land(m, 0) = 0

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

Recorded description

The bitwise AND of any natural number m with 0 is 0.

Formal statement

theorem Nat.land_zero_right : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.land x0 AxNat.zero) AxNat.zero)

Dependencies

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

1 direct dependencies 3 direct dependents

Evidence

kernel-Nat.land_zero_right

Kind: kernel-term
Status: checked

Supports: land(m, 0) = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. Unlike `land_zero_left`, `land m 0 = 0` is not `refl` at symbolic `m` -- the outer `Nat.rec` on the fuel (which is `m` itself) is stuck until `m`'s constructor shape is exposed -- so the proof is by induction on `m`. Both the base and step case close by `Eq.refl`, with the induction hypothesis unused in the step: the `n = 0` guard is tested OUTERMOST in `landAux`'s succ case (see `nat_prelude::land`'s module doc), so it collapses the whole term to `0` regardless of the (possibly symbolic) fuel predecessor. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe and the count is tested, so the exit status depends on what the run found. Anchored with `^Nat\.land_zero_right[[:space:]]` (verified exactly 1 row).

compute-Nat.land_zero_right

Kind: kernel-term
Status: checked

Supports: the equation instantiates correctly and states the value 0, not an arbitrary one

Checker command

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

Evidence notes

`Kernel::def_eq` reduces `land m 0` at m in {5, 1} to 0 directly (definition unfolding, no boundary theorem invoked), and separately confirms `Nat.land_zero_right`'s ADMITTED TYPE (via `Kernel::infer`, instantiated at m = 9) is exactly `Eq (land 9 0) 0` -- with a negative control asserting the inferred type must NOT def_eq `Eq (land 9 0) 1`, so this cannot pass on a statement claiming the wrong value.

footprint-Nat.land_zero_right

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

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

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration. The induction used here is ordinary `Nat.rec` over `Prop`, the same combinator every other induction in this prelude uses; this command is what makes the axiom-freedom claim falsifiable rather than asserted.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 215-nat-bitwise-2",
  "source": "one of four boundary/sanity theorems `declare_land_all` (`crates/axeyum-lean-kernel/src/nat_prelude/land.rs`) admits alongside the `Nat.land`/`Nat.landAux` definitions"
}