Mathlib v4.30 source proposition Nat.Prime.coprime_iff_not_dvd

Identifier: F:ml430-nat-prime-coprime-iff-not-dvd-a85d46a4
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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5 direct dependencies 1 direct dependents

Evidence

kernel-Nat.prime_coprime_iff_not_dvd

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`mp`: `gcd p n=1` and `p|n` give `p|gcd(p,n)` via `dvd_refl`+`dvd_gcd`, transported to `p|1` -- refuted by `not_dvd_one_of_two_le`. `mpr`: `g := gcd p n` divides `p` (`gcd_dvd_left`), so the divisor clause forces `g=1 or g=p`; `g=1` is the goal directly, `g=p` transports `g|n` (`gcd_dvd_right`) into `p|n`, contradicting the hypothesis.

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

Provenance

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