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Lean verifies a mathematical proof by checking a machine-generated proof term against a formally stated proposition. Tactics and automation help construct that term; elaboration translates convenient syntax into Lean’s core dependent type theory; a small trusted kernel then checks that the term has exactly the theorem’s declared type. A green checkmark therefore means “this formal statement follows from these definitions, imported declarations, and axioms”—not automatically that it captures the informal theorem a mathematician had in mind.
What Lean is actually verifying
Lean treats a theorem as a typed declaration. The statement is a proposition, and a proof is a term that inhabits that proposition. If the proposition is written as p, Lean’s task is to construct some term t for which the type checker can establish t : p.
This is the propositions-as-types viewpoint, usually associated with dependent type theory. Logical connectives receive type-theoretic interpretations. For example, an implication p → q behaves like a function type: a proof is a function that accepts evidence of p and returns evidence of q. A proof of p → p can therefore be the identity function, taking a proof of p and returning it unchanged.
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theorem identity (p : Prop) : p → p :=
fun h => hThe text above is not merely an explanation for a human reader. It denotes a term. Lean checks that the function’s input has type p, that its result also has type p, and that the whole function has the requested type p → p.
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That distinction matters. Lean does not decide whether a paragraph of ordinary mathematical prose sounds convincing. It checks a precise formal object against a precise type. Definitions, notation, quantifiers, structures, and imported theorems determine what the proposition means inside the Lean environment.
The four stages from source code to a checked theorem
A proof usually passes through four related stages. Keeping their responsibilities separate makes the system’s trust boundary easier to understand.
| Stage | What happens | What it does not establish |
|---|---|---|
| Lean source | You write a declaration, a direct term, or a tactic script beginning with by. |
The surface syntax itself is not yet a proof accepted by the kernel. |
| Elaboration | Lean resolves implicit arguments, notation, overloaded operations, expected types, and tactic commands, producing core expressions. | Successful elaboration is not the final logical certificate. |
| Proof construction | Tactics transform goals and local assumptions, assembling a term in the core theory. | A tactic’s own success does not bypass type checking. |
| Kernel checking | The trusted kernel checks the resulting term and confirms that it has the theorem’s declared type in the current environment. | It does not verify that the formal statement matches an informal intention or that every imported assumption is philosophically acceptable. |
1. You write a proposition and a proof
A proof can be a term written directly, as in the identity example, or a tactic script:
theorem identity (p : Prop) : p → p := by
intro h
exact hAfter intro h, the proof state contains a local assumption h : p and the remaining goal p. The exact h tactic closes that goal by supplying the assumption as the required term.
2. Elaboration turns friendly syntax into core expressions
Lean’s surface language is intentionally convenient. You can omit arguments that the expected type determines, use notation such as +, write pattern matches, and invoke tactics. The elaborator fills in implicit information, resolves names and type-class instances, checks that overloaded notation is meaningful, and constructs expressions in Lean’s smaller core language.
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Tactic execution is part of this elaboration process. A tactic reads the current proof state—goals plus local hypotheses—and asks Lean to build expressions that make progress. If a tactic cannot produce a valid expression for the goal, it fails or leaves goals unsolved.
3. Tactics construct, rather than replace, proof terms
Commands such as intro, exact, apply, rw, and arithmetic or algebraic automation are proof-building programs. They may split one goal into several, rewrite it using an equality, invoke a theorem, or search for a combination of lemmas. Their output is still an ordinary term in the core theory.
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4. The kernel checks the resulting term
The kernel performs the final type check. It verifies that every application supplies an argument of the required type, that dependent types line up, that variables are used within scope, and that the complete term has the theorem’s declared proposition. It also checks the definitions and declarations on which the term depends according to Lean’s core rules.
The kernel is deliberately much smaller and more rigid than the surrounding tooling. Editors, parsers, elaborators, tactics, decision procedures, and libraries make Lean usable, but acceptance ultimately depends on the kernel’s check of the generated term.
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Why “propositions as types” gives the checker a certificate
In ordinary mathematics, a reader follows a chain of definitions and inferences expressed in natural language and notation. In Lean, that chain is represented by a term whose type is the proposition being proved. Type checking is therefore a local, mechanical question: does this term inhabit this type under the declarations currently in scope?
For an implication, the term is function-like. For a conjunction, it contains evidence for both components. For an existential statement, it includes a witness together with evidence that the witness satisfies the property. Equality proofs connect terms that Lean’s conversion and equality rules recognize as equal. Dependent types allow later parts of a type to refer to earlier values, which lets specifications carry precise data-dependent constraints.
This certificate can be checked without replaying the tactic search or trusting the tactic’s explanation. A tactic may have used a complicated search procedure, but the stored result is a term that can be checked independently. Lean’s tactic documentation also describes proof terms as independently checkable evidence; an external checker can, in principle, validate the term against the same core rules.
What a green checkmark establishes—and what it does not
What it establishes
- The elaborated proof term has the proposition declared by the theorem.
