Mathlib v4.30 source proposition Nat.bitwise_bit'

Identifier: F:ml430-nat-bitwise-bit-4c4b28a8
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {f : Bool → Bool → Bool} (a : Bool) (m : ℕ) (b : Bool) (n : ℕ),
  (m = 0 → a = true) →
    (n = 0 → b = true) → Nat.bitwise f (Nat.bit a m) (Nat.bit b n) = Nat.bit (f a b) (Nat.bitwise f m n)

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

reconciliation-Nat.bitwise_bit

Kind: kernel-term
Status: checked

Supports: ∀ {f : Bool → Bool → Bool} (a : Bool) (m : ℕ) (b : Bool) (n : ℕ), (m = 0 → a = true) → (n = 0 → b = true) → Nat.bitwise f (Nat.bit a m) (Nat.bit b n) = Nat.bit (f a b) (Nat.bitwise f m n)

Evidence notes

Registers an independently constructed native theorem (`F:nat-bitwise-bit`, `Nat.bitwise_bit'` in `nat_prelude/bitwise.rs`) whose proposition matches this proof-free imported goal verbatim -- same binder order, same two side hypotheses, same conclusion. Our `Nat.bitwise` genuinely IS Mathlib's own general combinator (established for `bitwise_comm`/`bitwise_swap` already, unchanged here), and unlike the `testBit`-shaped siblings this mirror-flip criterion rules unflippable (Mathlib's `testBit` returns `Bool` against our `Nat`), this theorem's codomain is `Nat` throughout on both sides -- so the flip is honest in the strongest sense: Mathlib's `def Nat.bitwise`/`Nat.bit` and ours are the same functions, and this closes the same theorem about them.

Provenance

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