Mathlib v4.30 source proposition Nat.bitwise_comm

Identifier: F:ml430-nat-bitwise-comm-1a273bae
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {f : Bool → Bool → Bool}, (∀ (b b' : Bool), f b b' = f b' b) → ∀ (n m : ℕ), Nat.bitwise f n m = Nat.bitwise f m n

Dependencies

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1 direct dependencies 2 direct dependents

Evidence

reconciliation-Nat.bitwise_comm

Kind: kernel-term
Status: checked

Supports: ∀ {f : Bool → Bool → Bool}, (∀ (b b' : Bool), f b b' = f b' b) → ∀ (n m : ℕ), Nat.bitwise f n m = Nat.bitwise f m n

Evidence notes

Registers an independently constructed native theorem (`F:nat-bitwise-comm`, `Nat.bitwise_comm` in `nat_prelude/bitwise.rs`) whose proposition definitionally matches this proof-free imported goal, modulo the cosmetic `n m` / `m n` argument-name order in the two universally quantified naturals -- both sides state the SAME proposition (`bitwise` commutative for a commutative `f`), and `Eq` is symmetric regardless. Unlike `Nat.land_comm`/`Nat.lor_comm` (which needed a DIFFERENT route than Mathlib's own `bitwise_comm`, since our `land`/`lor` are hand-rolled fuel recursions rather than `bitwise` specializations at proof-construction time), this fact's native proof genuinely IS Mathlib's own general `Nat.bitwise` combinator, so the flip is honest in the strongest sense this repository's mirror-flip criterion recognizes: Mathlib's `def Nat.bitwise` and ours are the same function, and this closes the same theorem about it.

Provenance

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