The ceiling logarithm in base one is zero

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

Recorded description

For every n, clog_1(n) = 0.

Formal statement

theorem Nat.clog_one_left : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.clog (AxNat.succ AxNat.zero) x0) AxNat.zero)

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.clog_one_left

Kind: kernel-term
Status: checked

Supports: For every n, clog_1(n) = 0.

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- clog_one_left 2>/dev/null | grep -Ec '^Nat\.clog_one_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. The guard's OUTER cut here is `Nat.ble 2 b`; with `b = 1`, `ble (succ (succ zero)) (succ zero)` reduces (via `Nat.ble`'s recursive step) to `ble (succ zero) zero`, i.e. `Bool.false`, regardless of `n`, so the whole term collapses to `0` in every fuel case with a single rewrite. The induction on `n` exposes the fuel's constructor and nothing more -- the induction hypothesis is never used, and both minor premises are `Eq.refl`. `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. Controls run 2026-08-28: the real name exits 0 and `clog_one_leftXX` exits 1.

compute-Nat.clog_one_left

Kind: kernel-term
Status: checked

Supports: the fuel recursion terminates with the right value, not with an exhausted fuel

Checker command

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

Evidence notes

A fuel-based definition is only correct if the fuel always suffices, and an EXHAUSTED fuel returns `0` -- which is exactly what a wrong answer looks like for a logarithm. So the computation test is the fuel-sufficiency check: `Kernel::def_eq` reduces `clog 2 8` to `3`, `clog 2 7` to `3` (deliberately chosen to differ from the FLOOR logarithm `log 2 7 = 2`, so the test cannot pass by accidentally computing the wrong function), `clog 2 5` to `3` (three levels of the recursive `(n + b - 1) / b` step), `clog 5 4` to `1` and `clog 3 9` to `2`, with two negative controls differing by ONE successor (`clog 2 7 = 4` and `clog 3 9 = 1` must both be rejected) so that `def_eq` cannot pass vacuously.

footprint-Nat.clog_one_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. Mathlib's `Nat.clog` is defined by WELL-FOUNDED recursion on `(n + b - 1) / b`, and the Lean equation compiler's route to that carries `Quot.sound`/`propext`. The fuel-based structural definition in `nat_prelude/clog.rs` -- the same device `nat_prelude/log.rs` uses for `Nat.log` -- was chosen so that this row stays empty, and this command is what makes the claim falsifiable rather than asserted.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane 204-nat-clog",
  "source": "one of the four boundary equations `declare_clog_all` (`crates/axeyum-lean-kernel/src/nat_prelude/clog.rs`) admits alongside the `Nat.clogAux`/`Nat.clog` definitions"
}