The Tool Desk
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The best online resources for learning formal mathematics and proof assistants depend on what you want to build. For ordinary mathematics in a mature library, start with a short interactive Lean game and then work through Mathematics in Lean with Mathlib. Choose Theorem Proving in Lean for dependent type theory and prover foundations, Software Foundations or Rocq’s Mathematical Components route for programming-language verification, and Agda or Isabelle material when those systems match your project.
All of the core paths below are online and free to access. They are not interchangeable: each assumes a different mix of mathematical maturity, programming experience and prior logic. Use the comparison first, then follow the goal-specific route and version notes.
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At a glance
| Name | Best for | Key spec | Free plan / price | Platforms | Checked |
|---|---|---|---|---|---|
| Mathematics in Lean | Formalizing ordinary mathematics | Interactive Lean 4 and Mathlib textbook; basics through number theory and discrete mathematics | Free online | Web text with Lean in VS Code | November 2026 |
| Natural Number Game | First contact with tactic-based proofs | Interactive levels proving properties of natural-number operations | Free online | Web browser | November 2026 |
| The Mechanics of Proof | Gentler mathematics-first introduction | Lean community course aimed at readers with less mathematical experience | Free online | Web text and Lean environment | November 2026 |
| Theorem Proving in Lean | Type theory and Lean foundations | Dependent type theory, automation and interactive theorem proving | Free online | Web text with Lean | November 2026 |
| Logic and Proof | A first logic-and-Lean course | Official Lean learning-page recommendation combining logic with formal proofs | Free online | Web text with Lean | November 2026 |
| Software Foundations | Programming languages and verification | Machine-checked Rocq scripts and exercises; Logical Foundations is the entry point | Free online | Web text with Rocq | November 2026 |
| Rocq documentation and Mathematical Components | Rocq users from mathematics backgrounds | Official installation, tour and references plus a mathematics-oriented book route | Free documentation | Web text with Rocq | November 2026 |
| Agda documentation | Dependent types and Agda development | Getting started, installation, tutorials, troubleshooting and language reference | Free online | Web documentation and Agda editor workflow | November 2026 |
| Isabelle/HOL tutorials | Higher-order logic in the Isabelle ecosystem | University of Illinois formal-mathematics reading list points to tutorials | Free reading material | Web and local Isabelle | November 2026 |
Details checked November 2026.
“Best” here means best fit for a goal, not a universal ranking. The Lean community’s learning guide separates mathematics-oriented courses from foundations courses, while the Rocq project separates mathematics, programming and verification routes. That distinction is more useful than a single score.
Which resource should I start with for formalizing ordinary mathematics?
Start with Mathematics in Lean after an optional Natural Number Game warm-up. Lean’s official learning page identifies it as the main mathematics-focused resource, and the community guide describes it as the Mathlib-based path for mathematicians.
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1. Mathematics in Lean — the strongest default for Mathlib
Mathematics in Lean is an interactive textbook rather than a book to read passively. Its contents progress through Lean basics, logic, sets and functions, number theory, discrete mathematics and further subjects. You edit examples, ask Lean to check them and use the feedback to complete exercises.
The course is specifically about writing mathematics in the Lean 4 ecosystem with Mathlib. That makes it the most direct route if your destination is a formal proof that uses existing definitions, lemmas and tactics from Mathlib. It is less suitable as a first tour of functional programming or as a standalone explanation of every type-theory concept.
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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 matchWork through the chapters in order at first. Keep a Lean project open in VS Code, copy each example into a file and change one line at a time. When a proof fails, read the complete error—including the goal state—before searching for a tactic. This habit matters because the useful skill is translating a mathematical argument into a sequence of checkable terms, not memorizing isolated commands.
Version numbers require care. The live HTML documentation identified itself as v4.34.0 in this research, while the PDF identified itself as release v4.19.0. Those are not interchangeable environments: syntax, library names and tactic behavior can change. Prefer the live course and follow its current setup instructions, then make sure your installed Lean and Mathlib versions match the project files.
