A proof assistant verifies a mathematical argument by checking a formal proof term against a formal proposition. In Lean 4, you state the objects, assumptions, and conclusion precisely; interactive tactics help construct a proof; then Lean’s small kernel checks the resulting term. The check establishes that the formal statement follows from its assumptions under Lean’s foundations—not automatically that the formal statement captures everything you meant in ordinary mathematical language.
This guide uses Lean as the main tutorial path, with pointers to the official Mathematics in Lean and Theorem Proving in Lean materials. It also explains how Lean differs in broad outline from Rocq and Isabelle/HOL, and how to review a formally accepted theorem for semantic mistakes.
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What a proof assistant actually verifies
An informal proof is written for human readers. It may rely on familiar definitions, omit routine steps, or leave the scope of a word such as “number,” “continuous,” or “function” to context. A proof assistant cannot rely on that context. You must provide a formal proposition whose types, definitions, hypotheses, and conclusion are explicit.
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For example, the informal sentence “if P and Q are true, then Q and P are true” can be made completely explicit:
theorem and_swap (P Q : Prop) : P ∧ Q → Q ∧ P := by
intro h
exact ⟨h.2, h.1⟩The declaration introduces two propositions, states an implication, accepts a proof of the premise as h, and builds the required conjunction by taking its second and first components. The proof is not accepted because it looks plausible; it is accepted only if the resulting term has exactly the type Lean expects.
Lean’s FAQ describes the validation boundary this way: scripts and tactics produce an explicit proof term in the foundational logic, and a small trusted kernel checks that term (Lean FAQ). Tactics are therefore development tools, not a replacement for checking. A tactic may search, simplify, or automate a sequence of steps, but the kernel still verifies the proof object that remains.
Choose a learning route before writing the theorem
The official Lean learning page separates two complementary introductions. Theorem Proving in Lean focuses on Lean’s logical framework and language. Mathematics in Lean teaches interactive mathematical formalization on top of Mathlib. The same page also points beginners toward the Natural Number Game (Lean 4 documentation and learning resources).
For a reader whose goal is to formalize ordinary mathematics, Mathematics in Lean is the most direct route. Its introduction identifies the project as a hands-on course in expressing mathematics and proving results with Lean and Mathlib. The page currently identifies itself as version 4.19.0; Lean, Mathlib, editor integration, and tutorial files can change, so use the current instructions shown by the project when you begin.
Set up a reproducible Lean project
Keep theorem statements and proof code in project files rather than relying on an unrecorded interactive session. A project gives Lean a repeatable environment: the same source files, imports, definitions, and library versions can be checked again by you or another reader.
- Install Lean 4 and Visual Studio Code using the tutorial’s current instructions. Do not copy installation commands from an old post when the official project provides newer ones.
- Open the Mathematics in Lean project in VS Code. Work from the chapter files supplied by the project so that imports and supporting definitions match the exercises.
- Open a chapter file and run its examples. Lean’s editor feedback reports whether expressions are well formed, displays the current goal, and identifies errors near the code being edited.
- Save your own theorem in a project file. A saved declaration can be checked again after reopening the project, changing an import, or updating a proof.
- Keep the environment visible when sharing a result. A theorem can depend on imported definitions and lemmas, so retain the project context needed to reproduce the check.
The setup workflow and chapter-based exercises are described in the introduction to Mathematics in Lean. If your local editor reports a version or dependency mismatch, return to the project’s current setup instructions instead of silently changing declarations until they compile.
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Translate an informal argument into a formal proposition
Formalization is the central mathematical task. The assistant cannot decide which meaning you intended; you must choose the types and assumptions that make the claim precise.
Identify the objects
List every object in the informal argument and assign it a type. “A number” could mean a natural number, integer, rational, real, or an element of another algebraic structure. “A function” needs a domain and codomain. A statement about a sequence needs an index type and a rule assigning terms.
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Expose hidden assumptions
Rewrite phrases such as “for every,” “there exists,” “nonzero,” “positive,” or “is defined” as explicit hypotheses. If a cancellation step requires a denominator to be nonzero, include that condition. If the intended domain excludes a boundary case, state the restriction rather than relying on a convention a reader may not share.
Separate definitions from the claim
Introduce the objects and predicates that the theorem uses, then state the result in one proposition. This separation makes it easier to determine whether a failure comes from a definition, an imported lemma, or the final argument.