- The term is valid under the definitions, prior declarations, imported libraries, and axioms in the current environment.
- The kernel’s core typing rules accepted every dependency needed for that term.
What it does not establish by itself
- That the formal proposition expresses the informal statement you intended.
- That a definition models the intended mathematical object rather than a subtly different one.
- That every theorem imported from a library was stated with the assumptions you expected.
- That the result is unconditional if the environment contains additional axioms or incomplete proofs.
The most precise interpretation is: Lean’s kernel checked a proof of this formal statement from these declarations and assumptions. A human still has to review the formalization, the scope of its definitions, and the dependencies that give the statement its meaning.
Axioms, incomplete proofs, and dependency inspection
An axiom declaration postulates an inhabitant of a proposition instead of deriving one. Any theorem that uses that declaration is valid relative to the axiom. This is not necessarily a defect—mathematics routinely studies consequences of explicit assumptions—but the assumption must be visible when describing what has been verified.
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The extreme case makes the issue clear:
axiom impossible : False
theorem anything (p : Prop) : p := False.elim impossibleIf an environment assumes a proof of False, the eliminator for False can produce a proof of any proposition. The kernel can check the second theorem perfectly; the theorem is nevertheless conditional on the extra axiom.
Lean provides the command #print axioms for examining a theorem’s axiom dependencies. Use it after defining or importing a result:
#print axioms identityThe validation guidance in the Lean 4 documentation explains that sorryAx indicates an incomplete proof. Custom axioms likewise mean that the theorem is established relative to those assumptions. The exact list and presentation of built-in assumptions, and how certain automation is represented, can vary with the Lean version, so inspect the output in the version you are using rather than relying on an old screenshot or blanket claim.
Can a tactic bug make Lean accept a false theorem?
Under Lean’s intended trust model, an ordinary tactic bug cannot make the kernel accept an invalid term. The tactic might construct the wrong expression, but the kernel will reject it if its type does not match the goal. This separation is one of the central reasons to distinguish tactic programs from the kernel.
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That answer has boundaries. A theorem can still be surprising or mathematically inappropriate when its formal statement is wrong, when an imported theorem was misunderstood, or when an axiom or incomplete proof supplies an assumption that was not disclosed. The kernel protects the soundness of the core type theory relative to its environment; it does not audit the meaning of every declaration or the intentions of the person who wrote it.
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Lean is extensible. The Lean 4 system description by Leonardo de Moura and Sebastian Ullrich (2021) discusses extensions and examples of automation in mathlib, including a ring-theory solver and a Presburger-arithmetic decision procedure. Such procedures are valuable because they construct proof terms efficiently. Their implementation can be complex without becoming part of the trusted logical foundation, provided the resulting term is checked by the kernel.
The human trust boundary
It is misleading to summarize formal verification as “the computer proved the theorem” without qualification. Four questions remain relevant:
- What exactly is the statement? Read the types, quantifiers, hypotheses, and definitions, not only the theorem’s informal name.
- Which declarations does it use? Imported lemmas and definitions determine the environment in which the proof type-checks.
- Which axioms are present? Run
#print axiomsand investigate any custom assumptions orsorryAxdependency. - Does the formal model match the intended mathematics? Check that encodings of numbers, functions, limits, structures, and side conditions represent the claim you care about.
These checks complement, rather than compete with, kernel verification. The kernel supplies a precise mechanical guarantee; human review supplies semantic and foundational judgment.
A practical workflow for checking a Lean theorem
- Read the declaration. Identify the proposition after the colon and list its explicit hypotheses.
- Inspect the proof style. Determine whether the proof is a direct term or a
byblock whose tactics transform goals. - Follow the proof state. After each tactic, note which goals remain and which local assumptions are available. This reveals what the script actually proves.
- Check the environment. Confirm the Lean version, imported modules, namespace, notation, and type-class instances used by the declaration.
- Inspect dependencies. Run
#print axioms theoremNameand investigatesorryAxor custom axioms. - Review the formalization. Compare the theorem’s definitions and hypotheses with the intended informal claim. A successful check cannot repair a mismatch here.
For current behavior, consult the Lean Language Reference for the Lean release you are running. The reviewed reference identified version 4.35.0-rc4, while the maintained validation documentation includes version-specific notes around Lean 4.29.0 and 4.35.0. Trust guidance can change with releases, so version-sensitive details should always be checked against the documentation installed or published for your version.
The shortest accurate mental model
Think of Lean as a compiler-like pipeline with a strict final gate: you state a proposition; elaboration and tactics construct a core term; the kernel checks that the term inhabits the proposition; dependency inspection tells you which assumptions made that check possible; and a human decides whether the proposition and those assumptions express the mathematics intended.
That division of labor explains both Lean’s power and its limits. Automation can search and build evidence, while the kernel independently checks the evidence. The resulting guarantee is exact but conditional: it concerns the formal statement in its formal environment, not an unexamined translation from informal mathematical language.
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