Fix chapter 21

21-traversable/21-traversable.md

@@ -0,0 +1,169 @@
+# Chapter 21 Traversable
+
+## 21.3 sequenceA
+
+```haskell
+-- Why: (fmap . fmap) sum Just [1, 2, 3] = Just 6
+
+-- Let's check the types
+fmap :: Functor f => (a -> b) -> f a -> f b
+(fmap . fmap) :: (Functor f1, Functor f2) => (a -> b) -> f1 (f2 a) -> f1 (f2 b)
+(fmap . fmap) sum :: (Num a, Foldable t, Functor f1, Functor f2) =>
+    f1 (f2 (t a)) -> f1 (f2 a)
+(fmap . fmap) sum Just :: -- Just :: b -> Maybe b
+                          -- Just :: (->) b (Maybe b)
+                          -- for this to correspond with f1 (f2 (t a))
+                          -- f1 is '(->) b'
+                          -- f2 is Maybe
+                          -- b is 't a'
+                          -- so f1 (f2 a) is (->) (t a) (Maybe a)
+                          (Num a, Foldable t) => t a -> Maybe a
+
+-- A good way to see this is that we are lifting the 'sum' over the function
+-- functor and the Maybe functor produced as a result of the function functor
+--
+-- Just :: (->) a (Maybe a)
+--
+-- And we are lifting (sum :: Foldable t => t a -> a) over (->) a and Maybe.
+-- Notice that the a in Just needs to be specialized in a t a to function as
+-- the input for sum, i.e
+--
+-- Just' :: Foldable t => (->) (t a) (Maybe (t a))
+--
+-- We can now apply (fmap . fmap) sum and get:
+-- (fmap . fmap) sum Just :: Foldable t => (->) (t a) (Maybe (t a))
+
+-- It might be helpful to look at the generalized form:
+--
+-- (fmap . fmap) h1 h2 :: ?
+--
+(fmap . fmap) :: (Functor f1, Functor f2) => (a -> b) -> f1 (f2 a) -> f1 (f2 b)
+(fmap . fmap) h1 :: (Functor f1, Functor f2) => f1 (f2 a) -> f1 (f2 b)
+-- if we now want to add h2, with h2 being: b' -> c', we can deduce that
+-- (->) b' is the f1 functor and c' is 'f2 a'
+(fmap . fmap) h1 :: Functor f => (b' -> f a) -> (b' -> f b)
+-- A couple of points to remember:
+--   b' is an input of h2 and a is an input of h1,  so h2 needs to be a function
+--   that return returns an input from h1 in a functor (e.g. with Just this can
+--   take a 'Foldable t => t a' and returns a Maybe (t a))
+-- This implies that there are restrictions imposed on h2:
+--    h2 needs to return a functor which contains an input type of h1
+-- Applying h2 just gives:
+(fmap . fmap) h1 h2 :: (Functor f) => b' -> f b
+
+-- Does this restriction on h2 hold for 'Just'?
+-- Consider that sum :: Foldable t => t a -> a
+-- And Just :: a -> Maybe a, with a being more generic than 't a', we can
+-- indeed say that the resulting a in 'Maybe a' can be of type 't a'.
+
+-- We can rewrite (fmap . fmap) h1 h2 as:
+--
+-- fmap h1 . h2
+--
+-- So (fmap . fmap) sum Just can be written as fmap sum . Just
+
+-- Another example with different types accross the board:
+foo :: String -> Maybe [Bool]
+foo [] = Just []
+foo (x:xs) = case foo xs of
+    Just xs -> case x of
+        't' -> Just $ True : xs
+        'f' -> Just $ False : xs
+        otherwise -> Nothing
+    otherwise -> Nothing
+
+bar :: [Bool] -> Integer
+bar = foldr (\b c -> if b then c + 1 else c - 1) 0
+
+-- where (fmap . fmap) bar foo = fmap bar . foo
+```
+
+## 21.4 traverse
+
+Why is traverse not defined as:
+
+`traverse :: (Functor f, Traversable t) => (a -> f b) -> t a -> f (t b)`
+
+Maybe let's ask ourselves why sequenceA cannot be defined as such:
+
+`sequenceA :: (Traversable t, Functor f) => t (f a) -> f (t a)`
