$6.$ Let $||=={?}_{\text{t}\text{o}\text{t}}\{p_1\}S\{q_1\}$ and $||=={?}_{\text{t}\text{o}\text{t}}\{p_2\}S\{q_2\}.$ Decide whether $$ necessarily satisfies the following triples under

total correctness, justify your answer briefly.

a$.\{p_1^{{?}^{?}}{p}_2\}S\{q_{1}vv{q}_2\}$

b$.\{p_{1}vv{p}_2\}S\{q_1^{{?}^{?}}{q}_2\}$

c$.\{p_{1}vv{p}_2\}S\{q_{1}vv{q}_2\}$

$7.$ 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$.,\{not{p}_1{p}_2\}S\{not{q}_1{q}_2\}$

$8.$ 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 $10.$

$9.$ Let $S-=y$:$=y\%x$ and $q-=\sqrt[\phantom{2}]{y}>x.$

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

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

$10.$ 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 $wp(S,q).$