lor(0, n) = n — Axeyum

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

Recorded description

The bitwise OR of 0 with any natural number n is n.

Formal statement

theorem Nat.lor_zero_left : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.lor AxNat.zero x0) x0)

Dependencies

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0 direct dependencies 2 direct dependents

Evidence

kernel-Nat.lor_zero_left

Kind: kernel-term
Status: checked

Supports: lor(0, n) = n

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- lor_zero_left 2>/dev/null | grep -Ec '^Nat\.lor_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.lor m n := Nat.lorAux m m n` (fuel `m`); at `m = 0` the outer `Nat.rec` on the fuel argument is already at its base case, which -- unlike `Nat.landAux`'s constant-`0` base case -- is `lorAux`'s CORRECTED base case returning `n` (see `nat_prelude::lor`'s module doc for why OR, having no absorbing zero the way AND does, needs this), so `lorAux 0 0 n` iota-reduces directly to `n` 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\.lor_zero_left[[:space:]]` (verified exactly 1 row) so a future sibling name sharing the `lor_zero_left` prefix cannot inflate the count.

compute-Nat.lor_zero_left

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

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

footprint-Nat.lor_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.lorAux` uses the same structural fuel device (`Nat.rec` on the fuel argument) as `Nat.landAux`/`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.lor` 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 219-nat-lor",
  "source": "one of three boundary/sanity theorems `declare_lor_all` (`crates/axeyum-lean-kernel/src/nat_prelude/lor.rs`) admits alongside the `Nat.lor`/`Nat.lorAux` definitions"
}