Mathlib v4.30 source proposition Nat.dvd_two_of_totient_le_one

Identifier: F:ml430-nat-dvd-two-of-totient-le-one-3642bf31
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a : ℕ}, 0 < a → a.totient ≤ 1 → a ∣ 2

Dependencies

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17 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.dvd_two_of_totient_le_one

Kind: kernel-term
Status: checked

Supports: 0 < a -> totient a <= 1 -> a | 2

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- dvd_two_of_totient_le_one 2>/dev/null | grep -Ec '^Nat\.dvd_two_of_totient_le_one[[: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. `trichotomy` at c=2 on `a`: `a < 2` combined with `0 < a` forces `a = 1` (`dvd 1 2` via a concrete witness `2` and `one_mul`); `a = 2` is `dvd 2 2` (`dvd_refl`); `2 < a` contradicts `totient a <= 1` via `countRange_ge_two_of_two_witnesses` at witnesses `1` (`coprime_one_left_iff`, unconditional) and `pred a` (`coprime_succ_self`), refuted by peeling two `succ`s down to `not_succ_le_zero` (`totient_le_one_contradiction_above_two`, `totient_lemmas.rs`, shared with `totient_eq_one_iff`'s forward direction). `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.dvd_two_of_totient_le_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",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.dvd_two_of_totient_le_one` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dvd_two_of_totient_le_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"
    }
  ]
}