Q$4[20]$

Consider the following specification where a and $b$ are arrays of inte$-$

gers of length $n$

$f=($:AAiin$\{0,..n\}*b^{\text{'}}(i)=({\displaystyle \underset{?}{\overset{?}{}}}$jin$\{0,..,i\}*(-1{)}^{j}a(j))$:$)$

E$.$g$.a=[7,2,6,8],b^{\text{'}}=[7,5,11,3].$

Read the whole question before beginning to answer. The idea is to

develop a command of the form

$f{m}_0$; while $Ado{h}_0$

The command should run in $(n)$ time.

$($a$)[5]$ Propose a loop invariant $I$

Let $g=(($:I:$)=>f).$ Let $m=($:$I^{\text{'}}{?}^{{?}^{?}}{a}^{\text{'}}=a$:$)$ so that $fm$;$g.$

$($b$)[2]$ Give a command that refines $m.$

$($c$)[5]$ Give an expression A and such that

$I^{{?}^{?}}$notA$=>$AAiin$\{0,..n\}*b(i)=({\displaystyle \underset{?}{\overset{?}{}}}$jin$\{0,..i\}*(-1{)}^{j}a(j))$

is true for all states, i$.$e$.,$ so that $($:notA:$)=>g$ skip.

$A$ is

$($d$)[3]$ Propose a bound expression for the loop.

Let $h=($:$A^{{?}^{?}}I=>I^{\text{'}}{?}^{{?}^{?}}{a}^{\text{'}}=a$:$)$ so that

$g$ if A then $(h$;$g)$ else skip

$($e$)[5]$ Give a command that refines $h$ and that makes the bound

expression smaller.

$h$