Lean blueprints

This is a plasTeX plugin allowing to write blueprints for Lean 4 projects. You can learn what those blueprints are about by reading Terence Tao’s excellent blog post about it.

This infrastructure was originally created in 2020 for the Sphere Eversion Project. Since then, it has been used by many projects. The list below will give you many examples but is not intended to be exhaustive. It is in approximate chronological order.

• Sphere eversion project
• Liquid tensor experiment
• Fermat’s last theorem for regular primes
• Arithmetic Progressions - Almost Periodicity
• Polynomial Freiman-Ruzsa Conjecture
• Lower bounds for hypothesis testing based on information theory
• Prime number theorem and…
• Fermat's Last Theorem for 3
• Fermat’s last theorem
• Carleson Operators on Doubling Metric Measure Spaces
• Infinity Cosmos
• Analytic Number Theory Exponent Database
• Equational Theories

Installation

This package depends on plastexdepgraph which requires graphviz and its development libraries. If you have a user-friendly OS, it is as simple as sudo apt install graphviz libgraphviz-dev. See https://pygraphviz.github.io/documentation/stable/install.html otherwise.

Then, assuming you have a sane python environment, you only need to run:

pip install leanblueprint

Note this will automatically install plasTeX and the other needed python packages.

Upgrading

pip install -U leanblueprint

will upgrade to the latest version.

Starting a blueprint

This package provides a command line tool leanblueprint that automates in particular the creation of a blueprint for your Lean project. This tool is not mandatory in any way. Its goal is to make it easy to create a blueprint without worrying about choosing a file layout or a continuous integration and deployment setup. As every one–size-fits-all tool, it is fairly opinionated. It assumes in particular that your project repository is hosted on Github and you want to host its blueprint on github.io.

If you don’t want to use the leanblueprint command line tool, you can use this plugin as any other plasTeX plugin, using plastex --plugins leanblueprint my_file.tex (not recommended).

In order to use the leanblueprint tool, you need to already have a Lean project created using lake. In addition, your blueprint will be easier to configure if you have at least one commit in the git repository of your project and you have already configured its GitHub git remote (GitHub displays instructions allowing to do the remote setup when you create a new repository there). You should also tell GitHub that you want to use GitHub pages using GitHub actions. You can do that from the GitHub page of your repository by clicking on the Settings tab in the top menu, then the Pages link in the side menu and selecting GitHub Actions as the source, as in the following screenshot.

Assuming your project is ready and GitHub is configured, from your project folder run

leanblueprint new

You will then have to answer some questions to configure your blueprint. If unsure, accept all default answers by simply hitting Enter for each question. Only two questions will insist on having an explicit y/n answer: the question confirming you want to create the blueprint and the one proposing to commit to your git repository.

After running this creation script, you can push to GitHub and wait for GitHub Actions to build your blueprint. You can monitor this task in the Actions tab of the GitHub page of your repository. When building is done, the html version of your blueprint will be deployed to https://user_name.github.io/repo_name/blueprint/ (with the appropriate user or organization name and repository name). The pdf version will be at https://user_name.github.io/repo_name/blueprint.pdf. The API documentation will be at https://user_name.github.io/repo_name/docs/.

Local compilation

Assuming you used the leanblueprint command line tool to create your blueprint (or at least that you use the same file layout), you can use leanblueprint to build your blueprint locally. The available commands are:

• leanblueprint pdf to build the pdf version (this requires a TeX installation of course).
• leanblueprint web to build the web version
• leanblueprint checkdecls to check that every Lean declaration name that appear in the blueprint exist in the project (or in a dependency of the project such as Mathlib). This requires a compiled Lean project, so make sure to run lake build beforehand.
• leanblueprint all to run the previous three commands.
• leanblueprint serve to start a local webserver showing your local blueprint (this sounds silly but web browsers paranoia makes it impossible to simply open the generated html pages without serving them). The url you should use in your browser will be displayed and will probably be http://0.0.0.0:8000/, unless the port 8000 is already in use.

