The Tool Desk
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What “correctly translated” has to mean
There are three different translation questions, and they require different evidence:
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- Informal mathematics to formal theorem: Did the definitions, domains, quantifiers, assumptions, and exceptional cases in the proof assistant capture what the mathematician intended?
- Formal theorem to program specification: Does the specification describe the behavior the code is supposed to provide, rather than merely restating a convenient property?
- Specification or source program to generated code: Did a compiler, verification-condition generator, intermediate representation, or manual translation preserve the relevant semantics?
A kernel-checked proof answers only the question posed in the formal logic. If an assumption was omitted, a domain was narrowed, or a definition changed, the checker can still accept a flawless proof of the wrong claim. Likewise, a theorem about a specification is not an implementation proof until the program is shown to satisfy that specification.
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The practical test is therefore a chain of correspondence: document the intended claim, formalize and review it, prove the formal derivation, specify the program, prove implementation correctness, and validate every translation boundary that lies between the source and the executable result.
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Step 1: Freeze the intended mathematical claim
Start with a version-controlled statement written for a human reviewer, before writing proof-assistant syntax. Include every item that could alter the result:
- Definitions, including whether two apparently similar terms have different meanings.
- The type and domain of every variable: integers, real numbers, finite sequences, matrices of a particular dimension, and so on.
- Quantifiers and their scope. “For every input there exists an output” is not interchangeable with “there exists an output for every input.”
- Preconditions such as non-zero denominators, sorted input, positivity, termination measures, or dimensional constraints.
- Boundary and exceptional cases: empty collections, duplicate values, overflow, division by zero, non-termination, and invalid input.
- Whether the claim is total, partial, probabilistic, approximate, or conditioned on an external assumption.
Write a second version in plain language that explains what a successful result means. This prevents a formal statement from becoming the only interpretation of the theorem. A reviewer should be able to point to each phrase in the informal claim and find its formal counterpart.
Separate the theorem from a convenient lemma
Formal developments often prove a stronger or weaker intermediate result because it is easier to use. Label such lemmas explicitly. A proof of a restricted theorem—for example, one that assumes a list is non-empty—does not establish the advertised theorem for all lists. Keep the public claim and the assumptions required by the proof visible in the same documentation.
Step 2: Formalize the statement and review it independently
Encode the definitions and theorem in a proof assistant, then ask someone who understands the original mathematics to compare the formal statement with the human version. The reviewer should inspect the statement before concentrating on the proof script.
What to inspect in the formal statement
- Types: Check that coercions have not moved a value from exact arithmetic to an approximation, or from unbounded integers to a bounded machine type.
- Quantifier order: Read the generated goal in a form a mathematician can understand. A misplaced quantifier can make a theorem vacuous or substantially weaker.
- Definitions: Expand critical definitions at least once. A library function may have conventions about indexing, rounding, ordering, or empty inputs that differ from the informal proof.
- Assumptions: Record every axiom, imported theorem, type-class condition, and environmental assumption used by the proof.
- Edge cases: Test the statement mentally—or with small executable examples where appropriate—against zero, one, empty, maximal, and invalid cases.
Acceptance by the assistant is evidence about the formal statement, not about the act of translation from informal mathematics. Lean’s FAQ describes this boundary clearly: “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.” See the Lean FAQ for the stated workflow and independent proof-checking discussion.
Use adversarial statement review
Have a reviewer deliberately search for a way the formal theorem could be true while violating the intended result. Ask:
- Can I satisfy the assumptions with an object the informal theorem meant to exclude?
- Does the conclusion say anything useful for every permitted input?
- Did an implicit convention in the paper become an explicit but different convention in code?
- Would changing a strict inequality to a non-strict one alter the result at a boundary?
Keep the reviewer’s interpretation notes with the formal source. They are part of the assurance argument, not informal commentary to discard after the build passes.
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Step 3: Check the formal derivation
Once the statement is approved, let the proof assistant check the derivation. A successful check means the kernel accepted a proof term in its foundational logic under the imported definitions and assumptions.
Prefer small, independently checkable proof objects
Tactics and automation can generate proof terms, but the trust boundary is the component that checks those terms. Preserve the exact source, imports, tool version, library revision, and command used to run the checker. For generated or untrusted proofs, use the assistant’s independent-checking facilities when available. The Lean FAQ documents exporting and checking proof objects separately from the original proof-producing process.
