Mathlib v4.30 source proposition Nat.clog_of_left_le_one

Identifier: F:ml430-nat-clog-of-left-le-one-a6640cee
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {b : ℕ}, b ≤ 1 → ∀ (n : ℕ), Nat.clog b n = 0

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.clog_of_left_le_one

Kind: kernel-term
Status: checked

Supports: ∀ {b : ℕ}, b ≤ 1 → ∀ (n : ℕ), Nat.clog b n = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.clog_of_left_le_one` through the trusted `Kernel::add_declaration` gate (declared in `nat_prelude/log_clog_mirrors.rs`'s `declare_clog_of_left_le_one`). Route: `Le b 1` lifts to `Lt b 2` via `Nat.le_succ_succ`, then `nat_prelude/ops.rs`'s `cases_lt_bound` splits into `b = 0` and `b = 1`; each branch is `clog.rs`'s already-proved `Nat.clog_zero_left`/`Nat.clog_one_left` used directly at their full `∀ n, clog _ n = 0` type (no application needed -- `d.lemma(name, &[])` with an empty argument list returns the bare constant at its full quantified type). `nat_theorem_inventory`'s rendered type for `Nat.clog_of_left_le_one` is `(x0:AxNat)->(x1:Le x0 (succ zero))->(x2:AxNat)->Eq (clog x0 x2) zero`, matching this fact's `formal.statement` verbatim (`x0`=b, `x2`=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 (`clog_of_left_le_onexxxxx`) that this same pipeline reports `0`.

footprint-Nat.clog_of_left_le_one

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 clog_of_left_le_one` measured 0 directly.

Provenance

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