2. Natural Number Game — an optional interactive on-ramp
The Natural Number Game is a browser-based tutorial in which you prove elementary facts about natural-number operations. The Lean community also points to related games covering set theory, logic and undergraduate mathematics.
Use it when you want immediate feedback and small, clearly bounded goals. A level gives you a concrete target, and the game shows whether Lean accepts the proof. It does not replace a textbook: after the novelty wears off, move to Mathematics in Lean so that you learn how definitions, namespaces, imports and Mathlib’s existing theorems fit together.
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The Mechanics of Proof is the better first course if standard formalization texts feel too fast or assume too much mathematical experience. The Lean community describes it as mathematics-oriented and aimed at readers with less mathematical background.
Choose it instead of the game when you want sustained exposition, or use it before Mathematics in Lean if you need more explanation of why a proof step works. It remains Lean-specific, so the same version check applies: examples written for an older Lean release may need small updates.
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How do I learn proof-assistant and type-theory foundations?
Use Theorem Proving in Lean for the prover itself, and use Logic and Proof when you want a first course that introduces formal logic alongside Lean exercises.
4. Theorem Proving in Lean — for dependent type theory and prover mechanics
The official Lean learning page’s Theorem Proving in Lean material covers development and verification of proofs, dependent type theory, automated proof methods and Lean-specific interactive theorem proving. The community guide calls it the standard resource for type-theory foundations.
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This is not a replacement for Mathematics in Lean. The two books answer different questions. Mathematics in Lean asks how to formalize familiar mathematics with Mathlib; Theorem Proving in Lean asks how Lean’s logic and language represent propositions, data, functions and proofs. If you want to understand why dependent types work or why a tactic can close a goal, choose this route.
Expect a steeper conceptual curve. You will encounter universes, inductive types, dependent functions and proof terms before you have built a large mathematical development. That investment pays off when you need to define a new structure, diagnose elaboration errors or design reusable lemmas rather than merely apply existing ones.
5. Logic and Proof — a bridge between textbook logic and Lean
Lean’s official learning page lists Logic and Proof for readers seeking a first rigorous course combining logic and Lean. It is a good compromise when you want more formal reasoning than a game provides but are not yet ready for a deep dive into the implementation details of dependent type theory.
Use it to learn how propositions, quantifiers and logical connectives become machine-checked statements. Then decide whether your next step is the mathematics-focused Mathlib path or the foundations-focused theorem-proving text.
Which resources are best for programming languages and formal verification?
For Rocq, begin with Software Foundations if programming-language semantics and verification are your priority. If you come from mathematics, follow Rocq’s Mathematical Components recommendation and the project’s official documentation.
6. Software Foundations — a free, machine-checked verification curriculum
Software Foundations presents a series of books whose exposition, Rocq scripts and exercises are integrated. The project calls Logical Foundations the entry point. This makes it a strong choice for learning how operational semantics, program properties and proof automation fit together in a substantial development.
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The exercises are not merely end-of-chapter questions: the Rocq files are part of the learning workflow. You install the matching Rocq toolchain, compile the examples and complete proofs until the checker accepts them. Read the repository or book instructions for the version expected by the volume you choose; a current Rocq installation can require compatibility adjustments for older chapters.
Software Foundations is explicitly freely accessible, so it is the clearest free-first path for readers who want verification rather than conventional undergraduate mathematics. Its examples are in Rocq, not Lean, and concepts do not transfer line-for-line between systems.
7. Rocq documentation and Mathematical Components — choose by background
The Rocq documentation hub combines installation instructions, a tour, reference material and links to books. The project recommends Software Foundations for newcomers interested in programming languages and formal verification, and Mathematical Components for newcomers whose background is mathematics rather than computer science.
That split is a useful decision rule. Mathematical Components emphasizes a structured, algebraic style of formalization; Software Foundations builds toward semantics and verified programs. Read the short tour first, install the version it names and then commit to one book rather than mixing syntax from several editions.
