Mathlib v4.30 source proposition Nat.Coprime.dvd_mul_left

Identifier: F:ml430-nat-coprime-dvd-mul-left-e799d04c
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n k : ℕ}, k.Coprime m → (k ∣ m * n ↔ k ∣ n)

Dependencies

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4 direct dependencies 0 direct dependents

Evidence

kernel-Nat.coprime_dvd_mul_left

Kind: kernel-term
Status: checked

Supports: ∀ {m n k : ℕ}, k.Coprime m → (k ∣ m * n ↔ k ∣ n)

Checker command

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

Evidence notes

`Nat.coprime_dvd_mul_left` (nat_prelude/draw11_mirrors.rs, `declare_coprime_dvd_mul_left`, lane draw11-theorems-b) is a fresh construction: the forward (`mp`) direction is `Nat.gauss_lemma` verbatim (`gcd x y = 1 -> dvd x (mul y z) -> dvd x z`, this fact's exact hypothesis order once `Coprime` is unfolded to `gcd = 1`); the reverse (`mpr`) direction builds `n ∣ (m*n)` from `Nat.dvd_mul` (`a ∣ a*q`) transported across `Nat.mul_comm`, then closes with `Nat.dvd_trans`. `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : Eq.{1} AxNat (AxNat.gcd x0 x1) (AxNat.succ AxNat.zero)) -> Iff (AxNat.dvd x0 (AxNat.mul x1 x2)) (AxNat.dvd x0 x2)))))` -- reading `x0=k, x1=m, x2=n`, this is `gcd k m = 1 -> (dvd k (mul m n) <-> dvd k n)`, matching `formal.statement` (`k.Coprime m -> (k ∣ m*n ↔ k ∣ n)`) verbatim. `nat_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. Two discriminating unit tests in `nat_prelude_tests.rs` (`coprime_dvd_mul_left_states_and_proves_the_iff_at_a_concrete_point`) apply the theorem to a real coprime witness (`gcd 5 2 = 1` by `rfl`) at `(k,m,n)=(5,2,3)` and infer the resulting `Iff` proof term, not merely the arrow's shape.

footprint-Nat.coprime_dvd_mul_left

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q --release -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 2026-08-31: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds `Nat.coprime_dvd_mul_left`'s footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly (via `theorem_names`, which now lists it).

Provenance

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