[Haskell-cafe] How to define an operation in terms of itself (but of different type)?

[Ryan Ingram]

2009/1/24 Olex P <hoknamahn at gmail.com>:
> What I want to ask you guys can we define a function with arbitrary number
> of parameters? Actually not really arbitrary but just several possibilities
> (as we have with value constructors).
> For instance cross product can have 1, 2 or 3 vectors as inputs depends on
> the dimension. Is it 2d, 3d or 4d case.

You need type classes plus some language extension to do this
properly.  Here's an example using type families:

class Vector v => Cross v where
   type CrossResult v
   cross :: v -> CrossResult v

instance Cross Vector2 where
   type CrossResult v = Vector2
   cross (Vector2 x y) = Vector2 (-y) x  -- is this the right operation?

instance Cross Vector3 where
   type CrossResult v = Vector3 -> Vector3
   cross (Vector3 x1 y1 z1) (Vector3 x2 y2 z2) = Vector3 (y1*z2 -
z1*y2) (z1*x2 - x1*z2) (x1*y2 - y1*x2)

etc.

A difficult exercise is to define these operations in terms of each
other inductively so that it works for arbitrary vector types; you
need vectors indexed on some type-level natural, plus a bunch of
hackery in the instances to make it work.

  -- ryan