The documentation page listed Rocq Platform 2026.07.0 when checked. Platform bundles and package names can change, so use the live installation page rather than copying an old command from a blog post.
Is Coq’Art still useful?
Rocq’s official page also lists Coq’Art, a conventional textbook introducing Rocq and the Calculus of Inductive Constructions. It can suit readers who prefer a print-style narrative, but confirm the edition and current availability before buying; the online Software Foundations route is the safer free starting point.
Should I choose Agda or Isabelle/HOL instead?
Choose Agda when dependent types and Agda’s editor-oriented workflow are central to your project. Choose Isabelle/HOL when your work targets Isabelle’s higher-order-logic ecosystem, but verify the current official documentation and version before starting.
8. Agda’s official documentation — a system-first route
Agda’s documentation includes an overview, getting-started material, installation and troubleshooting help, tutorials and a language reference. It is the right starting point when you already know that your project will be written in Agda, because the tutorials explain the language and its tooling together.
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Use the overview to pick a tutorial that matches your background, then complete a small file locally before attempting a larger formalization. Agda’s documentation identified itself as version 2.10.0 in this research; check the live page because editor integrations and installation instructions are version-sensitive.
9. Isabelle/HOL tutorials — a promising lead that needs current verification
A University of Illinois formal-mathematics reading list includes an Isabelle/HOL tutorial and HOL4 tutorials. That is useful evidence that these are established teaching routes, but the official Isabelle documentation could not be inspected during the available data.
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How should I choose between Lean, Rocq, Agda and Isabelle?
Choose the ecosystem that matches your intended work and the examples you want to study; no proof assistant is universally best.
| Your priority | Best first route | Why | What to watch |
|---|---|---|---|
| Formalize school or university mathematics | Natural Number Game, then Mathematics in Lean | Interactive practice followed by Mathlib’s mathematics-focused textbook | Lean/Mathlib versions must match |
| Understand dependent type theory | Theorem Proving in Lean | Explains proof development, type theory and automation | More abstract than a mathematics-first course |
| Learn logic with a gentle formal start | Logic and Proof or The Mechanics of Proof | Combines exposition with machine-checked exercises | Choose based on whether logic or mathematical pacing matters more |
| Verify programs and study semantics | Software Foundations | Rocq scripts and exercises are integrated into the free series | Follow the volume’s expected Rocq version |
| Formalize mathematics in Rocq | Rocq documentation, then Mathematical Components | Official route aimed at mathematics-background newcomers | Learn Rocq’s syntax before importing a large library |
| Build an Agda project | Agda official tutorials | System-specific setup and language reference in one place | Editor and compiler versions matter |
| Work in Isabelle/HOL | Current Isabelle documentation plus a matching tutorial | Higher-order-logic tools and established tutorial material | Verify the current release; the reading-list link is not a substitute for official docs |
Use four questions to make the choice:
- What will you formalize? Ordinary mathematics points toward Mathlib or Mathematical Components; programs and semantics point toward Software Foundations.
- What background do you have? Programming experience helps with functional-programming and verification texts, while mathematical maturity helps with algebraic formalization. Neither automatically supplies proof-assistant skills.
- How much feedback do you need? Games offer immediate, small-step feedback; textbooks provide longer exercises; reference documentation is best when you already have a concrete problem.
- Which library or project must you support? A course written for Lean/Mathlib will not teach Rocq commands, and an Agda tutorial will not prepare you for Isabelle/HOL syntax.
What should I install before starting?
Install only the system used by your chosen course and copy its version requirements exactly. You generally need the proof assistant, an editor integration, the course files and a way to run the checker locally.
- Pick one route for the first month. Do not install four systems at once; switching syntax hides whether a difficulty is conceptual or merely tooling-related.
- Read the course’s setup page. For Lean, use the current Lean and Mathlib instructions linked from the official learning pages. For Rocq, use the project documentation and the Software Foundations volume’s prerequisites. For Agda or Isabelle, use their current project documentation.
