Merge$(A,p,q,r)$

$n_L=q-p+1,//$ length of $A[p$:$q]$

$n_R=r-q,//$ length of $A[q+1$:$r]$

let $L[0$:$n_L-1]$ and $R[0$:$n_R-1]$ be new arrays

for $i=0$ to $n_L-1\frac{,}{\frac{?}{?}}$ copy $A[p$:$q]$ into $L[0$:$n_L-1]$

$L[i]=A[p+i]$

for $j=0$ to $n_R-\frac{1}{\frac{?}{?}}$ copy $A[q+1$:$r]$ into $R[0$:$n_R-1]$

$R[j]=A[q+j+1]$

$i=0\frac{,}{\frac{?}{?}}$i indexes the smallest remaining element in $L$

$j=0\frac{,}{\frac{?}{?}}j$ indexes the smallest remaining element in $R$

$k=p\frac{,}{\frac{?}{?}}k$ indexes the location in $A$ to fill

$//$ As long as each of the arrays $L$ and $R$ contains an unmerged element,

$//$ copy the smallest unmerged element back into $A[p$:$r].$

while $L[i]R[j]A[k]=L[i]i=i+1A[k]=R[j]j=j+1k=k+1$LRA$[p$:$r]A[k]=L[i]i=i+1k=k+1A[k]=R[j]j=j+1k=k+1j$

$A[k]=R[j]$

$j=j+1$

$k=k+1i$

$A[k]=L[i]$

$i=i+1$

$k=k+1$

while $j$

$A[k]=R[j]$

$j=j+1$

$k=k+1j$

$ifL[i]R[j]$

$A[k]=L[i]$

$i=i+1$

else $A[k]=R[j]$

$j=j+1$

$k=k+1\frac{?}{\frac{?}{?}}$

Having gone through one $ofL$ and $R$ entirely, copy the$\frac{?}{\frac{?}{?}}$

remainder $of$ the other $to$ the end $ofA[p$:$r].$

while $i$

$A[k]=L[i]$

$i=i+1$

$k=k+1$

while $j$

$A[k]=R[j]$

$j=j+1$

$k=k+1$i and $j$

$ifL[i]R[j]$

$A[k]=L[i]$

$i=i+1$

else $A[k]=R[j]$

$j=j+1$

$k=k+1\frac{?}{\frac{?}{?}}$

Having gone through one $ofL$ and $R$ entirely, copy the$\frac{?}{\frac{?}{?}}$

remainder $of$ the other $to$ the end $ofA[p$:$r].$

while $i$

$A[k]=L[i]$

$i=i+1$

$k=k+1$

while $j$

$A[k]=R[j]$

$j=j+1$

$k=k+1$

State a loop invariant for the while loop of lines $12318$ of the MERGE procedure.

Show how to use it$,$ along with the while loops of lines $20323$ and $24327,$ to prove

that the MERGE procedure is correct.