Generate the citable premise database, and goal-driven retrieval over it

The database was the missing half: extraction and assembly existed, but
nothing turned the book's main-content statements into something an agent
could actually cite mid-proof, and nothing searched it. Two new scripts.

textbook/build_database.py takes the main-content statements and emits
both halves of a usable premise library:
- an importable Lean file where every entry is a named declaration, so a
  proof can cite goodman_1_6_20 by name;
- a JSONL search index carrying each entry's name, statement, mentioned
  identifiers and status.

An entry is emitted as an `axiom` (asserted from the textbook) until a
real lean-bench run proves it; passing --results upgrades those entries
to real theorems carrying the exact verified proof. This mirrors how the
book itself works -- its exercises are posed assuming its body results,
the same way using Mathlib does not mean re-proving Mathlib. The cost is
explicit: anything proven through an axiom inherits it, so a garbled
extraction asserted as an axiom makes downstream proofs worthless.
Guards: --require-checked refuses entries that have not passed
check_statements.py (all 21 section 1.6 entries have, and are now marked
accordingly), and the verifier's existing #print axioms check surfaces
what a finished proof rests on.

textbook/search_database.py is the retrieval step the proof loop calls at
each line: given the current goal, it ranks premises by identifier and
symbol overlap, weighting rare identifiers far above common ones
(IsGCDOfList matching means much more than Int matching), preferring a
proven premise over an asserted one, and formats the top few for the
prompt. Verified against three real goals: the exactly-applicable premise
tops each ranking by a wide margin (11.4 vs 1.6 for the gcd goal, 5.1 vs
3.7 for the prime-divides-product goal). It is a ranking heuristic, not a
decision procedure -- only Lean can say whether a premise applies.
Replacing score_premise() is the whole job of swapping in embeddings.

Also: renamed library_candidates/ to database/ to match what it is;
generated lib/Goodman/Chapter01Section06.lean (21 premises, 0 proven, 21
asserted) and database/goodman-1.6.index.jsonl; noted a latent bug in the
older build_library.py, which attaches each entry's non-import prelude
per-theorem and so would emit a shared `def` twice in one file --
build_database.py hoists and dedupes, and has a test pinning that.
13 new tests, all green (33 total, the one failure remains the
pre-existing unrelated lean_agent import error in test_dataset.py).

Still not done, and worth being clear about: nothing has been proven yet.
The bench targets have not been elaboration-checked, no lean-bench run
has happened over either file, and the goal-directed loop that would
consume this retrieval does not exist -- lean-bench's solve() is still
single-shot. Separately, lean-bench cannot currently run at all in this
checkout, because lean_bench/cli.py imports lean_agent at module load and
lean-agent is not installed.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017vbLFFJiHSZqGZHEkjvG5x

tests/test_build_database.py

@@ -0,0 +1,69 @@
+import unittest
+
+from textbook.build_database import build, decl_name, split_prelude, statement_identifiers
+
+ENTRIES = [
+    {"id": "goodman-1-6-1", "source_label": "Proposition 1.6.1", "kind": "proposition",
+     "statement": "\u2200 (a : \u2124), a + 0 = a", "prelude": "import Mathlib",
+     "natural_language": "Zero is an additive identity. [fix note: irrelevant]",
+     "requires": ["Int.add_zero"], "elaborates": True},
+    {"id": "goodman-1-6-24", "source_label": "Proposition 1.6.24", "kind": "proposition",
+     "statement": "\u2200 (as : List \u2124), IsGCDOfList 1 as", "prelude": "import Mathlib\ndef IsGCDOfList (d : \u2115) (as : List \u2124) : Prop := True",
+     "natural_language": "", "requires": [], "elaborates": True},
+    {"id": "goodman-1-6-99", "source_label": "Proposition 1.6.99", "kind": "proposition",
+     "statement": "\u2200 (as : List \u2124), IsGCDOfList 2 as", "prelude": "import Mathlib\ndef IsGCDOfList (d : \u2115) (as : List \u2124) : Prop := True",
+     "natural_language": "", "requires": [], "elaborates": True},
+]
+
+
+class BuildDatabaseTests(unittest.TestCase):
+    def test_decl_name_is_a_legal_lean_identifier(self):
+        self.assertEqual(decl_name('goodman-1-6-2-a'), 'goodman_1_6_2_a')
+
+    def test_split_prelude_separates_imports_from_defs(self):
+        imports, other = split_prelude('import Mathlib\ndef Foo := True')
+        self.assertEqual(imports, ['import Mathlib'])
+        self.assertEqual(other, 'def Foo := True')
+
+    def test_unproven_entries_become_axioms(self):
+        text, index, emitted, skipped = build(ENTRIES)
+        self.assertIn('axiom goodman_1_6_1 :', text)
+        self.assertEqual(len(emitted), 3)
+        self.assertEqual(skipped, [])
+        self.assertTrue(all(r['status'] == 'asserted' for r in index))
+
+    def test_proven_entries_become_real_theorems_with_their_proof(self):
+        results = {"goodman-1-6-1": {"id": "goodman-1-6-1", "success": True,
+                                      "attempts": [{"proof": "simp"}]}}
+        text, index, _emitted, _skipped = build(ENTRIES, results)
+        self.assertIn('theorem goodman_1_6_1 :', text)
+        self.assertNotIn('axiom goodman_1_6_1 :', text)
+        self.assertIn('  simp', text)
+        statuses = {r['id']: r['status'] for r in index}
+        self.assertEqual(statuses['goodman-1-6-1'], 'proved')
+        self.assertEqual(statuses['goodman-1-6-24'], 'asserted')
+
+    def test_shared_def_in_prelude_is_emitted_once_not_per_entry(self):
+        # Two entries share the same `def`; declaring it twice in one Lean file is an
+        # "already declared" error, so it must be hoisted and deduplicated.
+        text, _index, _emitted, _skipped = build(ENTRIES)
+        self.assertEqual(text.count('def IsGCDOfList'), 1)
+        self.assertEqual(text.count('import Mathlib'), 1)
+
+    def test_require_checked_skips_unverified_entries(self):
+        entries = [dict(ENTRIES[0]), dict(ENTRIES[1], elaborates=False)]
+        _text, index, emitted, skipped = build(entries, require_checked=True)
+        self.assertEqual(emitted, ['goodman-1-6-1'])
+        self.assertEqual(skipped, ['goodman-1-6-24'])
+        self.assertEqual(len(index), 1)
+
+    def test_index_carries_identifiers_and_strips_fix_notes(self):
+        _text, index, _emitted, _skipped = build(ENTRIES)
+        first = index[0]
+        self.assertIn('Int.add_zero', first['requires'])
+        self.assertNotIn('fix note', first['natural_language'])
+        self.assertNotIn('let', statement_identifiers('let x := 1'))
+
+
+if __name__ == '__main__':
+    unittest.main()