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December 4, 2014 03:22
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-- ROSE TREES, FUNCTORS, MONOIDS, FOLDABLES | |
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data Rose a = a :> [Rose a] deriving Show | |
-- =================================== | |
-- Ex. 0-2 | |
-- =================================== | |
root :: Rose a -> a | |
root (a :> _) = a | |
children :: Rose a -> [Rose a] | |
children (_ :> c) = c | |
xs = 0 :> [1 :> [2 :> [3 :> [4 :> [], 5 :> []]]], 6 :> [], 7 :> [8 :> [9 :> [10 :> []], 11 :> []], 12 :> [13 :> []]]] | |
ex2 = root . head . children . head . children . head . drop 2 $ children xs | |
-- =================================== | |
-- Ex. 3-7 | |
-- =================================== | |
size :: Rose a -> Int | |
size (_ :> []) = 1 | |
size (_ :> r) = 1 + sum (map size r) | |
leaves :: Rose a -> Int | |
leaves (_ :> []) = 1 | |
leaves (_ :> r) = sum (map leaves r) | |
ex7 = (*) (leaves . head . children . head . children $ xs) (product . map size . children . head . drop 2 . children $ xs) | |
-- =================================== | |
-- Ex. 8-10 | |
-- =================================== | |
instance Functor Rose where | |
fmap f (a :> []) = f a :> [] | |
fmap f (a :> r) = f a :> (map (fmap f) r) | |
ex10 = round . root . head . children . fmap (\x -> if x > 0.5 then x else 0) $ fmap (\x -> sin(fromIntegral x)) xs | |
-- =================================== | |
-- Ex. 11-13 | |
-- =================================== | |
class Monoid m where | |
mempty :: m | |
mappend :: m -> m -> m | |
newtype Sum a = Sum a | |
newtype Product a = Product a | |
instance Num a => Monoid (Sum a) where | |
mempty = Sum 0 | |
mappend (Sum m) (Sum n) = Sum (m+n) | |
instance Num a => Monoid (Product a) where | |
mempty = Product 1 | |
mappend (Product m) (Product n) = Product (m*n) | |
unSum :: Sum a -> a | |
unSum (Sum a) = a | |
unProduct :: Product a -> a | |
unProduct (Product a) = a | |
num1 = mappend (mappend (Sum 2) (mappend (mappend mempty (Sum 1)) mempty)) (mappend (Sum 2) (Sum 1)) | |
num2 = mappend (Sum 3) (mappend mempty (mappend (mappend (mappend (Sum 2) mempty) (Sum (-1))) (Sum 3))) | |
ex13 = unSum (mappend (Sum 5) (Sum (unProduct (mappend (Product (unSum num2)) (mappend (Product (unSum num1)) (mappend mempty (mappend (Product 2) (Product 3)))))))) | |
-- =================================== | |
-- Ex. 14-15 | |
-- =================================== | |
class Functor f => Foldable f where | |
fold :: Monoid m => f m -> m | |
foldMap :: Monoid m => (a -> m) -> (f a -> m) | |
foldMap g = fold . (fmap g) | |
instance Foldable Rose where | |
fold (a :> []) = a | |
fold (a :> r) = foldr mappend a (map fold r) | |
sumxs = Sum 0 :> [Sum 13 :> [Sum 26 :> [Sum (-31) :> [Sum (-45) :> [], Sum 23 :> []]]], Sum 27 :> [], Sum 9 :> [Sum 15 :> [Sum 3 :> [Sum (-113) :> []], Sum 1 :> []], Sum 71 :> [Sum 55 :> []]]] | |
ex15 = unSum (mappend (mappend (fold sumxs) (mappend (fold . head . drop 2 . children $ sumxs) (Sum 30))) (fold . head . children $ sumxs)) | |
-- =================================== | |
-- Ex. 16-18 | |
-- =================================== | |
ex17 = unSum (mappend (mappend (foldMap (\x -> Sum x) xs) (mappend (foldMap (\x -> Sum x) . head . drop 2 . children $ xs) (Sum 30))) (foldMap (\x -> Sum x) . head . children $ xs)) | |
ex18 = unSum (mappend (mappend (foldMap (\x -> Sum x) xs) (Sum (unProduct (mappend (foldMap (\x -> Product x) . head . drop 2 . children $ xs) (Product 3))))) (foldMap (\x -> Sum x) . head . children $ xs)) | |
-- =================================== | |
-- Ex. 19-21 | |
-- =================================== | |
fproduct, fsum :: (Foldable f, Num a) => f a -> a | |
fsum r = unSum $ fold $ fmap Sum r | |
fproduct r = unProduct $ fold $ fmap Product r | |
ex21 = ((fsum . head . drop 1 . children $ xs) + (fproduct . head . children . head . children . head . drop 2 . children $ xs)) - (fsum . head . children . head . children $ xs) | |
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