class Function ?

[Lloyd Allison]

Almost certainly this is either
 . easy and obvious or
 . unnecessary or
 . impossible
for some a well-known reason.  Which is it please?  ...


I would like to have a ``class Function'' which has the
operators ``$'', ``.'', etc. and *most* particularly ``'',
so that one can define sub-classes of Function
(e.g. functions having inverses, say) that can still
be applied in the usual way,  i.e. ``f x''.


Lloyd


PS. [How] can one make ``->'' an instance of some new class?
--
Lloyd ALLISON,  CSSE, Monash University, Victoria, Australia 3168.
web: http://www.csse.monash.edu.au/~lloyd/    tel: +61 3 9905 5205
--


This catches the spirit but falls short at  ``f x'' :-

module Main where
-- ------------------------------------------10/2002--L.A.--CSSE--Monash--.au--
-- Would like there to be a ``class Function'' having
-- an apply operator, why not ($), and perhaps others such as (.),
-- with ``->'' being an instance of class Function (as it is of Show 6.1.6)
-- (come to that, how do you make ``->'' an instance of anything new?),
-- and would like to define new instances and subclasses of class Function
-- along the lines of...

class Function fnType where         -- would like Function to be in Prelude and
  ($)   :: (fnType t u) -> t -> u     -- rather use Prelude's ($) or is it "" ?
  apply :: (fnType t u) -> t -> u

  f $ x = apply f x                        -- and then would like to write  f x
  apply f x = f Main.$ x


data Arrow t u = FN (t->u)           -- i.e.  ``->''

instance Function Arrow where        -- ? in Prelude ?
  apply (FN f) x = f x


class (Function fnType) => Invertible fnType where          -- a subclass, i.e.
  inverse :: fnType t u -> fnType u t                   -- invertible Functions


data IArrow t u = IFN (t->u) (u->t)

instance Function IArrow where
  apply (IFN f i) x = f x

instance Invertible IArrow where
  inverse (IFN f i) = IFN i f


successor = IFN (\x -> x+1) (\x -> x-1)  -- e.g. an Invertible Function


linRec p f x =                      -- e.g. yer typical linear recursive schema
  let x0 = x                        --      slightly contrived  (OK, a toy)
      up x = if p x then dn x else x : (up (apply f x))
      dn x = x : if x==x0 then [] else dn (apply (inverse f) x)
  in up x0


main = print "L.A., CSSE, Monash, 10/2002: Re a hypothetical  class Function"
 >> print( successor `apply` 6 )               -- prefer  successor 6
 >> print( successor Main.$ 6 )                -- prefer  successor $ 6
 >> print( (inverse successor) `apply` 6 )     -- prefer  (inverse successor) 6
 >> print( linRec  ((<=) 4)  successor 1 )
 >> print( linRec (\_->True) successor 1 )
-- ----------------------------------------------------------------------------