content: {-
   - Instant Insanity using Closed Type Families and DataKinds.
   -
   - See:  http://stackoverflow.com/questions/26538595
   -}
  
  {-# OPTIONS_GHC -freduction-depth=400 #-}
  
  {-# LANGUAGE MultiParamTypeClasses #-}
  {-# LANGUAGE TypeFamilies #-}
  {-# LANGUAGE UndecidableInstances #-}
  {-# LANGUAGE DataKinds, KindSignatures, PolyKinds #-}
  {-# LANGUAGE TypeOperators #-}
  
  import Prelude hiding (all, flip, map, filter )
  
  data Proxy (a :: k) = Proxy
  
  main = print (Proxy :: Proxy (Solutions Cubes))
  
  data Color = R | G | B | W
  
  data Cube = Cube Color Color Color Color Color Color
  
  type family And (b1 :: Bool) (b2 :: Bool) :: Bool where
      And True True = True
      And b1 b2 = False
  
  type family NE (x :: Color) (y :: Color) :: Bool where
      NE x x = False
      NE x y = True
  
  type family EQ (x :: Color) (y :: Color) :: Bool where
      EQ a a = True
      EQ a b = False
  
  type family All (l :: [Bool]) :: Bool where
      All '[] = True
      All (False ': xs) = False
      All (True ': xs) = All xs
  
  type family ListConcat (xs :: [k]) (ys :: [k]) :: [k] where
      ListConcat '[] ys = ys
      ListConcat (x ': xs) ys = x ': ListConcat xs ys
  
  type family AppendIf (b :: Bool) (a :: [Cube]) (as :: [[Cube]]) :: [[Cube]] where
      AppendIf False a as = as
      AppendIf True a as = a ': as
  
  data Transform = Rotate | Twist | Flip
  
  type family Apply (f :: Transform) (a :: Cube) :: Cube where
      Apply Rotate ('Cube u f r b l d) = ('Cube u r b l f d)
      Apply Twist  ('Cube u f r b l d) = ('Cube f r u l d b)
      Apply Flip   ('Cube u f r b l d) = ('Cube d l b r f u)
  
  type family Map (f :: Transform) (as :: [Cube]) :: [Cube] where
      Map f '[] = '[]
      Map f (a ': as) = (Apply f a) ': (Map f as)
  
  type family MapAppend (f :: Transform) (as :: [Cube]) :: [Cube] where
      MapAppend f xs = ListConcat xs (Map f xs)
  
  type family MapAppend2 (f :: Transform) (as :: [Cube]) :: [Cube] where
      MapAppend2 f xs = ListConcat xs (MapAppend f (Map f xs))
  
  type family MapAppend3 (f :: Transform) (as :: [Cube]) :: [Cube] where
      MapAppend3 f xs = ListConcat xs (MapAppend2 f (Map f xs))
  
  type family Iterate2 (f :: Transform) (as :: [Cube]) :: [Cube] where
      Iterate2 f '[] = '[]
      Iterate2 f (a ': as) = ListConcat [Apply f a, a] (Iterate2 f as)
  
  type family Iterate3 (f :: Transform) (as :: [Cube]) :: [Cube] where
      Iterate3 f '[] = '[]
      Iterate3 f (a ': as) =
          ListConcat [a, Apply f a, Apply f (Apply f a)] (Iterate3 f as)
  
  type family Iterate4 (f :: Transform) (as :: [Cube]) :: [Cube] where
      Iterate4 f '[] = '[]
      Iterate4 f (a ': as) =
          ListConcat [a, Apply f a, Apply f (Apply f a), Apply f (Apply f (Apply f a))]
                     (Iterate4 f as)
  
  type family Orientations (c :: Cube) :: [Cube] where
      Orientations c = MapAppend3 Rotate (MapAppend2 Twist (MapAppend Flip '[c]))
  
  type Cube1 = 'Cube B G W G B R
  type Cube2 = 'Cube W G B W R R
  type Cube3 = 'Cube G W R B R R
  type Cube4 = 'Cube B R G G W W
  
  type Cubes = [Cube1, Cube2, Cube3, Cube4]
  
  type family Compatible (c :: Cube) (d :: Cube) :: Bool where
      Compatible ('Cube u1 f1 r1 b1 l1 d1) ('Cube u2 f2 r2 b2 l2 d2) =
          All [NE f1 f2, NE r1 r2, NE b1 b2, NE l1 l2]
  
  type family Allowed (c :: Cube) (cs :: [Cube]) :: Bool where
      Allowed c '[] = True
      Allowed c (s ': ss) = And (Compatible c s) (Allowed c ss)
  
  type family MatchingOrientations (as :: [Cube]) (sol :: [Cube]) :: [[Cube]] where
      MatchingOrientations '[] sol = '[]
      MatchingOrientations (o ': os) sol =
          AppendIf (Allowed o sol) (o ': sol) (MatchingOrientations os sol)
  
  type family AllowedCombinations (os :: [Cube]) (sols :: [[Cube]]) where
      AllowedCombinations os '[] = '[]
      AllowedCombinations os (sol ': sols) =
          ListConcat (MatchingOrientations os sol) (AllowedCombinations os sols)
  
  type family Solutions (cs :: [Cube]) :: [[Cube]] where
      Solutions '[] = '[ '[] ]
      Solutions (c ': cs) = AllowedCombinations (Orientations c) (Solutions cs)
  