Mathlib v4.30 source proposition Nat.Abundant.of_dvd

Identifier: F:ml430-nat-abundant-of-dvd-686548ce
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n m : ℕ}, m.Abundant → m ∣ n → n ≠ 0 → n.Abundant

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

kernel-Nat.abundant_of_dvd

Kind: kernel-term
Status: checked

Supports: Abundant m -> dvd m n -> Not (Eq n zero) -> Abundant n

Checker command

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

Evidence notes

Destructs the `dvd` witness `q` (`n = mul m q`), derives `Lt zero q` from `Not (Eq n zero)` by contradiction (`q = 0` would force `n = 0` via `mul_zero`), then applies `Nat.sum_divisors_scale_le` (`divisor_sum_scale.rs`, `q * sumDivisors m <= sumDivisors (q*m)` for positive `q`) and multiplies the `Abundant m` hypothesis through by `q` (`Nat.mul_lt_mul_left`'s `mpr` direction) to get `Lt (mul q (mul 2 m)) (sumDivisors (mul q m))`, then reassociates `mul q (mul 2 m)` to `mul 2 n` via `mul_comm`/`mul_assoc` and rewrites `sumDivisors (mul q m)` to `sumDivisors n` via `mul_comm` (`n = mul m q = mul q m`).

footprint-Nat.abundant_of_dvd

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 divisor-sum-monotonicity",
  "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.of_dvd`",
      "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"
    }
  ]
}