Mathlib v4.30 source proposition Nat.coprime_of_dvd'

Identifier: F:ml430-nat-coprime-of-dvd-6f652673
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, (∀ (k : ℕ), Nat.Prime k → k ∣ m → k ∣ n → k ∣ 1) → m.Coprime n

Dependencies

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

Evidence

kernel-Nat.coprime_of_forall_prime_dvd

Kind: kernel-term
Status: checked

Supports: forall m n, (forall k, Nat.prime_condition k -> k | m -> k | n -> k | 1) -> Nat.gcd m n = 1

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.coprime_of_forall_prime_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, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Confirmed kernel type: `x0 x1, (x2, (2<=x2 /\ forall x3, x3|x2 -> x3=1 or x3=x2) -> x2|x0 -> x2|x1 -> x2|1) -> gcd x0 x1 = 1`, matching `m n, (k, Prime k -> k|m -> k|n -> k|1) -> Coprime m n` with the prelude's own inline primality spelling (no `Prime`/`Nat.Prime` predicate exists here; matches this file's established convention). Named `Nat.coprime_of_forall_prime_dvd` rather than `coprime_of_dvd` because that name is already taken by the unrelated `Nat.Coprime.of_dvd` (`dvd a1 a2 -> dvd b1 b2 -> Coprime a2 b2 -> Coprime a1 b1`), confirmed by grep of the whole `nat_prelude.rs` field list before naming. Proved via `declare_coprime_of_forall_prime_dvd` (`nat_prelude/primes.rs`): trichotomy on `g := gcd m n` via `lt_or_ge` twice. `g < 1` (so `g = 0` via `le_of_succ_le_succ`+`zero_le`+`le_antisymm`) forces `m = n = 0` (`zero_mul` on the `dvd 0 _` witness), and applying the hypothesis at `k = 2` (`prime_two`, already in this file) then contradicts itself (`refute_dvd_one_against_prime`, an existing private helper this file already had for exactly this `dvd p one -> False` shape). `1 <= g` and `g < 2` gives `g = 1` directly (`le_of_succ_le_succ` + `le_antisymm`) -- the goal. `1 <= g` and `2 <= g`: `exists_prime_dvd` gives a prime factor `pw` of `g`, hence of `m` and `n` (`dvd_trans`), so the hypothesis gives `pw | 1`, refuted the same way. No decidable equality or classical choice beyond the two order trichotomies (`lt_or_ge`) already proved in this prelude.

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