Do not describe a tactic as having “verified the mathematics” without naming the kernel, logic, axioms, and libraries involved. A checker can be small and highly scrutinized while the surrounding environment remains large. The meaningful claim is: this specific kernel accepted this proof term under these formal assumptions.
Check for accidental axioms and opaque shortcuts
Audit the dependency graph for:
- Explicit axioms introduced by the project.
- Imported theorems whose own assumptions are not acceptable for the intended assurance level.
- Admitted, sorry-like, or otherwise unchecked declarations.
- External decision procedures, SMT solvers, plugins, or native code whose results are trusted rather than reconstructed and checked.
- Definitions that hide partiality or non-computability behind an abstract interface.
Whether an assumption is acceptable depends on the project. The important point is to state it, not to imply that kernel checking removes every other trust dependency.
Step 4: Write a separate contract for the program
Translate the mathematical result into a program contract instead of assuming the theorem is already a code specification. The contract should state:
- Valid input representation and preconditions.
- Output representation and postconditions.
- State changes, side effects, and resource limits that matter.
- Loop or recursion invariants.
- Failure behavior, including errors, exceptions, partial results, and non-termination conditions.
- Numerical semantics: exact, fixed-point, floating-point, bounded integer, or approximate.
For an algorithm that computes a mathematical function, define the relationship between the machine representation and the mathematical value. For example, a theorem over arbitrary integers does not automatically cover 32-bit signed arithmetic with wraparound. You need either a range proof showing overflow cannot occur or a specification that includes the machine behavior.
Prove implementation correctness
Establish a refinement statement such as: for every input satisfying the precondition, execution of the implementation terminates with an output satisfying the postcondition. If the implementation is intentionally partial, state the permitted non-termination or failure cases.
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The Rocq Prover project describes mechanized program verification against specifications and mathematical developments, with examples including CompCert and formalized mathematics. That kind of development treats the program contract and the implementation proof as obligations distinct from proving an abstract theorem.
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Many projects contain several translations: a handwritten algorithm becomes a formal model; the model becomes verification conditions; source code becomes an intermediate language; an optimizer transforms that representation; and a compiler emits machine code. Draw this pipeline and mark exactly where a semantic correspondence is established.
Define source and target semantics
For each boundary, specify what a program means on both sides. A correspondence may preserve:
- Functional results only.
- Results plus termination.
- Exceptions, I/O, timing, memory safety, or concurrency behavior.
- Numerical behavior under a particular rounding mode.
Do not claim more preservation than the relation defines. If the validator covers only a language subset, list unsupported constructs and ensure the input really belongs to that subset.
Choose a validation design
Formal translation-validation work commonly distinguishes two designs:
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1Clear out junk files and repair common Windows errors2Scan for outdated or missing drivers - takes under a minute3Repair Windows errors before they cause bigger problems| Design | How it works | Trust and coverage questions |
|---|---|---|
| Proved validator | A validator is proved once, then checks each particular translation against formal source and target semantics. | Was the validator proof itself checked? Which transformations and language features does it cover? Is the validator implementation extracted or otherwise trusted? |
| Checked certificate | The translation produces a certificate or proof object, and a checker validates that artifact for each run. | What does the certificate establish? Is the checker independent of the translator? Are failed or malformed certificates handled safely? |
The dissertation Formally Validating Translations into an Intermediate Verification Language discusses these boundaries, including the fact that some approaches trust extracted validators or SMT solvers and that guarantees may apply only to selected translations or subsets. State those limits in the project’s assurance claim.
Validate manually translated proofs and algorithms
If a person converted a proof into code, treat the conversion as a translation boundary even without a compiler. Write a relation between mathematical objects and program data, then prove that each significant operation preserves that relation. Review the correspondence with someone who did not write the original proof or implementation. Independent review is especially valuable for indexing, termination arguments, and exceptional cases, where a mathematically sound idea is often altered by representation details.
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Step 6: Audit the trusted computing base and scope
Document what must be trusted for the final claim to hold:
- The proof assistant’s kernel and foundational logic.
- Compiler, interpreter, runtime, and operating-system assumptions relevant to execution.
- Imported libraries and any axioms they use.
- External solvers, native plugins, code generators, and extracted validators.
- The formal semantics of each source and target language.
- The exact language subset, optimization passes, and hardware model covered.
