lor(3, 5) = 7 — Axeyum

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

Recorded description

The bitwise OR of 3 and 5 is 7.

Formal statement

theorem Nat.lor_three_five : Eq.{1} AxNat (AxNat.lor (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero)))))) (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ (AxNat.succ AxNat.zero)))))))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.lor_three_five

Kind: kernel-term
Status: checked

Supports: lor(3, 5) = 7

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- lor_three_five 2>/dev/null | grep -Ec '^Nat\.lor_three_five[[: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. `3 = 0b011`, `5 = 0b101`, `3 ||| 5 = 0b111 = 7`; every numeral is concrete, so the fuel recursion, the div/mod halving, and the per-bit `max` (via `Nat.ble` + `bool_select_nat`) all reduce fully and the proof is `Eq.refl`. Deliberately discriminating against `F:nat-land-three-five`'s `3 &&& 5 = 1` (a wrong-way `bit_or` reusing the AND step, or an AND/OR combinator swap, cannot pass both facts' checkers at once). `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_three_five[[:space:]]` (verified exactly 1 row).

compute-Nat.lor_three_five

Kind: kernel-term
Status: checked

Supports: the theorem states the value 7, not an arbitrary or AND-shaped 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 3 5` directly to `7` (definition unfolding, no boundary theorem invoked), over a small table of concrete pairs including `(3, 5) -> 7` and `(6, 3) -> 7`, and separately confirms `Nat.lor_three_five`'s ADMITTED TYPE is exactly `Eq (lor 3 5) 7` -- with negative controls asserting the inferred type must NOT def_eq `Eq (lor 3 5) 1` (the AND-shaped wrong answer) or `Eq (lor 3 5) 5` (the first-operand-only wrong answer), so this cannot pass on a statement claiming a wrong value from either failure family.

footprint-Nat.lor_three_five

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 as `Nat.landAux` 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"
}