monad_laws.hs

-- Monad laws and examples
-- Thanks to http://learnyouahaskell.com/a-fistful-of-monads#monad-laws

-- Left identity
-- return x >>= f equals to f x

l1 = return 3 >>= (\x -> Just (x + 100000))
l2 = (\x -> Just (x + 100000)) 3

l3 = return "WoM" >>= (\x -> [x,x,x])
l4 = (\x -> [x,x,x]) "WoM"

-- Right identity
-- m >>= return equals to m

r1 = Just "Move on up" >>= return
r2 = Just "Move on up"

r3 = [1,2,3,4] >>= return
r4 = [1,2,3,4]

-- Left/Right identityはreturnのあるべきふるまいをきめる
-- Associativityは>>=の連鎖のあるべき振る舞い

-- Associativity
-- (m >>= f) >>= g equals to m >>= (\x -> f x >>= g)

type Birds = Int
type Pole = (Birds, Birds)

landLeft :: Birds -> Pole -> Maybe Pole
landLeft n (left, right) 
    | abs ((left + n) - right) < 4 = Just (left + n, right)
    | otherwise = Nothing

landRight :: Birds -> Pole -> Maybe Pole
landRight n (left, right)
    | abs ((left + n) - right) < 4 = Just (left, right + n)
    | otherwise = Nothing

a0 = return (0,0) >>= landRight 2 >>= landLeft 2 >>= landRight 2
a1 = ((return (0,0) >>= landRight 2) >>= landLeft 2) >>= landRight 2
a11 = ((return (0,0) >>= landRight 2) >>= (landLeft 2) >>= landRight 2)
a2 = return (0,0) >>= (\x -> 
    landRight 2 x >>= (\y ->
    landLeft 2 y >>= (\z ->
    landRight 2 z)))

a3 = return (0,0) >>= landRight 2 >>= landLeft 2 
a4 = return (0,0) >>= (\x -> 
    landRight 2 x >>= (\y ->
    landLeft 2 y)) 

(.) :: (b -> c) -> (a -> b) -> (a -> c)
f . g = (\x -> f (g x))

(<=<) :: (Monad m) => (b -> m c) -> (a -> m b) -> (a -> m c)
f <=< g = (\x -> g x >>= f)

f x = [x, -x]
g x = [x*3, x*2]
h = f <=< g
a5 = h 3

-- f <=< (g <=< h) equals to (f <=< g) <=< h

-- monad laws with <=<
-- Left identity: f <=< return === f
-- Right identity: return <=< f === return
-- associative law: f <=< (g <=< h) === (f <=< g) <=< h

モナド則→その言い換えの証明