bit-halving recursion is independent of the fuel, given two sufficient fuels

Identifier: F:nat-binary-rec-fuel-irrelevance
Proof route: kernel-lean
External status: unknown
Axiom footprint: Empty

Recorded description

For any carrier alpha, base value z, step function f, and naturals fuel1, n, fuel2: if n <= fuel1 and n <= fuel2, then binaryRecAux alpha z f fuel1 n = binaryRecAux alpha z f fuel2 n.

Formal statement

theorem Nat.binaryRecAux_agree_of_fuel : ((x0 : Sort (1)) -> ((x1 : x0) -> ((x2 : ((x2 : Bool) -> ((x3 : AxNat) -> ((x4 : x0) -> x0)))) -> ((x3 : AxNat) -> ((x4 : AxNat) -> ((x5 : AxNat) -> ((x6 : AxNat.le x4 x3) -> ((x7 : AxNat.le x4 x5) -> Eq.{1} x0 (AxNat.binaryRecAux x0 x1 x2 x3 x4) (AxNat.binaryRecAux x0 x1 x2 x5 x4)))))))))

Dependencies

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6 direct dependencies 1 direct dependents

Evidence

kernel-Nat.binaryRecAux_agree_of_fuel

Kind: kernel-term
Status: checked

Supports: Nat.binaryRecAux_agree_of_fuel is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory -- binaryRecAux_agree_of_fuel 2>/dev/null | grep -Ec '^Nat\.binaryRecAux_agree_of_fuel[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type. `nat_theorem_inventory` prints nothing for a name that does not exist, and `grep -Ec` consumes the pipe so the tested COUNT decides the exit status. The pattern is anchored `^Nat\.<name>[[:space:]]` and was verified with `/usr/bin/grep` (GNU grep), not the interactive `ugrep` shell function: 1 for the real name, 0 for a fabricated sibling. Run `--release`; the debug build SIGABRTs building the full constructed environment.

suite-nat-prelude

Kind: kernel-term
Status: checked

Supports: The whole nat_prelude suite, including the environment-derived coverage assertion, is green with this declaration present.

Checker command

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

Evidence notes

133 passed, 0 failed. `every_nat_declaration_is_checked_and_axiom_free` enumerates `kernel.environment()` and fails naming any `Nat.`-prefixed Definition or Theorem absent from the inventory lists, so this declaration cannot be present-but-unchecked. The suite is not feature-gated, so it cannot compile to zero tests.

footprint-nat-environment

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

Reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the FULL trusted surface rather than `Declaration::Axiom` alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-binaryrec",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}