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Proof Export

A Verum certificate is not a terminal artifact. It can be exported to Coq, Lean, Dedukti, or Metamath for cross-tool verification, long-term archival, and participation in the broader proof assistant ecosystem. This page specifies the certificate envelope, the export targets, and the round-trip guarantee each target offers.

Proof export turns a "Verum says this theorem is proved" claim into "here is the proof; check it yourself" — a social, not just technical, transition. When the exported proof re-verifies in Lean, the chain of trust extends from Verum's small audit-able kernel through Lean's independent kernel, providing cross-language cross-validation.

note

Statement-level export ships for all four targets — Lean / Coq / Dedukti / Metamath — via verum export --to <format> or the verum export-proofs --to <format> alias. Each target emits the theorem's signature and framework attribution as an admitted-proof scaffold; the target's own tactics close the proof on the re-check side. A weekly cross-tool re-check matrix exercises every target against its reference verifier.

Full proof-term export — each step of the kernel witness re-serialised in the target's native proof-step grammar — is the natural extension, gated on rule-specific conclusion types in the kernel's proof-tree replay (see Trusted kernel §5.2).

1. What gets exported​

Everything the kernel admits has a CoreTerm representation (§3 of Trusted kernel). Export is a printer from CoreTerm values to target-language source. Three granularities:

  1. Theorem skeleton. The theorem's signature (name, parameters, proposition) and stated framework dependencies. Re-checking in the target reduces to proving the same goal with the target's own tactics.
  2. Proof term. The full CoreTerm tree, with every step replayed in the target. Target re-checks by invoking its own kernel.
  3. SMT certificate. The underlying SMT-LIB obligation plus the solver's own proof-tree (native (proof …) or ALETHE), with the target invoking a solver-agnostic proof-checker (dkcheck, smtlib2proof, etc.).

The exporter picks the highest granularity the target supports. Lean imports theorems and proof terms; Dedukti imports SMT proof trees; Metamath imports theorem skeletons and framework axioms.

2. The certificate envelope​

Every exported proof ships inside a certificate envelope — a JSON structure that carries the proof plus all metadata needed for re-checking:

{
  "schema_version":     1,
  "verum_version":      "0.1.0",
  "kernel_version":     1,
  "obligation_hash":    "4f3a…",          // blake3 of the SMT obligation
  "source_span": {
    "file": "core/math/arith.vr",
    "line": 42, "col": 5,
    "end_line": 44, "end_col": 10
  },
  "theorem_name":       "safe_div_bounds",
  "proposition":        "forall a b: Int. b != 0 -> result == a / b",
  "framework_deps": [
    { "framework": "smt-backend", "rule": "smt_unsat", "citation": "obligation_4f3a…" },
    { "framework": "lurie_htt", "citation": "HTT §6.2.3" }
  ],
  "solver": {
    "backend": "smt-backend",
    "version": "4.12.2",
    "flags":   ["--timeout=60s", "--proofs=true"],
    "duration_ms": 43
  },
  "proof_term":         "…base64-encoded CoreTerm…",
  "proof_term_format":  "verum_core_term_v1",
  "smt_certificate":    { /* optional — if smt_unsat route used */ },
  "exporter_metadata": {
    "exported_at":   "2026-04-24T11:30:00Z",
    "exported_by":   "verum export-proofs --to lean",
    "signed_hash":   "…blake3 of envelope excluding this field…"
  }
}

2.1 Schema-versioning policy​

schema_version = 1 is stable across all 0.x Verum releases. Schema-breaking changes bump the version; readers negotiate with the schema_version field. The envelope format lives in the verum_core::certificate module so consumers can depend on it independently of the compiler.

2.2 Integrity​

The exporter_metadata.signed_hash is a blake3 hash over the envelope with the signed_hash field blanked — it protects against in-transit tampering of the metadata (the proof_term itself is content-addressed by obligation_hash, so tampering with the proof requires re-solving the problem).

For offline audit, the obligation_hash is a separately computed blake3 of the canonicalised SMT-LIB obligation (what the compiler asked the solver to solve). Any divergence between envelope and regenerated obligation is a mismatch and the envelope is rejected by the importer.

2.3 Framework dependency cone​

framework_deps lists every @framework-tagged axiom in the proof's transitive cone. For the importer, these are hypotheses that must be re-declared (or trusted verbatim) in the target system. Running

verum audit --framework-axioms --cone <module>

produces exactly this list.

3. Export targets​

3.1 Lean 4​

Granularity: proof term (full). Re-check path: Lean's own kernel. Coverage: statement-level emission ships; full proof-term emission is gated on kernel-side rule-specific conclusion types.

Export to .lean translates CoreTerm to Lean's Expr structure directly:

Verum CoreTermLean
Var(i)Expr.bvar i
Lam { name, ty, body }Expr.lam name ty body BinderInfo.default
App { func, arg }Expr.app func arg
Pi { name, ty, body }Expr.forallE name ty body .default
Inductive { path, args }Expr.const path [levels] applied to args
Ctor { ind, name, args }Expr.const "${ind}.${name}" …
Eq { ty, lhs, rhs }@Eq ty lhs rhs
Refl { ty, term }@rfl ty term
SmtProof { cert, claim }axiom smt_cert_<hash> : …
Axiom { name, ty, framework }axiom <name> : <ty>

Framework axioms are emitted as Lean axiom declarations; users can import Mathlib.X and #check to verify the theorem re-derives under their preferred framework formalization.

