Let $|==\{p_1\}S\{q_1\}$ and $|==\{p_2\}S\{q_2\}.$ Decide whether the following triples are valid under partial correctness,

justify your answer briefly.

a$.\{p_1^{{?}^{?}}{p}_2\}S\{q_1^{{?}^{?}}{q}_2\}$

b$.\{p_2\}S\{q_1{q}_2\}$

c$.\{not{p}_1{p}_2\}S\{not{q}_1{q}_2\}$

Let $w>wp(S,q),$ and let $S$ be a deterministic program. Decide whether each of the following statements is

true or false, justify your answer briefly. We assume $(q)$ for any well$-$formed state $.$

a$.|=={?}_{\text{t}\text{o}\text{t}}\{w\}S\{q\}$

b$.|==\{w^{{?}^{?}}q\}S\{q\}$

c$.$ There exists some state $$ such that $w$ but $M(S,)|==q.$

d$.$ If $|==w,$ then $.$

e$.$ If $w,$ then $|==\{notw\}S\{notq\}.$

You don't have to logically simplify your answers to questions $9$ and $10.$

Let $S-=y$:$=y\%x$ and $q-=\sqrt[\phantom{2}]{y}>x.$

a$.$ Calculate $wlp(S,q).$

b$.$ Calculate $wp(S,q).$

Let $S-=$ if $y0x$:$=\frac{y}{x},x0x$:$=\frac{x}{y}$ fi and $wlp(S,q)wp(S,q)q-=x.$

$a.$ Calculate $wlp(S,q).$

$b.$ Calculate $wp(S,q).$