Mathlib v4.30 source proposition Nat.Prime.not_perfect

Identifier: F:ml430-nat-prime-not-perfect-15c1235d
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {p : ℕ}, Nat.Prime p → ¬p.Perfect

Dependencies

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

2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.prime_not_perfect

Kind: kernel-term
Status: checked

Supports: Prime p -> Not (Perfect p)

Checker command

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

Evidence notes

`Perfect p` gives `Eq (sumDivisors p) (mul 2 p)`; transporting `prime_deficient`'s `Lt (sumDivisors p) (mul 2 p)` along it yields `Lt (mul 2 p) (mul 2 p)`, absurd via `lt_irrefl`.

footprint-Nat.prime_not_perfect

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": "2026-09-01",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-abundant-deficient-mirrors",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here through Kernel::add_declaration",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.prime_not_perfect`",
      "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"
    }
  ]
}