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Record and gate proofs, so "proven" means Lean verified it

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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
This commit is contained in:
Claude Bot
2026-09-18 19:43:54 +00:00
parent 69c5b788b9
commit fbcb0c3383
9 changed files with 269 additions and 49 deletions
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lib/Goodman/Chapter01Section06.lean
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@@ -8,10 +8,10 @@ import Mathlib
def IsGCDOfList (d : ℕ) (as : List ℤ) : Prop :=
(∀ a ∈ as, (d:ℤ) ∣ a) ∧ ∀ e : ℤ, (∀ a ∈ as, e ∣ a) → e ∣ (d:ℤ)
/-- Proposition 1.6.1 (Goodman, Algebra: Abstract and Concrete). Addition and multiplication on the integers are commutative and associative, 0 and 1 are their respective identities, the distributive law holds, and a product of integers is nonzero iff both factors are nonzero. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.1 (Goodman, Algebra: Abstract and Concrete). Addition and multiplication on the integers are commutative and associative, 0 and 1 are their respective identities, the distributive law holds, and a product of integers is nonzero iff both factors are nonzero. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_1 : ∀ (a b c : ℤ), (a + b = b + a) ∧ (a + (b + c) = (a + b) + c) ∧ (0 + a = a) ∧ (a + (-a) = 0) ∧ (a * b = b * a) ∧ (a * (b * c) = (a * b) * c) ∧ (1 * a = a) ∧ (a * (b + c) = a * b + a * c) ∧ (∀ (x y : ℤ), x * y ≠ 0 → x ≠ 0 ∧ y ≠ 0)
/-- Proposition 1.6.4 (Goodman, Algebra: Abstract and Concrete). Any natural number greater than 1 can be written as a product of prime numbers. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.4 (Goodman, Algebra: Abstract and Concrete). Any natural number greater than 1 can be written as a product of prime numbers. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_4 : ∀ (n : ℕ), n > 1 → ∃ (p : ℕ) (h : Nat.Prime p), p ∣ n
/-- Theorem 1.6.6 (Goodman, Algebra: Abstract and Concrete). There are infinitely many prime numbers. ASSERTED from the textbook, not proven here. -/
@@ -32,19 +32,19 @@ axiom goodman_1_6_11 : ∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → let d := Nat.gcd
/-- Corollary 1.6.13 (Goodman, Algebra: Abstract and Concrete). gcd(m,n) is the least positive element of I(m,n), and I(m,n) is exactly the set of integer multiples of gcd(m,n). ASSERTED from the textbook, not proven here. -/
axiom goodman_1_6_13 : ∀ (m n : ℤ), m ≠ 0 → n ≠ 0 → let I := {v : ℤ | ∃ a b : ℤ, v = a * m + b * n}; let d := Int.gcd m n; IsLeast {x : ℤ | 0 < x ∧ x ∈ I} (d : ℤ) ∧ I = {v : ℤ | ∃ k : ℤ, v = k * d}
/-- Corollary 1.6.15 (Goodman, Algebra: Abstract and Concrete). Two nonzero integers m and n are relatively prime if and only if there exist integers s and t such that sm + tn = 1. ASSERTED from the textbook, not proven here. -/
/-- Corollary 1.6.15 (Goodman, Algebra: Abstract and Concrete). Two nonzero integers m and n are relatively prime if and only if there exist integers s and t such that sm + tn = 1. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_15 : ∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → Int.gcd m n = 1 ↔ ∃ (s t : ℤ), s * m + t * n = 1
/-- Corollary 1.6.17 (Goodman, Algebra: Abstract and Concrete). If a and b are relatively prime natural numbers and both divide an integer x, then their product ab also divides x. ASSERTED from the textbook, not proven here. -/
/-- Corollary 1.6.17 (Goodman, Algebra: Abstract and Concrete). If a and b are relatively prime natural numbers and both divide an integer x, then their product ab also divides x. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_17 : ∀ (a b : ℕ) (x : ℤ), Nat.Coprime a b ∧ (a:ℤ) ∣ x ∧ (b:ℤ) ∣ x → (a:ℤ) * b ∣ x