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Informal prose often compresses “if,” “only if,” and “if and only if.” In Lean, these are different propositions. Likewise, a theorem proved for one fixed object is not automatically a theorem for all objects. Read every binder and connective in the generated goal.
A small example illustrates the translation process without hiding the logical structure:
theorem double_zero (n : Nat) : n + 0 = n := by
simpThe declaration says exactly which numbers are involved, what operation is used, and what equality must be proved. The simplifier supplies the routine derivation, but Lean checks the resulting proof term.
Use the library before rebuilding mathematics
Mathematics in Lean is built on Mathlib, a large collection of definitions and proved results. Reusing a suitable lemma usually produces a clearer and more maintainable proof than reproving a standard fact. The basics material and the project documentation describe ways to inspect available definitions and search for existing theorems (Mathematics in Lean, Chapter 2).
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- Inspect the candidate statement. Confirm its namespace, argument order, typeclass assumptions, and hypotheses. A theorem with a similar name may apply to a different number system or a stronger condition.
- Check imports. A declaration may exist in Mathlib but remain unavailable until the file imports the module that defines it.
- Test the smallest use. Apply the lemma in a short example or at the current goal. Read the resulting goal instead of assuming that elaboration inferred the intended arguments.
- Record why the lemma applies. In a larger development, a brief comment or a descriptive intermediate name can preserve the mathematical reasoning for future readers.
Library search is not merely a shortcut. It helps you align your formalization with established definitions and discover that a result may already have a more general, better-structured version.
Develop the proof interactively
Lean’s editor feedback supports a loop of proposing a step, reading the new goal, and revising the proof. Begin with the mathematical structure rather than trying to guess a single automation command.
Introduce assumptions and variables
For an implication, introduce its premise. For a universally quantified statement, introduce the arbitrary object. The local context then shows exactly what is available.
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Split structured goals
A conjunction requires both components; an existential goal requires a witness and a proof for that witness; an equality may become simpler after rewriting. Use proof steps that mirror those logical constructors so that the remaining goal stays understandable.
Apply known results
When a library theorem or earlier local lemma matches the goal, apply it and inspect any new side conditions. Those side conditions are often where an informal proof had hidden an assumption.
Automate only the routine part
Simplification and other tactics can discharge repetitive algebraic or logical steps. Treat their output as a proof-construction aid. The kernel does not trust the tactic merely because it ran; it checks the term the tactic produced.
For the conjunction example, the tactic proof and a direct term express the same logical construction:
theorem and_swap_term (P Q : Prop) : P ∧ Q → Q ∧ P :=
fun h => And.intro h.2 h.1The first version is often easier to develop interactively; the second makes the underlying term explicit. Both are accepted only when their types match the theorem statement.
Validate a Lean proof correctly
“How do I validate a Lean proof?” is a question addressed directly in the official Lean FAQ. Validation has several layers, and a green editor message covers only one of them.
1. Confirm that Lean accepts the declaration
Check the file in the intended project environment and ensure that the theorem has no errors or unsolved goals. A proof that works only after accidentally importing a broad module or relying on an undeclared local definition is not a reliable standalone result.
2. Understand what was checked
Lean transforms your proof script and tactics into a proof term in its foundational logic. The small kernel checks that term against the proposition. This is the machine-checkable derivation you can report.
3. Review the statement independently
Kernel acceptance does not establish that your formal proposition says what the informal theorem said. The Lean community’s “Did You Prove It?” guidance emphasizes checking the correspondence between the intended mathematical claim and the Lean statement, especially for a theorem not already stated in Mathlib (Did You Prove It?).
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4. Audit definitions and imported results
Read the definitions used by the theorem. A predicate can be weaker or stronger than the phrase chosen in an informal write-up. Also inspect important imported lemmas and the assumptions under which they were proved. Your theorem is only as informative as those formal foundations and dependencies.
5. Check boundary cases
Try values or structures that informal intuition might overlook: zero, one, empty collections, equal endpoints, negative values when the type permits them, or degenerate dimensions. A theorem that compiles but becomes vacuous under an unintended hypothesis needs a statement revision, not merely a shorter proof.
