{"id": "goodman-1-6-ex-1", "source_label": "Exercise 1.6.1", "kind": "exercise", "statement": "∀ (a d : ℤ), d ≥ 1 → ∃! qr : ℤ × ℤ, a = qr.1 * d + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < d", "prelude": "import Mathlib.Data.Int.Basic", "natural_language": "For all integers a and d, with d ≥ 1, there exist unique integers q and r such that a = qd + r and 0 ≤ r < d. [fix note: Exercise 1.6.1 literally asks to complete the proof of Proposition 1.6.7 via well-ordering; reused Proposition 1.6.7's own (already elaboration-checked) statement verbatim instead of re-deriving a weaker one -- the model's first draft dropped the uniqueness clause (`∃` instead of `∃!`), which is a materially different, weaker claim.]", "requires": ["Mathlib.Data.Int.Basic"], "references": ["Proposition 1.6.7"], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-2", "source_label": "Exercise 1.6.2", "kind": "exercise", "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.Data.Int.Basic", "natural_language": "For integers m and n, the set I(m, n) has the properties: (a) For x, y in I(m, n), x + y and -x are in I(m, n). (b) For all x in Z, xI(m, n) is a subset of I(m, n). (c) If b divides m and n, then b divides all elements of I(m, n). [fix note: Exercise 1.6.2 literally asks to prove Proposition 1.6.9; reused Proposition 1.6.9's own statement verbatim. The model's first draft used bare `I(m, n)` as if it were a function call, which is not valid Lean -- there is no such function.]", "requires": ["Mathlib.Data.Int.Basic"], "references": ["Proposition 1.6.9"], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-3", "source_label": "Exercise 1.6.3", "kind": "exercise", "statement": "∀ (p : ℕ), p > 1 → (∀ (a b : ℤ), a ≠ 0 → b ≠ 0 → (p:ℤ) ∣ (a * b) → ((p:ℤ) ∣ a ∨ (p:ℤ) ∣ b)) → Nat.Prime p", "prelude": "import Mathlib.Data.Nat.Prime", "natural_language": "If a natural number p > 1 has the property that for all nonzero integers a and b, if p divides the product ab, then p divides a or p divides b, then p is prime. [fix note: Fixed two type mismatches: bare `Prime p` (ambiguous which typeclass instance) changed to `Nat.Prime p` to match convention elsewhere; and `p ∣ (a * b)` for `p : ℕ`, `a b : ℤ` needed an explicit `(p:ℤ)` cast, same class of bug as the earlier library fixes.]", "requires": ["Mathlib.Data.Nat.Prime"], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-5", "source_label": "Exercise 1.6.5", "kind": "exercise", "statement": "∀ (m n : ℤ), m ≠ 0 → n ≠ 0 → Int.gcd m n = Int.gcd (|m|) (|n|)", "prelude": "import Mathlib.Data.Int.GCD", "natural_language": "For nonzero integers m and n, the greatest common divisor of m and n is equal to the greatest common divisor of their absolute values. [fix note: Original used bare `gcd m n` with `m n : ℤ`, ambiguous between the generic `GCDMonoid.gcd` (returns ℤ) and `Int.gcd` (returns ℕ); switched to `Int.gcd` explicitly (returns ℕ, matches the rest of the dataset's convention for integer gcd).]", "requires": ["Mathlib.Data.Int.GCD"], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-6", "source_label": "Exercise 1.6.6", "kind": "exercise", "statement": "∀ (m n : ℤ), m ≠ 0 → n ≠ 0 → IsGreatest {d : ℕ | (d:ℤ) ∣ m ∧ (d:ℤ) ∣ n} (Int.gcd m n)", "prelude": "import Mathlib.Data.Int.GCD", "natural_language": "For nonzero integers m and n, the greatest common divisor of m and n is the largest natural number dividing both m and n. [fix note: Original wrote `gcd m n = max {d : ℕ | ...