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For most readers who want to formalize mathematics, start with Lean 4 and mathlib. It combines a mathematics library, free learning materials, browser-accessible exercises, and an editor-based workflow. Rocq, Isabelle/HOL, Agda, and Metamath are also worthwhile, but they suit different goals and proof styles. These are free tools and learning routes, not services that judge whether an English or handwritten proof sounds convincing: you state a claim formally, construct a proof in the system’s language, and the system checks that formal result.
At a glance
| Name | Best for | Key spec | Free plan / price | Checked |
|---|---|---|---|---|
| Lean 4 + mathlib | Broad mathematical formalization | Math library; browser learning options and VS Code workflow | Free, open source | Nov 2026 |
| Rocq (formerly Coq) | Mathematics alongside program verification | Dependent types; Software Foundations and Mathematical Components learning routes | Free tool and learning materials | Nov 2026 |
| Isabelle/HOL | Checked developments, tutorial, and automation resources | Archive of Formal Proofs; Sledgehammer guide | Free tool and documentation | Nov 2026 |
| Agda | Constructive proof connected to dependently typed programming | Constructive setting; browser preview | Free tool | Nov 2026 |
| Metamath | Explicit proof steps and independent verification | Multiple verifiers and interactive assistants, including a browser-based assistant | Free project resources | Nov 2026 |
The comparison is about fit, learning resources, libraries, and proof style—not a benchmark of speed or ease. “Free” refers to the tools or resources described by their projects; it does not mean every optional book or every possible way of running them has the same cost.
What does a mathematical proof assistant actually check?
A proof assistant checks a proposition and proof that have been translated into its formal language. It does not automatically take an arbitrary textbook argument, a photo of handwritten work, or a paragraph of English and certify that argument as mathematically correct.
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The Lean community describes the system as checking proofs “down to their logical foundation.” That is a useful distinction from a general-purpose AI answer checker: the assistant validates a formal proof according to its rules, rather than estimating whether prose is plausible. The Lean community’s introduction to proof assistants explains this role. Metamath’s project materials likewise distinguish mechanically following proof steps from understanding their mathematical meaning.
Formalization is therefore both verification and translation. It can make assumptions and dependencies explicit, but the human still has to choose definitions and ensure the formal theorem matches the mathematical intent. These tools are most useful when a reader is willing to learn a formal language and wants machine-checked work—not when the only goal is to upload a handwritten proof and receive a yes-or-no verdict.
Which free proof assistant should you choose?
Lean is the broadest first stop for mathematics-oriented readers in this group. Choose another system when its particular logic, programming connection, archive, or explicit proof style better matches what you want to learn or formalize.
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1. Lean 4 with mathlib: the broad mathematical starting point
Choose it if: you want to formalize a range of mathematics and value an established library and a clear route from exercises to editor-based work.
Lean 4 is free and open source, and mathlib is a community-maintained body of formalized mathematics. A library matters because formal work often builds on existing definitions and theorems rather than starting every development from scratch. The Lean community site describes the project and points learners toward its materials.
There are two useful learning routes with different emphases. Mathematics in Lean teaches interactive mathematical formalization using mathlib. Theorem Proving in Lean focuses on core Lean. If your goal is to formalize familiar mathematics, the mathlib-oriented route is the more direct fit; if you want to understand the language’s foundations, start with the core-focused material.
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For a low-friction first encounter, try a browser-accessible exercise such as the Natural Number Game. If you want a local workflow, follow the official Lean installation and VS Code extension instructions, then work through Mathematics in Lean interactively. Browser access can help you explore without committing to an editor setup; local editing is a better fit when you are ready to work through a sustained development.
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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware matchTrade-off: Lean is a language and an environment to learn, not a button that checks informal proofs. Even with a substantial library, you must formulate the relevant statement and proof. Expect a learning process longer than an afternoon rather than instant mastery.
2. Rocq: mathematics and verified-program methods
Choose it if: you are interested in dependent types and want a system whose uses include both mathematical proof and program verification.
