Problem $4.$ Consider the following algorithm which computes the mean of a non$-$empty array of integers. The

mean of a nonempty list $a_1,$dots,$a_k$ of $k$ numbers is defined to be $\frac{1}{k}{\displaystyle \underset{i=1}{\overset{k}{}}}{a}_i.$ Please state any assumptions you

make.

procedure mean $($A$)$:

mu $=$ A$[0]$

n$\}=\backslash$operatorname$\{$len$\}($A

for $($i $=1$; i$i(i+1)5pts10pts$

$(1b)$ Provide the initialization component $of$ a proof $of$ correctness $of$ this algorithm that uses your loop invariant

from $(1a).[5pts]$

$(1c)$ Provide the maintenance component $of$ a proof $of$ correctness $of$ this algorithm that uses your loop invariant

from $(1a).$ That $is,$ assume that your loop invariant holds just before the $i-th$ iteration $of$ the for loop and

use this assumption $to$ show that your loop invariant holds just before iteration $(i+1).[5pts]$

$(1d)$ Provide the termination component $of$ a proof $of$ correctness $of$ this algorithm that uses your loop invariant

from $(1a).[5pts]$