Mathlib v4.30 source proposition Nat.Coprime.eq_of_mul_eq_zero

Identifier: F:ml430-nat-coprime-eq-of-mul-eq-zero-a2026bd5
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m.Coprime n → m * n = 0 → m = 0 ∧ n = 1 ∨ m = 1 ∧ n = 0

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.coprime_eq_of_mul_eq_zero

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, m.Coprime n → m * n = 0 → m = 0 ∧ n = 1 ∨ m = 1 ∧ n = 0

Checker command

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

Evidence notes

`Nat.coprime_eq_of_mul_eq_zero` (nat_prelude/draw11_mirrors.rs, `declare_coprime_eq_of_mul_eq_zero`, lane draw11-theorems-b) is a fresh construction: `Nat.mul_eq_zero` splits `m*n=0` into `m=0 ∨ n=0`; each disjunct is substituted into the coprimality hypothesis and collapsed with `Nat.gcd_zero_left` (the `n=0` case additionally uses `Nat.gcd_comm`, since the zero then sits on the wrong side for `gcd_zero_left` directly) to force the OTHER variable to `1`, then packaged with `And.intro`/`Or.inl`/`Or.inr`. `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Eq.{1} AxNat (AxNat.gcd x0 x1) (AxNat.succ AxNat.zero)) -> ((x3 : Eq.{1} AxNat (AxNat.mul x0 x1) AxNat.zero) -> Or (And (Eq.{1} AxNat x0 AxNat.zero) (Eq.{1} AxNat x1 (AxNat.succ AxNat.zero))) (And (Eq.{1} AxNat x0 (AxNat.succ AxNat.zero)) (Eq.{1} AxNat x1 AxNat.zero))))))`, matching `formal.statement` 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. A discriminating unit test in `nat_prelude_tests.rs` (`coprime_eq_of_mul_eq_zero_selects_the_correct_disjunct`) applies the theorem to real witnesses at `(m,n)=(0,1)` (both hypotheses `rfl`) all the way to a proof term inferred against the LEFT disjunct, and checks the type shape at `(m,n)=(1,0)` selects the RIGHT disjunct -- discriminating a swapped `Or.inl`/`Or.inr`.

footprint-Nat.coprime_eq_of_mul_eq_zero

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_eq_of_mul_eq_zero`'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.eq_of_mul_eq_zero` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Coprime.eq_of_mul_eq_zero`",
      "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"
    }
  ]
}