====== Lab 4. Algebraic Datatype definition ======

Below you will find the algebraic definition of the datatype ''IList'':
<code>
Void : IList
Cons : Int x IList -> IList
</code>

This definition has already been implemented in Scala, below. Please copy-paste this definition in your worksheet.
<code scala>
trait IList 
case object Void extends IList
case class Cons(x: Int, xs: IList) extends IList
</code>

**4.1.** Consider the following axioms for the operator ''isEmpty''.
<code>
isEmpty : IList -> Boolean
isEmpty(Void) = true
isEmpty(Cons(h,t)) = false.
</code>
Implement ''isEmpty'' in Scala:
<code scala>
def isEmpty(l: IList): Boolean = ???
</code>

**4.2.** Write down axioms for ''size : IList -> Int'' and implement the operator in Scala:
<code scala>
def size(l: IList): Int = ???
</code>

**4.3.** Implement ''contains'' which checks if an element is a member of a list.
<code scala>
def contains(e: Int, l: IList): Boolean = ???
</code>

**4.4.** Implement ''max'' which returns the largest integer from a list:
<code scala>
def max(l: IList): Int = ???
</code>

**4.5.** Implement ''take'' which returns a new list containing the first ''n'' elements of the original list:
<code scala>
def take(n: Int)(l: IList): IList = ???
</code>

**4.6.** Implement ''drop'' which returns a new list containing the original list without the first ''n'' elements:
<code scala>
def drop(n: Int)(l: IList): IList = ???
</code>

**4.7.** Implement ''append'' which concatenates two lists:
<code scala>
def append(l1: IList, l2: IList): IList = ???
</code>

**4.8.** (!) Implement ''last'' which returns the last element from a list:
<code scala>
def last(l: IList): Int = ???
</code>

**4.9.** (!) Implement ''reverse''. There are two different ways to implement reverse (with direct and with tail-end recursion). Try both implementations.
<code scala>
def reverse(l: IList): IList = ???
</code>

**4.10.** Implement ''isSorted'' which checks if a list is sorted:
<code scala>
def isSorted(l: IList): Boolean = ???
</code>

**4.11.** Implement ''merge'' which merges two sorted lists:
<code scala>
def merge(l1: IList, l2: IList): IList = ???
</code>

**4.12.** Implement ''mergeSort'' which sorts a list:
<code scala>
def mergesort(l: IList) IList = ???
</code>