## Expressions and types
Give OCaml expressions that have the following types.
* `int list`
* `string -> string list`
* `string list -> string`
Using pattern matching, write three functions, one for each of the following
properties. Your functions should return `true` if the input list
has the property and `false` otherwise.
* the list's first element is `"cs3110"`
* the list has exactly two or four elements; do not use the `length` function
* the first two elements of the list are equal
## Functions on lists
* Write a function that returns the product of all the elements in a list. (The
product of all the elements of an empty list is `1`.)
* Write a function that concatenates all the strings in a list.
(The concatenation of all the strings in an empty list is the empty string `""`.)
## Library functions
Consult the [`List` standard library](http://caml.inria.fr/pub/docs/manual-ocaml/libref/List.html)
to easily implement the following functions:
* Write a function that returns the last element of a list.
*Hint:* this can be solved by combining two library
functions, and without writing any pattern matching code of your own.
If the list is empty, your
function may throw an exception.
* Write a function `any_zeroes : int list -> bool` that returns `true` if and only if
the input list contains at least one `0`. *Hint:* this can be solved by using one library
function, and without writing any pattern matching code of your own.
Your solutions should need only about one line of code.
## Tail recursion
* Write a function `take : int -> 'a list -> 'a list` such that `take n lst` returns
the first `n` elements of `lst`. If `lst` has fewer than `n` elements, return all
* Write a function `drop : int -> 'a list -> 'a list` such that `drop n lst`
returns all but the first `n` elements of `lst`. If `lst` has fewer than `n` elements,
return the empty list.
Your functions must be tail recursive. Test them on long lists with large
values of `n` to see
whether they run out of stack space. Here is code to produce a long list:
let (--) i j =
let rec from i j l =
if i>j then l
else from i (j-1) (j::l)
in from i j 
let longlist = 0 -- 1_000_000
Note that code itself is tail recursive.
## More functions on lists
* Write a function `is_unimodal : int list -> bool` that takes an integer list and
returns whether that list is unimodal. A *unimodal list* is a list that
monotonically increases to some maximum value then monotonically decreases
after that value. Either or both segments (increasing or decreasing) may be empty.
A constant list is unimodal, as is the empty list.
* Write a function `powerset : int list -> int list list` that takes a set *S*
represented as a list and returns the set of all subsets of *S*. The order
of subsets in the powerset and the order of elements in the subsets do not matter.
*Hint:* Consider the recursive structure of this problem.
Suppose you already have `p`, such
that `p = powerset s`. How could you use `p` to compute `powerset (x::s)`?