For mathematical formalization, start with Mathematics in Lean (MIL), install Lean through VS Code and the official Lean 4 extension, and follow the toolchain specified by the tutorial or project you open. Use Theorem Proving in Lean 4 (TPIL) alongside MIL when you want a deeper grounding in Lean’s logic and proof system. If you are not sure which route fits, choose from the options below.
Which Lean learning path should you choose?
| Your goal | Start with | Why it fits |
|---|---|---|
| Formalize ordinary mathematics | Mathematics in Lean | It is aimed at mathematicians learning formalization with Mathlib and includes examples and exercises. |
| Try Lean with a low-friction, game-like introduction | Natural Number Game | The official Lean learning page recommends it for beginners and describes it as a gamified introduction to Lean 4. |
| Understand logic and theorem-proving foundations | Theorem Proving in Lean 4 | It covers dependent type theory, propositions and proofs, quantifiers, tactics, induction, and recursion. |
| Learn Lean as a programming language | Functional Programming in Lean | The official learning page identifies it as the main resource for programmers and says prior functional-programming experience is not assumed. |
| Look up syntax or language features once you have started | Lean Language Reference | It is a comprehensive reference, not a beginner tutorial. |
These resources serve different purposes rather than forming a single required sequence. For a mathematician, MIL is the most direct starting point; TPIL is useful when you want to understand what Lean is checking beneath the mathematical exercises.
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What Lean does when you formalize a proof
Lean is both a programming language and an interactive theorem prover. You describe mathematical objects and state a proposition in Lean’s formal language, then construct a proof that Lean checks. Mathlib, the mathematical library used in the MIL path, supplies existing definitions and results, so many exercises build on established mathematics rather than starting from scratch.
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Outbyte PC Repair FREEClear out junk files and repair common Windows errorsFree Scan →Outbyte Driver Updater FREEScan for outdated or missing drivers - takes under a minuteDriver Scan →A successful check means that the proof establishes the proposition as encoded, under Lean’s logic and kernel. It does not by itself establish that your encoding captures the theorem you intended to prove. You still need to choose appropriate definitions and assumptions and verify that the formal statement matches the mathematical claim.
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How to install Lean with VS Code
The official installation guide recommends using VS Code with the official Lean 4 extension. The extension provides a development environment that includes syntax highlighting and code completion, while Lean’s feedback helps you work through proof errors as you edit.
- Install Visual Studio Code if you do not already have it.
- Follow the official Lean installation guide to install the official Lean 4 extension and complete its guided setup. Manual installation is also covered there, but its steps can vary by environment.
- Open a Lean tutorial or project and wait for the extension to finish setting up the toolchain before interpreting missing feedback as a proof problem.
- Open a small Lean file, try one of its examples, and make a change. The Lean editor feedback shows whether the code and proof are accepted.
If you prefer not to install locally, the MIL repository describes browser access and cloud development options for learners who encounter installation difficulties.
How to work through your first formalization exercises
- Begin with a chapter in Mathematics in Lean that matches the mathematics you want to formalize.
- Use the Lean file associated with that chapter, then work through its examples in VS Code so you can see Lean’s feedback as you edit.
- Try the exercises rather than only reading the completed examples. The MIL repository recommends making a copy of the exercise folder so you can experiment without changing the originals.
- When a proof does not check, read the feedback and inspect the statement, definitions, and proof steps. If the editor is not providing normal feedback, first verify that the extension has finished loading the project’s toolchain.
- Use TPIL to explore concepts such as propositions, proof terms, tactics, induction, and dependent type theory when you want a more foundational explanation of what you are using.
Why Lean tutorial versions matter
Lean tutorials and projects can target different toolchain snapshots. The current TPIL page identifies Lean 4.33.0; the Lean Language Reference describes a public preview for Lean 4.35.0-rc3; and the MIL repository metadata describes a latest listed commit building on v4.30.0. Those numbers refer to different artifacts, not one universal Lean version.
When starting a project, use the toolchain it declares and follow the instructions attached to that project. Avoid combining setup steps or code from different tutorial versions without checking compatibility. If something fails, first check that you opened the project with its declared toolchain; a mismatch can look like a problem with your proof even when the tutorial and setup simply target different versions.
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