Mathlib v4.30 source proposition Nat.self_le_factorial

Identifier: F:ml430-nat-self-le-factorial-cfdffc69
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n : ℕ), n ≤ n.factorial

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.self_le_factorial

Kind: kernel-term
Status: checked

Supports: ∀ (n : ℕ), n ≤ n.factorial

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.self_le_factorial` through the trusted `Kernel::add_declaration` gate. Route (`desc_factorial.rs`, `declare_self_le_factorial`, independent of the `descFactorial`/`choose` bridge in the same file): direct induction on `n`. `n = 0` is `Nat.zero_le` directly. `n = succ j` scales `Nat.one_le_factorial` at `j` (`1 <= j!`, NOT the induction hypothesis, which only bounds `j` and is too weak) by `succ j` via `Nat.mul_le_mul_left`, rewrites `succ j * 1 = succ j` (`mul_one`) and commutes the right side (`mul_comm`) to `Le (succ j) (j! * succ j)`, then transports along `factorial_succ` (reversed) to the goal. `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.

footprint-Nat.self_le_factorial

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 (not merely the environment-wide bound), by name, in-tree.

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.self_le_factorial` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.self_le_factorial`",
      "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"
    }
  ]
}