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Repair common Windows errors and clear accumulated junk for a smoother, more stable PC - no reinstall needed.Free scan · no reinstallTo formalize a proof in Lean, state the mathematical claim as a theorem, construct a proof term for it—directly or with tactics—and let Lean’s kernel check that term. A practical start is to install Lean 4 with elan, use the official Lean extension for VS Code, and work in a Lake project; if you need Mathlib, set up that dependency in the project as well.
What formalizing a proof means in Lean
Lean is an interactive theorem prover based on dependent type theory. In the Curry–Howard correspondence used by Lean, a proposition is represented as a type and a proof is a term of that type. A theorem declaration therefore gives Lean a proposition to prove; formalization is complete only when Lean accepts a term that has the required type.
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This is more exacting than writing an informal argument. Definitions, assumptions, intermediate claims, and the proof itself must all be expressed in Lean’s language. Tactics can help build the proof, but the resulting proof term is checked by Lean’s kernel. The Lean Language Reference explains that tactic-generated terms are checked by the kernel, so a tactic bug does not by itself bypass that check. You still need to compile the actual project with its intended toolchain and dependencies.
Set up a Lean project before writing a substantial proof
For a first experiment, Lean’s official installation guide covers the editor setup and checking a saved file. For work that uses mathematical library results, use a Lake project configured for Mathlib rather than assuming a standalone file has those imports available.
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- Install the toolchain manager. Follow the official Lean installation guide to install elan, which manages Lean versions.
- Install the editor support. Install the official Lean 4 extension in VS Code. Open a Lean file in the configured project to see editor feedback as you work.
- Create or open a Lake project. Lake manages project configuration and packages. Keep the project’s toolchain and dependency configuration together so the Lean version and libraries are reproducible.
- Add Mathlib if the proof needs it. Follow the installation guide’s Mathlib project instructions. Fetching Mathlib for a new project can take time; the guide documents
lake exe cache getfor retrieving its cache. - Build the project. Run
lake buildfrom the project directory, especially after changing dependencies. A successful editor display is useful, but the build checks the project under its configured environment.
Use the toolchain configured by the project as authoritative. The official install guide discusses coordinating updates to the toolchain and Mathlib dependency. Documentation pages may describe different Lean versions: the reviewed Theorem Proving in Lean page states Lean 4.33.0, while the reviewed Language Reference states Lean 4.35.0-rc3. Those are page version labels, not a compatibility guarantee; check examples against the version your project actually uses.
State a small theorem and prove it
Begin with a claim whose mathematical structure you already understand. For example, the commutativity of addition on natural numbers can be stated as:
theorem add_comm_example (a b : Nat) : a + b = b + a := by
exact Nat.add_comm a bThe declaration names the theorem, introduces natural numbers a and b, and gives the proposition after the colon. The by keyword opens tactic mode. At that point Lean presents the proposition as a goal. Here exact Nat.add_comm a b supplies an existing proof term whose type matches the goal.
To formalize a result of your own, replace the library theorem with proof steps that establish your claim. Lean checks whether each step fits the current goal; if a step leaves an obligation unresolved or has the wrong type, the editor and compiler report an error. Keep the first examples small enough that you can identify what the goal is asking for at every step.
Choose term style, tactic style, or a mixture
Lean supports direct proof terms as well as tactic proofs. In the example, by introduces a sequence of instructions that builds a term. A direct proof can instead provide the term without opening tactic mode:
theorem add_comm_example' (a b : Nat) : a + b = b + a :=
Nat.add_comm a bNeither style is universally better. Direct terms make the object proving the proposition visible, while tactics are often convenient for decomposing a goal into manageable parts and using automation. A tactic proof can be shorter, but may take more work to read because a reader must infer how each instruction changed the goal. Choose based on the proof’s structure and the clarity you want for its intended readers; Lean lets you mix styles when that is useful.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Find the right resource for your next step
| Resource | Best fit | What it offers |
|---|---|---|
| Natural Number Game | Beginners | An interactive starting point recommended by Lean’s official Learn Lean page. |
| Mathematics in Lean | Mathematicians formalizing with Mathlib | The official Learn Lean page’s main resource for interactive, tactic-based mathematical formalization with Mathlib. |
| Theorem Proving in Lean | Readers seeking proof-development and language foundations | Covers proof development, dependent type theory, automation, and Lean-specific proof methods. |
| Lean Language Reference | Experienced users looking up precise behavior | Technical reference for syntax and language details, rather than a beginner course. |
These are online resources. The official Learn Lean page describes Lean as a tool for mathematical formalization, software verification, and general programming, so the same language and project tools can support work beyond mathematics. For a mathematics learner, starting with the Natural Number Game and then moving to Mathematics in Lean is a sensible route; consult the reference when you need authoritative detail on syntax or behavior.
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