Prompt 04 — Theorem, lemma, conjecture, and proof-card builder

Inputs

TARGET_PARTITION: {{TARGET_PARTITION}}
SOURCE_CANDIDATES: {{SOURCE_CANDIDATES_JSON}}
AVAILABLE_DEFINITIONS: {{AVAILABLE_DEFINITIONS_JSON}}
AVAILABLE_THEOREMS: {{AVAILABLE_THEOREMS_JSON}}
FORMALIZATION_TARGET: {{FORMALIZATION_TARGET_OR_NULL}}
CHECKER_RECEIPTS: {{CHECKER_RECEIPTS_JSON}}
CLEAN_ROOM_POLICY: {{CLEAN_ROOM_POLICY_JSON}}

Task

Create one object for each theorem-like statement and, when supplied, a separately linked proof object.

Statement card

Preserve:

• proposition kind: theorem, lemma, corollary, equivalence, existence, uniqueness, impossibility, bound, classification, or conjecture;
• quantifiers and variable domains;
• hypotheses and side conditions;
• conclusion;
• exactness class;
• dependency IDs;
• cases excluded by the source;
• counterexamples when assumptions are weakened or removed;
• known equivalent formulations only when supported by the input library.

Proof card

Return a concise proof certificate, not hidden chain-of-thought. Include:

• proof strategy label;
• ordered claims/lemmas used;
• key transformations with justifications;
• discharged and undischarged obligations;
• dependency graph;
• formal source or proof-term reference when supplied;
• checker name, version, result, and receipt ID when supplied;
• failure point if the proof is incomplete or invalid.

Status rules

• draft: generated statement or proof not independently checked.
• source_grounded: accurately extracted from a source, but not independently checked.
• bounded_tested: passed finite or randomized tests only.
• symbolically_verified: passed an independent symbolic checker for the declared assumptions.
• formally_verified: accepted by an external proof kernel with a supplied receipt.
• refuted: a valid counterexample is attached.
• inconclusive: unresolved obligations remain.

Never infer formally_verified from a source saying “proof” or from your own confidence.

Additional clean-room rule

For curriculum, output the problem statement and verifier interface in a separate object; do not emit the theorem name, proof, or target concept to learner-visible fields.

Return only a library-batch object.

Return JSON matching schemas/library-batch.schema.json.