✎ fp2023:lab04 [books]

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

==== 4.1. The type Nat ===== 
Consider the following type defined to represent natural numbers:
<code scala>
trait Nat 
case object Zero extends Nat
case class Succ(n: Nat) extends Nat
</code>

4.1.1. Implement the following function:
<code scala>
def isZero(n: Nat): Nat = 
   n match {
       case Zero => ???
       case Succ(np) => ???
   }
</code>

4.1.2. Implement the addition function over natural numbers:
<code scala>
def add(n: Nat, m: Nat): Nat = 
   n match {
       case Zero => ???
       case Succ(np) => ???
   }
</code>

4.1.3. Implement the subtraction function over natural numbers. If $math[n > m] then $math[m - n = 0].
<code scala>
def subtract(n: Nat, m: Nat): Nat = ???
</code>

4.1.4. Implement ''greater'' which checks if a natural number is strictly larger than the other:
<code scala>
def greater(n: Nat, m: Nat): Nat = 
   (n,m) match {
      ???
   }
</code>

4.1.5. Equality is already defined for all case classes, as structural equality (objects built in the same way are equal). However, re-implement equality over naturals:
<code scala>
def equal(n: Nat, m: Nat): Nat = ???
</code>

4.1.6. Implement a function which converts a Nat to a Scala Int:
<code scala>
def toInt(n: Nat): Int = ???
</code>

4.1.7. Implement a function which converts an Int to a Nat:
<code scala>
def fromInt(i: Int): Nat = ???
</code>
===== 4.2. The type IList (list over integers) =====

Consider the following type defined to represent lists of integers:
<code scala>
trait IList 
case object Void extends IList
case class Cons(x: Int, xs: IList) extends IList
</code>

4.2.1. Implement ''isEmpty'' which checks if a list is empty:
<code scala>
def isEmpty(l: IList): Boolean = ???
</code>

4.2.2. Implement ''size'' which returns the size of the list:
<code scala>
def size(l: IList): Int = ???
</code>

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

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

4.2.5. (!) 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>

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

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

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

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

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