Add filter, fold and summary slides for "Recursion patterns" section

VL2.tex

@@ -501,4 +501,158 @@ absList = map abs
 Cool, right? So now we have abstracted out the \textbf{recursion pattern} that is all the same for those 3 functions. \code{map} is actually part of the standard library (called \emph{Prelude}).
 \end{frame}
 
+\subsection{6.2. filter}
+
+\begin{frame}[fragile]
+\frametitle{6.2. filter}
+Imagine we want to filter all even numbers of a list and throw away all others. I'll give you the type signature:
+\begin{haskellcode}
+filterEven :: [Int] -> [Int]
+\end{haskellcode}
+Solution?
+\pause
+\begin{haskellcode}
+filterEven :: [Int] -> [Int]
+filterEven [] = []
+filterEven (x:xs)
+  | even x    = x : filterEven xs
+  | otherwise = filterEven xs
+\end{haskellcode}
+\pause
+Or: filter out all 0's, so we can count them later:
+\begin{haskellcode}
+filterZero :: [Int] -> [Int]
+filterZero [] = []
+filterZero (x:xs)
+  | x == 0    = x : filterZero xs
+  | otherwise = filterZero xs
+\end{haskellcode}
+Again: do you notice something?
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{6.2. filter (ctn.)}
+Let's abstract out the common pieces! This will be our type signature:
+\begin{haskellcode}
+filter :: (a -> Bool) -> [a] -> [a]
+\end{haskellcode}
+Solution?
+\pause
+\begin{haskellcode}
+filter :: (a -> Bool) -> [a] -> [a]
+filter f [] = []
+filter f (x:xs)
+  | f x       = x : filter f xs
+  | otherwise = filter f xs
+\end{haskellcode}
+Again: this function is part of the \emph{Prelude} as well.
+\end{frame}
+
+\subsection{6.3. fold}
+
+\begin{frame}[fragile]
+\frametitle{6.3. fold}
+There's one more important recursion pattern. Imagine you want the sum of all  numbers of a list, so the function type signature would be:
+\begin{haskellcode}
+sum :: [Int] -> Int
+\end{haskellcode}
+Solution?
+\pause
+\begin{haskellcode}
+sum :: [Int] -> Int
+sum [] = 0
+sum (x:xs) = x + sum xs
+\end{haskellcode}
+\pause
+Or the product:
+\begin{haskellcode}
+prod :: [Int] -> Int
+prod [] = 1
+prod (x:xs) = x * prod xs
+\end{haskellcode}
+\pause
+Or the length:
+\begin{haskellcode}
+length :: [a] -> Int
+length [] = 0
+length (x:xs) = 1 + length xs
+\end{haskellcode}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{6.3. fold (ctn.)}
+To cut the story short, the abstract solution looks like this:
+\begin{haskellcode}
+fold :: b -> (a -> b -> b) -> [a] -> b
+fold z f []     = z
+fold z f (x:xs) = f x (fold z f xs)
+\end{haskellcode}
+Whoa! What's going on here?\\
+Let's see...
+\begin{itemize}[<+->]
+\item \code{z} is what we return if the list is empty
+\item \code{f} is our function (e.g. \code{(*)} or \code{(+)})
+\item and the last remaining argument is the actual list we are working on
+\end{itemize}
+\onslide<+->
+The function application has the following form:\\
+\code{fold f z [a,b,c] == a `f` (b `f` (c `f` z))}
+\vspace{\baselineskip}
+\\
+This folds from the right, so the \emph{Prelude} already defines a function which is very similar to ours and called \textbf{foldr}.
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{6.3. fold (ctn.)}
+So how do our \code{sum}, \code{prod} and \code{length} functions look like if we use our \code{fold} abstraction?
+\pause
+\begin{haskellcode}
+sum :: [Int] -> Int
+sum xs = fold 0 (\x y -> x + y) xs
+-- a Haskeller would write
+sum = fold 0 (+)
+
+prod :: [Int] -> Int
+prod xs = fold 1 (\x y -> x * y) xs
+
+length :: [a] -> Int
+length xs = fold 0 (\x y -> 1 + y) xs
+\end{haskellcode}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{6.3. fold (ctn.)}
+There is also a function that folds from the \emph{left} which is also in the \emph{Prelude} and called \textbf{foldl}.\\
+To summarize:
+\begin{haskellcode}
+foldr f z [a,b,c] == a `f` (b `f` (c `f` z))
+foldl f z [a,b,c] == ((z `f` a) `f` b) `f` c
+\end{haskellcode}
+For \code{foldl} the \code{z} is sort of the starting value.
+\vspace{\baselineskip}
+\\
+\pause
+We can even express foldl in terms of foldr and vice versa. If you are interested, have a look here: \url{http://lambda.jstolarek.com/2012/07/expressing-foldl-in-terms-of-foldr/}
+\vspace{\baselineskip}
+\\
+You should definitely look them up in the Prelude and play with them: \url{https://hackage.haskell.org/package/base-4.8.0.0/docs/Prelude.html}
+\vspace{\baselineskip}
+\\
+GHCi...
+\begin{haskellcode}
+> foldr (-) 0 [1, 2, 3]
+> foldl (-) 0 [1, 2, 3]
+\end{haskellcode}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{6.3. summary}
+\begin{itemize}[<+->]
+\item if you find recurring patterns in your code, abstract them out! Experienced Haskellers avoid explicit recursion, unless the recursion pattern is so complex/specific that an abstraction doesn't make sense.
+\item map, filter, fold etc are all dependent on the data type (here: lists). For new data types (e.g. a tree) you can and will write your own recursion abstractions
+\item although these functions are so fundamental that they are already implemented for most data types out there
+\end{itemize}
+\end{frame}
+
+
 \end{document}