Mathlib v4.30 source proposition Nat.totient_eq_zero

Identifier: F:ml430-nat-totient-eq-zero-3be161d6
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, n.totient = 0 ↔ n = 0

Dependencies

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2 direct dependencies 3 direct dependents

Evidence

kernel-Nat.totient_eq_zero

Kind: kernel-term
Status: checked

Supports: n.totient = 0 iff n = 0

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- totient_eq_zero 2>/dev/null | grep -Ec '^Nat\.totient_eq_zero[[: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. Case-split on `n`: `n = 0` holds by the `countRange` base case directly (`Eq.refl`); `n = succ k` uses the top index `k` of the range `[0,n)` as the witness -- `Nat.coprime_succ_self k` (new, unregistered as its own fact: `gcd k (succ k) = 1` for every `k`) promotes the predicate at `k` to `true`, so `countRange`'s succ-case defining equation makes `totient (succ k)` defeq `succ (countRange f k)`, never `0` (`succ_ne_zero`), matching `succ k` itself never being `0` -- both `Iff` legs are then `ex_falso` from that contradiction. `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_zero

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 nat-totient",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.totient_eq_zero` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.totient_eq_zero`",
      "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"
    }
  ]
}