Mathlib v4.30 source proposition Nat.even_add

Identifier: F:ml430-nat-even-add-31386639
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, Even (m + n) ↔ (Even m ↔ Even n)

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.even_add

Kind: kernel-term
Status: checked

Supports: `Nat.even_add` 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: a four-way case split on `Nat.even_or_odd_exists m`/`Nat.even_or_odd_exists n`, relating `add m n`'s own Even/Odd witness to `m`'s and `n`'s via `Nat.add_add_add_comm`/`Nat.succ_add` (witness arithmetic, not `mod`, since this prelude's `Even`/`Odd` are existential).

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory even_add 2>/dev/null | awk -F'\t' '$1 == "Nat.even_add" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> Iff (AxNat.Even (AxNat.add x0 x1)) (Iff (AxNat.Even x0) (AxNat.Even x1))))"' | wc -l)" -ge 1

instance-Nat.even_add

Kind: kernel-term
Status: checked

Supports: Applies at a concrete `(m, n) := (2, 2)` (both Even, the EE case-split leg) with a REAL `Even 2` witness chained through both `mp` legs to confirm the whole combination type-checks, AND at genuinely free `m`, `n` (a symbolic restatement built via the same NatOps closure machinery `d.theorem` uses).

Checker command

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

footprint-Nat.even_add

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.even_add`. 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.even_add

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.even_add` 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": "2026-08-29",
  "established_by": "lane parity-finish (2026-08-30): built directly as a kernel declaration in the new file nat_prelude/even_add_family.rs, via a four-way case split on `Nat.even_or_odd_exists` at `m` and `n`, relating `add m n`'s own Even/Odd witness to `m`'s and `n`'s via `Nat.add_add_add_comm`/`Nat.succ_add`. Verified against the pinned Mathlib v4.30.0 source (commit c5ea00351c28e24afc9f0f84379aa41082b1188f) that this is a standard Nat parity lemma about the SAME `Nat.add`/`Nat.Even` 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.even_add` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.even_add`",
      "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"
    }
  ]
}