Give a full proof outline under partial correctness obtained by expansion of the partial proof outline below. Please use the Forward Assignment Axiom to prove all the assignment statements. Give a brief explanation on each logical implication used in the proof outline.

$\{y1\}x$:$=0$;$r$:$=1$

$\{$ inv $1r=2^{x}y\}$

while $2**rydor$:$=2**r$;$x$:$=x+1od$

$\{r=2^{x}y{2}^{x+1}\}$

Find a reasonable bound expression for the while loop in Question $3,$ then create a full proof outline under total correctness for the partial proof outline in Question $3.$ Please use the Backward Assignment Axiom to prove all the assignment statements. Give a brief explanation on each logical implication used in the proof outline.

Under total correctness, find a reasonable precondition $p$ and create a full proof outline for the following provable triple:

$\{p\}if\sqrt[\phantom{2}]{x}>y$ then $x$:$=b[x-y]$ else $y$:$=b[y-x]fi\{x=y\}$

Hint: Using Conditional Rule $2$ can avoid calculating the domain predicate for a conditional statement.

$6.$ Under total correctness, find a reasonable postcondition $q$ and create a full proof outline for the following provable triple:

$\{\sqrt[\phantom{2}]{x}y\}x$:$=x**y$;$x$:$=1x\{q\}$

Hint: Since the precondition and the statement are not safe, we should include their domain predicates in the precondition for total correctness. Be careful of the domain predicate for a sequence statement.

$7.$ Let $W-=\{invp\}\{bdt\}$ while $B$ do $S$ od and $|--{?}_{\text{t}\text{o}\text{t}}\{invp\}\{bdt\}||$ while $B$ do $S$ od $\{p^{{?}^{?}}$notB$\}.$ For each of the following statements about bound expression, decide true or false and justify your answer briefly.

a$.$ Let $|||==p,$ then $|{)}_{{?}_{?}}(d)!$inM$(W,).$

b$.$ The value of $t$ can be negative after the execution the last iteration of $W.$

c$.$