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A proof assistant accepting a file is not enough to show that an informal proof was faithfully formalized—or that a program implements the result. Check each link in the chain: the intended mathematics, the formal theorem, the proof term, the program specification, the implementation, and any translation or compiler stages between them. The result is only as strong as the correspondence you establish at each boundary and the assumptions you trust.
What “translated correctly” means
The phrase can describe several different checks, and they answer different questions. A mathematical argument may be written informally, encoded as a theorem in a proof assistant, represented by a checked proof term, used to specify a program, and then passed through compilers or other transformations. Correctness at one stage does not automatically establish correctness at the next.
- Faithful formalization: Does the formal theorem express the mathematical claim you intended?
- Proof validity: Does the proof assistant verify a derivation of that formal theorem from its stated assumptions?
- Implementation correctness: Does the program satisfy an explicit specification of its behavior?
- Translation correctness: Does each compiler pass, intermediate representation, or other transformation preserve the relevant meaning?
These obligations form a chain, not a single yes-or-no test. A kernel-checked proof gives strong evidence that a formal derivation follows within the assistant’s logic and assumptions. It does not by itself establish that a human encoded the intended theorem, that a program meets a specification, or that a compiler preserved behavior. Those claims require their own arguments.
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Before opening a proof assistant, write down the exact statement you want to establish in ordinary mathematical language. Include the definitions and scope that a reader might otherwise infer from context. Formalization makes omissions visible, but it cannot decide which interpretation you meant.
Make the statement specific
Record the following before encoding the result:
- Objects and domain: State whether the variables range over natural numbers, integers, real numbers, finite sequences, functions, or another domain. A claim over one domain may be false or require different hypotheses over another.
- Definitions: Write down what each term means, including any conventions for boundary cases. If a function is partial, specify what happens outside its intended input range.
- Assumptions: Include every condition used in the argument, such as positivity, finiteness, continuity, non-emptiness, or a relationship between variables.
- Quantifiers and order: Distinguish “for every input there exists an output” from “there exists one output that works for every input.” The order of quantifiers can change the theorem.
- Preconditions and exceptions: State restrictions on inputs, degenerate cases, and whether the claim includes zero, empty structures, equality at a boundary, or undefined operations.
- Conclusion: Say exactly what follows. Do not silently strengthen a claim from existence to uniqueness, from one case to all cases, or from a mathematical property to an algorithmic guarantee.
Keep this intended statement separate from the version that is convenient to prove. The formal statement may need equivalent notation or helper definitions, but every change should have a reason and should preserve the intended meaning. If the domain or definitions are ambiguous, resolve that ambiguity with the mathematical owner of the result before treating a successful proof as validation.
Review the formal theorem before trusting its proof
Encode the definitions and theorem in a selected assistant, then review the statement independently of the derivation. This is one of the most important checks: a proof can be completely valid and still prove a weaker, irrelevant, or incorrectly scoped proposition if the formal statement does not match the intended mathematics.
Compare both statements line by line
- Map each informal term to a formal definition. Verify that the encoded object has the intended meaning and that notation has not hidden a different convention.
- Match every hypothesis. For each assumption in the paper or problem statement, identify its formal counterpart. Also look for extra assumptions introduced to make the proof go through.
- Match quantifiers and domains. Check the types of all variables and the order and scope of quantifiers. Confirm that the formal theorem covers the intended range rather than only a convenient subset.
- Test the edge cases conceptually. Consider zero, empty inputs, equality boundaries, maximum or minimum values, and any exceptional cases relevant to the definitions. A formal statement that excludes a troublesome case may still be valid, but it does not establish a claim that includes that case.
- Ask a domain reviewer to check the correspondence. Have someone familiar with the mathematics compare the formal statement with the original intent, not merely inspect whether the proof is accepted.
This review addresses a boundary that proof checking does not: the human act of translating informal mathematics into a formal proposition. The Lean FAQ describes Lean’s checking of proof objects, not an automatic check that an author chose the right proposition. That distinction applies generally: the assistant checks the formal object presented to it.
