Run textbook pipeline pilot on section 1.6 (Divisibility in the Integers)
Generated with formalize.py against Qwen (qwen-max via DashScope), then hand-fixed before pushing, since these classes of bug pass elaboration but are still wrong and check_statements.py can't catch them without running Lean: - goodman-1-6-2 split into 4 separate theorems (1-6-2-a..d): the model's single merged statement had an operator-precedence bug (missing parens around an implicit right-associated AND after OR) that silently scoped 3 of the book's 4 independent claims inside the wrong disjunct. - goodman-1-6-1: dropped two vacuous "x + y : Nat is a Nat" clauses that confused Lean's type membership with set membership (element-of Nat on already-Nat-typed variables) and are automatically true by typing anyway. - goodman-1-6-9: swapped Mathlib's pointwise Set multiplication (x * I, which needs 'open scoped Pointwise' and is easy to get subtly wrong) for the equivalent elementwise statement. - goodman-1-6-13: replaced a misuse of min (a 2-argument function) applied to a Set with Mathlib's IsLeast predicate, matching what "least element" means in Int.gcd's characterization. - goodman-1-6-23: fixed Matrix index types (Fin n, not a bare Nat) and replaced an illegal literal '...' row with Matrix.vecMul; verified against the book's raw text that the n-by-n unimodular-matrix formulation is faithful, not a model invention. textbook/goodman-1.6.raw.jsonl keeps the unedited model output for reference. The remaining 13 statements are unedited and still need a real Lean elaboration pass (check_statements.py) plus a provability run (lean-bench) on a Mathlib-backed project -- neither ran here since this sandbox has no Lean.
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{"id": "goodman-1-6-1", "source_label": "Proposition 1.6.1", "kind": "proposition", "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)", "prelude": "import Mathlib", "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.", "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"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-4", "source_label": "Proposition 1.6.4", "kind": "proposition", "statement": "∀ (n : ℕ), n > 1 → ∃ (p : ℕ) (h : Nat.Prime p), p ∣ n", "prelude": "import Mathlib", "natural_language": "Any natural number greater than 1 can be written as a product of prime numbers.", "requires": ["Nat.exists_prime_factor"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-6", "source_label": "Theorem 1.6.6", "kind": "theorem", "statement": "∃ (p : ℕ → ℕ), ∀ (n : ℕ), Nat.Prime (p n) ∧ ∀ (i j : ℕ), i ≠ j → p i ≠ p j", "prelude": "import Mathlib", "natural_language": "There are infinitely many prime numbers.", "requires": ["Nat.infinite_primes"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-7", "source_label": "Proposition 1.6.7", "kind": "proposition", "statement": "∀ (a d : ℤ), d ≥ 1 → ∃! (q r : ℤ), a = q * d + r ∧ 0 ≤ r < d", "prelude": "import Mathlib", "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.", "requires": ["Int.division_theorem"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-9", "source_label": "Proposition 1.6.9", "kind": "proposition", "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)", "prelude": "import Mathlib", "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).", "requires": [], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-10", "source_label": "Lemma 1.6.10", "kind": "lemma", "statement": "∀ (m n d : ℤ), d ∣ m ∧ d ∣ n ∧ d ∈ {a * m + b * n | a b : ℤ} → ∀ (x : ℤ), x ∣ m ∧ x ∣ n → x ∣ d", "prelude": "import Mathlib", "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.", "requires": ["Int.gcd_is_greatest_common_divisor"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-11", "source_label": "Proposition 1.6.11", "kind": "proposition", "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → let d := Nat.gcd (Int.natAbs m) (Int.natAbs n) in d = Int.gcd m n ∧ d ∈ {a * m + b * n | a b : ℤ}", "prelude": "import Mathlib", "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).", "requires": ["Int.gcd_nat_abs", "Int.gcd_linear_combination"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-13", "source_label": "Corollary 