∀ {P : ℕ → ℕ → Sort u_1} (Ha0 : (m : ℕ) → P m 0) (H0b : (n : ℕ) → P 0 n)
  (H : (x y : ℕ) → P x y.succ → P x.succ y → P x.succ y.succ) (x x_1 : ℕ),
  Nat.pincerRecursion Ha0 H0b H x x_1 =
    match x, x_1 with
    | m, 0 => Ha0 m
    | 0, n => H0b n
    | n.succ, n_1.succ => H n n_1 (Nat.pincerRecursion Ha0 H0b H n n_1.succ) (Nat.pincerRecursion Ha0 H0b H n.succ n_1)