Mathlib v4.30 source proposition Nat.log_eq_one_iff

Identifier: F:ml430-nat-log-eq-one-iff-a89f8bab
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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16 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.log_eq_one_iff

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.log_eq_one_iff` through the trusted `Kernel::add_declaration` gate (declared in `nat_prelude/log_clog_mirrors.rs`'s `declare_log_eq_one_iff`, lane `nat-mirror-residue`) -- `F:ml430-nat-log-eq-one-iff-63d772fb`'s (`Nat.log_eq_one_iff'`) exact core (`log_eq_one_derive_base_le_n`, `log_eq_one_derive_sq_bound`, `log_eq_one_of_bounds`; see that fact's evidence for the full route), repackaged into Mathlib's STRONGER hypothesis set: `mpr` is given `1 < b` directly (no need for `derive_one_lt_base_from_bounds`) and `mp` additionally derives `1 < b` as an explicit output via `derive_one_lt_base_from_log_eq_one` (if `b <= 1`, `log_of_left_le_one` gives `Eq (log b n) 0` at this `n`, contradicting `Eq (log b n) 1` via `succ_ne_zero`) -- the SAME derivation `Nat.log_eq_one_iff'`'s `mp` needs anyway as an internal tool before it can even call `log_eq_one_derive_sq_bound`, just also surfaced here as part of the conclusion. `nat_theorem_inventory`'s rendered type for `Nat.log_eq_one_iff` is `(x0:AxNat)->(x1:AxNat)->Iff (Eq (log x0 x1) (succ zero)) (And (Lt x1 (mul x0 x0)) (And (Lt (succ zero) x0) (Le x0 x1)))`, matching this fact's `formal.statement` verbatim (`x0`=b, `x1`=n, nested `And` matching the `n < b*b \u2227 (1 < b \u2227 b \u2264 n)` right-associated form). `nat_theorem_inventory` exits non-zero for a name that does not exist (verified against `log_eq_one_iff_xxxx`: `error: no Nat theorem matches ...`, count 0), and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Note the substring hazard: `log_eq_one_iff` is a PREFIX of `log_eq_one_iff'`, so the query argument returns BOTH rows -- the anchored `grep -Ec '^Nat\.log_eq_one_iff[[:space:]]'` (whitespace after the exact name, not after the prime) is what discriminates the unprimed row from its primed sibling.

footprint-Nat.log_eq_one_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, total_trusted=0). `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 `p.log_eq_one_iff`) and `nat_prelude_tests::the_nat_prelude_declares_no_axioms` confirms zero `Axiom` declarations anywhere in the built environment. Full `nat_prelude::` sweep: 325 passed, 0 failed.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.log_eq_one_iff` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.log_eq_one_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"
    }
  ]
}