ldiff(5, 3) = 4

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

Recorded description

The bitwise AND-NOT of 5 and 3 is 4, not the 2 that ldiff(3, 5) gives -- ldiff is not commutative.

Formal statement

theorem Nat.ldiff_five_three : Eq.{1} AxNat (AxNat.ldiff (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero))))) (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero)))) (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero))))

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.ldiff_five_three

Kind: kernel-term
Status: checked

Supports: ldiff(5, 3) = 4

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- ldiff_five_three 2>/dev/null | grep -Ec '^Nat\.ldiff_five_three[[: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. `5 = 0b101`, `3 = 0b011`, `5 &~ 3 = 0b100 = 4` (bit 0 of 5 is cleared by bit 0 of 3; bit 2 of 5 survives since bit 2 of 3 is clear); every numeral is concrete, so the proof is `Eq.refl`. This is the ASYMMETRY theorem: the same two operands as `F:nat-ldiff-three-five`'s `3 &~ 5 = 2`, swapped, and NOT the same answer -- the sharpest negative control available for this definition, and one `Nat.land`/`Nat.lor` (both commutative) cannot even express. `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\.ldiff_five_three[[:space:]]` (verified exactly 1 row).

compute-Nat.ldiff_five_three

Kind: kernel-term
Status: checked

Supports: the theorem states the value 4, distinct from ldiff_three_five's 2

Checker command

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

Evidence notes

`Kernel::def_eq` reduces `ldiff 5 3` directly to `4` (definition unfolding, no boundary theorem invoked), and separately confirms `Nat.ldiff_five_three`'s ADMITTED TYPE is exactly `Eq (ldiff 5 3) 4` -- with a negative control asserting the inferred type must NOT def_eq `Eq (ldiff 5 3) 2` (that is `ldiff 3 5`, the swapped-operand result), so this cannot pass on a statement that fails to distinguish `ldiff`'s two argument orders.

footprint-Nat.ldiff_five_three

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.ldiffAux` uses the same structural fuel device as `Nat.landAux`/`Nat.lorAux` 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. Measured with `Nat.ldiff` landed: `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 226-nat-ldiff",
  "source": "one of four boundary/sanity theorems `declare_ldiff_all` (`crates/axeyum-lean-kernel/src/nat_prelude/ldiff.rs`) admits alongside the `Nat.ldiff`/`Nat.ldiffAux` definitions"
}