}`, but `max` is not a function on `Set` in Mathlib (same bug class as an earlier library fix that needed `IsLeast`); 'is the largest natural number dividing both' is `IsGreatest` on the set of common divisors.]", "requires": ["Mathlib.Data.Int.GCD"], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-7", "source_label": "Exercise 1.6.7", "kind": "exercise", "statement": "∀ (p : ℕ) (a : ℤ), Nat.Prime p ∧ a ≠ 0 → (p:ℤ) ∣ a ∨ IsCoprime (p:ℤ) a", "prelude": "import Mathlib.Data.Nat.Prime Mathlib.Data.Int.GCD", "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. [fix note: Exercise 1.6.7 is exactly Proposition 1.6.18's claim; reused Proposition 1.6.18's own statement verbatim rather than re-deriving (the model's draft had the same `Prime p` / untyped `gcd p a = 1` issues as Exercise 1.6.3).]", "requires": ["Mathlib.Data.Nat.Prime", "Mathlib.Data.Int.GCD"], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-8", "source_label": "Exercise 1.6.8", "kind": "exercise", "statement": "∀ (a b x : ℤ), IsCoprime a b → a ∣ (b * x) → a ∣ x", "prelude": "import Mathlib.Data.Int.GCD", "natural_language": "If a and b are relatively prime integers and x is an integer, then if a divides the product bx, a divides x. [fix note: Original wrote `gcd a b = 1`; switched to `IsCoprime a b` for consistency with how 'relatively prime' is formalized everywhere else in this dataset (Definition 1.6.14).]", "requires": ["Mathlib.Data.Int.GCD"], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-9", "source_label": "Exercise 1.6.9", "kind": "exercise", "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", "prelude": "import Mathlib.Data.Nat.Prime", "natural_language": "If a prime number p divides a product a₁ * a₂ * ... * aₙ of nonzero integers, then p divides one of the factors. [fix note: Exercise 1.6.9 is exactly Corollary 1.6.20's claim; reused Corollary 1.6.20's own statement verbatim. The model's draft used invalid variadic-ellipsis notation (`a₁ a₂ ... aₙ`) that is not valid Lean syntax at all -- an indexed list is required, matching how the body handles the same claim.]", "requires": ["Mathlib.Data.Nat.Prime"], "references": ["Proposition 1.6.19"], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-11-a", "source_label": "Exercise 1.6.11(a)", "kind": "exercise", "statement": "∀ (ns : List ℤ), let I := {v : ℤ | ∃ ms : List ℤ, ms.length = ns.length ∧ v = (List.zipWith (· * ·) ms ns).sum}; (∀ x ∈ I, ∀ y ∈ I, x + y ∈ I) ∧ (∀ x ∈ I, -x ∈ I) ∧ (∀ (c : ℤ), ∀ x ∈ I, c * x ∈ I)", "prelude": "import Mathlib", "natural_language": "For a list of nonzero integers n1..nk with I the set of integer combinations of them, I is closed under addition, negation, and scalar multiplication.", "requires": [], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-11-b", "source_label": "Exercise 1.6.11(b)", "kind": "exercise", "statement": "∀ (ns : List ℤ) (d : ℕ), IsGCDOfList d ns → IsLeast {v : ℤ | v ∈ {v : ℤ | ∃ ms : List ℤ, ms.length = ns.length ∧ v = (List.zipWith (· * ·) ms ns).sum} ∧ 0 < v} (d : ℤ)", "prelude": "import Mathlib\ndef IsGCDOfList (d : ℕ) (as : List ℤ) : Prop :=\n  (∀ a ∈ as, (d:ℤ) ∣ a) ∧ ∀ e : ℤ, (∀ a ∈ as, e ∣ a) → e ∣ (d:ℤ)", "natural_language": "The gcd of n1..nk is the least positive element of I (the set of their integer