claude-bot
fbcb0c3383
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
22 lines
12 KiB
JSON
22 lines
12 KiB
JSON
{"name": "goodman_1_6_1", "id": "goodman-1-6-1", "source_label": "Proposition 1.6.1", "kind": "proposition", "status": "proof_pending", "proof": "intro a b c\nrefine ⟨by ring, by ring, by ring, by ring, by ring, by ring, by ring, by ring, ?_⟩\nintro x y h\nexact ⟨fun hx => h (by simp [hx]), fun hy => h (by simp [hy])⟩", "statement": "∀ (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)", "natural_language": "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.", "identifiers": [], "requires": ["Int.add_comm", "Int.add_assoc", "Int.zero_add", "Int.add_left_neg", "Int.mul_comm", "Int.mul_assoc", "Int.one_mul", "Int.mul_add", "Int.mul_ne_zero"]}
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{"name": "goodman_1_6_4", "id": "goodman-1-6-4", "source_label": "Proposition 1.6.4", "kind": "proposition", "status": "proof_pending", "proof": "intro n hn\nobtain ⟨p, hp, hpd⟩ := Nat.exists_prime_and_dvd (show n ≠ 1 by omega)\nexact ⟨p, hp, hpd⟩", "statement": "∀ (n : ℕ), n > 1 → ∃ (p : ℕ) (h : Nat.Prime p), p ∣ n", "natural_language": "Any natural number greater than 1 can be written as a product of prime numbers.", "identifiers": ["Nat.Prime"], "requires": ["Nat.exists_prime_factor"]}
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{"name": "goodman_1_6_6", "id": "goodman-1-6-6", "source_label": "Theorem 1.6.6", "kind": "theorem", "status": "asserted", "proof": null, "statement": "∃ (p : ℕ → ℕ), ∀ (n : ℕ), Nat.Prime (p n) ∧ ∀ (i j : ℕ), i ≠ j → p i ≠ p j", "natural_language": "There are infinitely many prime numbers.", "identifiers": ["Nat.Prime"], "requires": ["Nat.infinite_primes"]}
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{"name": "goodman_1_6_7", "id": "goodman-1-6-7", "source_label": "Proposition 1.6.7", "kind": "proposition", "status": "asserted", "proof": null, "statement": "∀ (a d : ℤ), d ≥ 1 → ∃! qr : ℤ × ℤ, a = qr.1 * d + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < d", "natural_language": "Given integers a and d with d ≥ 1, there exist unique integers q and r such that a = qd + r and 0 ≤ r < d.", "identifiers": ["qr"], "requires": ["Int.division_theorem"]}
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{"name": "goodman_1_6_9", "id": "goodman-1-6-9", "source_label": "Proposition 1.6.9", "kind": "proposition", "status": "asserted", "proof": null, "statement": "∀ (m n : ℤ), let I := {v : ℤ | ∃ a b : ℤ, v = a * m + b * n}; (∀ x ∈ I, ∀ y ∈ I, x + y ∈ I) ∧ (∀ x ∈ I, -x ∈ I) ∧ (∀ x : ℤ, ∀ z ∈ I, x * z ∈ I) ∧ (∀ b : ℤ, b ∣ m ∧ b ∣ n → ∀ z ∈ I, b ∣ z)", "natural_language": "I(m,n) = {am+bn} is closed under addition and negation, absorbs multiplication by any integer, and any common divisor of m and n divides every element of I(m,n).", "identifiers": [], "requires": []}
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{"name": "goodman_1_6_10", "id": "goodman-1-6-10", "source_label": "Lemma 1.6.10", "kind": "lemma", "status": "asserted", "proof": null, "statement": "∀ (m n d : ℤ), d ∣ m ∧ d ∣ n ∧ d ∈ {v : ℤ | ∃ a b : ℤ, v = a * m + b * n} → ∀ (x : ℤ), x ∣ m ∧ x ∣ n → x ∣ d", "natural_language": "If a natural number d is a common divisor of m and n and an element of I(m, n), then d is the greatest common divisor of m and n.", "identifiers": [], "requires": ["Int.gcd_is_greatest_common_divisor"]}
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{"name": "goodman_1_6_11", "id": "goodman-1-6-11", "source_label": "Proposition 1.6.11", "kind": "proposition", "status": "asserted", "proof": null, "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → let d := Nat.gcd (Int.natAbs m) (Int.natAbs n); (d : ℤ) = Int.gcd m n ∧ (d : ℤ) ∈ {v : ℤ | ∃ a b : ℤ, v = a * m + b * n}", "natural_language": "The natural number d, which is the greatest common divisor of the absolute values of m and n, is also the greatest common divisor of m and n, and d is an element of I(m, n).", "identifiers": ["Nat.gcd", "Int.natAbs", "Int.gcd"], "requires": ["Int.gcd_nat_abs", "Int.gcd_linear_combination"]}
