Mathlib v4.30 source proposition Nat.abundant_iff_not_perfect_and_not_deficient

Identifier: F:ml430-nat-abundant-iff-not-perfect-and-not-deficient-9763e268
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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5 direct dependencies 0 direct dependents

Evidence

kernel-Nat.abundant_iff_not_perfect_and_not_deficient

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

With `x := mul 2 n`, `y := sumDivisors n`: `Lt x y <-> Not (Eq y x) /\ Not (Lt y x)`, pure trichotomy of Nat's order applied to the pair `(x, y)`. Forward: two contradiction lemmas (`Lt a b` + `Eq b a`, `Lt a b` + `Lt b a`) each collapse to `lt_irrefl`. Backward: `lt_or_ge` splits `Lt x y \/ Le y x`; the `Le y x` branch splits again via `lt_or_eq_of_le` into `Lt y x \/ Eq y x`, each contradicting one conjunct. The `0 != n` hypothesis is carried (matching Mathlib's guard) but unused: this holds unconditionally for this kernel's subtraction-free forms, including at `n = 0` (both sides are `Lt 0 0`).

footprint-Nat.abundant_iff_not_perfect_and_not_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.abundant_iff_not_perfect_and_not_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"
    }
  ]
}