Mathlib v4.30 source proposition Nat.div_two_mul_two_of_even

Identifier: F:ml430-nat-div-two-mul-two-of-even-9ccc5340
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Even n → n / 2 * 2 = n

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

4 direct dependencies 0 direct dependents

Evidence

kernel-Nat.div_two_mul_two_of_even

Kind: kernel-term
Status: checked

Supports: `Nat.div_two_mul_two_of_even` is admitted as a Theorem with EXACTLY the mirror's shape, pinned verbatim via the kernel's own renderer. Built directly against Mathlib's own name: `Even n -> n/2*2 = n`, via `Nat.even_iff_mod_two_eq_zero` (pinning `n % 2 = 0`), `Nat.div_mod_exec`'s reconstruction `n = 2*(n/2) + n%2`, then `Nat.add_zero`/`Nat.mul_comm`.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory div_two_mul_two_of_even 2>/dev/null | awk -F'\t' '$1 == "Nat.div_two_mul_two_of_even" && $3 == "((x0 : AxNat) -> ((x1 : AxNat.Even x0) -> Eq.{1} AxNat (AxNat.mul (AxNat.div x0 (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.succ (AxNat.succ AxNat.zero))) x0))"' | wc -l)" -ge 1

instance-Nat.div_two_mul_two_of_even

Kind: kernel-term
Status: checked

Supports: Applies at a concrete witness (Even 4, computing to Eq 4 4) AND at a genuinely free `n` (a symbolic restatement built via the same NatOps closure machinery `d.theorem` uses), plus a negative control confirming `Nat.div`'s truncation makes the `Even` hypothesis load-bearing: `5/2*2` computes to `4`, not `5`.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::div_two_mul_two_mirrors_apply_concretely_symbolically_and_reject_a_truncated_odd_control -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

footprint-Nat.div_two_mul_two_of_even

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.div_two_mul_two_of_even`. The `nat` prelude's trusted surface stays 0: no Axiom, no Opaque, no Quotient.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free nat

coverage-Nat.div_two_mul_two_of_even

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.div_two_mul_two_of_even` derived from `kernel.environment()` directly, not from an inventory list -- a Nat-namespace declaration missing from `theorem_names` fails this test by name.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane nat-parity-div (2026-08-30): built directly as a kernel declaration in the new file nat_prelude/parity_div.rs, over `Nat.even_iff_mod_two_eq_zero` (pins `n % 2`), `Nat.div_mod_exec` (the `n = 2*(n/2) + n%2` reconstruction) and `Nat.mul_comm`/`Nat.add_zero`. Verified against the pinned Mathlib v4.30.0 source (commit c5ea00351c28e24afc9f0f84379aa41082b1188f) that this is a standard Nat order/division lemma about the SAME Nat.div/Nat.mul this kernel already has (no new definition introduced), so mirroring it under Mathlib's own name is honest under the mirror-flip criterion.",
  "source": "statement-only extraction of `Nat.div_two_mul_two_of_even` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.div_two_mul_two_of_even`",
      "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"
    }
  ]
}