install Lean, but if you wish to contribute to an existing project, such as mathlib, the Lean installation will be handled automatically, and you can skip ahead to the
Project Management
section
To install the latest stable version of Lean 4, use
$ elan toolchain install leanprover/lean4:stable
However, many applications (including using mathlib4) require development versions of Lean 4. For instance, the build of Lean that mathlib4 is using can be found in the file
[1]
. To install the desired build, run
$ elan toolchain install leanprover/lean4:
build
where
build
should be replaced with the desired build.
The installed versions of Lean can be listed with
$ elan show
and a default can be selected with
$ elan default
build
When using Lean 4, the default build can be overridden for a given project by
cd
-ing to the project directory and creating a file
lean-toolchain
build
Replace
build
with the appropriate build identifier.
To get the path to an executable in the current Lean build, use
$ elan which
name_of_executable
Lean 3 via the AUR
Install
lean-community-bin
AUR
or
lean-community
AUR
.
To install mathlib, you must first install
python-mathlibtools
AUR
, which will install 'leanproject'.
Note
python-mathlibtools
AUR
will not install mathlib, just the tools needed to download mathlib.
To install mathlib, you must create a Lean project:
$ cd
/path/to/project
$ leanproject new
name_of_project
which will download mathlib to
/path/to/project/name_of_project/
_target/deps/mathlib
.
Project management
In this section, we will describe how to create and contribute to Lean 4 projects, including mathlib4.
Structure of a project
A Lean 4 project will contain several files that are used to manage the project.
lakefile.lean
: contains information on how to build the project, as well as a list of dependencies.
lean-toolchain
: a single-line file specifying the build to be used for the project.
lake-manifest.json
: an automatically generated file that contains information on the specific version of each of the installed dependencies. If you have a
lake-manifest.json
file, running
lake update
will download the versions of the dependencies specified in
lake-manifest.json
. This is useful to avoid incompatible dependency versions.
.lake
(or, in earlier versions of Lean 4,
lake-packages
): this is where dependencies are installed.
build
: directory where built files are stored.
Lean files, containing the Lean code.
Creating a project
To create a project, install elan and a Lean build, as described in the
Lean 4 using elan
section above.
Next, run
$ lake new
path/to/project
This will create a default
lakefile.lean
and
lean-toolchain
file.
This will not download mathlib, which you may want to do.
Mathlib
is the official math library of the Lean project, and contains a lot of definitions, lemmas and theorems that you may need for your project. If you want to add mathlib to your project, you must add it as a dependency in the lakefile. See below for details.
Managing a project
If you are working with an existing project, for instance one you have downloaded from the internet (e.g. github), you do not need to install Lean separately, as long as you have elan installed, see the
Lean 4 using elan
section above. As long as the appropriate
lean-toolchain
file is present in the project, you can simply run
$ lake update
which will download the required version of Lean, as well as all the dependencies for the project.
Dependencies can be added by editing the
lakefile.toml
file.
For example, to add mathlib as a dependency, add the following
lakefile.toml
[require]]
name = "mathlib"
scope = "leanprover-community"
to
lakefile.toml
. Next run
$ curl
https://raw.githubusercontent.com/leanprover-community/mathlib4/master/lean-toolchain
-o lean-toolchain
$ lake update
to download mathlib, and
$ lake exe cache get
to download the pre-compiled library (see section
Caches and oleans
below).
Contributing to Mathlib
If you wish to participate in the ongoing effort to formalize much of mathematics, you may want to make contributions to
Mathlib
, which is the official math library of the Lean project.
To do so, you will first need to download mathlib:
$ git clone
https://github.com/leanprover-community/mathlib4.git
and
cd
to the downloaded directly and run
$ lake update
$ lake exe cache get
If you wish to submit your work, you will need to get a branch on the mathlib
Git
server, and submit a
pull request
. To get a branch, contact the community on the
leanprover Zulip chat
, introduce yourself and your project, and ask for a branch.
Automatically generating documentation
It is possible to automatically generate documentation, in the same style as the
mathlib documentation
. This is done using the
doc-gen4
package. To install it, add it to your
lakefile.lean
:
lakefile.lean
meta if get_config? env = some "dev" then require «doc-gen4» from git "
https://github.com/leanprover/doc-gen4
"@"main"
(the
"dev"
part of the code ensures that the doc can only be built when running
lake
with
-Kenv=dev
, as below, so as to avoid generating the doc when merely trying to build the package) and download it:
$ lake -Kenv=dev update
The documentation can be generated using
$ lake -Kenv=dev build
NameOfProject
:docs
where
NameOfProject
should be replaced by the name of your project. This will generate the documentation and place it in the
build/doc
directory. You can then navigate through the documentation by opening
build/doc/index.html
in a browser.
Note
opening the documentation files locally using a browser is not officially supported. Instead, it is recommended to start an http server, for instance with
python -m http.server
Note
doc-gen4
assumes the project builds properly. If there are any errors at build time, the documentation will not be functional. Make sure that
lake build
exits without errors before generating the documentation
The documentation also includes a search function. To search for a term in the documentation, use
file:///path/to/doc/find/index.html?pattern='search_term'#doc
Text editors
Using Lean with Emacs
Install lean-mode
To use Lean with emacs, you must install the
lean4-mode
package which is currently not available on MELPA and must be installed manually (there is also the
lean3-mode
package for lean3).
Automatic unicode input
Lean uses unicode symbols as part of its language.
lean-mode
can handle this input easily by translating standard text to unicode characters, for instance, typing
\R
will resolve to the unicode character
ℝ
.
To do this, the
Lean
input method needs to be selected.
This can be done automatically by adding the following to your
Emacs configuration file
:
.emacs
(add-hook 'lean-mode-hook
  (lambda () (set-input-method "Lean")))
Viewing goal in separate buffer
Keeping track of the "goal" is very useful when writing proofs in Lean.
The goal can be seen directly in emacs using the
C-c C-b
keybinding, which will open a new buffer in which the goal is shown.
This buffer can also be opened automatically when editing a
.lean
file by adding:
.emacs
(add-hook 'lean-mode-hook 'lean-toggle-show-goal)
to your
Emacs configuration file
.
Using Lean with Neovim
Install lean.nvim
To use Lean with
Neovim
, you must install the
lean.nvim
plugin. This can be done easily using
vim-plug
byadding the following to your
Neovim Configuration file
:
init.vim
call plug#begin()

