Mathlib v4.30 source proposition Nat.totient_even

Identifier: F:ml430-nat-totient-even-28e0415f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, 2 < n → Even n.totient

Dependencies

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31 direct dependencies 2 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.totient_even

Kind: kernel-term
Status: checked

Supports: 2 < n -> Even (totient n)

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- totient_even 2>/dev/null | grep -Ec '^Nat\.totient_even[[: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. Route: peel index 0 off `[0,n)` via `countRange_split(f,1,n-1)` (`f 0 = false` since `n > 1`, via `gcd_zero_left` + `beq_eq_false_of_ne`), reducing the goal to `Even (countRange h (n-1))` for the shifted predicate `h(k) := f(1+k)`. Apply the general, `totient`-independent `Nat.countRange_reversal_even` (`count_range_reversal.rs`, landed by the prior `totient-even-exec` lane) at `L := n-1`: the reflection-invariance hypothesis chains `gcd(k1,n) = 1 <-> gcd(k2,n) = 1` for `k1+k2=n` through THREE already-declared `Iff`s (`coprime_self_add_right` twice plus `coprime_symmetric`, no new gcd fact); the no-fixed-point hypothesis derives `False` from a would-be fixed point directly -- `gcd k1 n = 1` plus `k1 | (k1+k1) = n` gives `k1 | gcd k1 n = 1` so `k1 = 1`, forcing `n = 2`, contradicting `2 < n` via `lt_irrefl`. `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, 0 for a fabricated one).

footprint-Nat.totient_even

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