====== Lab 4. Data types in Scala ======

Objectives:
  * get familiar with **algebraic data types**
  * get familiar with **pattern matching** and **recursion** with them

==== 4.1 Natural Numbers ====
Given the following implementation of the natural numbers, solve the next few exercices.
<code scala>
trait NaturalNumber
case object Zero extends NaturalNumber
case class Successor(x: NaturalNumber) extends NaturalNumber
</code>

**4.1.1** Write a function which takes two natural numbers, and return their sum.
<code scala>
def add(x: NaturalNumber, y: NaturalNumber): NaturalNumber = ???
</code>

**4.1.2** Write a function which takes two natural numbers, and return their product.
<code scala>
def multiply(x: NaturalNumber, y: NaturalNumber): NaturalNumber = ???
</code>

**4.1.3** Write a function which takes an int and converts it to a NaturalNumber.
<code scala>
def toNaturalNumber(x: Int): NaturalNumber = ???
</code>

==== 4.2 Binary Trees ====
Given the following implementation of binary trees, solve the next few exercices.
<code scala>
trait BTree
case object EmptyTree extends BTree
case class Node(value: Int, left: BTree, right: BTree) extends BTree
</code>

**4.2.1** Write a function which takes a BinaryTree and returns its depth.
<code scala>
def depth(tree: BTree): Int = ???
</code>

**4.2.2** Write a function which takes a BinaryTree and returns the number of nodes with even number of children.
<code scala>
def evenChildCount(tree: BTree): Int = ???
</code>

**4.2.3** Write a function which takes a BinaryTree and flattens it (turns it into a list containing the values of the nodes).
<code scala>
def flatten(tree: BTree): List[Int] = ???
</code>

**4.2.4** Write a function which takes a BinaryTree and return the number of nodes whose values follow a ceratain rule.
<code scala>
def countNodes(tree: BTree, cond: Int => Boolean): Int = ???
</code>

**4.2.5** Write a function which takes a BinaryTree and return mirrored BTree.
<code scala>
def mirror(tree: BTree): BTree= ???
</code>

==== 4.3 Matrix manipulation ====
We shall represent matrices as //lists of lists//, i.e. values of type ''[ [Integer ] ]''. Each element in the outer list represents a line of the matrix. 
Hence, the matrix

$math[ \displaystyle \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right)] 

will be represented by the list ''[ [1,2,3],[4,5,6],[7,8,9] ]''.

To make signatures more legible, add the //type alias// to your code:
<code scala> type Matrix = List[List[Int]] </code>
which makes the type-name ''Matrix'' stand for ''[ [Integer] ]''.

**4.3.1** Write a function that computes the scalar product with an integer:

$math[ \displaystyle 2 * \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \\ \end{array}\right)] 

<code scala>
def vprod(m: Matrix)(v: Int): Matrix = ??? 
</code>

**4.3.2** Write a function which adjoins two matrices by extending rows:

$math[ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right)  hjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 & 5 & 6 \\ 3 & 4 & 7 & 8\\\end{array}\right) ]

<code scala>
def join(m1: Matrix, m2: Matrix): Matrix = ???
</code>

**4.3.3** Write a function which adjoins two matrices by adding new rows:

$math[ \displaystyle \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\\end{array}\right)  vjoin \left(\begin{array}{cc} 5 & 6 \\ 7 & 8\\\end{array}\right) = \left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6\\ 7 & 8\\ \end{array}\right) ]

<code scala>
def vjoin(m1: Matrix, m2: Matrix): Matrix = ???
</code>

**4.3.4** Write a function which adds two matrices.

<code scala>
def msum(m1: Matrix, m2: Matrix): Matrix = ???
</code>

**4.3.5** Write a function which computes the transposition of a matrix:

$math[ tr \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right) = \left(\begin{array}{ccc} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ \end{array}\right) ]

<code scala>
def tr(m: Matrix): Matrix = ???
</code>

**4.3.6** Write a function which computes the vectorial product of two matrices.
  * (Hint: start by writing a function which computes $math[a_{ij}] for a given line $math[i] and column $math[j] (both represented as lists))
  * (Hint: write a function which takes a line of matrix m1 and the matrix m2 and computes the respective line from the product)

<code scala>
def mprod(m1: Matrix, m2: Matrix): Matrix = ???
</code>