Rocq is the current name of the prover formerly called Coq. Its project highlights both mathematical proofs and verified programs, making it a natural choice for readers whose interest crosses that boundary. As of October 30, 2026, the project site listed Rocq Prover 9.2.0 and Rocq Platform 2026.07.0; version labels matter because editor and language-server compatibility can depend on matching components.
The learning resource should match your emphasis. Software Foundations: Logical Foundations is a free route oriented toward programming languages and logical reasoning. For a mathematics-first approach, the free Mathematical Components book introduces the SSReflect style. The official Rocq documentation and learning page identifies these resources and provides project guidance.
For installation, the project points readers to the Rocq Platform. Before following editor instructions, check that the VsRocq extension and language-server versions fit the Rocq version you install; mixing instructions for different releases can cause setup problems unrelated to the proof itself. Rocq is a stronger fit than a general math-first recommendation if program verification is part of your objective, but the programming-oriented materials may be less direct for someone who only wants a first introduction to formalized mathematics.
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3. Isabelle/HOL: an established archive and automation resources
Choose it if: you want to study existing checked developments, use an official tutorial, or explore the system’s automation resources.
Isabelle is a generic proof assistant; Isabelle/HOL is the setting highlighted by its tutorial, Programming and Proving in Isabelle/HOL. The Archive of Formal Proofs contains Isabelle-checked work, giving learners examples of completed formal developments as well as a place to explore material beyond a first tutorial. The Isabelle documentation also includes a Sledgehammer guide, which is useful to know about if you are investigating the system’s proof-automation resources.
The project’s official site listed Isabelle2026 in October 2026, and the corresponding documentation page provides the current tutorial and guides. Use documentation for the release you are installing rather than assuming an older setup guide still applies.
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4. Agda: constructive proof and dependently typed programming
Choose it if: you specifically want to explore mathematical proving in a constructive setting tied to dependently typed programming.
Agda’s documentation presents mathematical proving in a constructive setting. That makes it a meaningful alternative when the foundations and style of the proof matter to you, not just the ability to have a theorem accepted. Constructive reasoning is a different orientation from simply seeking a convenient checker; make sure it matches the kind of mathematics and programming connection you want to study.
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A browser preview is available as a way to explore Agda before choosing an installation route. That is useful for deciding whether to continue, but a preview is not necessarily the right environment for a larger project. The available official information does not establish a single preferred editor setup or a comparable library inventory here, so do not infer those details from the browser preview alone.
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Choose it if: you want to examine proof steps closely, learn how a verifier checks them, or distinguish interactive proof creation from verification of existing work.
Metamath’s ecosystem includes several verifiers and interactive assistants, including a browser-based assistant. Those tools do different jobs: a verifier checks a proof, while an interactive assistant helps a person create one. If you are browsing archived proofs, that is not the same activity as constructing a new proof interactively. The project’s Metamath site explains the project and its available tools.
Metamath rewards explicitness. Its proof style can require more visible proof work than a reader expects from systems with automation resources, and mechanical verification should not be confused with intuitive understanding of why a theorem matters. A verifier can establish that steps conform to formal rules; the mathematical interpretation and explanatory insight remain separate questions.
The project offers a free PDF of Metamath: A Computer Language for Mathematical Proofs. An optional hardcover is also documented: the second edition by Norman Megill and David A. Wheeler, published in 2019, is 248 pages and has ISBN 9780359702237. The book is not the proof checker, and buying it is not required to use the project. Live price and stock information were not verified, so treat the identifier and edition details—not any assumed availability—as the useful facts.
How should you make your first choice?
Choose based on what you want to learn, then use the lowest-friction route that still lets you practice the system’s real proof workflow.
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- If your goal is broad mathematical formalization, begin with Lean. Try a browser exercise, then move to Mathematics in Lean with the official VS Code extension if you want a sustained local workflow.
- If software correctness is part of the goal, look at Rocq. Start with Software Foundations: Logical Foundations; if mathematics is the center, use Mathematical Components instead.