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{"name": "goodman_1_6_13", "id": "goodman-1-6-13", "source_label": "Corollary 1.6.13", "kind": "corollary", "status": "asserted", "proof": null, "statement": "∀ (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}", "natural_language": "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).", "identifiers": ["Int.gcd", "IsLeast"], "requires": ["IsLeast", "Int.gcd"]}
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{"name": "goodman_1_6_15", "id": "goodman-1-6-15", "source_label": "Corollary 1.6.15", "kind": "corollary", "status": "proof_pending", "proof": "intro m n _\nrw [Int.gcd_eq_one_iff_coprime]\nconstructor\n· intro h\n obtain ⟨u, v, huv⟩ := h\n exact ⟨u, v, huv⟩\n· intro h\n obtain ⟨s, t, hst⟩ := h\n exact ⟨s, t, hst⟩", "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → Int.gcd m n = 1 ↔ ∃ (s t : ℤ), s * m + t * n = 1", "natural_language": "Two nonzero integers m and n are relatively prime if and only if there exist integers s and t such that sm + tn = 1.", "identifiers": ["Int.gcd"], "requires": ["Int.coprime_iff_gcd_eq_one"]}
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{"name": "goodman_1_6_17", "id": "goodman-1-6-17", "source_label": "Corollary 1.6.17", "kind": "corollary", "status": "proof_pending", "proof": "intro a b x h\nexact IsCoprime.mul_dvd (Nat.isCoprime_iff_coprime.mpr h.1) h.2.1 h.2.2", "statement": "∀ (a b : ℕ) (x : ℤ), Nat.Coprime a b ∧ (a:ℤ) ∣ x ∧ (b:ℤ) ∣ x → (a:ℤ) * b ∣ x", "natural_language": "If a and b are relatively prime natural numbers and both divide an integer x, then their product ab also divides x.", "identifiers": ["Nat.Coprime"], "requires": ["Nat.coprime_mul_dvd"]}
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{"name": "goodman_1_6_18", "id": "goodman-1-6-18", "source_label": "Proposition 1.6.18", "kind": "proposition", "status": "asserted", "proof": null, "statement": "∀ (p : ℕ) (a : ℤ), Nat.Prime p ∧ a ≠ 0 → (p:ℤ) ∣ a ∨ IsCoprime (p:ℤ) a", "natural_language": "If p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime.", "identifiers": ["Nat.Prime", "IsCoprime"], "requires": ["Int.prime_or_coprime"]}
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{"name": "goodman_1_6_19", "id": "goodman-1-6-19", "source_label": "Proposition 1.6.19", "kind": "proposition", "status": "proof_pending", "proof": "intro p a b h hdvd\nexact Int.Prime.dvd_mul' h.1 hdvd", "statement": "∀ (p : ℕ) (a b : ℤ), Nat.Prime p ∧ a ≠ 0 ∧ b ≠ 0 → (p:ℤ) ∣ a * b → (p:ℤ) ∣ a ∨ (p:ℤ) ∣ b", "natural_language": "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.", "identifiers": ["Nat.Prime"], "requires": ["Int.prime_dvd_mul"]}
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{"name": "goodman_1_6_20", "id": "goodman-1-6-20", "source_label": "Corollary 1.6.20", "kind": "corollary", "status": "proof_pending", "proof": "intro p as hp _ h\nobtain ⟨a, ha, hpa⟩ := (Nat.Prime.prime_int hp).dvd_prod_iff.mp h\nexact ⟨a, ha, hpa⟩", "statement": "∀ (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", "natural_language": "If a prime number p divides a product of nonzero integers, then p divides one of the factors.", "identifiers": ["as", "List", "Nat.Prime", "List.prod"], "requires": ["Int.prime_dvd_prod"]}
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{"name": "goodman_1_6_21", "id": "goodman-1-6-21", "source_label": "Theorem 1.6.21", "kind": "theorem", "status": "asserted", "proof": null, "statement": "∀ (n : ℕ), n > 1 → ∃! (ps : List ℕ), (∀ (p : ℕ), p ∈ ps → Nat.Prime p) ∧ n = List.prod ps", "natural_language": "The prime factorization of a natural number is unique.", "identifiers": ["ps", "List", "Nat.Prime", "List.prod"], "requires": ["Nat.unique_prime_factorization"]}
