Mathlib v4.30 source proposition Nat.odd_totient_iff_eq_one

Identifier: F:ml430-nat-odd-totient-iff-eq-one-d0491d84
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Odd n.totient ↔ n.totient = 1

Dependencies

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

9 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.odd_totient_iff_eq_one

Kind: kernel-term
Status: checked

Supports: Odd (totient n) iff totient n = 1

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. Route: the same `trichotomy(two, n)` split `totient_eq_one_iff` uses. Forward (`Odd (totient n) -> totient n = 1`): `n < 2` splits into `n = 0` (transports the hypothesis to `Odd (totient 0)`, defeq `Odd 0`, refuted by `even_not_odd` against a fresh `Even 0` witness) or `n = 1` (`totient 1 = 1` by defeq, closes by `Eq.refl` transported); `n = 2` closes the same way (`totient 2 = 1` by defeq); `2 < n` derives `Even (totient n)` from `totient_even` and refutes it against the hypothesis via `odd_not_even`. Reverse: transports a fresh `Odd 1` witness (`Exists.intro` at `k = 0`, `Eq.refl`) along the hypothesised `totient n = 1`. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` requires the admitted declaration to be printed (confirmed 1 for the real name).

footprint-Nat.odd_totient_iff_eq_one

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": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane totient-even-finish",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.odd_totient_iff_eq_one` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.odd_totient_iff_eq_one`",
      "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"
    }
  ]
}