- Confirm the version in a terminal or editor. Record the exact release so an error report can include it. The Mathematics in Lean materials showed v4.34.0 in live HTML and v4.19.0 in a PDF; that discrepancy is a reason to follow the live instructions.
- Run one supplied example unchanged. If it fails before you edit anything, fix the installation or version mismatch first.
- Keep a small practice file. Re-type definitions and proofs instead of only reading rendered pages. Machine feedback is the point of these resources.
You do not need a paid plan or specialized hardware for these learning paths. A normal computer, an editor and enough time to compile examples are sufficient.
How do I study effectively when proofs keep failing?
Reduce the failing proof to the smallest goal you can explain, then ask whether the problem is mathematical, syntactic, imported-library related or version related.
- Read the goal state. Identify the variables, hypotheses and exact target. A theorem that looks obvious on paper may have a different type than you expected.
- Check imports and namespaces. A tactic or lemma may be unavailable because the example omitted an import, used a namespace qualifier or belongs to a different library version.
- Replace automation with a short proof. If a large tactic fails, prove an intermediate lemma or apply one constructor at a time. This reveals what the checker is asking for.
- Compare versions before searching. Search results often show commands for older releases. A current error may be a renamed theorem rather than a flaw in the mathematics.
- Work every exercise. Reading a completed proof gives recognition, not recall. Reconstruct it from the statement, then compare your term with the supplied solution when available.
The Lean community explicitly warns that proof assistants remain difficult and that proficiency does not arrive after one afternoon. Plan for regular sessions, expect periods of confusion and measure progress by the size and clarity of the proofs you can explain, not by how quickly you finish a chapter.
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How we picked these resources
The selection follows the distinctions made in the official learning pages and teaching projects:
- Learning goal: mathematics formalization, logic and prover foundations, programming or software verification.
- Audience: mathematical maturity, programming experience and prior exposure to logic.
- Format: interactive game, worked textbook, reference documentation or course reading list.
- System and library: Lean with Mathlib, Rocq, Agda or Isabelle/HOL, including version sensitivity.
- Practice and feedback: whether learners can edit examples, run a checker and complete exercises.
The result is a goal-based list rather than a claim that one assistant wins every category. The official sources themselves describe different audiences and purposes.
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Which path is the best fit for you?
If your immediate aim is to formalize ordinary mathematics, use the Natural Number Game only as a warm-up and make Mathematics in Lean your main course. If you are more interested in the machinery of proofs, choose Theorem Proving in Lean; if the first logic course feels too abstract, start with The Mechanics of Proof or Logic and Proof.
For verified programming, begin with Software Foundations and Rocq. For a mathematics-oriented Rocq project, follow the Mathematical Components recommendation. Agda and Isabelle/HOL are sensible choices when a project, supervisor or existing development already uses them; in those cases, current system documentation matters more than a generic ranking.
Whichever route you choose, keep the first project small, run every example locally and treat version information as part of the proof. That combination—clear goal, matching ecosystem and sustained practice—is more reliable than collecting links or changing assistants whenever a proof becomes difficult.
Frequently Asked Questions
Can I learn a proof assistant without knowing advanced mathematics?
Yes. The Mechanics of Proof and Logic and Proof provide gentler entry points, and the Natural Number Game starts with elementary arithmetic. You still need patience with formal definitions, but advanced university mathematics is not a prerequisite for learning the basic workflow.
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Is Mathlib available for Rocq, Agda or Isabelle?
Mathlib is the library associated with Lean in the mathematics route described here. Rocq, Agda and Isabelle have their own libraries and conventions, so a theorem or tactic from one ecosystem cannot usually be pasted into another unchanged.
How long does it take to become productive with Lean?
There is no reliable universal timeline. The Lean community cautions that proficiency will not come after one afternoon; regular practice with small exercises is a better plan than setting a fixed number of days.
Do I need a powerful computer to run these courses?
No special hardware is specified by the resources in this guide. You need a normal development computer, the relevant proof assistant and editor, and enough local storage and time for projects to compile.
