ldiff(0, n) = 0

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

Recorded description

The bitwise AND-NOT of 0 with any natural number n is 0.

Formal statement

theorem Nat.ldiff_zero_left : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.ldiff AxNat.zero x0) AxNat.zero)

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-Nat.ldiff_zero_left

Kind: kernel-term
Status: checked

Supports: ldiff(0, n) = 0

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- ldiff_zero_left 2>/dev/null | grep -Ec '^Nat\.ldiff_zero_left[[: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. `Nat.ldiff m n := Nat.ldiffAux m m n` (fuel `m`); at `m = 0` the outer `Nat.rec` on the fuel argument is already at its base case, which is `ldiffAux`'s constant-`0` row (the SAME shape `Nat.land` uses, not `Nat.lor`'s corrected `n`-returning row -- see `nat_prelude::ldiff`'s module doc for why: `m` is both the fuel-sized operand AND `ldiff`'s absorbing-zero operand, `ldiff 0 n = 0` for every `n`, so ignoring the leftover `n`-argument is exactly as safe as it is for `land`), so `ldiffAux 0 0 n` iota-reduces directly to `0` regardless of `n`'s shape -- the proof is `Eq.refl`, no induction and no case-split combinator. `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_zero_left[[:space:]]` (verified exactly 1 row) so a future sibling name sharing the `ldiff_zero_left` prefix cannot inflate the count.

compute-Nat.ldiff_zero_left

Kind: kernel-term
Status: checked

Supports: the equation instantiates correctly and states the value 0, not an arbitrary one

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 0 n` at n in {0, 5, 1} to 0 directly (definition unfolding, no boundary theorem invoked), and separately confirms `Nat.ldiff_zero_left`'s ADMITTED TYPE (via `Kernel::infer`, instantiated at n = 7) is exactly `Eq (ldiff 0 7) 0` -- with a negative control asserting the inferred type must NOT def_eq `Eq (ldiff 0 7) 7` (the pass-through wrong answer, i.e. `lor`'s shape at this same boundary), so this cannot pass on a statement claiming the wrong (OR-shaped) value.

footprint-Nat.ldiff_zero_left

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 (`Nat.rec` on the fuel argument) as `Nat.landAux`/`Nat.lorAux`/`Nat.logAux`/`Nat.testBitAux`/`Nat.sizeAux` 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"
}