bitwise or_fn m n = lor m n, for all m and n

Identifier: F:nat-bitwise-or-eq-lor
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The general bitwise combinator instantiated at Bool.or agrees with the specialized bitwise-OR at every pair of natural numbers.

Formal statement

theorem Nat.bitwise_or_eq_lor : ((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.bitwise (fun (x2 : Bool) => fun (x3 : Bool) => Bool.rec.{1} (fun (x4 : Bool) => Bool) x3 Bool.true x2) x0 x1) (AxNat.lor x0 x1)))

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.bitwise_or_eq_lor

Kind: kernel-term
Status: checked

Supports: for all m n, bitwise or_fn m n = lor m n

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- bitwise_or_eq_lor 2>/dev/null | grep -Ec '^Nat\.bitwise_or_eq_lor[[: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. The anchor `^Nat\.bitwise_or_eq_lor[[:space:]]` matters: the same filter also prints `Nat.bitwise_or_eq_lor_three_five` (the single-point predecessor this theorem supersedes), and the trailing `[[:space:]]` excludes it -- verified exactly 1 row. `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.

compute-Nat.bitwise_or_eq_lor

Kind: kernel-term
Status: checked

Supports: the equivalence holds at symbolic operands and at seven concrete pairs, and depends on lor's own guard values

Checker command

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

Evidence notes

Same two-way check as the `land` twin: the statement is re-declared over genuinely bound variables with this theorem as the entire proof (so it must apply at non-numeral operands), and instantiated at seven concrete pairs including `(3,5)`, `(5,3)` and `(6,3)` where AND and OR values differ. The same test also mutation-verifies that `lorAux`'s guard values are load-bearing rather than incidental: replacing `lor`'s `n = 0` branch -- which returns `m`, since OR has no absorbing zero on the fuel operand -- with the constant `0` that `land` uses makes the kernel refuse the declaration and the whole prelude build fails (0 passed, 121 failed).

footprint-Nat.bitwise_or_eq_lor

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.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-rec-agreement",
  "source": "`declare_rec_agreement_all` (`crates/axeyum-lean-kernel/src/nat_prelude/rec_agreement.rs`)"
}