Mathlib v4.30 source proposition Nat.coprime_one_left_iff

Identifier: F:ml430-nat-coprime-one-left-iff-45945e80
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n : ℕ), Nat.Coprime 1 n ↔ True

Dependencies

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

2 direct dependencies 4 direct dependents

Evidence

kernel-Nat.coprime_one_left_iff

Kind: kernel-term
Status: checked

Supports: Nat.Coprime 1 n iff True

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the constructed proof term against the stated type: `gcd 1 n` divides `1` directly (`gcd_dvd_left`), so `eq_one_of_dvd_one` gives `gcd 1 n = 1` unconditionally; the `Iff` with `True` is immediate (`True.intro` forward, the equation itself backward). `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` requires the admitted declaration to be printed.

footprint-Nat.coprime_one_left_iff

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 every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-gcd",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.coprime_one_left_iff` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.coprime_one_left_iff`",
      "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"
    }
  ]
}