Mathlib v4.30 source proposition Nat.dvd_mul

Identifier: F:ml430-nat-dvd-mul-ebd102e2
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {k m n : ℕ}, k ∣ m * n ↔ ∃ k₁ k₂, k₁ ∣ m ∧ k₂ ∣ n ∧ k₁ * k₂ = k

Dependencies

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

Evidence

kernel-Nat.dvd_mul_split

Kind: kernel-term
Status: checked

Supports: ∀ {k m n : ℕ}, k ∣ m * n ↔ ∃ k₁ k₂, k₁ ∣ m ∧ k₂ ∣ n ∧ k₁ * k₂ = k

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dvd_mul_split` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type. New proof, lane dvd-mul-split: `nat_prelude/dvd_mul_split.rs`, wired in via one `declare_dvd_mul_split` call after `declare_gcd_mul_right_mirrors`/`declare_dvd_add_iff_left`. NOT named `Nat.dvd_mul`: that kernel name is already taken by the unrelated trivial lemma `∀ a q, dvd a (a*q)` (`nat_prelude.rs`'s pre-existing `dvd_mul` field) -- declaring under Mathlib's literal name would hit `DeclarationExists`. Route: forward direction sets k1 := gcd(k,m), k2 := k/gcd(k,m) via `gcd_dvd_left`/`gcd_dvd_right`/`dvd_gcd`/`Nat.gcd_mul_right` (the distributive law landed by lane gcd-mul-right the same day) plus a positive-factor cancellation (`one_le_of_dvd_pos`, `mul_left_cancel_of_pos`); the k=0 case is handled by a DIRECT case split on `m=0 ∨ n=0` (`mul_eq_zero`), not by the general gcd formula, which does not reproduce a valid witness pair there (`gcd(0,m)=m`, `0/m=0`, forcing `k2=0` and needing `0∣n`, false in general). Reverse direction is uniform four-factor regrouping, no case split. `nat_theorem_inventory`'s rendered type is `(x0:AxNat)->(x1:AxNat)->(x2:AxNat)->Iff (dvd x0 (mul x1 x2)) (Exists x3, Exists x4, And (dvd x3 x1) (And (dvd x4 x2) (Eq (mul x3 x4) x0)))`, matching this fact's `formal.statement` exactly (x0/x1/x2/x3/x4 = k/m/n/k1/k2). `nat_theorem_inventory` exits non-zero for a name that does not exist (verified: `dvd_mul_split_bogus_xyz` -> exit 1, `0 theorems` printed, `error: no Nat theorem matches ... -- an absent theorem is a failed check, not an empty report`), and the anchored `grep -cE` (tested `-ge 1`, not piped through `grep -q`) requires the exact name followed by whitespace.

footprint-Nat.dvd_mul_split

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: `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, exit 0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds `Nat.dvd_mul_split`'s own 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` (it is listed there), and a dedicated test `dvd_mul_split_applies_at_a_concrete_discriminating_instance_and_a_free_degenerate_one` applies it at a concrete discriminating triple (k,m,n)=(6,4,9) via `Iff.mpr` with the algorithm's own witnesses (k1,k2)=(2,3) -- checking both the statement shape and the logical content, since the mpr application actually produces a real proof of `dvd 6 36` -- and at the degenerate `k=0,m=0` case with a genuinely FREE `n` (pushed into an explicit `LocalContext`) via `Iff.mp`.

Provenance

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