Note: plasTeX does not call BibTeX. If you have a bibliography, you should use leanblueprint pdf before leanblueprint web to get it to work in the web version (and redo it when you add a reference).

Editing the blueprint

Assuming you used the leanblueprint command line tool to create your blueprint (or at least that you use the same file layout), the source of your blueprint will be in the blueprint/src subfolder of your Lean project folder.

Here you will find two main TeX files: web.tex and print.tex. The first one is intended for plasTeX while the second one is intended for a traditional TeX compiler such as xelatex or lualatex. Each of them includes macros/common.tex for all TeX macros that make sense for both kinds of outputs (this should be most of your macros). Macros that should behave differently depending on the target format should go to either macros/web.tex or macros/print.tex. All those files already exist and contains comments reminding you about the above explanations.

The main content of your blueprint should live in content.tex (or in files imported in content.tex if you want to split your content).

The main TeX macros that relate your TeX code to your Lean code are:

• \lean that lists the Lean declaration names corresponding to the surrounding definition or statement (including namespaces).
• \leanok which claims the surrounding environment is fully formalized. Here an environment could be either a definition/statement or a proof.
• \uses that lists LaTeX labels that are used in the surrounding environment. This information is used to create the dependency graph. Here an environment could be either a definition/statement or a proof, depending on whether the referenced labels are necessary to state the definition/theorem or only in the proof.

The example below show those essential macros in action, assuming the existence of LaTeX labels def:immersion, thm:open_ample and lem:open_ample_immersion and assuming the existence of a Lean declaration sphere_eversion.

\begin{theorem}[Smale 1958]
  \label{thm:sphere_eversion}
  \lean{sphere_eversion}
  \leanok
  \uses{def:immersion}
  There is a homotopy of immersions of $𝕊^2$ into $ℝ^3$ from the inclusion map to
  the antipodal map $a : q ↦ -q$.
\end{theorem}

\begin{proof}
  \leanok
  \uses{thm:open_ample, lem:open_ample_immersion}
  This obviously follows from what we did so far.
\end

Note that the proof above is abbreviated in this documentation. Be nice to you and your collaborators and include more details in your blueprint proofs!

By default, the dependency graph will collect the environments definition, lemma, proposition, theorem and corollary. You can change this list using the thms option which expects a list of environment names separated by + signs. For instance you can write

\usepackage[thms=dfn+lem+prop+thm+cor]{blueprint}

if you like short environment names. See the plastexdepgraph documentation for other dependency graph options having nothing to do with Lean.

The above macros are by far the most important, but there are a couple more.

• \notready which claims the surrounding environment is not ready to be formalized, typically because it requires more blueprint work.
• \discussion gives a GitHub issue number where the surrounding definition or statement is discussed.
• \proves inside a proof environment gives the LaTeX label of the LaTeX statement being proved. This is necessary only when the proof does not immediately follow the statement.

Blueprint configuration

Most of the configuration is handled during the blueprint creation if you used the leanblueprint client. But some of it can be changed by LaTeX macros in the web version of LaTeX preamble (in the file web.tex if you use the default layout).

• \home{url} defines the url of the home page of the project.
• \github{url} defines the url of the git repository of the project.
• \dochome{url} defines the url of the doc-gen API documentation of the project.

• \graphcolor{node_type}{color}{description} sets a color in the dependency graph and its description in the legend. The default values are

  • stated, green, Green
  • can_state, blue, Blue
  • not_ready, #FFAA33, Orange
  • proved, #9CEC8B, Green
  • can_prove, #A3D6FF, Blue
  • defined, #B0ECA3, Light green
  • fully_proved, #1CAC78, Dark green

  In particular you can use the above color descriptions to interpret the node type by comparison with the default legend.

Acknowledgments

The continuous integration configuration template used by leanblueprint new is based on the work of many people who had to write such a configuration by hand in the past, including Yakov Pechersky, Ben Toner, Sky Wilshaw, Yaël Dillies and Utensil Song. It also got contributions from Pietro Monticone.