6. Preserve a reproducible record
Keep the source file, imports, project configuration, and version information together. If another checker or an independent build can verify the proof term, that provides additional confidence in the checking process, while still leaving the statement-to-meaning review as a human responsibility.
“In return, Lean provides feedback and information, interprets expressions and guarantees that they are well-formed, and ultimately certifies the correctness of our proofs.”
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Here “correctness” is correctness of the formal proof and formal statement under Lean’s foundations and trusted checker. It is not an automatic certification that an informal paragraph was translated without omission.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Understand the trust boundary
A proof assistant reduces the amount of software and reasoning that must be trusted, but it does not eliminate trust assumptions. Lean’s kernel, its foundational logic, the definitions and axioms available in the environment, and the imported library results form the context in which the proof is certified.
- Kernel checking: the kernel verifies that the proof term has the declared proposition as its type.
- Proof-producing tactics: tactics may search or automate, but their produced terms are still checked by the kernel.
- Formal foundations: the result is relative to Lean’s logic and the axioms or principles used by the development.
- Statement fidelity: humans must decide whether the proposition models the intended informal theorem.
- Dependency review: imported definitions and lemmas should be appropriate, understood, and available in the claimed environment.
This explains why “the file compiles” and “the mathematical claim was correctly formalized” are related but distinct conclusions.
Lean, Rocq, and Isabelle/HOL: choose by project needs
The evidence supports a high-level foundational comparison, not a universal ranking. Lean and Rocq (formerly Coq) have much in common through dependent type theory, although their technical details differ. Isabelle/HOL is based on higher-order logic and the LCF approach. The Lean FAQ discusses these distinctions.
| System | Broad foundation described in the sources | Questions to answer before choosing |
|---|---|---|
| Lean | Dependent type theory; proofs are checked by a small kernel. | Does Mathlib cover the target mathematics? Do the Lean tutorials and editor workflow fit the team? |
| Rocq | Dependent type theory, with technical differences from Lean. | Do the project’s existing developments and community resources target Rocq? |
| Isabelle/HOL | Higher-order logic using an LCF-style approach. | Does the project’s intended logic, library coverage, and proof workflow match Isabelle/HOL? |
For a new mathematical formalization, compare the actual definitions and maintained libraries needed by the project, the learning material available to contributors, the editor and build workflow, and the requirements of any existing codebase. Do not infer usability, library size, or automation quality from the foundational label alone.
Common failure modes and fixes
The theorem is weaker than the informal claim
Symptom: Lean accepts a statement, but it applies only to a special case or becomes trivial because a strong hypothesis was added accidentally.
Fix: Rewrite the informal claim in plain language, compare every quantifier and hypothesis with the Lean declaration, and test boundary cases. Follow the statement-review questions in “Did You Prove It?” rather than changing the proof tactic.
The theorem is stronger than intended
Symptom: A definition silently includes objects that the informal argument excluded, such as zero denominators or empty structures.
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A familiar lemma does not apply
Symptom: The theorem search finds a result with a similar name, but Lean leaves unsolved typeclass or side-condition goals.
Fix: Inspect the lemma’s full type, verify the imported module, and prove or supply each required hypothesis. Check whether your object belongs to the structure the lemma expects.
Automation hides the mathematical reason
Symptom: A tactic closes the goal, but you cannot explain which theorem or assumptions made the step valid.
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The proof works in one file but not another
Symptom: Moving the theorem changes elaboration or produces missing-identifier errors.
Fix: Compare imports, namespaces, local variables, notation, and project configuration. Keep a minimal reproducible file and retain the exact environment used for the accepted proof.
A practical checklist for a formally verified argument
- Have you chosen the correct Lean type for every mathematical object?
- Are all quantifiers, definitions, domain restrictions, and nonzero or positivity assumptions explicit?
- Does the formal statement match the informal theorem in both scope and direction?
- Did you search Mathlib before re-proving a standard result?
- Can you explain the purpose of each major proof step or imported lemma?
- Does the project check the file without errors or unsolved goals?
- Have you tested boundary and degenerate cases?
- Have you recorded the project environment and relevant version information?
- Have you distinguished kernel acceptance from the human review of statement meaning?
Following this process turns a mathematical argument into a reproducible, machine-checked artifact while keeping the essential human responsibility in view: deciding exactly which claim the formal statement should express.
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