...

Plug 'Julian/lean.nvim'
Plug 'neovim/nvim-lspconfig'
Plug 'nvim-lua/plenary.nvim'

...

call plug#end()
open nvim, and type
:PlugInstall
If you are using Lean 3, you will also need to install
lean-language-server
AUR
. (The language server is built into Lean 4, so it is not necessary for Lean 4 installs).
Next, create a configuration file at
$XDG_CONFIG_HOME/nvim/plugin/lean.lua
:
$XDG_CONFIG_HOME/nvim/plugin/lean.lua
require('lean').setup{
 abbreviations = { builtin = true },
 mappings = true,
}
abbreviatons
enables abbreviating unicode characters, and
mappings
enables a number of useful keybindings, see
https://github.com/Julian/lean.nvim/
for a full list.
More configuration options are available at
https://github.com/Julian/lean.nvim/
.
Tips and tricks
Caches and oleans
In order to use the theorems and lemmas in mathlib, mathlib needs to be built (i.e. compiled).
This is a long process.
To avoid this, the mathlib community provides a repository of pre-compiled mathlib files (also called "oleans").
To download them using Lean 4, use
$ lake exe cache get
and using Lean 3, use
$ leanproject get-cache
Note
In order for the oleans to be used, they have to have been pre-compiled with the same version of Lean (i.e. the same nightly build if you are using those) as is used by the project.
Documentation
A good resource to learn theorem proving in Lean 4 is
[2]
.
Of particular note, a succinct "cheatsheet" for useful commands for theorem proving in Lean 4 is available from
[3]
.
The documentation for mathlib in Lean 4 is available at
[4]
.
Since the documentation is still rather sparse, it is often more productive to address questions to the lean community directly.
The most popular forum for this is the
leanprover Zulip chat
.