[
  {
    "label": "Definition 1.6.3",
    "concept": "prime number",
    "mathlib_ref": "Nat.Prime",
    "lean_code": null,
    "notes": "The definition in the book is consistent with Mathlib's definition of a prime number."
  },
  {
    "label": "Definition 1.6.8",
    "concept": "greatest common divisor (gcd) of two integers",
    "mathlib_ref": "Nat.gcd",
    "lean_code": null,
    "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`."
  },
  {
    "label": "Definition 1.6.14",
    "concept": "relatively prime integers",
    "mathlib_ref": "IsCoprime",
    "lean_code": null,
    "notes": "Corrected from an earlier hallucinated `Int.Coprime` (no such constant exists). Mathlib's `IsCoprime a b` (in a commutative semiring, so usable for ℤ) captures relative primality via Bezout-style coefficients and is definitionally the right notion here."
  },
  {
    "label": "Definition 1.6.22",
    "concept": "greatest common divisor (gcd) of several integers",
    "mathlib_ref": null,
    "lean_code": "def IsGCDOfList (d : ℕ) (as : List ℤ) : Prop :=\n  (∀ a ∈ as, (d:ℤ) ∣ a) ∧ ∀ e : ℤ, (∀ a ∈ as, e ∣ a) → e ∣ (d:ℤ)",
    "notes": "No single ready-made Mathlib constant for 'gcd of a `List ℤ`' matches the book's definition verbatim, so this concept gets a real, hand-written Lean `def` (see lean_code) instead of only a JSON pointer -- it is compiled into textbook/lib/Definitions.lean by build_definitions.py and inlined into any statement's `prelude` that needs it, so Lean can actually elaborate proofs that use it, not just read a description of it."
  },
  {
    "label": "Definition 1.6.25",
    "concept": "pairwise relatively prime integers",
    "mathlib_ref": "List.Pairwise IsCoprime",
    "lean_code": null,
    "notes": "Corrected from an earlier hallucinated `Int.PairwiseCoprime` (no such constant exists). Pairwise relative primality of a list is expressed as `List.Pairwise IsCoprime as` (or `List.Pairwise (IsCoprime ·) as`), combining the general `List.Pairwise` relation with `IsCoprime`, rather than via a single dedicated name."
  }
]