Prove the following predicate logic sequents using natural deduction:

$($a$)$xA$($x$)$x$$A$($x$)$

$($b$)$xA$($x$)->$ S $$x$($A$($x$)->$ S$)$

$($c$)$x$($A$($x$)$ B$($x$))$xA$($x$)$xB$($x$)$

$($d$)$x$$y$($A$($x$)$ B$($y$))$y$$x$($A$($x$)$ B$($y$))$