bitwise is commutative for a commutative combinator

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

Recorded description

For every Bool -> Bool -> Bool function f such that f is commutative, and every natural m, n: bitwise(f, m, n) = bitwise(f, n, m).

Formal statement

theorem Nat.bitwise_comm : ((x0 : ((x0 : Bool) -> ((x1 : Bool) -> Bool))) -> ((x1 : ((x1 : Bool) -> ((x2 : Bool) -> Eq.{1} Bool (x0 x1 x2) (x0 x2 x1)))) -> ((x2 : AxNat) -> ((x3 : AxNat) -> Eq.{1} AxNat (AxNat.bitwise x0 x2 x3) (AxNat.bitwise x0 x3 x2)))))

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.bitwise_comm

Kind: kernel-term
Status: checked

Supports: Nat.bitwise_comm is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe so the tested count decides the exit status. Anchored with `^Nat\.bitwise_comm[[:space:]]` so a sibling name cannot satisfy this row. Run `--release` -- the debug build of this inventory example SIGABRTs on stack depth (unrelated to this proof).

compute-Nat.bitwise_comm

Kind: kernel-term
Status: checked

Supports: bitwise(xor, 3, 5) = bitwise(xor, 5, 3) = 6 (0b011 xor 0b101 = 0b110), a DISCRIMINATING instance -- a symmetric pair like (5, 5) cannot catch a transposed argument.

Checker command

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

Evidence notes

Applies `Nat.bitwise_comm` symbolically (against free `m`, `n` at a fixed concrete `f = xor_fn`) and at the concrete, bit-differing instance `(m, n) = (3, 5)`, confirming both that the admitted type matches `Eq (bitwise xor 3 5) (bitwise xor 5 3)` and that both sides compute to `6`. The `hf` commutativity witness for `xor_fn` is itself built and kernel-checked in the same test (`bool_fn_comm`, a four-leaf `Bool.rec` case split). A companion negative control shows `bitwise_aux_comm_of_fuel`'s UNCONDITIONAL analogue is false for `f = or_fn` at insufficient fuel (`bitwiseAux or 0 0 1 = 1` against `bitwiseAux or 0 1 0 = 0`), confirmed by a Python simulation before any Rust was written (`docs/plan/status/256-nat-bitwise-comm.md`).

footprint-Nat.bitwise_comm

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

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

Evidence notes

Reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the FULL trusted surface rather than `Declaration::Axiom` alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-bitwise-comm",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}