Mathlib v4.30 source proposition Nat.Dvd.dvd.nat_lcm_left

Identifier: F:ml430-nat-dvd-dvd-nat-lcm-left-6143311e
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.Dvd.dvd.nat_lcm_left` in the pinned Mathlib v4.30 source.

Formal statement

∀ {a b : ℕ}, a ∣ b → ∀ (c : ℕ), a ∣ c.lcm b

Dependencies

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

2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dvd_lcm_of_dvd_right

Kind: kernel-term
Status: checked

Supports: ∀ {a b : ℕ}, a ∣ b → ∀ (c : ℕ), a ∣ c.lcm b

Checker command

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

Evidence notes

This mirror was ALREADY PROVED in this ledger under a different name: `Nat.dvd_lcm_of_dvd_right` (nat_prelude/primes.rs, `declare_dvd_lcm_of_dvd`) states `dvd a b -> dvd a (lcm c b)` for universally quantified `a b c`, which is exactly Mathlib's `a ∣ b → ∀ c, a ∣ c.lcm b` up to Pi-binder order (the `c` sits before the hypothesis here rather than after, which does not change the proposition). `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.dvd x0 x1) -> AxNat.dvd x0 (AxNat.lcm x2 x1)))))` -- reading `x0=a, x1=b, x2=c`, this is `dvd a b -> dvd a (lcm c b)`, matching `formal.statement` verbatim. No new Rust was needed; `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.dvd_lcm_of_dvd_right

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.dvd_lcm_of_dvd_right`'s footprint by [].

Provenance

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