Haskell

Haskell

1. Functors

   • class Functor f where     fmap :: (a -> b) -> f a -> f b
   • First functor law
     • fmap id = id
   • Second functor law
     • fmap (f . g) = fmap f . fmap g
   • Examples
     • List
     • ZipList
     • Maybe
     • Either

2. Applicative functors

   • class (Functor f) => Applicative f where     pure :: a -> f a     (<*>) :: f (a -> b) -> f a -> f b
   • Laws
     • pure f <*> = fmap f x pure id <*> v = v pure (.) <*> u <*> v <*> w = u <*> (v <*> w) pure f <*> pure x = pure (f x) u <*> pure y = pure ($ y) <*> u

3. Monoids

   • mempty `mappend` x = x x `mappend` mempty = x (x `mappend` y) `mappend` z = x `mappend` (y `mappend` z)
   • Lists
     • []
   • Any
     • False
   • All
     • True
   • Product
     • 1
   • Sum
     • 0
   • Ordering
     • instance Monoid Ordering where       mempty = EQ       LT `mappend` _ = LT       EQ `mappend` y = y       GT `mappend` _ = GT 
   • Maybe
     • instance Monoid a => Monoid (Maybe a) where       mempty = Nothing       Nothing `mappend` m = m       m `mappend` Nothing = m       Just m1 `mappend` Just m2 = Just (m1 `mappend` m2)  
   • First
     • instance Monoid (First a) where       mempty = First Nothing       First (Just x) `mappend` _ = First (Just x)       First Nothing `mappend` x = x 
   • Last

4. Foldable