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{"name": "goodman_1_6_23", "id": "goodman-1-6-23", "source_label": "Lemma 1.6.23", "kind": "lemma", "status": "asserted", "proof": null, "statement": "∀ (n : ℕ) (a : Fin n → ℤ), 2 ≤ n → (∀ i, a i ≠ 0) → ∃ (d : ℕ) (Q : Matrix (Fin n) (Fin n) ℤ), IsUnit Q.det ∧ ∀ j : Fin n, Matrix.vecMul a Q j = if j.val = 0 then (d : ℤ) else 0", "natural_language": "Given n >= 2 nonzero integers a_1..a_n, there is a natural number d and a unimodular (det a unit, i.e. det = +-1) integer n-by-n matrix Q such that the row vector a times Q equals (d, 0, ..., 0).", "identifiers": ["Fin", "Matrix", "IsUnit", "Q.det", "Matrix.vecMul", "j.val"], "requires": ["Matrix.vecMul", "Matrix.det", "IsUnit"]}
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{"name": "goodman_1_6_24", "id": "goodman-1-6-24", "source_label": "Proposition 1.6.24", "kind": "proposition", "status": "asserted", "proof": null, "statement": "∀ (as : List ℤ), as.length ≥ 2 → (∀ (a : ℤ), a ∈ as → a ≠ 0) → ∃ (d : ℕ) (cs : List ℤ), IsGCDOfList d as ∧ (d:ℤ) = List.sum (List.zipWith (· * ·) cs as)", "natural_language": "The greatest common divisor of a list of at least two nonzero integers exists and is an integer linear combination of the integers in the list.", "identifiers": ["as", "List", "as.length", "cs", "IsGCDOfList", "List.sum", "List.zipWith"], "requires": ["GCD.gcd_exists", "GCD.gcd_linear_combination"]}
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{"name": "goodman_1_6_26", "id": "goodman-1-6-26", "source_label": "Lemma 1.6.26", "kind": "lemma", "status": "asserted", "proof": null, "statement": "∀ (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", "natural_language": "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.", "identifiers": ["as", "List", "as.length", "IsCoprime", "List.prod"], "requires": ["Int.pairwise_coprime_mul_dvd"]}
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{"name": "goodman_1_6_2_a", "id": "goodman-1-6-2-a", "source_label": "Proposition 1.6.2", "kind": "proposition", "status": "proof_pending", "proof": "intro u v h\nexact Int.mul_eq_one_iff_eq_one_or_neg_one.mp h", "statement": "∀ (u v : ℤ), u * v = 1 → (u = 1 ∧ v = 1) ∨ (u = -1 ∧ v = -1)", "natural_language": "If two integers multiply to 1, both are 1 or both are -1.", "identifiers": [], "requires": ["Int.eq_one_or_self_of_prime"]}
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{"name": "goodman_1_6_2_b", "id": "goodman-1-6-2-b", "source_label": "Proposition 1.6.2", "kind": "proposition", "status": "proof_pending", "proof": "intro a b h\nexact Int.associated_iff.mp (associated_of_dvd_dvd h.1 h.2)", "statement": "∀ (a b : ℤ), a ∣ b ∧ b ∣ a → a = b ∨ a = -b", "natural_language": "If two integers divide each other, they are equal or negatives of each other.", "identifiers": [], "requires": ["Int.dvd_antisymm"]}
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{"name": "goodman_1_6_2_c", "id": "goodman-1-6-2-c", "source_label": "Proposition 1.6.2", "kind": "proposition", "status": "proof_pending", "proof": "intro a b c h\nexact dvd_trans h.1 h.2", "statement": "∀ (a b c : ℤ), a ∣ b ∧ b ∣ c → a ∣ c", "natural_language": "Divisibility of integers is transitive.", "identifiers": [], "requires": ["Int.dvd_trans"]}
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{"name": "goodman_1_6_2_d", "id": "goodman-1-6-2-d", "source_label": "Proposition 1.6.2", "kind": "proposition", "status": "proof_pending", "proof": "intro a b c s t h\nexact dvd_add (h.1.mul_left s) (h.2.mul_left t)", "statement": "∀ (a b c s t : ℤ), a ∣ b ∧ a ∣ c → a ∣ (s * b + t * c)", "natural_language": "If an integer divides two others, it divides any integer linear combination of them.", "identifiers": [], "requires": ["dvd_add", "Dvd.dvd.mul_left"]}
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