land is commutative

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

Recorded description

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

Formal statement

theorem Nat.land_comm : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.land x0 x1) (AxNat.land x1 x0)))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

3 direct dependencies 1 direct dependents

Evidence

kernel-Nat.land_comm

Kind: kernel-term
Status: checked

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

compute-Nat.land_comm

Kind: kernel-term
Status: checked

Supports: land(3, 6) = land(6, 3) = 2 (0b011 & 0b110 = 0b010), 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::land_comm_applies_at_a_concrete_discriminating_instance

Evidence notes

Applies `Nat.land_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 (land 3 6) (land 6 3)` and that both sides compute to `2`.

footprint-Nat.land_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-fuel-transport",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}