/-- Proposition 1.6.18 (Goodman, Algebra: Abstract and Concrete). If p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime. ASSERTED from the textbook, not proven here. -/
axiom goodman_1_6_18 : ∀ (p : ℕ) (a : ℤ), Nat.Prime p ∧ a ≠ 0 → (p:ℤ) ∣ a ∨ IsCoprime (p:ℤ) a
/-- Proposition 1.6.19 (Goodman, Algebra: Abstract and Concrete). If p is a prime number and a and b are nonzero integers, and p divides the product ab, then p divides a or p divides b. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.19 (Goodman, Algebra: Abstract and Concrete). If p is a prime number and a and b are nonzero integers, and p divides the product ab, then p divides a or p divides b. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_19 : ∀ (p : ℕ) (a b : ℤ), Nat.Prime p ∧ a ≠ 0 ∧ b ≠ 0 → (p:ℤ) ∣ a * b → (p:ℤ) ∣ a ∨ (p:ℤ) ∣ b
/-- Corollary 1.6.20 (Goodman, Algebra: Abstract and Concrete). If a prime number p divides a product of nonzero integers, then p divides one of the factors. ASSERTED from the textbook, not proven here. -/
/-- Corollary 1.6.20 (Goodman, Algebra: Abstract and Concrete). If a prime number p divides a product of nonzero integers, then p divides one of the factors. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_20 : ∀ (p : ℕ) (as : List ℤ), Nat.Prime p → (∀ (a : ℤ), a ∈ as → a ≠ 0) → (p:ℤ) ∣ List.prod as → ∃ (a : ℤ), a ∈ as ∧ (p:ℤ) ∣ a
/-- Theorem 1.6.21 (Goodman, Algebra: Abstract and Concrete). The prime factorization of a natural number is unique. ASSERTED from the textbook, not proven here. -/
@@ -59,15 +59,15 @@ axiom goodman_1_6_24 : ∀ (as : List ℤ), as.length ≥ 2 → (∀ (a : ℤ),
/-- Lemma 1.6.26 (Goodman, Algebra: Abstract and Concrete). If a list of integers is pairwise relatively prime and each integer in the list divides an integer z, then the product of the integers in the list also divides z. ASSERTED from the textbook, not proven here. -/
axiom goodman_1_6_26 : ∀ (as : List ℤ) (z : ℤ), as.length ≥ 2 → (∀ (i j : ℕ), i ≠ j → i < as.length → j < as.length → IsCoprime (as[i]) (as[j])) → (∀ (a : ℤ), a ∈ as → a ≠ 0) → (∀ (a : ℤ), a ∈ as → a ∣ z) → List.prod as ∣ z
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). If two integers multiply to 1, both are 1 or both are -1. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). If two integers multiply to 1, both are 1 or both are -1. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_2_a : ∀ (u v : ℤ), u * v = 1 → (u = 1 ∧ v = 1) ∨ (u = -1 ∧ v = -1)
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). If two integers divide each other, they are equal or negatives of each other. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). If two integers divide each other, they are equal or negatives of each other. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_2_b : ∀ (a b : ℤ), a ∣ b ∧ b ∣ a → a = b ∨ a = -b
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). Divisibility of integers is transitive. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). Divisibility of integers is transitive. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_2_c : ∀ (a b c : ℤ), a ∣ b ∧ b ∣ c → a ∣ c
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). If an integer divides two others, it divides any integer linear combination of them. ASSERTED from the textbook, not proven here. -/
/-- Proposition 1.6.2 (Goodman, Algebra: Abstract and Concrete). If an integer divides two others, it divides any integer linear combination of them. ASSERTED from the textbook; a proof is recorded but not yet verified. -/
axiom goodman_1_6_2_d : ∀ (a b c s t : ℤ), a ∣ b ∧ a ∣ c → a ∣ (s * b + t * c)