A loop invariant of the inner for$-$loop is

$$min$\_$idx is the index of the minimum value of arr$[$i$],$ arr$[$i$+1],...,$ arr$[$j$-1].$

At termination of the inner for$-$loop,

$$j$=$n$.$

Thus, the property of the inner for$-$loop is $($repeat of termination$)$:

$$min$\_$idx is the index of the smallest value in arr$[$i$],$ arr$[$i$+1],...,$ arr$[$n$-1].$

Answer the following three questions. Be sure your answers are readable.

What is the loop invariant of the outer for$-$loop $($Hint: one piece of the invariant is that the beginning of the array is sorted, but the other piece is to describe what we know about the other elements in the array$)?$

What is the Initialization $($i$.$e$.,$ show that the loop invariant is true before the first pass through the loop$)?$

What is the maintenance step $($i$.$e$.,$ for iteration $$i$$ through the loop, assume the loop invariant before the pass through the loop; then show it is still true after the pass$)?$