Mathlib v4.30 source proposition Nat.log_eq_zero_iff

Identifier: F:ml430-nat-log-eq-zero-iff-819cea74
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.log_eq_zero_iff` in the pinned Mathlib v4.30 source.

Formal statement

∀ {b n : ℕ}, Nat.log b n = 0 ↔ n < b ∨ b ≤ 1

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.log_eq_zero_iff

Kind: kernel-term
Status: checked

Supports: ∀ {b n : ℕ}, Nat.log b n = 0 ↔ n < b ∨ b ≤ 1

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.log_eq_zero_iff` through the trusted `Kernel::add_declaration` gate (declared in `nat_prelude/log_clog_mirrors.rs`'s `declare_log_eq_zero_iff`). Route, `mpr` (`Or (Lt n b) (Le b 1) -> Eq (log b n) 0`): case-split the disjunction -- `Lt n b` closes by `log.rs`'s already-proved `Nat.log_of_lt`, `Le b 1` closes by this file's new `Nat.log_of_left_le_one` applied at `n` (the same boundary-split recipe as `Nat.clog_of_left_le_one`, over `Nat.log_zero_left`/`Nat.log_one_left`). Route, `mp` (`Eq (log b n) 0 -> Or (Lt n b) (Le b 1)`): case-split `Nat.lt_or_ge n b`; the `Lt n b` side is `Or.inl` directly; the `Le b n` side splits AGAIN on `Nat.lt_or_ge 1 b` -- `Le b 1` is `Or.inr` directly, `Lt 1 b` is refuted by this file's new `Nat.log_pos` (the same recipe as `Nat.clog_pos` with the two guard cuts' roles swapped), which gives `Lt 0 (log b n)`; transporting that along the hypothesis `Eq (log b n) 0` gives `Lt 0 0`, absurd via `Nat.lt_irrefl`, closing the goal by `False.elim`. `nat_theorem_inventory`'s rendered type for `Nat.log_eq_zero_iff` is `(x0:AxNat)->(x1:AxNat)->Iff (Eq (log x0 x1) zero) (Or (Lt x1 x0) (Le x0 (succ zero)))`, matching this fact's `formal.statement` verbatim (`x0`=b, `x1`=n). `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed; verified against a deliberately misspelled name (`log_eq_zero_iffxxxxx`) that this same pipeline reports `0`.

footprint-Nat.log_eq_zero_iff

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 (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly (via `theorem_names`, which now lists it) and `nat_prelude_tests::the_nat_prelude_declares_no_axioms` confirms zero `Axiom` declarations anywhere in the built environment. `theorem_axiom_footprint log_eq_zero_iff` measured 0 directly.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.log_eq_zero_iff` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.log_eq_zero_iff`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}