====== Lab 8. Intro to Haskell ======

===== 8.1 Getting to know Haskell =====

**Prequisites**: having a working haskell environment ([[pp:haskell-environment|Haskell Environment]])

**Haskell** is a general-purpose, purely functional programming language, that we will use for the rest of the semester to showcase functional patterns and programming styles. \\

{{ :pp:2023:haskell:haskell.png?nolink&200 |}}

This section is designed for us to get comfortable with haskell syntax, we will use several concept that we learned in Scala, such as tail-recursion, folds and maps, but this time in a purely functional context.

==== A trip through time ====
Remember: [[pp:2024:scala:l01|Lab 1. Introduction to Scala]]

**8.1.1.** Implement a tail-recursive function that computes the factorial of a natural number.
<code haskell>
fact :: Int -> Int
fact = undefined
</code>
**8.1.2.** Implement a tail-recursive function that computes the greatest common divisor of two natural numbers.
<code haskell>
mygcd :: Int -> Int -> Int
mygcd a b = undefined
</code>
**8.1.3.** Implement the function ''mySqrt''  which computes the square root of an integer $ a $.

==== Lists ====
The following Scala syntax for working with lists, can be translated to Haskell as follows:
^ Scala ^ Haskell cases ^ Haskell pattern matching ^ Haskell guards ^
|<code scala>
def f(l: List[Int]) = l match {
  case Nil => ...
  case (x::xs) => ...
}
</code>|<code haskell> 
f l = case l of
  [] -> ...
  (x:xs) -> ...
</code> | <code haskell>
f [] = ...
f (x:xs) = ...
</code> | <code haskell>
f l | l == [] = ...
    | otherwise = ...
</code> |

**8.1.4.** Implement funtions ''mymin'' and ''mymax'' that take a list of ints, and return the smallest/biggest value in the list.

**8.1.5.** Implement a function ''unique'' that takes a list of ints, and removes all duplicates.

**8.1.6.** Given a list of ints, return a list of strings where for each element, return:
  * **'Fizz'** if the number is divisible by 3
  * **'Buzz'** if the number is divisible by 5
  * **'FizzBuzz'** if the number is divisible by 3 **and** 5
  * a string representation of the number otherwise

**8.1.7.** Extend the function from **8.1.6.** with the following rules:
  * **'Bazz'** if the number is divisible by 7
  * **'FizzBazz'** if the number is divisible by 21
  * **'BuzzBazz'** if the number is divisible by 35
  * **'FizzBuzzBazz'** if the number is divisible by 105

\\
<hidden>
You can test **8.1.6** and **8.1.7** with the following snippet, if your function is $ f $:
<code haskell>
f [1..n]
</code>
</hidden>

<note>
In Haskell, the list data type is denote by the type the list holds surrounded by square paranthesis.
<code haskell>
[Int] -- list of ints
[Double] -- list of doubles
[[Int]] -- list of lists of ints (matrices)

[] -- !!! not a data type, represents the empty list (Nil in Scala)
</code>
</note>

==== Types in Haskell ====

In Haskell, functions are curried by default, **i.e.** a function:
<code haskell>
f a b = ...
</code>
is the same as:
<code haskell>
f = \a -> \b -> ...
</code>

So, if $ a $ is a ''Int'' and $ b $ a ''Double'', and $ f $ returns a ''Char'', it would have the following type:
<code haskell>
f :: Int -> Double -> Char
</code>

**8.1.8.** Check the type signature of the following functions:
  * ''foldl''
  * ''foldr''
  * ''filter''
  * ''map''

<note important>
If a function is not ambigous, ''ghc'' can infer the type signature, for **educational** purposes, going forward you will have to write signatures for all functions you define, this is considered good practice and helps prevent bugs.
</note>

<note tip>
In ''ghci'', you can check the type of a expression with: '':t''
</note>

===== 8.2 A predicate-based implementation for sets =====
**8.2.1.** Consider **sets** represented as characteristic functions with signature ''s :: Integer -> Bool'', where ''s x'' is true if ''x'' a member in the set. Examples:
<code haskell>
s1 1 = True
s1 2 = True
s1 _ = False

s2 x = mod x 2 == 0

s3 _ = False

</code>
Above, ''s1'' is the set $math[\{1,2\}], ''s2'' is the set of even integers and ''s3'' is the empty-set. Write a function which tests if an element is a member of a set:
<code haskell>
mem :: (Integer -> Bool) -> Integer -> Bool
mem = ...
</code>

**8.2.2.** Define the set $math[\{2^n \mid n\in\mathbb{N}\}]. 

**8.2.3.** Define the set of natural numbers.

**8.2.4.** Implement the intersection of two sets. Use lambdas.
<code haskell>
intersection :: (Integer -> Bool) -> (Integer -> Bool) -> (Integer -> Bool)
</code>

**8.2.5.** Write intersection in another way, (without using lambdas). 
<code haskell>
intersection' :: (Integer -> Bool) -> (Integer -> Bool) -> Integer -> Bool
</code>

**8.2.6.** Write a function which takes a list of integers, and returns the set which contains them.
<code haskell>
toSet :: [Integer] -> (Integer -> Bool)
</code>

**8.2.7.** Implement a function which takes a list of sets and computes their intersection. Use fold.

<code haskell>
capList :: [Integer -> Bool] -> Integer -> Bool
</code>

**8.2.8.** Implement the function described in **8.2.7** but without using fold.

<code haskell>
capList' :: [Integer -> Bool] -> Integer -> Bool
</code>

**8.2.9** Write a function that takes a list of sets and an operation over sets (like intersection or union) and applies on the given list. Use fold.

<code haskell>
setsOperation :: [Integer -> Bool] -> ((Integer -> Bool) -> (Integer -> Bool) -> (Integer -> Bool)) -> (Integer -> Bool)
</code>

**8.2.10** Write a function that applies a set to a list of integers, resulting a list containing the elements from the given list that are a part of the set.
<code haskell>
applySet :: (Integer -> Bool) -> [Integer] -> [Integer]
</code>

**8.2.11.** Implement a function that takes a set and a list of integers and returns a tuple of list. The first element of the tuple is a list that contains the elements from the given list that are part of the set, and the second element contains the elements that are not part of the set. After implementing your version, check out `partition` from `Data.List` and use it!

<code haskell>
partitionSet :: (Integer -> Bool) -> [Integer] -> ([Integer], [Integer])
</code>

===== 8.3 Brain Twisters =====

**8.3.1.** Implement ''map'' using ''foldl'' and ''foldr''.
<code haskell>
mymapl :: (a -> b) -> [a] -> [b]
mymapr :: (a -> b) -> [a] -> [b]
</code>

**8.3.2.** Implement ''filter'' using ''foldl'' and ''foldr''.
<code haskell>
myfilterl :: (a -> Bool) -> [a] -> [a]
myfilterr :: (a -> Bool) -> [a] -> [a]
</code>

**8.3.3.** Implement ''foldl'' using ''foldr''.
<code haskell>
myfoldl :: (a -> b -> a) -> a -> [b] -> a
</code>

**8.3.4.** Implement ''bubbleSort''.
<code haskell>
bubbleSort :: [Int] -> [Int]
</code>

**8.3.5.** Implement ''quickSort''.
<code haskell>
quickSort :: [Int] -> [Int]
</code>