Choose the proof assistant that best fits the mathematics you need to formalize—not the one with the strongest general reputation. First check whether its maintained libraries contain the definitions and neighboring results your project needs; then compare foundations, proof-development workflow, learning resources, and any software-verification requirements. Lean is worth an early trial when Mathlib fits your subject, but the available evidence does not establish a universal winner over Rocq or Isabelle.
Start with the mathematics and its existing library
For a real formalization, being able to reuse a suitable definition or lemma can matter more than a feature checklist. Search each candidate’s library for the objects and results closest to your intended work, then check that their assumptions and abstractions match your project. A result with a similar name is not necessarily reusable if it formalizes a different definition.
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- Lean: Mathlib’s documentation overview lists material in areas including analysis, category theory, group theory, linear algebra, measure theory, ring theory, and topology. The list is a useful starting point, not proof that a particular theorem or formulation is already available. Browse the Mathlib documentation overview.
- Isabelle: The Archive of Formal Proofs (AFP) is a place to inspect Isabelle developments and look for related formalized mathematics. Search the Archive of Formal Proofs.
- Rocq: Begin with the project’s official documentation, then investigate whether the project’s required definitions and results are available in the libraries relevant to your work. Rocq documentation.
The available sources do not establish a matched inventory of mathematical library coverage across all three systems. Compare candidates against your own theorem or a close representative example rather than inferring that one ecosystem is broader from these starting points.
Compare the foundations your project needs
Lean and Rocq (formerly Coq) share dependent type theory foundations, although they differ technically. Isabelle/HOL is based on higher-order logic and the LCF approach. Those distinctions matter if the project has explicit requirements about its foundational assumptions, constructive content, or use of dependent types; they do not automatically determine which assistant will be more convenient for a particular proof. Lean’s FAQ discusses these distinctions.
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When constructive reasoning matters
Do not assume that choosing Lean means ordinary library work is automatically constructive. Lean’s foundational logic is not inherently classical, and the axiom of choice is optional; however, the FAQ notes that Mathlib and its tactics use choice freely. If constructive content is a project requirement, inspect the actual axioms and dependencies used by the development rather than relying on the name of the proof assistant.
When a foundation is prescribed
If a course, collaboration, or verification project requires a particular logic or foundational style, treat that as a hard constraint and confirm that the system’s current documentation supports the needed approach. For a project without such a constraint, compare the foundations alongside library fit and maintainability instead of treating one as a standalone quality ranking.
Use a like-for-like trial before committing
A short trial can reveal practical differences that a general comparison cannot. Pick a small, representative piece of your planned work—one definition and a theorem that exercises the style of reasoning you expect—and attempt it in each serious candidate. Keep the mathematical statement and intended assumptions fixed so the comparison is meaningful.
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- Check for reuse. Search the relevant library or archive for the definition, lemma, or theorem. Record whether existing material matches your intended abstraction and assumptions, not just whether a similar result exists.
- Formalize the representative example. Note where the system’s proof style feels clear or awkward, what automation you rely on, and whether the resulting definitions and proof are understandable to the people who will maintain them.
- Try the expected project workflow. Check that collaborators can build and review the development using the project’s intended tools, and assess how the work will be kept reproducible. The documentation entry points below can help you explore each system; they do not establish a matched ranking of editor integration or maintenance.
- Compare the trade-offs together. Weigh library reuse, clarity, automation, workflow, learning support, and the team’s ability to maintain the formalization. Prefer the system that fits the complete project rather than the one that wins on a single feature.
Compare the systems on the evidence available
| System | Mathematical starting point | Foundation noted in the sources | Learning and documentation starting point |
|---|---|---|---|
| Lean | Mathlib’s documentation lists multiple mathematical areas, including analysis, category theory, group theory, linear algebra, measure theory, ring theory, and topology. Check for your specific results. Mathlib documentation | Dependent type theory; its technical differences from Rocq include proof irrelevance, universe hierarchy, and recursion and termination checking. Its logic is not inherently classical, but Mathlib and its tactics use choice freely. Lean FAQ | The official learning page points to Mathematics in Lean, tutorials, references, and interactive games. Mathematics in Lean is aimed at mathematicians using Mathlib. Lean learning resources |
| Rocq (formerly Coq) | A comparable, matched inventory of mathematical library coverage is not established by the sources reviewed. Search the resources relevant to your project. Rocq documentation | Dependent type theory, with technical differences from Lean. Lean FAQ | The official documentation landing page is the starting point for Rocq manuals and resources. Rocq documentation |
| Isabelle/HOL | The Archive of Formal Proofs is a searchable place to inspect Isabelle developments; equivalent coverage across the three systems has not been established here. Archive of Formal Proofs | Higher-order logic and the LCF approach. Lean FAQ | The official documentation page is the starting point for Isabelle materials. Isabelle documentation |
This table is a comparison framework, not a scorecard: the evidence is uneven, and none of these sources supplies a controlled comparison of performance, ease of use, or overall ecosystem size.
Choose learning resources that match the system
Learning materials are system-specific, so inspect the resources for the candidates you are considering and try the teaching style before making a long-term commitment.
- For Lean: The project identifies Mathematics in Lean as its main resource for mathematicians learning formalization through interactive, tactic-based theorem proving with Mathlib. Its learning page also links tutorials, references, and interactive games. Explore Lean learning resources.
- For Rocq: Use the official documentation landing page to find its documentation and manuals. Explore Rocq documentation.
- For Isabelle: Start at its official documentation page and select materials relevant to your intended use. Explore Isabelle documentation.
These pages establish where to begin, not which system is easiest to learn. Your mathematical background, preferred proof style, available mentoring, and the demands of your project all affect that judgment.
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Account for software verification if the project includes it
If the same project must formalize mathematics and verify software, include a small software-verification task in your trial as well as a mathematical theorem. The Lean project explicitly describes Lean as useful for formalizing mathematics and formal verification, so it is a relevant candidate for mixed work. The sources here do not provide a matched comparison of Lean, Rocq, and Isabelle on a defined verification task; test the requirements your project actually has rather than assuming an overall advantage. Lean learning page and Lean FAQ.
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Documentation is a starting point, not enough evidence to rank future maintenance or contributor availability. For a long-lived project, check the current releases, library activity, build reproducibility, contribution practices, and whether collaborators can sustain the development. The sources linked here do not provide a matched assessment of those factors across the three systems.
Use older comparisons as historical context
A 2017 paper compares Isabelle/HOL and Coq through expressiveness, limitations, usability, and proof examples. It can help frame questions about those systems, but it predates the current project context, does not compare Lean, and should not be treated as a current performance or ecosystem ranking. Read the 2017 Isabelle/HOL and Coq comparison.
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