Even (xor m n) iff Even m iff Even n

Identifier: F:nat-even-xor
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural m, n: Even(xor(m, n)) holds iff Even(m) holds iff Even(n) holds.

Formal statement

theorem Nat.even_xor : ((x0 : AxNat) -> ((x1 : AxNat) -> Iff (AxNat.Even (AxNat.xor x0 x1)) (Iff (AxNat.Even x0) (AxNat.Even x1))))

Dependencies

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12 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.even_xor

Kind: kernel-term
Status: checked

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

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- even_xor 2>/dev/null | grep -Ec '^Nat\.even_xor[[: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 -- producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe and the tested count is what decides the exit status. Anchored with `^Nat\.even_xor[[:space:]]` so a future sibling cannot satisfy this row.

compute-Nat.even_xor

Kind: kernel-term
Status: checked

Supports: even_xor(4, 6): both even, xor(4, 6) = 2 (even) -- mp/mpr round-trip through a hand-built Even 2/Even 4/Even 6. even_xor(3, 5): both odd, xor(3, 5) = 6 (even) -- mpr applied to a constructed Iff (Even 3) (Even 5) (built from Not (Even 3)/Not (Even 5)) lands on Even 6. Both instances exercise the genuinely-bitwise (both-nonzero) branch of the proof, not just the m=0/n=0 boundary shortcuts.

Checker command

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

Evidence notes

A swapped mp/mpr, a wrong remainder in the even_iff_mod_two_eq_zero bridge, or a sign error in the per-bit XOR combine would make one of these applications either fail to type-check or land on a type that is not defeq to the expected side.

footprint-Nat.even_xor

Kind: instance-pin
Status: checked

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

Checker command

cargo run -q -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-parity-lowbit",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}