combinations).", "requires": [], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-11-c", "source_label": "Exercise 1.6.11(c)", "kind": "exercise", "statement": "∀ (ns : List ℤ) (d : ℕ), IsGCDOfList d ns → {v : ℤ | ∃ ms : List ℤ, ms.length = ns.length ∧ v = (List.zipWith (· * ·) ms ns).sum} = {v : ℤ | (d:ℤ) ∣ v}", "prelude": "import Mathlib\ndef IsGCDOfList (d : ℕ) (as : List ℤ) : Prop :=\n  (∀ a ∈ as, (d:ℤ) ∣ a) ∧ ∀ e : ℤ, (∀ a ∈ as, e ∣ a) → e ∣ (d:ℤ)", "natural_language": "I (the set of integer combinations of n1..nk) equals the set of multiples of their gcd d.", "requires": [], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-12", "source_label": "Exercise 1.6.12(b)", "kind": "exercise", "statement": "∀ (ns : List ℤ), (∃ d : ℕ, IsGCDOfList d ns ∧ d = 1) ↔ (1 : ℤ) ∈ {v : ℤ | ∃ ms : List ℤ, ms.length = ns.length ∧ v = (List.zipWith (· * ·) ms ns).sum}", "prelude": "import Mathlib\ndef IsGCDOfList (d : ℕ) (as : List ℤ) : Prop :=\n  (∀ a ∈ as, (d:ℤ) ∣ a) ∧ ∀ e : ℤ, (∀ a ∈ as, e ∣ a) → e ∣ (d:ℤ)", "natural_language": "(a) Determine if the integers n₁, n₂, ..., nₖ are relatively prime if and only if they are pairwise relatively prime. (b) Show that n₁, n₂, ..., nₖ are relatively prime if and only if 1 ∈ I(n₁, n₂, ..., nₖ). [fix note: Part (a) ('is relatively prime the same as pairwise relatively prime?') is a discussion question, not a fixed claim to formalize -- and its unconditional-identity reading is actually FALSE in general (e.g. 6, 10, 15 are jointly coprime but not pairwise coprime), so it cannot be stated as an unconditional theorem. Moved to the excluded file with a reason. This entry keeps only part (b), the genuine 'relatively prime iff 1 in I' claim, rewritten with `ns : List ℤ` / `IsGCDOfList` instead of the model's invalid variadic `gcd(n₁, n₂, ..., nₖ)` and `I(n₁, ..., nₖ)` notation.]", "requires": ["Mathlib.Data.Int.GCD"], "references": [], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-14", "source_label": "Exercise 1.6.14", "kind": "exercise", "statement": "∀ (m n : ℤ), m ≠ 0 → n ≠ 0 → (IsCoprime m n ↔ ∀ (p : ℕ), Nat.Prime p → (p:ℤ) ∣ m → ¬((p:ℤ) ∣ n))", "prelude": "import Mathlib.Data.Int.GCD Mathlib.Data.Nat.Prime", "natural_language": "Show that two nonzero integers are relatively prime if and only if they have no common prime factors. Use this to give a different proof of Corollary 1.6.17, relying on the uniqueness of the prime factorization of an integer. [fix note: Fixed the same `Prime p` / untyped `gcd m n = 1` / missing ℤ casts as the other exercises; used `IsCoprime m n` for consistency.]", "requires": ["Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.Prime"], "references": ["Corollary 1.6.17"], "include_in_bench": true, "exclude_reason": null}
{"id": "goodman-1-6-ex-15", "source_label": "Exercise 1.6.15", "kind": "exercise", "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", "prelude": "import Mathlib.Data.Int.GCD", "natural_language": "Let a₁, a₂, ..., aₙ be pairwise relatively prime integers, and let a = a₁ * a₂ * ... * aₙ. If an integer z is divisible by each aᵢ, then z is divisible by a. [fix note: Exercise 1.6.15 literally asks to prove Lemma 1.6.26; reused Lemma 1.6.26's own statement verbatim instead of the model's invalid variadic-ellipsis draft.]", "requires": ["Mathlib.Data.Int.GCD"], "references": ["Lemma 1.6.26"], "include_in_bench": true, "exclude_reason": null}