- If you want to learn from an archive and official tutorial, consider Isabelle/HOL. Read Programming and Proving in Isabelle/HOL, browse the Archive of Formal Proofs, and check the hardware guidance before attempting large work.
- If constructive proof is your specific interest, preview Agda. Use the browser preview to see whether its setting suits your goal before selecting an installation approach.
- If you want to understand explicit proof steps, explore Metamath. Decide whether you want to inspect and verify existing proofs or use an interactive assistant to create new ones; those are related but distinct tasks.
Do not select solely by the number of listed resources or by a release number. The practical fit depends on the mathematical domain, the system’s logical setting, the libraries you can build on, how much proof construction versus automation you want, the editor and installation path, and—for ambitious work—the available hardware. Those are selection criteria, not measured speed or ease-of-use rankings.
What can go wrong when learning a proof assistant?
Most early confusion comes from mixing up the informal theorem, its formal statement, and the proof script that establishes it. Keeping those layers separate makes troubleshooting more productive.
- The accepted statement is not the intended theorem. Revisit the definitions and assumptions, then compare the formal proposition with the informal claim. Acceptance establishes the formal result, not the correctness of the translation from ordinary mathematics.
- A tutorial does not match the installed version. Use official documentation for the current release. This is especially relevant for Rocq, where the project cautions readers to match VsRocq and language-server versions, and for Isabelle, whose project provides version-specific documentation.
- Setup feels harder than the mathematics. Try a browser-accessible learning option first where one is available. For a local setup, follow the project’s official installation instructions rather than combining steps from different release generations.
- A large project is slow or difficult to run. Check the project’s hardware guidance before assuming a small-experiment setup is suitable for a substantial development. Isabelle’s published 2026 recommendations scale from 4 GB RAM and 2 cores for small experiments to 16–32 GB and 8 cores for large projects.
- Automation is mistaken for explanation. Automation resources can help with proof construction, but a checked result is not necessarily an intuitive explanation. Read the theorem, its assumptions, and the proof context rather than treating acceptance as a substitute for understanding.
- A verifier is mistaken for an interactive assistant. In Metamath, for example, distinguish tools that check existing proof steps from assistants intended to help create proofs.
How we picked
This comparison draws on the supplied project materials; it is not a hands-on test or a speed ranking. The five tools were selected because the supplied project materials document distinct mathematical or proof-engineering goals and provide a concrete starting route: Lean’s mathlib and learning materials; Rocq’s mathematics and verified-program focus; Isabelle’s checked archive, tutorial, and automation guide; Agda’s constructive setting and browser preview; and Metamath’s explicit proofs, verifiers, and interactive assistants. The comparison also considers existing libraries, proof construction versus automation, editor and installation friction, teaching resources, and hardware guidance where supplied.
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The date-sensitive release references reflect project listings checked October 30, 2026: Isabelle2026 and Rocq 9.2.0 with Rocq Platform 2026.07.0. Tool versions, extension compatibility, documentation, and the availability of physical books can change; consult the project pages linked above when setting up a current installation.
Frequently Asked Questions
Can I check a handwritten proof with Lean or Rocq?
Not directly: these systems work on formal statements and proofs written in their respective languages. Turning handwritten mathematics into that form is itself a formalization task, and the translation still needs to preserve the claim you meant to prove.
Do I need to know programming before learning a proof assistant?
Prior programming experience can help with the language-like workflow, but the resources serve different audiences: for example, the Lean community offers mathematics-focused learning, while Rocq’s Software Foundations route is explicitly programming-language oriented. Choose a tutorial aligned with your goal rather than assuming every proof assistant requires the same background.
Can a proof assistant discover a proof for me?
Some systems include automation resources, but proof assistants still operate on formal claims and their accepted proof procedures. Automation should not be taken to mean that a tool can interpret any informal theorem and independently supply the intended mathematical argument.
Is the Metamath book required to use Metamath?
No. The project provides a free PDF, and the checker tools are distinct from the optional book. The hardcover’s 2019 second-edition details identify a publication, not a required purchase or a confirmed current stock status.