Check the derivation and understand the trusted base
Once the statement is reviewed, ask the proof assistant to check the derivation. In Lean, user-written scripts and tactics produce an explicit proof term, which is then checked by a small trusted kernel. The official Lean FAQ puts it this way: “Lean works with proof objects: The user-written proof scripts and tactics are executed to produce an explicit term in the foundational logic that represents that proof, which is then passed to a small trusted component (the kernel) for verification.”
This architecture matters because a tactic script is not itself the final justification: it constructs a proof object for the kernel to check. The trust claim is still bounded. It concerns the formal theorem, its logic and assumptions, and the components relied on to produce or check the proof. A checked proof does not validate the informal-to-formal correspondence or the semantics of unrelated program code.
Practical checks for a proof artifact
- Run the assistant’s checker on the complete development, not just a fragment copied from a larger context. Confirm that the reported theorem is the one you intended to prove.
- Inspect the theorem’s assumptions and dependencies. Record axioms, imported results, and any external tools or components that affect the claim.
- For generated or untrusted proof scripts, consider checking the resulting proof object independently where the assistant supports that workflow. Lean’s FAQ and reference material describe proof-object export and independent checking.
- Preserve the exact source, tool and library versions, configuration, and proof artifact. These details make it possible to reproduce what was checked and to distinguish a later proof change from the original result.
A “proof accepted” result is meaningful only when you can identify what proposition was checked, what assumptions it used, and what checker or trusted components were involved. It is not a general certification of all surrounding code or documentation.
Specify what the program must do
A mathematical theorem and a program contract are different objects. The theorem may establish a property of a mathematical function or structure; a program is an implementation with inputs, outputs, state, and possible failures. To claim that code is correct, define the expected behavior and prove that the implementation satisfies it.
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For the relevant program or function, specify its inputs and valid input range, its outputs, its preconditions, and its postconditions. Add invariants where the implementation changes state or proceeds through repeated steps. Describe relevant failure behavior as well: whether invalid input is rejected, an error is returned, or the program makes no guarantee outside the precondition.
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Be precise about the connection to the theorem. If the theorem says a solution exists, does the program need to find one? If the theorem says a result is unique, does the implementation return that result or merely a value satisfying a weaker property? If the code uses finite machine values while the theorem uses mathematical integers or real numbers, explain how those representations correspond and what happens on overflow, rounding, or out-of-range inputs. These are specification questions; the theorem alone does not settle them.
Prove the implementation meets the contract
Formalize the contract in a verification environment and establish that the implementation satisfies it. This may involve proving properties of individual functions, invariants of an algorithm, or a relation between a program state and a mathematical model. The Rocq Prover project describes mechanized program verification against specifications, alongside mathematical formalization. The key point is not a particular tool: the specification-to-implementation proof is a separate obligation from proving the underlying mathematical result.
Keep the claim at the level actually proved. A proof about an abstract algorithm does not automatically cover a particular implementation unless the implementation is connected to that algorithm. Likewise, a proof about a function under preconditions does not guarantee behavior on inputs that violate them. Make the boundary explicit in the contract and in any report of the verification result.
Validate every translation boundary
When a proof, program, or specification moves between languages or representations, establish that the transformation preserves the property you care about. This applies to manual translations, generated verification conditions, compiler passes, and intermediate languages. It is not enough to prove the source-level claim and assume that later stages retain its meaning.
Connect source and target semantics
Describe what the source representation means and what the target representation means, then define the relation between them. A validation argument should state which programs or constructs are covered and what property is preserved. If the source and target use different notions of arithmetic, memory, exceptions, or evaluation, those differences need to be addressed rather than hidden by similar-looking syntax.
Translation validation focuses on a particular translation: a validator checks that this output corresponds to its input under the formalized semantics. One design proves a validator once and runs it on each translation. Another produces a certificate for a translation and checks that certificate. These approaches move the trust boundary in different ways. A proved validator still depends on the correctness of its proof and trusted checking path; a certificate-based approach depends on the certificate checker and on whether the certificate establishes the stated property.
The dissertation Formally Validating Translations into an Intermediate Verification Language discusses these validation approaches and their boundaries. It notes that some methods rely on trusting extracted validators or SMT solvers, and that guarantees may cover selected translations or subsets rather than every construct in a language. Therefore, record the exact transformation, supported subset, and trusted components instead of describing a validator as proving all translations correct.