3.2 Coq​

Granularity: proof term. Re-check path: Coq's kernel. Coverage: statement-level emission ships; full proof-term emission is gated on kernel-side rule-specific conclusion types.

Export to .v uses the same CoreTerm → Expr mapping adapted for Coq's Gallina syntax. The main differences from Lean:

  • Cumulative universes are handled via Type_i annotations.
  • Cubical primitives (HoTT layer) require the HoTT plugin; a fallback path proves the axiom ua opaquely.
  • @framework axioms become Axiom <name> : <ty>.

3.3 Dedukti​

Granularity: SMT certificate (for smt_unsat traces). Re-check path: dkcheck rewrite system. Coverage: statement-level emission ships; full proof-term emission is gated on kernel-side rule-specific conclusion types.

Dedukti is a shallow logical framework — it is the best target for exporting SMT proofs because the .dk format is explicitly designed as an interchange for automated theorem provers. Each adapter proof rule has a Dedukti signature; the exporter re-plays the proof tree as applications of those signatures.

3.4 Metamath​

Granularity: theorem skeleton + framework axioms. Re-check path: community .mm database. Coverage: statement-level emission ships; full proof-term emission is gated on kernel-side rule-specific conclusion types.

Metamath is optimized for foundational certainty: its proofs are low-level, long, and completely explicit. Full proof-term export to Metamath is impractical for SMT-discharged theorems (would produce megabyte-size proofs for trivial obligations), so the exporter emits only the theorem statement plus framework dependencies. Useful for capture into the community set.mm database with a note that the underlying SMT proof lives in the certificate archive.

4. Round-trip guarantees​

Not all targets offer the same trust guarantee on import. The table below compares:

TargetProof checked byTCB deltaPreserves framework deps?
Lean 4Lean kernelIndependent kernel; disagreement is a bug to fileYes (as Lean axiom)
CoqCoq kernelIndependent kernel; HoTT layer imports pluginYes (as Coq Axiom)
DeduktidkcheckShallowest TCB; explicit rewrite rulesDependencies must be in scope
MetamathMetamath checkerSmallest kernel of any targetFramework axioms emitted as Metamath $a statements

Pick Lean for the widest community of re-users, Coq for formalization-heavy contexts, Dedukti for SMT-heavy batches, and Metamath for smallest-TCB archival.

5. CLI​

verum export-proofs target/proofs/ --to lean --out target/lean/
verum export-proofs target/proofs/ --to coq --out target/coq/
verum export-proofs target/proofs/ --to dedukti --out target/dk/
verum export-proofs target/proofs/ --to metamath --out target/mm/

Additional flags:

FlagEffect
--selective FILEExport only the proofs listed in FILE (one per line).
--include-frameworkEmit framework axioms as target-language axioms (default on).
--bundlePack all exported proofs into a tarball.
--verify-afterAfter export, invoke target's kernel to re-check.
--on-mismatch {error,warn,ignore}What happens if re-check disagrees.

6. Round-trip test suite​

The exporter is tested with a round-trip harness: for each of core/math/hott.vr, representative external proof corpora, and a sample of refinement obligations:

  1. Verify with --strategy certified.
  2. Export to each target.
  3. Invoke target's kernel (Lean: lean file.lean; Coq: coqc file.v; Dedukti: dkcheck file.dk; Metamath: metamath -c file.mm).
  4. Record pass / fail / unknown per target.

CI publishes a weekly "interop matrix" showing which targets re-check which parts of the corpus. Deterioration of the matrix is treated as a regression.

7. Limitations​

  1. Cubical proofs (HoTT layer) require a HoTT-aware target. Lean supports HoTT via Mathlib.Cubical; Coq requires the HoTT plugin; Dedukti and Metamath do not handle HoTT and will emit the theorem as an opaque axiom with the framework tag hott:native.

  2. SMT proof-tree fidelity. Native solver (proof …) formats are under-specified and change between releases. ALETHE is more stable. For the Certified strategy we prefer ALETHE-shaped proofs when both agree, falling back to a native format only when ALETHE cannot represent the proof. This preference is configurable via --smt-proof-preference {auto}.

  3. Framework axiom drift. If a framework's formalization in the target (e.g., Mathlib's version of Lurie HTT) changes, re-verification may fail even though the Verum proof is still sound. The envelope carries the framework ID but not a specific version; consumers must pin their version.

  4. Proof size. Large refinement proofs can produce megabyte- size Lean files. The exporter emits a warning when a single proof exceeds 1 MB. Bundle mode (--bundle) compresses the output with zstd for transport.

8. Why export matters​

Q: If the kernel already proved it, why export? A: Because your trust should not depend on our kernel.

Three scenarios where export is the right answer:

  1. Cross-tool verification: your project uses Coq for downstream proofs; Verum is just one source of lemmas among many. Export lets you combine them in Coq's environment.
  2. Regulatory archival: a medical-device proof must be auditable by a third-party reviewer whose tooling is Metamath. Emit the proof once, deposit the archive, hand the reviewer the smallest-kernel target.
  3. Supply-chain audit: a library claims its invariants are proved. Run verum export-proofs --verify-after --to lean on every release; publish the re-check log. Consumers see "Verum + Lean both accept this proof" — a concrete cross-validation.

Verification in Verum is a means, not an end. Export is how you put the result to use outside Verum.

9. See also​