bitwise f m 0 = if f true false then m else 0

Identifier: F:nat-bitwise-zero-right
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every bit-combining function f and every natural m, the general bitwise combinator applied to (f, m, 0) equals m when f true false holds and 0 otherwise.

Formal statement

theorem Nat.bitwise_zero_right : ((x0 : ((x0 : Bool) -> ((x1 : Bool) -> Bool))) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.bitwise x0 x1 AxNat.zero) (Bool.rec.{1} (fun (x2 : Bool) => AxNat) AxNat.zero x1 (x0 Bool.true Bool.false))))

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-Nat.bitwise_zero_right

Kind: kernel-term
Status: checked

Supports: forall f m, bitwise f m 0 = if f true false then m else 0

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- bitwise_zero_right 2>/dev/null | grep -Ec '^Nat\.bitwise_zero_right[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. Universally quantified over `f : Bool -> Bool -> Bool` and `m : Nat`, holds unconditionally. Proved by induction on `m`: every step is `Eq.refl` (the `n = 0` guard, tested outermost with the theorem's own literal `0`, collapses the whole succ-step term regardless of the fuel predecessor -- exactly `land_zero_right`'s shape). The BASE case is the one piece of proof content beyond `land`/`lor`/`ldiff`'s zero-right theorems: at `m = 0` the LHS reduces to `bitwiseAux`'s fuel-exhaustion row conditioned on `f false true` while the RHS is conditioned on `f true false` -- two different, generally non-defeq `Bool` terms for symbolic `f` -- both provably `0` there via a small `Bool`-case-split lemma (`bool_select_same`, `bitwise.rs`) rather than by `refl` alone. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe and the count is tested, so the exit status depends on what the run found. Anchored with `^Nat\.bitwise_zero_right[[:space:]]` (verified exactly 1 row).

compute-Nat.bitwise_zero_right

Kind: kernel-term
Status: checked

Supports: the theorem specializes correctly at both an absorbing and a non-absorbing f

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::bitwise_computes_and_its_boundary_theorems_apply

Evidence notes

Instantiates the universal theorem at two concrete `f` witnesses (`and_fn`, `or_fn`, built inline) and a concrete `m = 9`, confirming the ADMITTED TYPE at each instantiation `def_eq`s the expected specialization: at `and_fn` (absorbing) `bitwise and_fn 9 0` reduces to `0`, at `or_fn` (identity) it reduces to `9` -- with negative controls that each instantiation must NOT also state the other's value.

footprint-Nat.bitwise_zero_right

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration, including the `bool_select_same` case-split helper this theorem's base case needs. Measured with `Nat.bitwise` landed: `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-bitwise-general",
  "source": "one of two f-general boundary theorems `declare_bitwise_all` (`crates/axeyum-lean-kernel/src/nat_prelude/bitwise.rs`) admits alongside the `Nat.bitwise`/`Nat.bitwiseAux` definitions"
}