Rebuild lean-bench: pages in, verified indexed Lean out
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Replaces the previous tree entirely. The pipeline is four separable pieces, each
testable on its own:

1. leanbench/llm.py      provider-neutral OpenAI-compatible connector, stdlib only,
                        with retries, JSON coercion and a scripted client so the
                        rest of the code is testable without a network.
2. leanbench/detect.py   per-page detection of formalizable mathematics. Case 1 is
                        mathematics asserted as fact (definition/theorem/lemma/...,
                        deliberately not sub-classified); case 2 is a problem left
                        open to the reader. Hits come back as spans -- start page
                        and index, end page and index -- so several per page is
                        normal. A hit cut off by the page end makes the detector
                        re-read that page together with the next, repeatedly, so an
                        item spread over three pages is found whole.
                        Offsets come from verbatim quotes resolved by
                        leanbench/anchors.py (exact -> whitespace-insensitive ->
                        fuzzy), because models quote well and count badly.
3. leanbench/formalize.py + leanbench/verify.py
                        each span becomes one Lean 4 declaration, judged by the real
                        compiler: full elaboration for case 1, statement legality
                        only for case 2, which has no proof to check. Lean's own
                        diagnostics feed back for a repair attempt. "verified" (Lean
                        accepted it) and "proved" (accepted with no sorry) are kept
                        strictly apart.
4. leanbench/db.py       indexed SQLite plus JSONL export. Every item is indexed by
                        the types of its inputs, or by its own type when it is not a
                        proof, and records the source book (sha256) and the page it
                        starts on -- so `drop --source H --pages A-B` followed by a
                        re-ingest redoes exactly those pages and nothing else.

Also: lean/ Lake project pinned to Mathlib, docker/ image carrying Lean 4 and a
prebuilt Mathlib (the Mathlib revision and the toolchain are resolved against each
other at build time rather than both hardcoded), and .gitea/workflows for tests,
image build and push to the Gitea registry, and a manual "submit a book" run.

58 tests; they need neither network nor Lean.
This commit is contained in:
Zayden-Jung
2026-09-19 07:05:52 +00:00
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1.6 Divisibility in the Integers
Definition 1.6.1. Let a and b be integers. We say that a divides b, written a | b,
if there exists an integer k such that b = a k.
Theorem 1.6.2. Let a, b and c be integers. If a divides b and a divides c, then a
divides b + c.
Proof. Write b = a k and c = a m for integers k and m. Then b + c = a (k + m), so a
divides b + c.
Theorem 1.6.3. Let a and b be integers, not both zero. Then the set of common
divisors of a and b has a greatest element, written gcd(a, b), and there exist
integers s and t with gcd(a, b) = s a + t b.
Exercise 1.6.4. Show that if a divides b and b divides a, then a = b or a = -b.
Exercise 1.6.5. Prove that the sum of two odd integers is even.
Exercise 1.6.6. Determine whether every integer greater than 2 is the sum of two
primes.