Mathlib v4.30 source proposition Nat.land_bit

Identifier: F:ml430-nat-land-bit-b9ab7475
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (a : Bool) (m : ℕ) (b : Bool) (n : ℕ), Nat.bit a m &&& Nat.bit b n = Nat.bit (a && b) (m &&& n)

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

reconciliation-Nat.land_bit

Kind: kernel-term
Status: checked

Supports: ∀ (a : Bool) (m : ℕ) (b : Bool) (n : ℕ), Nat.bit a m &&& Nat.bit b n = Nat.bit (a && b) (m &&& n)

Evidence notes

Registers an independently constructed native theorem (`F:nat-land-bit`, `Nat.land_bit` in `nat_prelude/bit_decode.rs`) whose proposition definitionally matches this proof-free imported goal -- Mathlib's `Nat.land` is `Nat.bitwise and`, and our `Nat.land` is proved equal to the `bitwise and_fn` specialization by `Nat.bitwise_and_eq_land`, so this closes the SAME function's `bit`-decode identity, 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 (the `Nat.bit` decode bridge -- an artificially chosen sufficient fuel, plus `Nat.bit_div_two`/`Nat.bit_mod_two` decoding, plus a `Bool`-first case tree resolving the two fuel-recursion guards) is NOT Mathlib's own route via `bitwise_bit'`, which is why this fact's `depends_on` stays empty rather than naming `F:ml430-nat-bitwise-bit-4c4b28a8` -- our proof never used the general `bitwise` recursion equations.

Provenance

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