land(1, 1) = 1

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

Recorded description

The bitwise AND of 1 and 1 is 1.

Formal statement

theorem Nat.land_one_one : Eq.{1} AxNat (AxNat.land (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero)) (AxNat.succ AxNat.zero)

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.land_one_one

Kind: kernel-term
Status: checked

Supports: land(1, 1) = 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- land_one_one 2>/dev/null | grep -Ec '^Nat\.land_one_one[[: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. `land 1 1 = landAux 1 1 1`: one fuel step exposes the guard (`n = 0`? no; `m = 0`? no), computes `2 * landAux 0 (1/2) (1/2) + (1%2)*(1%2) = 2*0 + 1 = 1`, entirely by delta+iota -- the proof is `Eq.refl`. `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_one_one[[:space:]]` (verified exactly 1 row).

compute-Nat.land_one_one

Kind: kernel-term
Status: checked

Supports: the definition genuinely computes the AND of matching bits, not a constant-zero stub

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 1 1` to `1` directly (definition unfolding, no boundary theorem invoked; part of a 9-pair concrete computation table also covering `land 0 0`, `land 6 3 = 2`, and the self-AND `land 7 7 = 7`), and separately confirms `Nat.land_one_one`'s ADMITTED TYPE (via `Kernel::infer`) is exactly `Eq (land 1 1) 1` -- with a negative control asserting the inferred type must NOT def_eq `Eq (land 1 1) 0`, so this cannot pass on a statement claiming the wrong value.

footprint-Nat.land_one_one

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. 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"
}