module Data.Comp.Ops where
import Data.Foldable
import Data.Traversable
import Control.Applicative
import Control.Monad hiding (sequence, mapM)
import Prelude hiding (foldl, mapM, sequence, foldl1, foldr1, foldr)
infixr 6 :+:
data (f :+: g) e = Inl (f e)
| Inr (g e)
fromInl :: (f :+: g) e -> Maybe (f e)
fromInl = caseF Just (const Nothing)
fromInr :: (f :+: g) e -> Maybe (g e)
fromInr = caseF (const Nothing) Just
caseF :: (f a -> b) -> (g a -> b) -> (f :+: g) a -> b
caseF f g x = case x of
Inl x -> f x
Inr x -> g x
instance (Functor f, Functor g) => Functor (f :+: g) where
fmap f (Inl e) = Inl (fmap f e)
fmap f (Inr e) = Inr (fmap f e)
instance (Foldable f, Foldable g) => Foldable (f :+: g) where
fold (Inl e) = fold e
fold (Inr e) = fold e
foldMap f (Inl e) = foldMap f e
foldMap f (Inr e) = foldMap f e
foldr f b (Inl e) = foldr f b e
foldr f b (Inr e) = foldr f b e
foldl f b (Inl e) = foldl f b e
foldl f b (Inr e) = foldl f b e
foldr1 f (Inl e) = foldr1 f e
foldr1 f (Inr e) = foldr1 f e
foldl1 f (Inl e) = foldl1 f e
foldl1 f (Inr e) = foldl1 f e
instance (Traversable f, Traversable g) => Traversable (f :+: g) where
traverse f (Inl e) = Inl <$> traverse f e
traverse f (Inr e) = Inr <$> traverse f e
sequenceA (Inl e) = Inl <$> sequenceA e
sequenceA (Inr e) = Inr <$> sequenceA e
mapM f (Inl e) = Inl `liftM` mapM f e
mapM f (Inr e) = Inr `liftM` mapM f e
sequence (Inl e) = Inl `liftM` sequence e
sequence (Inr e) = Inr `liftM` sequence e
infixl 5 :<:
infixl 5 :=:
data Pos = Here | GoLeft Pos | GoRight Pos | Sum Pos Pos
data Emb = NotFound | Ambiguous | Found Pos
type family GetEmb (f :: * -> *) (g :: * -> *) :: Emb where
GetEmb f f = Found Here
GetEmb f (g1 :+: g2) = Pick f (g1 :+: g2) (GetEmb f g1) (GetEmb f g2)
GetEmb f g = NotFound
type family Pick f g (e1 :: Emb) (r :: Emb) :: Emb where
Pick f g (Found x) (Found y) = Ambiguous
Pick f g Ambiguous y = Ambiguous
Pick f g x Ambiguous = Ambiguous
Pick f g (Found x) y = Found (GoLeft x)
Pick f g x (Found y) = Found (GoRight y)
Pick f g x y = Split f g
type family Split (f :: * -> *) (g :: * -> *) :: Emb where
Split (f1 :+: f2) g = Pick2 (GetEmb f1 g) (GetEmb f2 g)
Split f g = NotFound
type family Pick2 (e1 :: Emb) (r :: Emb) :: Emb where
Pick2 (Found x) (Found y) = Found (Sum x y)
Pick2 Ambiguous y = Ambiguous
Pick2 x Ambiguous = Ambiguous
Pick2 NotFound y = NotFound
Pick2 x NotFound = NotFound
data EmbD (e :: Emb) (f :: * -> *) (g :: * -> *) where
HereD :: EmbD (Found Here) f f
GoLeftD :: EmbD (Found p) f g -> EmbD (Found (GoLeft p)) f (g :+: g')
GoRightD :: EmbD (Found p) f g -> EmbD (Found (GoRight p)) f (g' :+: g)
SumD :: EmbD (Found p1) f1 g -> EmbD (Found p2) f2 g -> EmbD (Found (Sum p1 p2)) (f1 :+: f2) g
class GetEmbD (e :: Emb) (f :: * -> *) (g :: * -> *) where
getEmbD :: EmbD e f g
instance GetEmbD (Found Here) f f where
getEmbD = HereD
instance GetEmbD (Found p) f g => GetEmbD (Found (GoLeft p)) f (g :+: g') where
getEmbD = GoLeftD getEmbD
instance GetEmbD (Found p) f g => GetEmbD (Found (GoRight p)) f (g' :+: g) where
getEmbD = GoRightD getEmbD
instance (GetEmbD (Found p1) f1 g, GetEmbD (Found p2) f2 g)
=> GetEmbD (Found (Sum p1 p2)) (f1 :+: f2) g where
getEmbD = SumD getEmbD getEmbD
data SimpPos = SimpHere | SimpLeft SimpPos | SimpRight SimpPos
data Res = CompPos SimpPos Pos | SingPos SimpPos
type family DestrPos (e :: Pos) :: Res where
DestrPos (GoLeft e) = ResLeft (DestrPos e)
DestrPos (GoRight e) = ResRight (DestrPos e)
