Mathlib v4.30 source proposition Nat.even_xor

Identifier: F:ml430-nat-even-xor-78a39432
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

reconciliation-Nat.even_xor

Kind: kernel-term
Status: checked

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

Evidence notes

Registers an independently constructed native theorem (`F:nat-even-xor`, `Nat.even_xor` in `nat_prelude/xor_parity.rs`) whose proposition definitionally matches this proof-free imported goal -- Mathlib's `Nat.xor` is `Nat.bitwise xor`, and our `Nat.xor` is DEFINED the same way (`nat_prelude/xor.rs`: `Nat.xor := Nat.bitwise xor_fn`), so this closes the SAME function's parity characterization, not a lookalike about a different definition (the criterion this repository uses to decide when flipping an `ml430` mirror is honest: Mathlib's `def` is the same function, not merely a theorem about a different one). No Autogenesis operation produced the theorem; the route taken (a fresh parity <-> `mod _ 2` bridge plus one step of `bitwiseAux`'s recursor) is NOT necessarily Mathlib's own proof route.

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.even_xor` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.even_xor`",
      "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"
    }
  ]
}