xor a b != 0 iff a != b

Identifier: F:nat-xor-ne-zero-iff
Proof route: kernel-lean
External status: Not recorded
Axiom footprint: Empty

Recorded description

For every natural a, b: xor(a, b) is not equal to 0 if and only if a is not equal to b.

Formal statement

theorem Nat.xor_ne_zero_iff : ((x0 : AxNat) -> ((x1 : AxNat) -> Iff (Not (Eq.{1} AxNat (AxNat.xor x0 x1) AxNat.zero)) (Not (Eq.{1} AxNat x0 x1))))

Dependencies

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5 direct dependencies 1 direct dependents

Evidence

kernel-Nat.xor_ne_zero_iff

Kind: kernel-term
Status: checked

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

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory -- xor_ne_zero_iff 2>/dev/null | grep -Ec '^Nat\.xor_ne_zero_iff[[:space:]]'

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 (expected >= 1) decides the exit status, never `grep -q` under pipefail. Anchored with `[[:space:]]`, never a literal tab, per this ledger's own grep-dialect gotcha. `--release` is mandatory: this tool builds `creal`/`complex`/`cpoint`, which overflow the default debug thread stack.

compute-Nat.xor_ne_zero_iff

Kind: kernel-term
Status: checked

Supports: At the discriminating concrete pair (a, b) = (3, 5), xor(3, 5) def-eq's to 6, so Not (Eq (xor 3 5) 0) is genuinely inhabited (via Nat.succ_ne_zero at 5, since succ 5 def-eq's to 6); the Iff.mp direction lands on Not (Eq 3 5) (and NOT on the negative control Not (Eq 3 3)), and the Iff.mpr direction lands back on Not (Eq (xor 3 5) 0). The theorem also re-derives against a genuinely free (a, b) pair, not merely a concrete instance.

Checker command

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

Evidence notes

A concrete instantiation alone can hide a defect a symbolic build exposes (and vice versa), so both are checked. (3, 5) is chosen because xor(3,5)=6 is neither operand, giving the negative control (Eq 3 3) something to discriminate against, and because Nat.succ_ne_zero at 5 supplies a genuine, cheap proof of Not (Eq (xor 3 5) 0) without needing a separate numeral-inequality lemma.

footprint-Nat.xor_ne_zero_iff

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-xor-ne-zero",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}