inductive Vector (α : Type u): Nat → Type u where
| nil : Vector α 0
| cons (head : α) (tail : Vector α n) : Vector α (n+1)

theorem Nat.lt_of_add_lt_add_right {a b c : Nat} (h : a + b < c + b) : a < c := sorry

def Vector.nth : ∀{n}, Vector α n → Fin n → α
| n+1, Vector.cons x xs, ⟨  0, _⟩ => x
| n+1, Vector.cons x xs, ⟨k+1, h⟩ => xs.nth ⟨k, Nat.lt_of_add_lt_add_right h⟩