Prove the following about Algorithm 9.17: a) The IN's and OUT's never grow; that is, successive values of these sets are subsets (not necessarily proper) of their previous values. b) If expression e...

Prove the following about Algorithm 9.17:

a) The IN's and OUT's never grow; that is, successive values of these sets are subsets (not necessarily proper) of their previous values.

b) If expression e is removed from IN[B] or OUTpH], then there is a path from the entry of the flow graph to the beginning or end of block B, respectively, along which e is either never computed, or after its last computation, one of its arguments might be redefined.

c) If expression e remains in IN[B] or OUTpB], then along every path from the entry of the flow graph to the beginning or end of block B, respectively, e is computed, and after the last computation, no argument of e could be redefined.