land decodes through a bit-appended argument

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

Recorded description

For every Bool a, Nat m, Bool b, Nat n: land(bit(a, m), bit(b, n)) = bit(a && b, land(m, n)).

Formal statement

theorem Nat.land_bit : ((x0 : Bool) -> ((x1 : AxNat) -> ((x2 : Bool) -> ((x3 : AxNat) -> Eq.{1} AxNat (AxNat.land (AxNat.bit x0 x1) (AxNat.bit x2 x3)) (AxNat.bit ((fun (x4 : Bool) => fun (x5 : Bool) => Bool.rec.{1} (fun (x6 : Bool) => Bool) Bool.false x5 x4) x0 x2) (AxNat.land x1 x3))))))

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.land_bit

Kind: kernel-term
Status: checked

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

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- land_bit 2>/dev/null | grep -Ec '^Nat\.land_bit[[: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 decides the exit status. Anchored with `^Nat\.land_bit[[:space:]]` so a future sibling (e.g. `land_bit_of_le`) cannot satisfy this row.

compute-Nat.land_bit

Kind: kernel-term
Status: checked

Supports: land(bit(true, 2), bit(false, 3)) = land(5, 6) = 4 (0b101 & 0b110 = 0b100), against bit(true && false, land(2, 3)) = bit(false, 2) = 4 (land(2,3)=2, 0b10 & 0b11 = 0b10) -- a DISCRIMINATING instance: a wrong-way `&&` or a swapped `bit` encoding does not land on 4 here.

Checker command

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

Evidence notes

Applies `Nat.land_bit` both against fully free (a, m, b, n) -- by re-declaring the statement as a fresh closed theorem over Pi/lambda-bound variables and requiring the kernel admit it -- and at the concrete, bit-differing instance (a=true, m=2, b=false, n=3), confirming both instantiations compute to the stated values.

footprint-Nat.land_bit

Kind: instance-pin
Status: checked

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

Checker command

cargo run -q -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": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-bit-decode",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}