Mathlib v4.30 source proposition Nat.factorial_dvd_ascFactorial

Identifier: F:ml430-nat-factorial-dvd-ascfactorial-44a4e641
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n k : ℕ), k.factorial ∣ n.ascFactorial k

Dependencies

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

7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.factorial_dvd_ascFactorial

Kind: kernel-term
Status: checked

Supports: ∀ (n k : ℕ), k.factorial ∣ n.ascFactorial k

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.factorial_dvd_ascFactorial` through the trusted `Kernel::add_declaration` gate. Route: `Nat.ascFactorial_succ_eq_factorial_mul_choose : (succ m).ascFactorial k = k! * (m+k).choose k` (declared alongside, `asc_factorial.rs`), reindexed by `n := succ m` so no `Nat.sub` is ever needed (unlike the falling-factorial bridge, which truncates). Proved by induction on `k`, `m` fixed, chaining `asc_factorial_succ`, the IH, `mul_left_comm`, `Nat.succ_add`, `Nat.succ_mul_choose_eq`, `Nat.add_succ`, `mul_assoc` and `factorial_succ`. `factorial_dvd_ascFactorial` case-splits `n`: `n = 0` via `Nat.zero_ascFactorial_succ` (ascFactorial 0 (succ k) = 0, by induction on `k`) + `dvd_zero` (`k = 0` via `dvd_refl`); `n = succ m` via the bridge + `Nat.dvd_mul`, transported along the bridge equation. `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.factorial_dvd_ascFactorial

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 (not merely the environment-wide bound), by name, in-tree.

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.factorial_dvd_ascFactorial` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.factorial_dvd_ascFactorial`",
      "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"
    }
  ]
}