Mathlib v4.30 source proposition Nat.descFactorial_of_lt

Identifier: F:ml430-nat-descfactorial-of-lt-fbcf5d26
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n k : ℕ}, n < k → n.descFactorial k = 0

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.descFactorial_of_lt

Kind: kernel-term
Status: checked

Supports: ∀ {n k : ℕ}, n < k → n.descFactorial k = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.descFactorial_of_lt` through the trusted `Kernel::add_declaration` gate. Route (`desc_factorial.rs`, `declare_desc_factorial_of_lt`): induction on `k` with `n` held fixed; `k = 0` is `not_lt_zero`; `k = succ j` splits `n < succ j` (`le_of_lt_succ` then `lt_or_eq_of_le`) into `n < j` (the IH collapses the product via `mul_zero`) and `n = j` (transport to the diagonal `descFactorial n (succ n) = (n-n)*descFactorial n n`, collapsed via `sub_self`/`zero_mul`) -- the truncated-`Nat.sub` boundary that gives this declaration its own module doc comment ("this definition's highest-risk seam"). This declaration predates the current session; the source-controlled construction is what `nat_theorem_inventory` reports admitted, not new proof work. `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.descFactorial_of_lt

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.descFactorial_of_lt` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.descFactorial_of_lt`",
      "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"
    }
  ]
}