each bit of a bitwise AND is the product of the two operand bits

Identifier: F:nat-testbit-land
Proof route: kernel-lean
External status: Not recorded
Axiom footprint: Empty

Recorded description

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

Formal statement

theorem Nat.testBit_land : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> Eq.{1} AxNat (AxNat.testBit (AxNat.land x0 x1) x2) (AxNat.mul (AxNat.testBit x0 x2) (AxNat.testBit x1 x2)))))

Dependencies

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26 direct dependencies 3 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.testBit_land

Kind: kernel-term
Status: checked

Supports: Nat.testBit_land is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory -- testBit_land 2>/dev/null | grep -Ec '^Nat\.testBit_land[[: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. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe so the tested count (expected >= 1) decides the exit status, never `grep -q` under pipefail. Anchored with `[[:space:]]`, never a literal tab. `--release` is mandatory: this tool builds `creal`/`complex`/`cpoint`, which overflow the default debug thread stack.

concrete-Nat.testBit_land

Kind: kernel-term
Status: checked

Supports: Checked at the concrete discriminating instance (m, n) = (3, 5) (binary 011/101, land 3 5 = 1 = 001) across all three of its meaningfully differing bits (bit 0: 1/1 -> AND 1; bit 1: 1/0 -> AND 0; bit 2: 0/1 -> AND 0), each with a negative control that the OTHER bit value does not also def-eq -- and symbolically against a genuinely free (m, n, i) triple.

Checker command

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

Evidence notes

A single bit position could not discriminate a swapped combine (e.g. an OR-shaped step), so all three bits at (3,5) are checked.

footprint-Nat.testBit_land

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

cargo run -q --release -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": "2026-08-30",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane blocked-mirror-divergences",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}