In Prolog, suppose we have a predicate supervisor $(x,Y)$ that is true just when $x$ is the direct supervisor of $Y.$

a$)$ Define a predicate higherup $(x,Y)$ that is true just when $x$ is above or equal to $Y$ in a chain of supervisors. For example if supervisor$($a$,$b$),$ supervisor$($b$,$c$)$ and supervisor$($c$,$d$)$ are true then higherup$($a$,$a$),$ higherup$($a$,$b$),$ higherup$($a$,$c$)$ and higherup$($a$,$d$)$ would all be true.

b$)$ Define a predicate shared$\_$boss $(x,Y,Z)$ that is true just when $Z$ is higherup than both $x$ and $Y.$

c$)$ Define a predicate closest$\_$shared$\_$boss $(x,Y,Z)$ that's true just when $Z$ is a shared$\_$boss of $x$ and $Y$ and there is no $W,$ distinct from $Z,$ such that $W$ is a shared boss of $x,$ and $Y$ and $Z$ is higher up than $W..$

d$)$ Draw a tree diagram of a hierarchy in which sup$($a$,$b$),$ sup$($b$,$c$),$ sup$($c$,$d$),$ sup$($c$,$e$),$ sup$($e$,$f$)$ are all true. Then explain how your rules for shared$\_$boss would find all the shared bosses $d$ and $f.$ Also show how your rules would find the nearest shared boss of $d$ and $f.$