∀ {motive : ℕ → ℕ → Sort u} (x y : ℕ) (ind : (x y : ℕ) → 0 < y ∧ y ≤ x → motive (x - y) y → motive x y)
  (base : (x y : ℕ) → ¬(0 < y ∧ y ≤ x) → motive x y),
  Nat.div.inductionOn x y ind base =
    if h : 0 < y ∧ y ≤ x then ind x y h (Nat.div.inductionOn (x - y) y ind base) else base x y h