A list of elements is alternating if:
- two consecutive elements are not equal;
- for each sublist [a;b;c] of three elements, the ordering relation between a and b is the inverse of the one between b and c.
For instance consider the following lists:
let l0 = [1;5;2;5;1;6];;
let l1 = [1;5;2;5;4;3];;
let l2 = [1;5;2;3;2;4];;
let l3 = [3;1;4;2;5;3];;Here, l0, l2 and l3 are alternating, while l1 is not.
A list of integers is ping pong if there is some central element (the "net") around which the others alternate, without touching the net.
For instance, the list l0 above is ping pong (with net 3 or 4), while the others are not. In particular, l1 is not alternating; in l2, the net should be between 2 and 3, but there are no integers within that interval; in l3, the net is shifting towards the right: it is 2 at the first step, 3 at the second step, and 4 at the third.
Write functions to detect if a list is alternating and ping pong, with the following types:
val alt : 'a list -> bool = <fun>
val ping_pong : int list -> bool = <fun>