land(3, 5) = 1

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

Recorded description

The bitwise AND of 3 (0b011) and 5 (0b101) is 1 (0b001).

Formal statement

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

Dependencies

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

0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.land_three_five

Kind: kernel-term
Status: checked

Supports: land(3, 5) = 1

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- land_three_five 2>/dev/null | grep -Ec '^Nat\.land_three_five[[: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. `3 = 0b011`, `5 = 0b101`; `land 3 5 = landAux 3 3 5` unfolds through two real fuel steps (`3 & 5 -> 2*(1 & 2) + 1*1`, `1 & 2 -> 2*(0 & 1) + 1*0`, `0 & 1 -> 0` since `m = 0`) to `2*(2*0 + 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_three_five[[:space:]]` (verified exactly 1 row).

compute-Nat.land_three_five

Kind: kernel-term
Status: checked

Supports: the recursive step genuinely computes bitwise AND, including at DIFFERING bit positions, not merely matching ones

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 3 5` to `1` directly (definition unfolding, no boundary theorem invoked), with two explicit negative controls asserting `land 3 5` must NOT def_eq `5` (the wrong-way OR-of-first-operand answer) and must NOT def_eq `7` (the OR of the two operands) -- distinguishing this from a mis-implemented step (e.g. an OR-shaped or negated bit combinator) that `land_one_one` alone, whose single bit position matches on both operands, cannot rule out. Also confirms `Nat.land_three_five`'s ADMITTED TYPE (via `Kernel::infer`) is exactly `Eq (land 3 5) 1`, with the same `5` negative control on the inferred type.

footprint-Nat.land_three_five

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"
}