[Haskell-cafe] Creating a type for a subset of the integers

[Don Stewart]

jules:
> Brad Larsen wrote:
> >Hi there list,
> >
> >How would one go about creating a new type for a subset of the integers, 
> >for (contrived) example just the even integers?  I was thinking of 
> >making a new type
> >
> >newtype EvenInt = EvenInt Integer
> >
> >but the problem with this is that it accepts any integer, even odd 
> >ones.  So to prevent this, the module does not export the constructor 
> >for it---rather, the module exports a function `mkEvenInt' that creates 
> >an EvenInt if the given value is acceptable or raises an error otherwise.
> >
> >
> >What's the right way to do this?  Thanks!
> 
> There are two ways:
> 
> (1) Have a representation which admits invalid values, and provide 
> combinators which only perfect validity, and prove that consumers using 
> your combinators can't produce invalid values.
> 
> (2) Have a cleverly designed representation such that every 
> representation is valid.
> 
> An example here, for (2) would be to store n/2; there is a bijection 
> between 'Integer' and 'EvenInt' given by n/2.
> 
> In real, more complex problems, (2) often isn't possible and we resort 
> to (1). E.g. the representation of balanced trees (AVL? RedBlack?) 
> admits invalid values (both unbalanced trees and out-of-order trees) and 
>  we rely on the reduced set of combinators never to generate one.

1) is always a challenge to particular types (heh) of people to do 2)

We've a page on the wiki about this idiom,

    http://www.haskell.org/haskellwiki/Smart_constructors

including  links to type level enforcement.

Last time I tried type level decimals they were still a little bit
hairy. This might be easier now with type families though.

-- Don