Mathlib v4.30 source proposition Nat.add_one_mul_choose_eq

Identifier: F:ml430-nat-add-one-mul-choose-eq-d364de16
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n k : ℕ), (n + 1) * n.choose k = (n + 1).choose (k + 1) * (k + 1)

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Nat.add_one_mul_choose_eq

Kind: kernel-term
Status: checked

Supports: ∀ (n k : ℕ), (n + 1) * n.choose k = (n + 1).choose (k + 1) * (k + 1)

Checker command

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

Evidence notes

`Nat.add_one_mul_choose_eq` (nat_prelude/draw11_mirrors.rs, `declare_add_one_mul_choose_eq`, lane draw11-theorems-b) is a fresh construction: `Nat.succ_mul_choose_eq(n,k) : (succ k) * choose(succ n)(succ k) = (succ n) * choose n k` read backwards (`symm`) already states the target equation with the two products' factor order swapped on the right-hand side; one `Nat.mul_comm` on that product finishes it. This codebase spells `n+1` as `Nat.succ n`, definitionally equal to `Nat.add n 1`. `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> Eq.{1} AxNat (AxNat.mul (AxNat.succ x0) (AxNat.choose x0 x1)) (AxNat.mul (AxNat.choose (AxNat.succ x0) (AxNat.succ x1)) (AxNat.succ x1))))` -- reading `x0=n, x1=k`, this is `mul (succ n) (choose n k) = mul (choose (succ n)(succ k)) (succ k)`, matching `formal.statement` (`(n+1) * n.choose k = (n+1).choose(k+1) * (k+1)`) verbatim. `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. A discriminating unit test in `nat_prelude_tests.rs` (`add_one_mul_choose_eq_holds_at_a_concrete_point`) checks the RENDERED type at `(n,k)=(3,1)` (not just the reduced value, which the dropped-`mul_comm` variant would also satisfy at `4*3=2*6=12` if compared only numerically) and confirms both sides reduce to `12`.

footprint-Nat.add_one_mul_choose_eq

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q --release -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 2026-08-31: 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 `Nat.add_one_mul_choose_eq`'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).

Provenance

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