Mathlib v4.30 source proposition Nat.Prime.deficient

Identifier: F:ml430-nat-prime-deficient-89e0badf
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Nat.Prime n → n.Deficient

Dependencies

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5 direct dependencies 2 direct dependents

Evidence

kernel-Nat.prime_deficient

Kind: kernel-term
Status: checked

Supports: Prime n -> Deficient n

Checker command

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

Evidence notes

`prime_one_lt` gives `Lt one n`; `add_lt_add_left` lifts it to `Lt (add n one) (add n n)`, defeq `Lt (succ n) (add n n)`. A local bridge lemma (`two_mul_eq_add_self`, needed because `Nat.mul` recurses on its RIGHT argument so `mul 2 n` is stuck for symbolic `n`) relates `add n n` to `mul 2 n`, and `sum_divisors_prime` (`perfect.rs`: `sumDivisors p = succ p`) relates `succ n` to `sumDivisors n`, both via `transport` along a `symm`'d equation.

footprint-Nat.prime_deficient

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_deficient`",
      "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"
    }
  ]
}