Types, named (was: "Name that type!"), plus two more questions

Here are the answers to the previous quiz.

The first datatype holds any factorial number of values. Here it is again, with one small change and a few de-obfuscations:

> data FacHelp (f :: * -> *) (y :: *) = 
>      Stop y
>    | More (FacHelp (OneMore f) (f y))
> newtype OneMore (f :: * -> *) (y :: *) = 
>      OneMore (y, f y)
> newtype Identity y = Identity y
> type Factorial y = FacHelp Identity y

The second is lists of length more than two containing lists of length more than two. This means that no inhabitant contains a prime number of elements. As far as I know, it is unknown whether there is a datatype with every inhabitant containing a prime number of elements. (GADTs can do it, of course, but I'm ignoring those for now, since they can do anything. (mirror, mirror 2, mirror 3, mirror 4)

Here's that datatype again, renamed:

> data LongList x = Two x x
>                 | AndAnother x (LongList x)
> data Composite x = Composite (LongList (LongList x))

Update:
Chung-chieh Shan pointed out to me that I got that datatype wrong. (The other text above is correct, I think.) Anyway, here's his counterexample:

> test :: Composite Char
> test = Composite (Two (Two 'a' 'b')
>                       (AndAnother 'c' (Two 'd' 'e')))

This is a correct (I think!) implementation of a container that can't be of a prime size:

> data Product f g a = Times (f (g a))
>                    | LeftIncr (Product (OneMore f) g a)
>                    | RightIncr (Product f (OneMore g) a)
> newtype Pair a = Pair (a,a)
> type Comp = Product Pair Pair

End update

The third datatype uses the type H, which encodes containers with a square number of elements. H is based on the idea that the sequence of partial sums of the sequence of odd natural numbers is the same as the sequence of squares. That is, {0, 1, 1+3, 1+3+5, …} = {0², 1², 2², 3², …}

We can now use Lagrange's four-square theorem to see that question 3(b) is, strangely, a container that can hold any number of elements. An extension of this by Legendre showed [cite] that every number not of the form 4^k(8m + 7) is representable by the sum of three squares, and this answers question 3(a).

Here are both parts again, renamed and cleaned up a bit:

> data SqHelp x y = End
>                 | Extra x (SqHelp (y,y,x) y)
> type Square a = SqHelp a a
> data Legendre x = Legendre (Square x) (Square x) (Square x)
> data Lagrange x = Lagrange (Square x) (Legendre x)

Here are two more questions:
4. Can we write something analogous to 3(a) in a more direct way?
5.

> data M a b = N b
>            | O (M (a,b) a)
> newtype P a = P (M a a)