Prove the following: a) If a definition d is in IN [B], then there is some acyclic path from the block containing d to B such that d is in the IN's and OUT 's all along that path.  b) If an expression...

Prove the following:


a) If a definition d is in IN [B], then there is some acyclic path from the block containing d to B such that d is in the IN's and OUT 's all along that path.

b) If an expression x + y is not available at the entrance to block B, then there is some acyclic path that demonstrates that fact; either the path is from the entry node and includes no statement that kills or generates x + y, or the path is from a block that kills x + y and along the path there is no subsequent generation of x + y.

c) If x is live on exit from block B, then there is an acyclic path from B to a use of x, along which there are no definitions of x.