Mathlib v4.30 source proposition Nat.log_anti_left

Identifier: F:ml430-nat-log-anti-left-b72490ec
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {b c n : ℕ}, 1 < c → c ≤ b → Nat.log b n ≤ Nat.log c n

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.log_anti_left

Kind: kernel-term
Status: checked

Supports: ∀ {b c n : ℕ}, 1 < c → c ≤ b → Nat.log b n ≤ Nat.log c n

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.log_anti_left` through the trusted `Kernel::add_declaration` gate (declared in `nat_prelude/log_clog_mirrors.rs`'s `declare_log_anti_left`). Route: `Nat.log_antitone_left` (`nat_prelude/log_clog_order.rs`) already proves the same conclusion but takes `1 < b` as an EXPLICIT third hypothesis (that file's diagonal `AntitoneOn`-unfold framing states it pointwise rather than deriving it); Mathlib's `log_anti_left` needs only `1 < c` and `c ≤ b`. `1 < b` follows from those two via `Nat.lt_of_lt_of_le`, so this fact's proof is a one-line wrapper: derive `1 < b`, then re-dispatch to `log_antitone_left`. `nat_theorem_inventory`'s rendered type for `Nat.log_anti_left` is `(x0:AxNat)->(x1:AxNat)->(x2:AxNat)->(x3:Lt (succ zero) x1)->(x4:Le x1 x2)->Le (log x2 x0) (log x1 x0)`, matching this fact's `formal.statement` verbatim (`x0`=n, `x1`=c, `x2`=b). `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_anti_leftxxxxx`) that this same pipeline reports `0`.

footprint-Nat.log_anti_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 (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_anti_left` measured 0 directly.

Provenance

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