Mathlib v4.30 source proposition Nat.Prime.dvd_factorial

Identifier: F:ml430-nat-prime-dvd-factorial-5ace903f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n p : ℕ}, Nat.Prime p → (p ∣ n.factorial ↔ p ≤ n)

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.prime_dvd_factorial_iff_le

Kind: kernel-term
Status: checked

Supports: ∀ {n p : ℕ}, Nat.Prime p → (p ∣ n.factorial ↔ p ≤ n)

Checker command

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

Evidence notes

`declare_prime_dvd_factorial_iff_le` (`nat_prelude/prime_dvd_factorial_lcm.rs`) admits `Nat.prime_dvd_factorial_iff_le` through the trusted `Kernel::add_declaration` gate. Primality is spelled inline (`2 <= p /\ forall d, d | p -> d = 1 \/ d = p`), matching `euclid_lemma`'s convention, since this prelude has no separate `Nat.Prime` predicate. `mpr` is `dvd_factorial_of_le` fed `prime_one_le`; `mp` splits `Nat.lt_or_ge n p` and refutes the `n < p` branch: `Nat.coprime_factorial_of_lt_prime` (`gauss_lemma.rs`) gives `gcd p n! = 1`, `p | n!` (the hypothesis) plus `p | p` (`dvd_refl`) gives `p | gcd p n!` (`dvd_gcd`), transported along the coprimality equation into `p | 1` -- refuted by `prime_not_dvd_one`.

footprint-Nat.prime_dvd_factorial_iff_le

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 [].

Provenance

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