Mathlib v4.30 source proposition Int.gcd_dvd_iff

Identifier: F:ml430-int-gcd-dvd-iff-66fa03b3
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b : ℤ} {n : ℕ}, a.gcd b ∣ n ↔ ∃ x y, ↑n = a * x + b * y

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

9 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Int.gcd_dvd_iff

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Int.gcd_dvd_iff` in the pinned Mathlib v4.30 source.

Checker command

cargo test -p axeyum-lean-kernel --lib int_prelude::

Evidence notes

`declare_gcd_dvd_iff` (`crates/axeyum-lean-kernel/src/int_prelude/gcd_dvd_iff.rs`): both directions route through the already-checked Bezout identity at the named computable witnesses `Int.gcdA`/`Int.gcdB` (`gcd_eq_gcd_ab_witnesses`), so only the fact's own quantifiers need elimination/introduction -- two nested `Exists.rec` applications for the reverse direction (`mpr`), and `Nat.dvd`'s own witness scaling both Bezout coefficients for the forward direction (`mp`), closed by `dvd_trans`/`dvd_add`/`nat_abs_dvd_nat_abs_of_dvd` and `add_mul`/`mul_assoc` respectively.

footprint-Int.gcd_dvd_iff

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- this theorem's trusted closure is empty

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Int.gcd_dvd_iff 2>/dev/null | grep -Ec 'integer[[:space:]]Int\.gcd_dvd_iff[[:space:]]0[[:space:]]*$')" -ge 1

Evidence notes

The matched row's trailing footprint column is EMPTY, so a nonzero footprint or an absent declaration both give a count of 0 and fail. Pass exactly one name per invocation -- this tool keeps only the first.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.gcd_dvd_iff` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.gcd_dvd_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"
    }
  ]
}