lean - 证明递归函数的性质

Question

试图证明正数之和是正数。

-- some function
variable {w : nat -> int}

-- sum of w's from 0 to i
def sw : nat -> int
| 0     := w 0
| (i+1) := sw i + w (i+1)

-- w is always positive
axiom w_pos : ∀ i : nat, 0 < w i

-- sum of w's is always positive
lemma sw_pos : ∀ i : nat, 0 < sw i // error here
| 0     := w_pos 0
| (i+1) :=
have h₁ : 0 + w (i+1) < sw (i+1), from add_lt_add_right (sw_pos i) (w (i+1)),
have h₂ : 0 < 0 + w (i+1), from add_lt_add_left (w_pos (i+1)) 0,
show 0 < sw (i+1), from lt_trans h₂ 

在声明中sw_pos，Lean 抱怨说它“不知道如何合成占位符上下文” sw。

如何解决此错误？