+
+Let's try to implement it:
+
+```haskell
+-- Without applicative, but functor allowed
+instance Traversable [] where
+    sequenceA [] = ??? -- we have no way of putting this in a functor, we
+                       -- would need pure at least...
+    sequenceA (x:xs) = ??? -- we have an x :: f a
+                           -- and an xs :: [f a]
+                           -- We can turn the xs into f [a] via sequenceA:
+                           -- sequenceA xs, but now we need to concatenate
+                           -- 'f a' and 'f [a]' with only using fmap and this
+                           -- is impossible. We can fmap (:) into 'f a', but
+                           -- then we need (<*>) to apply this to 'f [a]'
+
+-- with applicative it works:
+instance Traversable [] where
+    sequenceA [] = pure []
+    sequenceA (x:xs) = (:) <$> x <*> sequenceA xs
+```
+
+## 21.6 morse code revisited
+
+Why `(sequence .) . fmap` and not `sequence . fmap`
+
+```haskell
+fmap :: Functor f => (a -> b) -> (f a -> f b)
+sequence :: (Monad m, Traversable t) => t (m a) -> m (t a)
+(.) :: (b -> c) -> (a -> b) -> (a -> c)
+(.) sequence :: (Monad m, Traversable t) => (a -> t (m b)) -> (a -> m (t b))
+(.) sequence fmap :: -- a :: (a' -> b')
+                     -- t (m b) :: (->) (f a') (f b')
+                     --    t :: (->) (f a') i.e. (->) (f a') must be Traversable
+                     --    m b :: f b' i.e f shoulde be a Monad
+                     (Monad m, Traversable (->) (f a)) =>
+                     (a -> b) -> m (m a -> b)
+-- Which is not really wat we want...
+-- However, if you check (.) sequence, then it looks close to what we want
+(.) ((.) sequence) :: (Monad m, Traversable t) =>
+    (a -> (b -> t (m c))) -> (a -> (b -> m (t c)))
+(.) ((.) sequence) fmap :: -- a = (a' -> b')
+                           -- b -> t (m c) = (f a') -> f b'
+                           -- b = f a'
+                           -- t (m c) = f b'
+                           --   t = f (A traversable t is also a functor, see
+                           --          its definition)
+                           --   b' :: m c
+                           (Monad m, Traversable t) =>
+                           (a' -> m c) -> (t a' -> m (t c))
+```
+
+## 21.10 Traversable Laws
+
+```haskell
+Compose :: f (g a) -> Compose f g a
+f :: Applicative f => a -> f b
+g :: Applicative g => a -> f b -- bascause of traverse g
+fmap g :: (Applicative f1, Functor f2) => f2 a -> f2 (f1 b)
+fmap g . f :: (Applicative f1, Applicative f2) => a -> f2 (f1 b)
+Compose . (fmap g . f) :: (Applicative f1, Applicative f2) =>
+    a -> Compose f g b
+traverse (Compose . fmap g . f) ::
+    (Applicative f1, Applicative f2, Traversable t) =>
+    t a -> Compose f1 f2 (t b)
+
+traverse f :: (Applicative f, Traversable t) => t a -> f (t b)
+fmap (traverse g) :: (Applicative f1, Applicative f2, Traversable t) =>
+    f1 (t a) -> f1 (f2 (t b))
+fmap (traverse g) . traverse f ::
+    (Applicative f1, Applicative f2, Traversable t) =>
+    t a -> f1 (f2 (t b))
+Compose . fmap (traverse g) . traverse f ::
+    (Applicative f1, Applicative f2, Traversable t) =>
+    t a -> Compose f1 f2 (t b)
+```
+
+## 21.12 Chapter Exercises
+
+see [src/chapter.hs](./src/chapter.hs)

21-traversable/21.12-chapter-exercises.md

@@ -1,2 +0,0 @@
-# Chapter Exercises
-see src/chapter.hs