Mathlib v4.30 source proposition Nat.Abundant.mul_left

Identifier: F:ml430-nat-abundant-mul-left-4de4fbe7
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n m : ℕ}, n.Abundant → m ≠ 0 → (m * n).Abundant

Dependencies

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5 direct dependencies 0 direct dependents

Evidence

kernel-Nat.abundant_mul_left

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

A direct instance of `Nat.sum_divisors_scale_le` (`divisor_sum_scale.rs`) at scale factor `q := m`: `m * sumDivisors n <= sumDivisors (m*n)`. Multiplies the `Abundant n` hypothesis through by `m` (`Nat.mul_lt_mul_left`'s `mpr` direction) to get `Lt (mul m (mul 2 n)) (sumDivisors (mul m n))`, then reassociates `mul m (mul 2 n)` to `mul 2 (mul m n)` via `mul_comm`/`mul_assoc`.

footprint-Nat.abundant_mul_left

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.mul_left`",
      "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"
    }
  ]
}