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5. Applicative Functors

20/02/23

Generalising fmap​

fmap0 :: a -> f a 
fmap1 :: (a -> b) -> f a -> f b
fmap2 :: (a -> b -> c) -> f a -> f b -> f c
fmap3 :: (a -> b -> c -> d) -> f a -> f b -> f c -> f d
pure :: a-> f a -- Turn a value into a data strucutre. 
((*)) :: f (a -> b) -> f a -> f b --() is a prefix operator. Need to give brackets when defining it as a type. Generalised form of function applicaiton

pure g (*) x (*) y (*) z -- applicative style

(*) -- does it to the left

fmap0 :: a -> f a
fmap0 = pure 

fmap1 :: (a -> b) -> f a -> f b
fmap1 g x = pure g (*) x

fmap2 :: (a -> b -> c) -> f a -> f b -> f c
fmap2 g x y = (pure g (*) x ) (*) y

Applicative Functors​

class Functor f => Applicative f where
    pure :: a -> f a
    ((*)) :: f(a->b) -> f a -> f b

Example: Maybe​

instance Applicative Maybe where
    -- pure :: a -> Maybe a
    pure x = Just x
    -- ((*)) :: maybe (a->b) -> Maybe a -> Maybe b
    Nothing (*) mx = Nothing
    (Just g) (*) mx = fmap g mx

Example: Lists​

instance Applicative [] where
    -- pure :: a -> [a]
    pure x = [x]
    -- ((*)) :: [a->b] -> [a] -> [b]
    gs (*) xs = [g x | g <- gs, x <- xs]