The floor logarithm of zero is zero, in every base

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

Recorded description

For every base b, log_b(0) = 0.

Formal statement

theorem Nat.log_zero_right : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.log x0 AxNat.zero) AxNat.zero)

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.log_zero_right

Kind: kernel-term
Status: checked

Supports: For every base b, log_b(0) = 0.

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- log_zero_right 2>/dev/null | grep -Ec '^Nat\.log_zero_right[[: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.log b n` is `Nat.logAux b n n` -- the fuel is `n` itself -- so at `n = 0` the fuel is already exhausted and `logAux`'s zero minor premise fires. The proof is `Eq.refl`; the kernel closes it by iota-reduction alone. `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 four real names exit 0 and `log_zero_rightXX` exits 1.

compute-Nat.log_zero_right

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::log_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 `log 2 8` to `3`, `log 2 7` to `2` and `log 3 9` to `2`, with two negative controls differing by ONE successor (`log 2 8 = 4` and `log 3 9 = 1` must both be rejected) so that `def_eq` cannot pass vacuously.

footprint-Nat.log_zero_right

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. This matters more than usual here: Mathlib's `Nat.log` is defined by WELL-FOUNDED recursion on `n / b`, and the Lean equation compiler's route to that carries `Quot.sound`/`propext`. The fuel-based structural definition in `nat_prelude/log.rs` 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 199-nat-log",
  "source": "one of the four boundary equations `declare_log_all` (`crates/axeyum-lean-kernel/src/nat_prelude/log.rs`) admits alongside the `Nat.logAux`/`Nat.log` definitions"
}