Mathlib v4.30 source proposition Int.dvd_mul_gcd_iff_dvd_mul

Identifier: F:ml430-int-dvd-mul-gcd-iff-dvd-mul-22d6488e
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {k n m : ℤ}, k ∣ n * ↑(k.gcd m) ↔ k ∣ n * m

Dependencies

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

5 direct dependencies 1 direct dependents

Evidence

kernel-Int.dvd_mul_gcd_iff_dvd_mul

Kind: kernel-term
Status: checked

Supports: ∀ {k n m : ℤ}, k ∣ n * ↑(k.gcd m) ↔ k ∣ n * m

Checker command

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

Evidence notes

build_int_prelude admits Int.dvd_mul_gcd_iff_dvd_mul through the trusted Kernel::add_declaration gate, so producing this row at all is a machine-checked proof. New proof, lane int-gcd-mul-transport: int_prelude/gcd_scaled_mirrors.rs. Built by applying the same shape as Int.dvd_gcd_mul_iff_dvd_mul at (b,c) := (m,n) (scaling factor on the left) and commuting both sides into place with Int.mul_comm, mirroring nat_prelude/gcd_mul_right_mirrors.rs's dvd_mul_gcd_iff_dvd_mul one layer up. int_theorem_inventory's rendered type is `(x0:Int)->(x1:Int)->(x2:Int)->Iff (Int.dvd x0 (Int.mul x1 (Int.ofNat (Int.gcd x0 x2)))) (Int.dvd x0 (Int.mul x1 x2))`, matching this fact's formal.statement character for character (x0/x1/x2 = k/n/m). int_theorem_inventory exits non-zero for a name that does not exist (verified: dvd_mul_gcd_iff_dvd_mul_bogus_xyz -> exit 1, "no Int declaration matches"), and the anchored grep -c (tested -ge 1, not piped through grep -q) requires the exact name followed by whitespace, so it does not also match Int.dvd_gcd_mul_iff_dvd_mul or Int.dvd_gcd_mul_gcd_iff_dvd_mul.

footprint-Int.dvd_mul_gcd_iff_dvd_mul

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_mul_gcd_iff_dvd_mul` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.dvd_mul_gcd_iff_dvd_mul`",
      "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"
    }
  ]
}