Mathlib v4.30 source proposition Nat.add_choose

Identifier: F:ml430-nat-add-choose-eb49fa11
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (i j : ℕ), (i + j).choose j = (i + j).factorial / (i.factorial * j.factorial)

Dependencies

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

8 direct dependencies 0 direct dependents

Evidence

kernel-Nat.add_choose

Kind: kernel-term
Status: checked

Supports: ∀ (i j : ℕ), (i + j).choose j = (i + j).factorial / (i.factorial * j.factorial)

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.add_choose` through the trusted `Kernel::add_declaration` gate, declared in new module `nat_prelude/add_choose_div.rs`. It is the division-normal form of the already-proved `Nat.add_choose_mul_factorial_mul_factorial` (`(i+j).choose j * i! * j! = (i+j)!`, `F:ml430-nat-add-choose-mul-factorial-mul-factorial-26ba01ef`): converted to `(i+j).choose j = (i+j)! / (i! * j!)` by the standard "exact factor divided back out" route (a `div_eq_of_mul_eq` helper, copied per this crate's per-file local-helper convention -- the same technique `coprime_lemmas.rs`/`lcm_gcd_lemmas.rs` already use for other targets), with the divisor's positivity (`Le 1 (i! * j!)`) from `one_le_factorial` on each factor plus `one_le_mul`. `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.choose (AxNat.add x0 x1) x1) (AxNat.div (AxNat.factorial (AxNat.add x0 x1)) (AxNat.mul (AxNat.factorial x0) (AxNat.factorial x1)))))`, matching this fact's `formal.statement`. The grep anchor `^Nat\.add_choose[[:space:]]` is deliberately anchored so it does not also match `Nat.add_choose_mul_factorial_mul_factorial` (verified: the same command against the un-anchored substring returns 2, against this exact anchored pattern returns 1); the `-ge 1` count-based test (not `grep -q`) requires the admitted declaration to actually be printed, and grepping a made-up name (`Nat.add_choose_bogus`) greps to `0`.

footprint-Nat.add_choose

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` (this declaration's `p.add_choose` entry was added there in the same commit that declared it), and `the_build_is_deterministic`/`every_promised_name_is_admitted_with_the_expected_kind` (both re-run on every `nat_prelude::` sweep) confirm the declaration is admitted at the exact rendered type checked above. `nat_prelude/add_choose_div_tests.rs` independently checks this declaration's `axiom_footprint` too, plus a concrete instantiation at i=2,j=3 (5.choose 3 = 10, 5! = 120, 2!*3! = 12, 120/12 = 10, all confirmed as actual reductions) with a genuinely-discriminating negative control (5.choose 1 = 5). Full `nat_prelude::` sweep: 251 passed, 0 failed.

Provenance

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