Mathlib v4.30 source proposition Nat.coprime_two_left

Identifier: F:ml430-nat-coprime-two-left-1b47e7c4
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Nat.Coprime 2 n ↔ Odd n

Dependencies

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10 direct dependencies 4 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.coprime_two_left

Kind: kernel-term
Status: checked

Supports: ∀ {n : ℕ}, Nat.Coprime 2 n ↔ Odd n

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.coprime_two_left` 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. Route: `2` is prime (a private `prime_two` helper rebuilding `prime_condition(2)`, since a divisor of `2` is `1` or `2`), so `coprime_or_dvd_of_prime` splits `gcd 2 n = 1 ∨ dvd 2 n`, and `prime_dvd_iff_not_coprime` relates `dvd 2 n` to `Not (gcd 2 n = 1)`. A private bridge connects `dvd 2 n` and `Even n` via the `2*k = k+k` identity, and `even_or_odd_exists`/`even_not_odd` rule out the even case in each direction of the `Iff`. A concrete-witness swap-detecting test (`coprime_two_left_applies_at_a_concrete_odd_witness_and_is_axiom_free`) round-trips a hand-built `Odd 5` through `mpr` then `mp` and confirms it lands back on `Odd 5`, which only type-checks if `mp`/`mpr` were wired to `iff_intro` in the correct order.

footprint-Nat.coprime_two_left

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": "axeyum-lean-kernel build_nat_prelude, lane parity-coprime",
  "source": "statement-only extraction of `Nat.coprime_two_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.coprime_two_left`",
      "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"
    }
  ]
}