Mathlib v4.30 source proposition Nat.add_choose_mul_factorial_mul_factorial

Identifier: F:ml430-nat-add-choose-mul-factorial-mul-factorial-26ba01ef
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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4 direct dependencies 1 direct dependents

Evidence

kernel-Nat.add_choose_mul_factorial_mul_factorial

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.add_choose_mul_factorial_mul_factorial` through the trusted `Kernel::add_declaration` gate, declared in `nat_prelude/choose_factorial_add.rs` (a new module: the falling-factorial/choose bridge already in `desc_factorial.rs` gave `descFactorial(i+j,j) = j! * choose(i+j,j)`, but nothing tied `descFactorial` back to plain `factorial` by the complementary `i!` -- this module supplies exactly that missing piece by induction on `j`, then assembles the two). `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.mul (AxNat.mul (AxNat.choose (AxNat.add x0 x1) x1) (AxNat.factorial x0)) (AxNat.factorial x1)) (AxNat.factorial (AxNat.add x0 x1))))`, matching this fact's `formal.statement`. `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 both ways: the real name greps to a count `-ge 1`; grepping a made-up name (`Nat.add_choose_mul_factorial_mul_factorial_bogus`) greps to `0`.

footprint-Nat.add_choose_mul_factorial_mul_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 via `theorem_names` (this declaration's `p.add_choose_mul_factorial_mul_factorial` entry was added there in the same session), 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/choose_factorial_add_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, both confirmed as actual reductions) with a genuinely-discriminating negative control.

Provenance

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