1.6.13", "kind": "corollary", "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}", "prelude": "import Mathlib", "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).", "requires": ["IsLeast", "Int.gcd"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-15", "source_label": "Corollary 1.6.15", "kind": "corollary", "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → Int.gcd m n = 1 ↔ ∃ (s t : ℤ), s * m + t * n = 1", "prelude": "import Mathlib", "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.", "requires": ["Int.coprime_iff_gcd_eq_one"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-17", "source_label": "Corollary 1.6.17", "kind": "corollary", "statement": "∀ (a b : ℕ) (x : ℤ), Nat.Coprime a b ∧ a ∣ x ∧ b ∣ x → a * b ∣ x", "prelude": "import Mathlib", "natural_language": "If a and b are relatively prime natural numbers and both divide an integer x, then their product ab also divides x.", "requires": ["Nat.coprime_mul_dvd"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-18", "source_label": "Proposition 1.6.18", "kind": "proposition", "statement": "∀ (p a : ℤ), Nat.Prime p ∧ a ≠ 0 → p ∣ a ∨ Int.Coprime p a", "prelude": "import Mathlib", "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.", "requires": ["Int.prime_or_coprime"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-19", "source_label": "Proposition 1.6.19", "kind": "proposition", "statement": "∀ (p a b : ℤ), Nat.Prime p ∧ a ≠ 0 ∧ b ≠ 0 → p ∣ a * b → p ∣ a ∨ p ∣ b", "prelude": "import Mathlib", "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.", "requires": ["Int.prime_dvd_mul"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-20", "source_label": "Corollary 1.6.20", "kind": "corollary", "statement": "∀ (p : ℤ) (as : List ℤ), Nat.Prime p ∧ ∀ (a : ℤ), a ∈ as → a ≠ 0 → p ∣ List.prod as → ∃ (a : ℤ), a ∈ as ∧ p ∣ a", "prelude": "import Mathlib", "natural_language": "If a prime number p divides a product of nonzero integers, then p divides one of the factors.", "requires": ["Int.prime_dvd_prod"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-21", "source_label": "Theorem 1.6.21", "kind": "theorem", "statement": "∀ (n : ℕ), n > 1 → ∃! (ps : List ℕ), (∀ (p : ℕ), p ∈ ps → Nat.Prime p) ∧ n = List.prod ps", "prelude": "import Mathlib", "natural_language": "The prime factorization of a natural number is unique.", "requires": ["Nat.unique_prime_factorization"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-23", "source_label": "Lemma 1.6.23", "kind": "lemma", "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 = 0 then (d : ℤ) else 0", "prelude": "import Mathlib", "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).", "requires": ["Matrix.vecMul", "Matrix.det", "IsUnit"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-24", "source_label": "Proposition 1.6.24", "kind": "proposition", "statement": "∀ (as : List ℤ), as.length ≥ 2 → (∀ (a : ℤ), a ∈ as → a ≠ 0) → ∃ (d : ℕ) (cs : List ℤ), d = GCD.gcd as ∧ d = List.sum (List.zipWith (*) cs as)", "prelude": "import Mathlib", "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.", "requires": ["GCD.gcd_exists", "GCD.gcd_linear_combination"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-26", "source_label": "Lemma 1.6.26", "kind": "lemma", "statement": "∀ (as : List ℤ) (z : ℤ), as.length ≥ 2 → (∀ (i j : ℕ), i ≠ j → i < as.length → j < as.length → Int.Coprime (as[i]) (as[j])) → (∀ (a : ℤ), a ∈ as → a ≠ 0) → (∀ (a : ℤ), a ∈ as → a ∣ z) → List.prod as ∣ z", "prelude": "import Mathlib", "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.", "requires": ["Int.pairwise_coprime_mul_dvd"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-2-a", "source_label": "Proposition 1.6.2", "kind": "proposition", "statement": "∀ (u v : ℤ), u * v = 1 → (u = 1 ∧ v = 1) ∨ (u = -1 ∧ v = -1)", "prelude": "import Mathlib", "natural_language": "If two integers multiply to 1, both are 1 or both are -1.", "requires": ["Int.eq_one_or_self_of_prime"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-2-b", "source_label": "Proposition 1.6.2", "kind": "proposition", "statement": "∀ (a b : ℤ), a ∣ b ∧ b ∣ a → a = b ∨ a = -b", "prelude": "import Mathlib", "natural_language": "If two integers divide each other, they are equal or negatives of each other.", "requires": ["Int.dvd_antisymm"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-2-c", "source_label": "Proposition 1.6.2", "kind": "proposition", "statement": "∀ (a b c : ℤ), a ∣ b ∧ b ∣ c → a ∣ c", "prelude": "import Mathlib", "natural_language": "Divisibility of integers is transitive.", "requires": ["Int.dvd_trans"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-2-d", "source_label": "Proposition 1.6.2", "kind": "proposition", "statement": "∀ (a b c s t : ℤ), a ∣ b ∧ a ∣ c → a ∣ (s * b + t * c)", "prelude": "import Mathlib", "natural_language": "If an integer divides two others, it divides any integer linear combination of them.", "requires": ["dvd_add", "Dvd.dvd.mul_left"], "include_in_bench": true, "exclude_reason": null}
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[
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{
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"label": "Definition 1.6.3",
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"concept": "prime number",
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"mathlib_ref": "Nat.Prime",
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"lean_code": null,
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"notes": "The definition in the book is consistent with Mathlib's definition of a prime number."
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},
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{
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"label": "Definition 1.6.8",
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"concept": "greatest common divisor (gcd) of two integers",
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"mathlib_ref": "Nat.gcd",
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"lean_code": null,
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"notes": "The definition in the book is consistent with Mathlib's definition of gcd for natural numbers, and can be extended to integers using `Int.gcd`."
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},
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{
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"label": "Definition 1.6.14",
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"concept": "relatively prime integers",
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"mathlib_ref": "Int.Coprime",
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"lean_code": null,
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"notes": "The definition in the book is consistent with Mathlib's definition of coprime integers."
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},
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{
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"label": "Definition 1.6.22",
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"concept": "greatest common divisor (gcd) of several integers",
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"mathlib_ref": "GCD.gcd",
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"lean_code": null,
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"notes": "The definition in the book is consistent with Mathlib's generalization of gcd to multiple integers."
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},
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{
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"label": "Definition 1.6.25",
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"concept": "pairwise relatively prime integers",
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"mathlib_ref": "Int.PairwiseCoprime",
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"lean_code": null,
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"notes": "The definition in the book is consistent with Mathlib's definition of pairwise coprime integers."
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}
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]
