Mathlib v4.30 source proposition Nat.totient_eq_one_iff

Identifier: F:ml430-nat-totient-eq-one-iff-68d883a0
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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16 direct dependencies 3 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.totient_eq_one_iff

Kind: kernel-term
Status: checked

Supports: totient n = 1 iff n = 1 or n = 2

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- totient_eq_one_iff 2>/dev/null | grep -Ec '^Nat\.totient_eq_one_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. Reverse direction: `totient 1 = 1` and `totient 2 = 1` both hold by pure `countRange` reduction (`Eq.refl`, like `totient_computes_on_small_numerals`). Forward direction shares `dvd_two_of_totient_le_one`'s `trichotomy` shape at c=2 on `n`: `n < 2` splits again (`lt_or_eq_of_le`) into `n = 0` (contradicts the hypothesis `totient n = 1` since `totient 0 = 0` by defeq, refuted via `succ_ne_zero`) or `n = 1` (`or_inl` directly); `n = 2` is `or_inr` directly; `2 < n` is refuted by the same shared `totient_le_one_contradiction_above_two` (`totient_lemmas.rs`) after converting the hypothesis `Eq (totient n) 1` to `Le (totient n) 1` via `le_refl` transported along the equation. `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.totient_eq_one_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",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.totient_eq_one_iff` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.totient_eq_one_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"
    }
  ]
}