Mathlib v4.30 source proposition Nat.dvd_gcd_mul_gcd_iff_dvd_mul

Identifier: F:ml430-nat-dvd-gcd-mul-gcd-iff-dvd-mul-07fec722
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.dvd_gcd_mul_gcd_iff_dvd_mul` in the pinned Mathlib v4.30 source.

Formal statement

∀ {k n m : ℕ}, k ∣ k.gcd n * k.gcd m ↔ k ∣ n * m

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dvd_gcd_mul_gcd_iff_dvd_mul

Kind: kernel-term
Status: checked

Supports: ∀ {k n m : ℕ}, k ∣ k.gcd n * k.gcd m ↔ k ∣ n * m

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- dvd_gcd_mul_gcd_iff_dvd_mul 2>/dev/null | grep -Ec '^Nat\.dvd_gcd_mul_gcd_iff_dvd_mul[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.dvd_gcd_mul_gcd_iff_dvd_mul` through the trusted `Kernel::add_declaration` gate. New proof, lane gcd-mul-right, in `nat_prelude/gcd_mul_right_mirrors.rs`, wired in via `declare_gcd_mul_right_mirrors` (declared AFTER `dvd_mul_gcd_iff_dvd_mul`, which it depends on). Route: the general shape `dvd_gcd_scaled_iff` applied at (a,b,c):=(k,n,gcd(k,m)) gives `Iff (dvd k (gcd(k,n)*gcd(k,m))) (dvd k (n*gcd(k,m)))`; the right-hand side IS `dvd_mul_gcd_iff_dvd_mul`'s left-hand side, so one `iff_trans` against that already-proved fact (applied at k,n,m) lands on `dvd k (n*m)`. `nat_theorem_inventory`'s rendered type is `(x0:AxNat)->(x1:AxNat)->(x2:AxNat)->Iff (dvd x0 (mul (gcd x0 x1) (gcd x0 x2))) (dvd x0 (mul x1 x2))`, matching this fact's `formal.statement` exactly. `nat_theorem_inventory` exits non-zero for a name that does not exist (measured: exit 1, `0 theorems` on stdout, so `test -ge 1` fails). Verified no substring overlap against the sibling facts `Nat.dvd_gcd_mul_iff_dvd_mul`/`Nat.dvd_mul_gcd_iff_dvd_mul`: the anchored pattern matched exactly its own row (count 1) in a combined inventory of all three.

footprint-Nat.dvd_gcd_mul_gcd_iff_dvd_mul

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly, and `gcd_mul_right_mirrors_apply_at_concrete_and_symbolic_instances` checks it applies at a concrete discriminating triple (k,n,m)=(6,4,9) -- both the statement shape and, by applying `Iff.mp` to a real `dvd_refl` proof witness, the logical content -- and symbolically at a genuinely free triple via a fresh restating theorem.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.dvd_gcd_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 `Nat.dvd_gcd_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"
    }
  ]
}