Mathlib v4.30 source proposition Int.dvd_gcd_iff

Identifier: F:ml430-int-dvd-gcd-iff-5c30733d
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.dvd_gcd_iff` in the pinned Mathlib v4.30 source.

Formal statement

∀ {a b : ℤ} {c : ℕ}, c ∣ a.gcd b ↔ ↑c ∣ a ∧ ↑c ∣ b

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

kernel-Int.dvd_gcd_nat_iff

Kind: kernel-term
Status: checked

Supports: ∀ {a b : ℤ} {c : ℕ}, c ∣ a.gcd b ↔ ↑c ∣ a ∧ ↑c ∣ b

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example int_theorem_inventory -- dvd_gcd_nat_iff 2>/dev/null | /usr/bin/grep -cE '^theorem[[:space:]]+Int\.dvd_gcd_nat_iff[[:space:]]')" -ge 1

Evidence notes

`build_int_prelude` admits `Int.dvd_gcd_nat_iff` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. New proof, lane int-dvd-mirrors: `int_prelude/dvd_gcd_mirrors.rs`'s `declare_dvd_gcd_nat_iff` -- Mathlib's `Int.dvd_gcd_iff`. `mp`: `Nat.gcd_dvd_left`/`Nat.gcd_dvd_right` (up to the same `Int.gcd` unfold `declare_dvd_gcd_nat` uses) plus `Nat.dvd_trans` give `Nat.dvd c (natAbs a)`/`Nat.dvd c (natAbs b)`, and `dvd_of_nat_abs_dvd` lifts each back to `ofNat c ∣ a`/`ofNat c ∣ b` (again erasing the cast by `rfl`). `mpr` is exactly `declare_dvd_gcd_nat`'s body. `int_theorem_inventory`'s rendered type matches this fact's `formal.statement` (`∀ {a b : ℤ} {c : ℕ}, c ∣ a.gcd b ↔ ↑c ∣ a ∧ ↑c ∣ b`). `int_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed; the anchor requires the exact name followed by whitespace, so it cannot match a longer sibling name sharing the same prefix. Verified both ways: the real name greps to a count `-ge 1`; grepping a fabricated name (`Int.dvd_gcd_nat_iff_bogus_xyz`) makes `int_theorem_inventory` fail closed (exit 1, "no Int declaration matches").

footprint-Int.dvd_gcd_nat_iff

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Int prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example prelude_axiom_inventory -- --require-axiom-free integer

Evidence notes

`prelude_axiom_inventory --require-axiom-free integer` enumerates the built Int environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: integer axiom=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by []. int_prelude_tests::every_int_declaration_is_checked_and_axiom_free and derived_laws_have_no_axiom_footprint (both re-run on every int_prelude:: sweep) additionally confirm this theorem's own Kernel::axiom_footprint directly.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.dvd_gcd_iff` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.dvd_gcd_iff`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}