{
 Instant Insanity using Closed Type Families and DataKinds.

 See: http://stackoverflow.com/questions/26538595
}
{# OPTIONS_GHC freductiondepth=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)