Ask what the guarantee actually covers
- Is the exact translation or compiler output checked, or is only the compiler itself proved under specific conditions?
- Which source and target languages, constructs, and transformation passes are included?
- Does the validation cover the entire program, or only a generated proof obligation or selected intermediate representation?
- Are external solvers, extracted code, runtime libraries, or unverified stages included in the trusted base?
- Does the preserved property match the program contract, or merely a narrower property such as equivalence for a subset of inputs?
A practical end-to-end validation workflow
- Freeze the claim. Write the intended theorem, definitions, assumptions, domains, quantifiers, and exceptional cases in a form the mathematical owner can approve. Do not let the proof’s convenient shape silently redefine the goal.
- Formalize and review the statement. Encode the theorem in the chosen assistant. Compare it with the approved claim, including types and boundary cases, and obtain a domain review of that comparison.
- Check the proof object. Run the assistant’s checker on the complete formal development. Identify the theorem, assumptions, axioms, dependencies, and trusted components behind the result.
- Define the program contract. Specify permitted inputs, outputs, preconditions, postconditions, invariants, and failure behavior. State how the mathematical objects in the theorem map to the program’s representations.
- Prove implementation correctness. Establish that the specific implementation satisfies the contract, rather than relying on the theorem as a substitute for an implementation proof.
- Validate transformations. For each compiler pass, generated verification condition, manual translation, or intermediate representation, identify the semantics and prove or check the relevant correspondence. State coverage and limitations for each stage.
- Audit trust and scope. Record axioms, trusted tools, external solvers, language subsets, compiler stages, and unverified code. Phrase the assurance claim so it does not exceed those boundaries.
- Preserve a reproducible record. Keep the formal source, proof artifact, exact code or translation, tool and library versions, and validator or compiler configuration together. This is practical reproducibility guidance; it should not be mistaken for a guarantee supplied automatically by any proof assistant.
Choosing a tool or validation method
Lean, Rocq, and Isabelle are examples of established environments relevant to theorem proving and formal verification, but they are not interchangeable and there is no universal ranking. Choose based on the language and semantics you need to verify, the formal libraries available, the translation boundary, the assurance required, and the team’s ability to maintain the development.
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| Option | What the cited source establishes | Decision points |
|---|---|---|
| Lean | The official FAQ describes it as a functional programming language and theorem prover; explicit proof terms are checked by a small trusted kernel. | Consider the target language, library fit, proof-object checking workflow, trusted boundary, and team familiarity. |
| Rocq | The project site describes mechanized program verification against specifications and mathematical developments, with examples including CompCert and mathematical formalizations. | Consider specification style, library support, extraction or compiler boundary, and the team’s experience maintaining proofs. |
| Isabelle | The translation-validation dissertation uses Isabelle in research on formal translation validation and verification. | Check support for the semantics and transformations in scope, the proof or certificate workflow, library fit, and project requirements. |
| Translation validation | A validator checks a particular translation against formalized source and target semantics; approaches can prove a validator once or check certificates for translations. | Compare per-translation coverage, validator and checker trust, supported transformations, and the cost of producing or checking evidence. |
The relevant question is not which name is most familiar, but whether the selected environment can express the claim, model the program and transformations that matter, and provide an assurance boundary your project can explain and maintain.
What acceptance does—and does not—let you claim
A carefully checked development can support a strong, precise statement: a specified formal theorem was proved from stated assumptions; a specified implementation was proved to satisfy a stated contract; and specified translations were validated under stated semantics and coverage. It cannot support a broader claim unless the missing correspondences are also established.
For example, “the assistant accepted my proof” is not equivalent to “the informal argument was formalized faithfully.” “The algorithm is mathematically correct” is not equivalent to “this program meets its requirements.” And “the source program is verified” is not equivalent to “every compiler and runtime stage preserves the verified behavior.” Each sentence crosses a different boundary and needs evidence for that boundary.
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There is no directly comparable success-rate or error-rate statistic in the cited sources for translating mathematical proofs into code. Microsoft Research’s Lean project page reports that the community had digitized “over half of the undergraduate mathematics curriculum” and had “over one million lines of code” in its community library. Those page claims have no publication year stated there; the line count changes over time, and neither figure measures how often proof translations are correct. Library size or project examples should not be presented as a general reliability percentage.
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