land(0, n) = 0

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

Recorded description

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

Formal statement

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

Dependencies

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0 direct dependencies 4 direct dependents

Evidence

kernel-Nat.land_zero_left

Kind: kernel-term
Status: checked

Supports: land(0, n) = 0

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- land_zero_left 2>/dev/null | grep -Ec '^Nat\.land_zero_left[[: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. `Nat.land m n := Nat.landAux m m n` (fuel `m`); at `m = 0` the outer `Nat.rec` on the fuel argument is already at its base case, so `landAux 0 0 n` iota-reduces directly to `0` regardless of `n` -- the proof is `Eq.refl`, no induction and no case-split combinator. `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_left[[:space:]]` (verified exactly 1 row) so a future sibling name sharing the `land_zero_left` prefix cannot inflate the count.

compute-Nat.land_zero_left

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 0 n` at n in {0, 5} to 0 directly (definition unfolding, no boundary theorem invoked), and separately confirms `Nat.land_zero_left`'s ADMITTED TYPE (via `Kernel::infer`, instantiated at n = 7) is exactly `Eq (land 0 7) 0` -- with a negative control asserting the inferred type must NOT def_eq `Eq (land 0 7) 1`, so this cannot pass on a statement claiming the wrong value.

footprint-Nat.land_zero_left

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. `Nat.landAux` uses the structural fuel device (`Nat.rec` on the fuel argument, exactly `Nat.logAux`/`Nat.testBitAux`/`Nat.sizeAux`'s shape) rather than Mathlib's well-founded `Nat.bitwise`, so nothing here needs `WellFounded`/`Quot.sound`/`propext`; 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"
}