Lean 4 with Mathlib is the best default starting point if you want an open-source environment that can check formal mathematics down to its logical foundations. Isabelle is the strongest alternative when its generic proof-assistant ecosystem or an existing Isabelle project fits your work. For natural-language assistance, OpenProof and research tools from EPFL add language-model and tactic-search layers around Lean, but they do not replace the proof checker.
The crucial distinction is between creating a candidate solution and checking a formal proof. An assistant can suggest mathematics or translate a word problem into Lean; Lean or Isabelle can then verify only the formal statement and proof that you submit. You must still confirm that the formal statement means what the original problem intended.
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| Job | Best starting point | Why | What still needs human attention |
|---|---|---|---|
| Solve an ordinary exercise and explain the result | Lean 4 with Mathlib plus a general-purpose reasoning model | Mathlib supplies formal definitions and lemmas, while a model can help draft an informal approach or Lean code. | Check arithmetic, assumptions, and whether the formal theorem matches the exercise. |
| Write and interactively debug a machine-checked proof | Lean 4 with Mathlib | It has an active community, a large mathematics library, browser try-out access, tutorials, and an introductory book. | Learn Lean syntax, types, tactics, and library conventions. |
| Use a different general-purpose proof assistant | Isabelle | It is a mature generic proof assistant with free open-source distributions for Linux, Windows, and macOS. | Choose the logic, libraries, and workflow that suit your project; do not assume Lean and Isabelle files are interchangeable. |
| Generate candidate Lean proofs conversationally | OpenProof | Its agents combine language-model suggestions with tactic search and send candidates to Lean’s type checker. | Install the required Lean and Mathlib versions, configure a model provider, and inspect the generated theorem statement. |
| Formalize papers or long mathematical documents | EPFL’s LeanFlow, LeanProbe, or Lean Interact projects | These released tools focus on document-level autoformalization and agent feedback. | Project maturity, setup, and suitability vary; verify each repository before using it for a production workflow. |
What counts as an open-source AI math tool?
A proof assistant is software with a formal language for defining objects, stating properties, and proving those properties. Its kernel checks whether a proof term has the required type. Lean and Isabelle belong to this category. They can solve or verify mathematics, but they are not chatbots: you normally express the theorem in a formal language and guide the system with definitions, lemmas, and tactics.
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An AI-assisted prover adds a workflow layer. A language model can turn natural-language instructions into a candidate theorem, suggest tactics, search a library, or repair a failed proof. The candidate is useful only when the proof assistant accepts it. This architecture gives you a valuable audit trail: the generated text is disposable, while the checked formal artifact can be rerun and reviewed.
Open source applies to the software project, not automatically to every service it calls. An agent may invoke a hosted model through a provider selected during setup. The available documentation does not establish that every model call is local, free, or covered by the same license as the prover. Treat model-provider choice, data handling, and usage limits as separate decisions.
Tool-by-tool comparison
| Tool | Best fit | Evidence-backed capabilities | Limitations and setup questions |
|---|---|---|---|
| Lean 4 + Mathlib | Formalizing mathematics and checking proofs | Open-source development environment; community mathematical library; online try-out, installation guidance, tutorials, and Theorem Proving in Lean 4 are available through the Lean community. | You must learn Lean’s formal language. Library names, theorem statements, and tactic behavior require precise syntax and type information. |
| Isabelle | A separate, general-purpose proof-assistant ecosystem | The project describes Isabelle as a generic proof assistant. Isabelle2025-2 was announced as available in January 2026, with free distributions for Linux, Windows, and macOS under open-source licenses. | Use it when its documentation, logic, libraries, or an existing formalization fit your needs. No common benchmark in the reviewed material proves it faster or more accurate than Lean. |
| OpenProof | Conversational attempts at Lean 4 theorems | Its repository describes natural-language theorem handling by language-model agents and tactic search, with candidate proofs checked by Lean’s type checker. It reports a searchable corpus of more than 190,000 Mathlib declarations. | Lean 4 and Mathlib are prerequisites. Model-provider selection is part of setup, so local execution and zero cost are not guaranteed. |
| LeanFlow, LeanProbe, and Lean Interact | Research-oriented formalization and agent feedback | EPFL lists these as released software for document-level autoformalization and verification using Lean. | They target research workflows rather than a beginner’s first theorem. Confirm current repositories, dependencies, and maintenance status before adoption. |
Lean 4 with Mathlib: the strongest default
Lean is an open-source development environment for formal mathematics, also known as machine-checkable mathematics. The Lean community describes a proof assistant as software that lets you define objects, specify their properties, and prove those specifications. That combination makes Lean suitable both for learning formal methods and for building a reusable body of verified mathematics.
Mathlib is the community-built library that makes Lean practical for everyday mathematics. Microsoft Research described it as exceeding one million lines of code and said the community had digitized more than half of the undergraduate mathematics curriculum into Lean on the project page accessed in 2026. Those are project descriptions rather than independently audited counts, and the totals will change as contributions land, but they indicate the breadth of material available.
