Mathlib v4.30 source proposition Nat.odd_totient_iff

Identifier: F:ml430-nat-odd-totient-iff-b6a6596f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Odd n.totient ↔ n = 1 ∨ n = 2

Dependencies

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

2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.odd_totient_iff

Kind: kernel-term
Status: checked

Supports: Odd (totient n) iff n = 1 or n = 2

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. Route: `Nat.odd_totient_iff_eq_one` (`Odd (totient n) <-> totient n = 1`) composed with `Nat.totient_eq_one_iff` (`totient n = 1 <-> n = 1 \/ n = 2`) by direct `mp`/`mpr` function composition, no general `iff_trans` helper. `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, distinguished from `Nat.odd_totient_iff_eq_one` by the anchored `[[:space:]]` after the name).

footprint-Nat.odd_totient_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": "2026-08-29",
  "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` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.odd_totient_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"
    }
  ]
}