The original Ershov numbers were used for a machine that allowed the right operand of an expression to be in memory, rather than a register. How would you modify the tree-labeling scheme to generate...

The original Ershov numbers were used for a machine that allowed the right operand of an expression to be in memory, rather than a register. How would you modify the tree-labeling scheme to generate optimal code for this machine model?

Prove by induction on the number of iterations of the for-loop of lines (4) through (6) of Algorithm 9.11 that none of the IN's or OUT 's ever shrinks. That is, once a definition is placed in one of these sets on some round, it never disappears on a subsequent round.