Mathlib v4.30 source proposition Nat.eq_zero_of_lcm_eq_zero

Identifier: F:ml430-nat-eq-zero-of-lcm-eq-zero-d09b7af7
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m.lcm n = 0 → m = 0 ∨ n = 0

Dependencies

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

3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.eq_zero_of_lcm_eq_zero

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, m.lcm n = 0 → m = 0 ∨ n = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.eq_zero_of_lcm_eq_zero` through the trusted `Kernel::add_declaration` gate (declared in the new `nat_prelude/lcm_gcd_lemmas.rs`'s `declare_eq_zero_of_lcm_eq_zero`). Route: `gcd_mul_lcm : gcd m n * lcm m n = m * n`, transported along the hypothesis `lcm m n = 0`, collapses the right side to `m * n = gcd m n * 0 = 0`; `mul_eq_zero` then splits the product into the disjunction -- no induction. `nat_theorem_inventory`'s rendered type for `Nat.eq_zero_of_lcm_eq_zero` is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Eq.{1} AxNat (AxNat.lcm x0 x1) AxNat.zero) -> Or (Eq.{1} AxNat x0 AxNat.zero) (Eq.{1} AxNat x1 AxNat.zero))))`, matching this fact's `formal.statement` verbatim (`x0`/`x1`/`x2` are the universally quantified variables in order). `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.

footprint-Nat.eq_zero_of_lcm_eq_zero

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). 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 []. `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), and `lcm_gcd_lemmas_apply_at_concrete_discriminating_instances` re-checks it at a concrete instantiation chosen to discriminate a swapped argument or a wrong disjunct/conclusion order via `def_eq` plus a genuinely false negative control rejected by `declare_theorem`.

Provenance

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