Mathlib v4.30 source proposition Nat.lcm_div

Identifier: F:ml430-nat-lcm-div-eb5d8892
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n k : ℕ}, k ∣ m → k ∣ n → (m / k).lcm (n / k) = m.lcm n / k

Dependencies

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18 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.lcm_div

Kind: kernel-term
Status: checked

Supports: ∀ {m n k : ℕ}, k ∣ m → k ∣ n → (m / k).lcm (n / k) = m.lcm n / k

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.lcm_div` through the trusted `Kernel::add_declaration` gate (declared in the new `nat_prelude/lcm_gcd_lemmas.rs`'s `declare_lcm_div`). Route: induction on the divisor `k`. At `k = 0`, `div _ 0 = 0` (this kernel's totality convention) collapses every term on both sides to `0`. At `k = succ k'`, write `m = k*m1`, `n = k*n1` (`dvd_elim` on the two hypotheses) and let `q := (lcm m n)/k`; the same mutual-divisibility technique `lcm_assoc` uses shows `lcm m1 n1 = q` via two small local cancellation helpers (`scale_dvd`/`dvd_cancel_left_of_pos`), then `div_eq_of_mul_eq` rewrites the conclusion in terms of `div m k`/`div n k`. `nat_theorem_inventory`'s rendered type for `Nat.lcm_div` is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.dvd x2 x0) -> ((x4 : AxNat.dvd x2 x1) -> Eq.{1} AxNat (AxNat.lcm (AxNat.div x0 x2) (AxNat.div x1 x2)) (AxNat.div (AxNat.lcm x0 x1) x2))))))`, 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.lcm_div

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