The index had no proof field and every entry was an axiom, so the
database asserted its premises rather than proving them. Adds the proof
half, with the verification gate that makes the status trustworthy.
- Database records now carry `proof` and `proof_checked`, and the search
index exposes the proof, with status one of proved / proof_pending /
asserted. build_database.py emits a real `theorem` only when a proof
has been verified (by check_proofs.py, or by a lean-bench run via
--results); a recorded but unverified proof stays an `axiom` and is
reported as proof_pending, so the library never claims more than Lean
has confirmed. A test pins that gating in both directions.
- textbook/check_proofs.py verifies proofs by compiling
`theorem ... := by <proof>`. Stricter than check_statements.py in three
ways: it rejects sorry/admit/native_decide outright, and it reads
`#print axioms` to reject any proof depending on more than Lean's own
three axioms (propext, Classical.choice, Quot.sound) -- which in
particular catches a proof leaning on one of this database's own
book-asserted axioms, since that is not an independent proof. With
--update it writes proof_checked back, promoting entries on the next
build.
- Ten candidate proofs recorded, all currently proof_checked=false
pending a Lean run: goodman-1-6-1 (the eight algebraic identities are
`ring`, plus a short argument for the nonzero conjunct), the four parts
of Proposition 1.6.2 (Int.mul_eq_one_iff_eq_one_or_neg_one,
associated_of_dvd_dvd with Int.associated_iff, dvd_trans, dvd_add with
mul_left), 1-6-4 (Nat.exists_prime_and_dvd), 1-6-15
(Int.gcd_eq_one_iff_coprime, where IsCoprime unfolds to exactly the
target existential), 1-6-17 (IsCoprime.mul_dvd via
Nat.isCoprime_iff_coprime), 1-6-19 (Int.Prime.dvd_mul') and 1-6-20
(Prime.dvd_prod_iff). These are citations of existing Mathlib results,
which is the realistic way to populate a textbook premise library --
the remaining 11 (1-6-6, -7, -9, -10, -11, -13, -18, -21, -23, -24,
-26) are the ones needing actual work and carry no proof yet.
Written without Lean available here, so expect a real failure rate on
first compile -- run check_proofs.py --update and send back the FAILED
ids with diagnostics, same loop as the statement fixes.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017vbLFFJiHSZqGZHEkjvG5x
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