lor is commutative

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

Recorded description

For every natural m, n: lor(m, n) = lor(n, m).

Formal statement

theorem Nat.lor_comm : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.lor x0 x1) (AxNat.lor x1 x0)))

Dependencies

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3 direct dependencies 1 direct dependents

Evidence

kernel-Nat.lor_comm

Kind: kernel-term
Status: checked

Supports: Nat.lor_comm 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 -- lor_comm 2>/dev/null | grep -Ec '^Nat\.lor_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 -- 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\.lor_comm[[:space:]]` so a future sibling (e.g. `lor_aux_comm_of_fuel`) cannot satisfy this row.

compute-Nat.lor_comm

Kind: kernel-term
Status: checked

Supports: lor(3, 6) = lor(6, 3) = 7 (0b011 | 0b110 = 0b111), a DISCRIMINATING instance -- a symmetric pair like (5, 5) cannot catch a transposed argument, since both orderings agree regardless of whether the proof is right.

Checker command

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

Evidence notes

Applies `Nat.lor_comm` symbolically (against free `m`, `n`) and at the concrete, bit-differing instance `(m, n) = (3, 6)`, confirming both that the admitted type matches `Eq (lor 3 6) (lor 6 3)` and that both sides compute to `7`. Also applies `Nat.lor_aux_comm_of_fuel` at the shared fuel `m + n = 9` and checks an insufficient-fuel negative control (`lorAux 0 0 1 = 1` against `lorAux 0 1 0 = 0`) confirming the `Le` hypotheses this theorem carries -- unlike `Nat.land_aux_comm_of_fuel`, which needs none -- are load-bearing.

footprint-Nat.lor_comm

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