Mathlib v4.30 source proposition Nat.Prime.not_coprime_iff_dvd

Identifier: F:ml430-nat-prime-not-coprime-iff-dvd-c83110ca
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, ¬m.Coprime n ↔ ∃ p, Nat.Prime p ∧ p ∣ m ∧ p ∣ n

Dependencies

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

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

Evidence

kernel-Nat.prime_not_coprime_iff_dvd

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, ¬m.Coprime n ↔ ∃ p, Nat.Prime p ∧ p ∣ m ∧ p ∣ n

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.prime_not_coprime_iff_dvd` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist (verified against a fabricated name), and the anchored `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Primality is spelled inline (`2 ≤ p ∧ ∀ c, c ∣ p → c = 1 ∨ c = p`, matching `prime_even_iff`/`euclid_lemma`'s own convention) rather than through a named `Nat.Prime` predicate, since this prelude has none, and the existential witness pairs primality with `dvd p m ∧ dvd p n` (right-associated, matching Mathlib's `Prime p ∧ p ∣ m ∧ p ∣ n`) -- this repository's established precedent for flipping a Mathlib `Nat.Prime` mirror (`prime_even_iff`, `prime_odd_of_ne_two`, `prime_dvd_of_dvd_pow` already do the same substitution).

footprint-Nat.prime_not_coprime_iff_dvd

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.not_coprime_iff_dvd` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.Prime.not_coprime_iff_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"
    }
  ]
}