Why Mathlib changes the experience
- Existing definitions: common structures such as groups, rings, topological spaces, and real-valued functions can be reused instead of rebuilt.
- Reusable lemmas: once you find the right theorem, a proof often becomes a short application or a controlled tactic invocation.
- Machine-checked dependencies: imported results are checked as part of the environment, so a proof does not rely on an unverified citation.
- Community guidance: the project provides tutorials, an online try-out, installation instructions, and Theorem Proving in Lean 4.
Start with the Lean community resources for the current browser environment, supported platforms, installation links, and learning materials. Microsoft Research’s Lean project overview explains the project’s purpose and its relationship to Mathlib.
A practical first session
- Try the browser version first. Use the community’s online environment to compile a tiny theorem without changing your operating system.
- Learn the type system before tactics. Practice declaring variables, reading the goal state, and distinguishing propositions from data. Many apparent “AI mistakes” are type mismatches.
- Install a local Lean toolchain for serious work. Follow the current instructions linked by the community rather than copying an old version number from a tutorial. Match the Mathlib revision required by your project.
- Create a small project. Keep theorem statements, imports, and examples in version control so that a successful proof can be reproduced after libraries update.
- Formalize the statement before searching for a proof. Decide domains, quantifiers, side conditions, and definitions explicitly. A theorem that omits a nonzero denominator or a positivity assumption can be formally true for the wrong reason.
- Use library search and automation in that order. Find existing definitions and lemmas, then try tactics. Automation is easier to debug when you understand the objects it is manipulating.
- Compile after each meaningful change. The error location and goal state tell you whether the problem is syntax, elaboration, a missing import, or a genuinely incomplete argument.
Isabelle: a substantive alternative
Isabelle is a generic proof assistant, not merely a lighter version of Lean. Its project distributes builds for Linux, Windows, and macOS at no charge under open-source licenses. Isabelle2025-2 was listed as available in January 2026. If your group already has Isabelle theories, expertise, or documentation, switching systems can cost more than staying with the established ecosystem.
Indicative hardware guidance
The Isabelle project gives requirements drawn from small experiments through extra-large projects rather than a single minimum specification.
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|---|---|---|---|
| Small experiments | 4 GB | 2 cores | A starting point for modest theories; editor responsiveness depends on imports and background checking. |
| Extra-large projects | 64 GB | 16 cores | Useful headroom for substantial developments; not a universal requirement for every Isabelle user. |
Choose Isabelle when its logic, Isar proof language, libraries, platform support, or an existing project is the deciding factor. The available evidence does not provide a common task benchmark, so calling either Isabelle or Lean universally faster would overstate what is known.
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OpenProof and AI-assisted Lean workflows
OpenProof is designed for conversational theorem-proving attempts in Lean 4. Its documented workflow lets an agent read a natural-language theorem, propose Lean code, search for tactics, and submit candidates to the Lean checker. The checker rejects terms that do not type-check; accepted terms become formal proofs inside the chosen environment.
What you need before trying it
- A compatible Lean 4 installation.
- The Mathlib revision expected by the project.
- A configured model provider. The project documentation does not establish that the provider is local or free.
- A way to inspect and preserve the generated Lean source, not just the agent’s conversational explanation.
The repository reports a searchable corpus of more than 190,000 Mathlib declarations. Because that number is project-stated and can change, treat it as an indication of search scope, not a permanent specification.
How to use an agent without surrendering control
- Write the informal problem in your own words, including all assumptions and the desired conclusion.
- Ask the agent to restate the theorem formally before asking for a proof. Compare every quantifier, domain, equality, inequality, and side condition with the original.
- Let it propose a short proof first. If that fails, request one diagnostic change at a time rather than an opaque rewrite of the whole file.
- Read the resulting Lean declaration and imports. A proof that succeeds only because of an unexpectedly strong imported theorem deserves review.
- Run the file locally in a clean environment and retain the exact versions used.
- Explain the checked proof in ordinary mathematics separately. Lean confirms validity of the formal artifact; it does not certify that your translation captured the intended question.
EPFL’s document-level tools
EPFL’s AI for Math project lists LeanFlow, LeanProbe, and Lean Interact as released software. Their focus is broader than solving one exercise: they support autoformalizing mathematical documents and giving agents verifiable feedback from Lean.
This distinction matters for papers, lecture notes, and collections of theorems. A document-level system must identify definitions, references, scopes, and dependencies across many statements. A beginner who only wants to check one algebra exercise may find such tooling excessive. Before adopting one, inspect its current repository for supported Lean versions, model requirements, installation steps, examples, and issue activity; those details change faster than a general project description.
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How to turn a word problem into a checked proof
The following workflow works whether you write Lean manually or use an AI layer to draft code.
