Mathlib v4.30 source proposition Nat.Prime.dvd_or_dvd

Identifier: F:ml430-nat-prime-dvd-or-dvd-4ae88221
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p m n : ℕ}, Nat.Prime p → p ∣ m * n → p ∣ m ∨ p ∣ n

Dependencies

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

Evidence

kernel-Nat.euclid_lemma

Kind: kernel-term
Status: checked

Supports: ∀ {p m n : ℕ}, Nat.Prime p → p ∣ m * n → p ∣ m ∨ p ∣ n

Checker command

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

Evidence notes

This mirror is `Nat.euclid_lemma` (`nat_prelude/bezout.rs`) verbatim, up to the bound-variable names (`a,b` vs `m,n`): `euclid_lemma : forall p a b, prime_condition p -> p|(a*b) -> p|a or p|b`. No new declaration was made for this fact; the checker below confirms `Nat.euclid_lemma` itself, and `prime_dvd_mirrors.rs`'s module doc records the identification explicitly so a later reader does not go looking for a separate `Nat.prime_dvd_or_dvd` declaration that does not exist.

footprint-Nat.euclid_lemma

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. A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds `Nat.euclid_lemma`'s footprint by [].

Provenance

{
  "date": "2026-08-30",
  "established_by": "axeyum-lean-kernel build_nat_prelude (declare_euclid_lemma, bezout.rs); identified by lane prime-dvd-mirrors",
  "source": "statement-only extraction of `Nat.Prime.dvd_or_dvd` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Prime.dvd_or_dvd`",
      "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"
    }
  ]
}