I'm trying to calculate $\text{pred}\, c_0$, where $\text{pred}$ is the previous church encoded number and $c_0$ is the number $0$ ($\lambda s. \lambda z. z$).
The formula for $\text{pred}$ is $\lambda n. \lambda s. \lambda z. n (\lambda g. \lambda h. h (g s))(\lambda u.z)(\lambda v.v)$
My $\beta$ reduction looks like this so far (I've underlined the function and the applied parameter)
$$ \text{pred}\, c_0 \to \lambda s. \lambda z. c_0 \underline{(\lambda g. \lambda h. h (g s))}\text{ }\underline{(\lambda u.z)}(\lambda v.v) \to $$
$$\lambda s. \lambda z. c_0 (\lambda h. h (\underline{(\lambda u.z)}\text{ } \underline{s}))(\lambda v.v) \to$$
$$\lambda s. \lambda z. c_0 \underline{(\lambda h. h z)}\text{ }\underline{(\lambda v.v)} \to$$
$$\lambda s. \lambda z. \underline{c_0}\text{ } \underline{z} \to$$
$$\lambda s. \lambda z. (???)$$