Mathlib v4.30 source proposition Nat.even_iff

Identifier: F:ml430-nat-even-iff-024826e9
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n : ℕ}, Even n ↔ n % 2 = 0

Dependencies

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7 direct dependencies 14 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.even_iff_mod_two_eq_zero

Kind: kernel-term
Status: checked

Supports: `Nat.even_iff_mod_two_eq_zero` is admitted as a Theorem with EXACTLY the mirror's shape (`Even n <-> n % 2 = 0`), pinned verbatim via the kernel's own renderer. Pre-existing in this prelude before this lane started -- `nat_prelude/parity.rs`'s `declare_even_iff_mod_two_eq_zero` -- and is the SAME proposition as Mathlib's `Nat.even_iff` under a different internal name: this prelude builds `Nat.Even`/`Nat.Odd` from the same `∃k, n = k+k` definition Mathlib uses, so flipping this mirror onto the pre-existing theorem is honest under the mirror-flip criterion (same definition, not a different one Mathlib merely proves a theorem about).

Checker command

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

footprint-Nat.even_iff_mod_two_eq_zero

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.even_iff_mod_two_eq_zero`. 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_iff_mod_two_eq_zero

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.even_iff_mod_two_eq_zero` 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 nat-parity-div (2026-08-30): NOT newly proved by this lane -- flipped onto a PRE-EXISTING kernel theorem, `Nat.even_iff_mod_two_eq_zero` (`nat_prelude/parity.rs`, landed before this lane's session, over `Nat.Even n := ∃k, n = k+k`, Mathlib's own definition). Verified the fact's formal.statement (`Even n <-> n % 2 = 0`) matches the kernel-rendered type of that theorem exactly via `nat_theorem_inventory`, and that Mathlib's `Nat.even_iff` is the same proposition about the same `Even`/`Odd` definition (checked at Mathlib v4.30.0 commit c5ea00351c28e24afc9f0f84379aa41082b1188f) -- so flipping is honest under the mirror-flip criterion (same definition, not a different one Mathlib merely proves a theorem about).",
  "source": "statement-only extraction of `Nat.even_iff` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.even_iff`",
      "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"
    }
  ]
}