Whats the syntax for the coproduct (disjoint union) of types in Haskell?

Question

consider the following

data Point=Point{x::Float,y::Float}
data Shape=Circle{centre::Point,radius::Float}
           |Rectangle {uleft::Point,bRight::Point}

Here the type Shape is a coproduct of two types Circle and Rectangle. I may want to reuse the types Circle and Rectangle elsewhere. So it would be useful to do this instead:

data Point=Point{x::Float,y::Float}
data Circle=Circle{centre::Point,radius::Float}
data Rectangle=Rectangle {uleft::Point,bRight::Point}
data Shape =Circle | Rectangle

but I get a compilation error when I do this: Circle is declared twice. Whats the correct syntax for attempting this, or this not possible?

Answer 1

The coproduct of types in Haskell is commonly denoted by Either:

data Either a b = Left a | Right b

type Shape = Either Circle Rectangle
-- so you have shapes as either Left c for some Circle c
-- or Right r for some Rectangle r

This works quite nicely, although for technical reasons it isn't exactly a coproduct. Another common way would be to define a type like so:

data Shape = CircleShape Circle | RectangleShape Rectangle

so that CircleShape :: Circle -> Shape and RectangleShape :: Rectangle -> Shape are your two injections.

It's wrong to say as you do in your question that the original Shape is a coproduct of types Circle and Rectangle, because the latter two aren't types. If you want to set things up so that Circle p r is both a value of type Circle and a value of type Shape, then that's really contrary to the spirit of Haskell's type system (although something similar might be possible with sufficiently many type system extensions).