Mathlib v4.30 source proposition Nat.Coprime.lcm_eq_mul

Identifier: F:ml430-nat-coprime-lcm-eq-mul-edf52888
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m.Coprime n → m.lcm n = m * n

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

kernel-Nat.coprime_lcm_eq_mul

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, m.Coprime n → m.lcm n = m * n

Checker command

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

Evidence notes

This mirror was ALREADY PROVED in this ledger under a different name before the ml430 extraction landed: `Nat.coprime_lcm_eq_mul` (nat_prelude/lcm.rs, `declare_coprime_lcm_eq_mul`) states `gcd a b = 1 -> lcm a b = a * b`, which IS Mathlib's `Nat.Coprime.lcm_eq_mul` -- this codebase never gives `Coprime` its own predicate, spelling it `gcd _ _ = 1` directly everywhere (see nat_prelude/coprime_lemmas.rs's module doc for the same convention). `nat_theorem_inventory`'s rendered type for `Nat.coprime_lcm_eq_mul` is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Eq.{1} AxNat (AxNat.gcd x0 x1) (AxNat.succ AxNat.zero)) -> Eq.{1} AxNat (AxNat.lcm x0 x1) (AxNat.mul x0 x1))))`, matching this fact's `formal.statement` (`m.Coprime n -> m.lcm n = m*n`) verbatim once `Coprime` is unfolded to `gcd = 1`. No new Rust was written for this fact; `draw11-theorems-b` lane confirmed the existing declaration's rendered type and wrote this evidence. `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.coprime_lcm_eq_mul

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_lcm_eq_mul`'s footprint by [].

Provenance

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