A smaller checker can make proof objects easier to audit, but “small kernel” is not the same as “no assumptions.” The claim should identify the checker boundary and everything outside it that could invalidate the result.
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Step 7: Preserve a reproducible artifact
Archive a complete bundle rather than only a screenshot of a successful build:
- The human-readable theorem and definitions.
- The formal source, proof object or certificate, and dependency manifest.
- Proof-assistant, library, compiler, solver, and validator versions.
- The exact code, intermediate representation, and translation configuration checked.
- Scripts that reproduce checking, including command-line options and environment settings.
- A written scope statement listing assumptions, unsupported features, and known unverified components.
Run the checker from a clean environment and record whether it reproduces the result. If a dependency changes, re-check rather than assuming compatibility. A dated artifact makes it possible to distinguish a theorem that remains valid from a build that merely happened to pass in an undocumented environment.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to review the result before calling it correct
| Question | Evidence you should require |
|---|---|
| Is this the intended theorem? | Human-readable statement, formal statement, and independent domain review showing definitions, domains, quantifiers, and edge cases correspond. |
| Is the derivation valid? | A proof term or certificate accepted by the stated kernel and a record of axioms and imported results. |
| Does the code meet the theorem’s promise? | A program specification plus a proof that the implementation satisfies it under stated machine and runtime assumptions. |
| Did translation preserve meaning? | Formal source and target semantics, a proved validator or checked certificate, and an explicit coverage statement. |
| Can another party verify it? | Reproducible sources, versions, commands, and independently checkable artifacts. |
Common false positives and how to correct them
“The assistant accepted it, so the informal proof was translated correctly.”
Acceptance proves only the formal theorem. Reopen the statement, compare it line by line with the intended mathematics, and obtain an independent domain review.
“The theorem is true, so any implementation of the idea is correct.”
Prove an implementation contract. Check representation invariants, overflow, indexing, mutation, termination, and failure behavior. A mathematical function and a machine procedure are different artifacts.
“The compiler produced code, so semantics were preserved.”
Require a compiler-correctness theorem, a translation validator, or a checked certificate covering the actual source features and optimization passes. A guarantee for an intermediate-language subset does not automatically cover arbitrary source code.
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“Automation found a proof, so the solver is part of the mathematics.”
Identify whether the solver’s result is reconstructed by the kernel or trusted directly. If it is trusted, include the solver in the trusted base and record its version and configuration.
“A passing example demonstrates correctness.”
Examples and tests can expose a bad statement or implementation, but they do not replace a proof of the quantified contract. Use them as review aids and regression checks, not as the formal guarantee.
Choosing a proof assistant or validation approach
Lean, Rocq, and Isabelle are not interchangeable rankings. Choose according to the language and semantics you need to model, the available libraries, the translation boundary, the acceptable trusted base, and the team’s ability to maintain the development.
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| Option | Established role | Decision factors |
|---|---|---|
| Lean | A functional programming language and theorem prover whose explicit proof terms are checked by a small trusted kernel. | Target language, mathematical and software libraries, proof-object workflow, independent checking, and team familiarity. |
| Rocq | Mechanized program verification against specifications and mathematical developments; the project highlights CompCert and formalized mathematics. | Specification style, libraries, extraction or compiler boundary, supported semantics, and team experience. |
| Isabelle | Used in research on formal translation validation and verification. | Supported semantics, proof and certificate workflow, library fit, and project-maintenance skills. |
| Translation validation | A method for checking a particular translation against formalized source and target semantics, using a proved validator or checked certificates. | Per-translation coverage, validator trust, supported transformations, certificate cost, and failure handling. |
The right choice is the one that lets you state and check the relevant semantics with an assurance boundary your team can explain. Tool popularity or a large library does not, by itself, prove that a particular translation is faithful.
What a successful check does—and does not—establish
A complete result can support a precise statement such as: “Under these definitions, axioms, machine assumptions, and language-subset restrictions, this checked proof establishes the formal theorem, and this separately checked implementation or translation satisfies the stated correspondence.” That is substantially stronger than saying “the code came from a proof.”
There is no generally applicable success-rate or error-rate percentage for translating mathematical proofs into code established by the sources above. Project size, library line counts, or prominent case studies should not be converted into a reliability statistic. Assurance comes from the explicit chain of statements, proofs, semantic relations, and trust assumptions for the artifact being checked.
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