- Normalize the question. List known quantities, unknowns, domains, units, and the exact claim to prove. Separate definitions from assumptions.
- Check satisfiability informally. Look for contradictory assumptions, missing conditions, or an answer that depends on a hidden convention.
- Choose formal types. Decide whether a quantity is a natural number, integer, rational, real, finite set, function, or another structure. Coercions between these types are a common source of errors.
- State the theorem with explicit hypotheses. Include nonzero denominators, positivity, finiteness, continuity, or other conditions that the informal argument uses.
- Search the relevant library. Reuse Mathlib or Isabelle lemmas where possible. Search by concept and type, not just by the wording of the original prompt.
- Build a minimal proof. Establish elementary facts first, then apply higher-level lemmas. Small intermediate claims make failures local and explanations clearer.
- Compile in a reproducible environment. A green result means the exact formal statement and proof passed the checker under those imports and versions.
- Audit the translation. Compare the formal theorem line by line with the original problem and test edge cases that informal wording might conceal.
- Record provenance. Keep the source text, formal statement, imports, toolchain version, and any model-generated suggestions together so another person can reproduce the check.
Common failure modes and fixes
| Symptom | Likely cause | Fix |
|---|---|---|
| The theorem compiles but answers the wrong question | The natural-language prompt was translated incorrectly or assumptions were strengthened silently. | Review quantifiers, domains, definitions, and edge cases before trusting the proof. |
| An AI suggestion produces many unknown identifiers | The declaration name, namespace, import, or Mathlib version is wrong. | Search the installed library, check imports, and ask for one corrected identifier at a time. |
| A tactic works in one file but not another | Different imports, toolchain revisions, local simp lemmas, or options change the environment. | Reduce to a minimal file and pin the project’s versions. |
| Automation stalls on an easy-looking goal | The goal lacks a type annotation, has coercions, or needs an intermediate lemma. | Inspect the goal state, add explicit types, and prove the missing fact separately. |
| Proof search finds a surprisingly short proof | A powerful imported theorem may already encode most of the argument. | Inspect the theorem and its hypotheses; confirm that using it is mathematically appropriate. |
| Model output is unavailable or expensive | The AI layer depends on an external provider whose limits and pricing are separate from the open-source prover. | Check provider settings, privacy terms, quotas, and whether a local model is supported before sending sensitive work. |
| A project breaks after an update | Lean, Mathlib, Isabelle, or an agent dependency changed. | Record versions, read migration notes, and update in a separate branch with regression examples. |
How to evaluate a claimed proof
- Check the formal statement first. A valid proof of an unintended theorem has no bearing on the original problem.
- Inspect assumptions. Look for hypotheses that make the conclusion trivial or exclude the difficult cases.
- Read imported results. Understand what each major lemma says and whether its conditions are discharged correctly.
- Test boundary cases. Try zero, negative, empty, singular, or minimal values where the informal statement permits them.
- Separate explanation from evidence. A fluent model explanation is not a proof; the checker’s acceptance is evidence only for the formal artifact.
- Reproduce the result. Another machine or a clean checkout should be able to compile the same file with the recorded versions.
Learning path for beginners
Stage 1: learn the language of propositions
Use the Lean community’s browser try-out and introductory materials to practice variables, functions, propositions, implication, conjunction, equality, and quantifiers. Concentrate on reading goal states; this skill transfers directly to debugging AI-generated code.
Stage 2: formalize familiar mathematics
Translate elementary algebra, finite sums, and simple induction proofs. The aim is not difficulty but accurate modeling: state exactly what is known and what must be shown.
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Stage 3: navigate the library
Learn namespaces, imports, theorem application, simplification, rewriting, and typeclass inference. Library fluency usually saves more time than asking a model to produce a longer proof.
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Stage 4: add AI assistance selectively
Use an agent to suggest names, sketch a proof, or repair a local error. Require it to show the formal statement and preserve a compilable file. Do not let it hide imports or silently alter assumptions.
Stage 5: work on a maintained project
Once a small development is reproducible, adopt version control, review conventions, and automated checks. This is where the difference between a one-off answer and dependable formal mathematics becomes visible.
What the available evidence can and cannot establish
The projects document capabilities, releases, libraries, and workflows, but no common benchmark score was verified for Lean, Isabelle, OpenProof, or the EPFL tools on the same set of problems. You should therefore choose by task coverage, library fit, verification boundary, setup, and learning resources rather than by an unsupported universal performance ranking.
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Bottom line
Choose Lean 4 with Mathlib for the broadest default path into open-source, machine-checked mathematics and pair it with an AI layer only when you want help drafting or searching. Choose Isabelle when its generic proof-assistant ecosystem or an existing formalization is the better fit. Use OpenProof or EPFL’s tools as workflow accelerators, never as substitutes for reading the formal statement and rerunning the proof checker.
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