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{"id": "goodman-1-6-1", "source_label": "Proposition 1.6.1", "kind": "proposition", "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 : ℕ), x + y ∈ ℕ) ∧ (∀ (x y : ℕ), x * y ∈ ℕ) ∧ (∀ (x y : ℤ), x * y ≠ 0 → x ≠ 0 ∧ y ≠ 0)", "prelude": "import Mathlib", "natural_language": "Addition and multiplication on the integers are commutative and associative, 0 and 1 are identity elements for addition and multiplication respectively, the distributive law holds, the sum and product of positive integers are positive, and the product of non-zero integers is non-zero.", "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", "Nat.add_pos", "Nat.mul_pos", "Int.mul_ne_zero"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-2", "source_label": "Proposition 1.6.2", "kind": "proposition", "statement": "∀ (u v : ℤ), u * v = 1 → (u = 1 ∧ v = 1) ∨ (u = -1 ∧ v = -1) ∧ ∀ (a b : ℤ), a ∣ b ∧ b ∣ a → a = b ∨ a = -b ∧ ∀ (a b c : ℤ), a ∣ b ∧ b ∣ c → a ∣ c ∧ ∀ (a b c s t : ℤ), a ∣ b ∧ a ∣ c → a ∣ (s * b + t * c)", "prelude": "import Mathlib", "natural_language": "If the product of two integers is 1, then both integers are either 1 or -1. If an integer divides another and vice versa, then they are equal or negatives of each other. Divisibility is transitive. If an integer divides two other integers, it divides any linear combination of them.", "requires": ["Int.mul_eq_one_iff", "Int.dvd_antisymm", "Int.dvd_trans", "Int.dvd_linear_combination"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-4", "source_label": "Proposition 1.6.4", "kind": "proposition", "statement": "∀ (n : ℕ), n > 1 → ∃ (p : ℕ) (h : Nat.Prime p), p ∣ n", "prelude": "import Mathlib", "natural_language": "Any natural number greater than 1 can be written as a product of prime numbers.", "requires": ["Nat.exists_prime_factor"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-6", "source_label": "Theorem 1.6.6", "kind": "theorem", "statement": "∃ (p : ℕ → ℕ), ∀ (n : ℕ), Nat.Prime (p n) ∧ ∀ (i j : ℕ), i ≠ j → p i ≠ p j", "prelude": "import Mathlib", "natural_language": "There are infinitely many prime numbers.", "requires": ["Nat.infinite_primes"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-7", "source_label": "Proposition 1.6.7", "kind": "proposition", "statement": "∀ (a d : ℤ), d ≥ 1 → ∃! (q r : ℤ), a = q * d + r ∧ 0 ≤ r < d", "prelude": "import Mathlib", "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.", "requires": ["Int.division_theorem"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-9", "source_label": "Proposition 1.6.9", "kind": "proposition", "statement": "∀ (m n : ℤ), let I := {a * m + b * n | a b : ℤ} in (∀ (x y : I), x + y ∈ I) ∧ (∀ (x : I), -x ∈ I) ∧ (∀ (x : ℤ), x * I ⊆ I) ∧ (∀ (b : ℤ), b ∣ m ∧ b ∣ n → ∀ (z : I), b ∣ z)", "prelude": "import Mathlib", "natural_language": "For integers m and n, the set I(m, n) = {am + bn | a, b ∈ ℤ} is closed under addition, negation, and multiplication by any integer. If an integer b divides both m and n, then b divides every element of I(m, n).", "requires": ["Int.linear_combination_closure", "Int.linear_combination_div"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-10", "source_label": "Lemma 1.6.10", "kind": "lemma", "statement": "∀ (m n d : ℤ), d ∣ m ∧ d ∣ n ∧ d ∈ {a * m + b * n | a b : ℤ} → ∀ (x : ℤ), x ∣ m ∧ x ∣ n → x ∣ d", "prelude": "import Mathlib", "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.", "requires": ["Int.gcd_is_greatest_common_divisor"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-11", "source_label": "Proposition 1.6.11", "kind": "proposition", "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → let d := Nat.gcd (Int.natAbs m) (Int.natAbs n) in d = Int.gcd m n ∧ d ∈ {a * m + b * n | a b : ℤ}", "prelude": "import Mathlib", "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).", "requires": ["Int.gcd_nat_abs", "Int.gcd_linear_combination"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-13", "source_label": "Corollary 1.6.13", "kind": "corollary", "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → let d := Int.gcd m n in d = min {x ∈ ℕ | x ∈ {a * m + b * n | a b : ℤ}} ∧ {a * m + b * n | a b : ℤ} = {k * d | k : ℤ}", "prelude": "import Mathlib", "natural_language": "The greatest common divisor d of m and n is the least element of N ∩ I(m, n), and I(m, n) is the set of all integer multiples of d.", "requires": ["Int.gcd_minimal", "Int.linear_combination_set"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-15", "source_label": "Corollary 1.6.15", "kind": "corollary", "statement": "∀ (m n : ℤ), m ≠ 0 ∧ n ≠ 0 → Int.gcd m n = 1 ↔ ∃ (s t : ℤ), s * m + t * n = 1", "prelude": "import Mathlib", "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.", "requires": ["Int.coprime_iff_gcd_eq_one"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-17", "source_label": "Corollary 1.6.17", "kind": "corollary", "statement": "∀ (a b : ℕ) (x : ℤ), Nat.Coprime a b ∧ a ∣ x ∧ b ∣ x → a * b ∣ x", "prelude": "import Mathlib", "natural_language": "If a and b are relatively prime natural numbers and both divide an integer x, then their product ab also divides x.", "requires": ["Nat.coprime_mul_dvd"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-18", "source_label": "Proposition 1.6.18", "kind": "proposition", "statement": "∀ (p a : ℤ), Nat.Prime p ∧ a ≠ 0 → p ∣ a ∨ Int.Coprime p a", "prelude": "import Mathlib", "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.", "requires": ["Int.prime_or_coprime"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-19", "source_label": "Proposition 1.6.19", "kind": "proposition", "statement": "∀ (p a b : ℤ), Nat.Prime p ∧ a ≠ 0 ∧ b ≠ 0 → p ∣ a * b → p ∣ a ∨ p ∣ b", "prelude": "import Mathlib", "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.", "requires": ["Int.prime_dvd_mul"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-20", "source_label": "Corollary 1.6.20", "kind": "corollary", "statement": "∀ (p : ℤ) (as : List ℤ), Nat.Prime p ∧ ∀ (a : ℤ), a ∈ as → a ≠ 0 → p ∣ List.prod as → ∃ (a : ℤ), a ∈ as ∧ p ∣ a", "prelude": "import Mathlib", "natural_language": "If a prime number p divides a product of nonzero integers, then p divides one of the factors.", "requires": ["Int.prime_dvd_prod"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-21", "source_label": "Theorem 1.6.21", "kind": "theorem", "statement": "∀ (n : ℕ), n > 1 → ∃! (ps : List ℕ), (∀ (p : ℕ), p ∈ ps → Nat.Prime p) ∧ n = List.prod ps", "prelude": "import Mathlib", "natural_language": "The prime factorization of a natural number is unique.", "requires": ["Nat.unique_prime_factorization"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-23", "source_label": "Lemma 1.6.23", "kind": "lemma", "statement": "∀ (as : List ℤ), as.length ≥ 2 → (∀ (a : ℤ), a ∈ as → a ≠ 0) → ∃ (d : ℕ) (Q : Matrix ℤ as.length as.length), Q.det ≠ 0 ∧ Q.inverse.det ≠ 0 ∧ (d, 0, ..., 0) = (as[0], as[1], ..., as[as.length - 1]) * Q", "prelude": "import Mathlib", "natural_language": "Given a list of at least two nonzero integers, there exists a natural number d and an invertible integer matrix Q such that (d, 0, ..., 0) = (a1, a2, ..., an) * Q.", "requires": ["Matrix.int_matrix_gcd"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-24", "source_label": "Proposition 1.6.24", "kind": "proposition", "statement": "∀ (as : List ℤ), as.length ≥ 2 → (∀ (a : ℤ), a ∈ as → a ≠ 0) → ∃ (d : ℕ) (cs : List ℤ), d = GCD.gcd as ∧ d = List.sum (List.zipWith (*) cs as)", "prelude": "import Mathlib", "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.", "requires": ["GCD.gcd_exists", "GCD.gcd_linear_combination"], "include_in_bench": true, "exclude_reason": null}
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{"id": "goodman-1-6-26", "source_label": "Lemma 1.6.26", "kind": "lemma", "statement": "∀ (as : List ℤ) (z : ℤ), as.length ≥ 2 → (∀ (i j : ℕ), i ≠ j → i < as.length → j < as.length → Int.Coprime (as[i]) (as[j])) → (∀ (a : ℤ), a ∈ as → a ≠ 0) → (∀ (a : ℤ), a ∈ as → a ∣ z) → List.prod as ∣ z", "prelude": "import Mathlib", "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.", "requires": ["Int.pairwise_coprime_mul_dvd"], "include_in_bench": true, "exclude_reason": null}
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Reference in New Issue
Block a user