module Numeric.Addition.Partitionable ( -- * Partitionable Additive Semigroups Partitionable(..) ) where import Prelude ((-),Bool(..),(\$),id,(>>=)) import Numeric.Semigroup.Additive import Numeric.Natural import Data.List.NonEmpty (NonEmpty(..), fromList) concat :: NonEmpty (NonEmpty a) -> NonEmpty a concat m = m >>= id class Additive m => Partitionable m where -- | partitionWith f c returns a list containing f a b for each a b such that a + b = c, partitionWith :: (m -> m -> r) -> m -> NonEmpty r instance Partitionable Bool where partitionWith f False = f False False :| [] partitionWith f True = f False True :| [f True False, f True True] instance Partitionable Natural where partitionWith f n = fromList [ f k (n - k) | k <- [0..n] ] instance Partitionable () where partitionWith f () = f () () :| [] instance (Partitionable a, Partitionable b) => Partitionable (a,b) where partitionWith f (a,b) = concat \$ partitionWith (\ax ay -> partitionWith (\bx by -> f (ax,bx) (ay,by)) b) a instance (Partitionable a, Partitionable b, Partitionable c) => Partitionable (a,b,c) where partitionWith f (a,b,c) = concat \$ partitionWith (\ax ay -> concat \$ partitionWith (\bx by -> partitionWith (\cx cy -> f (ax,bx,cx) (ay,by,cy)) c) b) a instance (Partitionable a, Partitionable b, Partitionable c,Partitionable d ) => Partitionable (a,b,c,d) where partitionWith f (a,b,c,d) = concat \$ partitionWith (\ax ay -> concat \$ partitionWith (\bx by -> concat \$ partitionWith (\cx cy -> partitionWith (\dx dy -> f (ax,bx,cx,dx) (ay,by,cy,dy)) d) c) b) a instance (Partitionable a, Partitionable b, Partitionable c,Partitionable d, Partitionable e) => Partitionable (a,b,c,d,e) where partitionWith f (a,b,c,d,e) = concat \$ partitionWith (\ax ay -> concat \$ partitionWith (\bx by -> concat \$ partitionWith (\cx cy -> concat \$ partitionWith (\dx dy -> partitionWith (\ex ey -> f (ax,bx,cx,dx,ex) (ay,by,cy,dy,ey)) e) d) c) b) a