defining lambda expressions

I am studying for my exam and I wanted to do some extra excercises, but I have some problems with solving :) Can anyone please help or give me some advice where to start? Thank you!

We can represent every list of lambda expressions $x_0, x_1, ..., x_k$ with lambda expression $[x_0, x_1, ..., x_k]$ , defined as: $[x_0, x_1, ... , x_k] = \lambda c n. c \, x_0 \, (c \, x_1 \, (... (c \, x_k \, n) ...))$

Define lambda expression s, satisfying the equation $s([\underline{n_0}, ..., \underline{n_k}]) = \underline{n_0 + ... + n_k}$ , where $\underline{n}$ is a Church numeral, representing natural number $n$.

Define lambda expression $r$ , satisfying the equation $r([x_0, ..., x_k]) = [x_k, ..., x_0]$ .

Define lambda expression $h$ and $t$ , satisfying the equation $h([x_0, ..., x_k]) = x_0 in t([x_0, x_1, ..., x_k]) = [x_1, ..., x_k]$ .

• Actually I don't know where to start, because I am not sure if I understand what I have to do. I don't understand if at the beginning we get the list, or we have to calculate the numbers from expression given. Also, I know how to define sum for example, but I don't know how to get first element from the list. Jun 21, 2013 at 18:16