Mathlib v4.30 source proposition Nat.deficient_iff_not_abundant_and_not_perfect

Identifier: F:ml430-nat-deficient-iff-not-abundant-and-not-perfect-18bbe30a
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, n ≠ 0 → (n.Deficient ↔ ¬n.Abundant ∧ ¬n.Perfect)

Dependencies

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5 direct dependencies 0 direct dependents

Evidence

kernel-Nat.deficient_iff_not_abundant_and_not_perfect

Kind: kernel-term
Status: checked

Supports: n != 0 -> (Deficient n <-> Not (Abundant n) /\ Not (Perfect n))

Checker command

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

Evidence notes

The mirror image of `abundant_iff_not_perfect_and_not_deficient`: with `x := mul 2 n`, `y := sumDivisors n`, `Lt y x <-> Not (Lt x y) /\ Not (Eq y x)`. Same technique, conjunct order and `Eq` direction swapped to match this fact's statement, exactly as Mathlib's own two lemmas differ.

footprint-Nat.deficient_iff_not_abundant_and_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.deficient_iff_not_abundant_and_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"
    }
  ]
}