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
74 lines
8.6 KiB
Lean4
74 lines
8.6 KiB
Lean4
-- Generated by textbook/build_database.py -- do not hand-edit.
|
||
-- Citable premise database from the textbook's main content.
|
||
-- `axiom` entries are asserted from the book, NOT proven here: anything
|
||
-- proven through one inherits it, so check `#print axioms` downstream.
|
||
|
||
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; 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; 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. -/
|
||
axiom goodman_1_6_6 : ∃ (p : ℕ → ℕ), ∀ (n : ℕ), Nat.Prime (p n) ∧ ∀ (i j : ℕ), i ≠ j → p i ≠ p j
|
||
|
||
/-- Proposition 1.6.7 (Goodman, Algebra: Abstract and Concrete). Given integers a and d with d ≥ 1, there exist unique integers q and r such that a = qd + r and 0 ≤ r < d. ASSERTED from the textbook, not proven here. -/
|
||
axiom goodman_1_6_7 : ∀ (a d : ℤ), d ≥ 1 → ∃! qr : ℤ × ℤ, a = qr.1 * d + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < d
|
||
|
||
/-- Proposition 1.6.9 (Goodman, Algebra: Abstract and Concrete). 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). ASSERTED from the textbook, not proven here. -/
|
||
axiom goodman_1_6_9 : ∀ (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)
|
||
|
||
/-- Lemma 1.6.10 (Goodman, Algebra: Abstract and Concrete). 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. ASSERTED from the textbook, not proven here. -/
|
||
axiom goodman_1_6_10 : ∀ (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
|
||
|
||
/-- Proposition 1.6.11 (Goodman, Algebra: Abstract and Concrete). 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). ASSERTED from the textbook, not proven here. -/
|
||
axiom goodman_1_6_11 : ∀ (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}
|
||
|
||
/-- 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; 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; 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; 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; 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. -/
|
||
axiom goodman_1_6_21 : ∀ (n : ℕ), n > 1 → ∃! (ps : List ℕ), (∀ (p : ℕ), p ∈ ps → Nat.Prime p) ∧ n = List.prod ps
|
||
|
||
/-- Lemma 1.6.23 (Goodman, Algebra: Abstract and Concrete). 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). ASSERTED from the textbook, not proven here. -/
|
||
axiom goodman_1_6_23 : ∀ (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
|
||
|
||
/-- Proposition 1.6.24 (Goodman, Algebra: Abstract and Concrete). 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. ASSERTED from the textbook, not proven here. -/
|
||
axiom goodman_1_6_24 : ∀ (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)
|
||
|
||
/-- 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; 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; 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; 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; 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)
|
||
|