DestrPos (Sum e1 e2) = ResSum (DestrPos e1) e2
DestrPos Here = SingPos SimpHere
type family ResLeft (r :: Res) :: Res where
ResLeft (CompPos p e) = CompPos (SimpLeft p) (GoLeft e)
ResLeft (SingPos p) = SingPos (SimpLeft p)
type family ResRight (r :: Res) :: Res where
ResRight (CompPos p e) = CompPos (SimpRight p) (GoRight e)
ResRight (SingPos p) = SingPos (SimpRight p)
type family ResSum (r :: Res) (e :: Pos) :: Res where
ResSum (CompPos p e1) e2 = CompPos p (Sum e1 e2)
ResSum (SingPos p) e = CompPos p e
type family Or x y where
Or x False = x
Or False y = y
Or x y = True
type family In (p :: SimpPos) (e :: Pos) :: Bool where
In SimpHere e = True
In p Here = True
In (SimpLeft p) (GoLeft e) = In p e
In (SimpRight p) (GoRight e) = In p e
In p (Sum e1 e2) = Or (In p e1) (In p e2)
In p e = False
type family Duplicates' (r :: Res) :: Bool where
Duplicates' (SingPos p) = False
Duplicates' (CompPos p e) = Or (In p e) (Duplicates' (DestrPos e))
type family Duplicates (e :: Emb) where
Duplicates (Found p) = Duplicates' (DestrPos p)
class NoDup (b :: Bool) (f :: * -> *) (g :: * -> *)
instance NoDup False f g
inj_ :: EmbD e f g -> f a -> g a
inj_ HereD x = x
inj_ (GoLeftD e) x = Inl (inj_ e x)
inj_ (GoRightD e) x = Inr (inj_ e x)
inj_ (SumD e1 e2) x = case x of
Inl y -> inj_ e1 y
Inr y -> inj_ e2 y
type f :<: g = (GetEmbD (GetEmb f g) f g, NoDup (Duplicates (GetEmb f g)) f g)
inj :: forall f g a . (f :<: g) => f a -> g a
inj = inj_ (getEmbD :: EmbD (GetEmb f g) f g)
type f :=: g = (f :<: g, g :<: f)
proj_ :: EmbD e f g -> g a -> Maybe (f a)
proj_ HereD x = Just x
proj_ (GoLeftD p) x = case x of
Inl y -> proj_ p y
_ -> Nothing
proj_ (GoRightD p) x = case x of
Inr x -> proj_ p x
_ -> Nothing
proj_ (SumD p1 p2) x = case proj_ p1 x of
Just y -> Just (Inl y)
_ -> case proj_ p2 x of
Just y -> Just (Inr y)
_ -> Nothing
proj :: forall f g a . (f :<: g) => g a -> Maybe (f a)
proj = proj_ (getEmbD :: EmbD (GetEmb f g) f g)
spl :: (f :<: f1 :+: f2) => (f1 a -> b) -> (f2 a -> b) -> f a -> b
spl f1 f2 x = case inj x of
Inl y -> f1 y
Inr y -> f2 y
infixr 8 :*:
data (f :*: g) a = f a :*: g a
ffst :: (f :*: g) a -> f a
ffst (x :*: _) = x
fsnd :: (f :*: g) a -> g a
fsnd (_ :*: x) = x
infixr 7 :&:
data (f :&: a) e = f e :&: a
instance (Functor f) => Functor (f :&: a) where
fmap f (v :&: c) = fmap f v :&: c
instance (Foldable f) => Foldable (f :&: a) where
fold (v :&: _) = fold v
foldMap f (v :&: _) = foldMap f v
foldr f e (v :&: _) = foldr f e v
foldl f e (v :&: _) = foldl f e v
foldr1 f (v :&: _) = foldr1 f v
foldl1 f (v :&: _) = foldl1 f v
instance (Traversable f) => Traversable (f :&: a) where
traverse f (v :&: c) = liftA (:&: c) (traverse f v)
sequenceA (v :&: c) = liftA (:&: c)(sequenceA v)
mapM f (v :&: c) = liftM (:&: c) (mapM f v)
sequence (v :&: c) = liftM (:&: c) (sequence v)
class DistAnn s p s' | s' -> s, s' -> p where
injectA :: p -> s a -> s' a
projectA :: s' a -> (s a, p)
class RemA s s' | s -> s' where
remA :: s a -> s' a
instance (RemA s s') => RemA (f :&: p :+: s) (f :+: s') where
remA (Inl (v :&: _)) = Inl v
remA (Inr v) = Inr $ remA v
instance RemA (f :&: p) f where
remA (v :&: _) = v
instance DistAnn f p (f :&: p) where
injectA c v = v :&: c
projectA (v :&: p) = (v,p)
instance (DistAnn s p s') => DistAnn (f :+: s) p ((f :&: p) :+: s') where
injectA c (Inl v) = Inl (v :&: c)
injectA c (Inr v) = Inr $ injectA c v
projectA (Inl (v :&: p)) = (Inl v,p)
projectA (Inr v) = let (v',p) = projectA v
in (Inr v',p)