Problem $2(4$ pts$.,1$ pt$.$ each$)$: Hoare Triples

State whether each Hoare triple is valid. If it is invalid, explain why and show how you would

modify the postcondition to make it the strongest for the given precondition. Unless otherwise

stated, assume all referenced variables are defined as integers.

$(1)\{$ x $>0\}$

y $=-2*$ x ;

$\{$ y $>0$ y $\%2=0\}$

$(2)\{$

$$

x $1>=$ k $\}$

x $=$ x $+1$;

$\{$ k $>0\}$

$(3)\{$ i $+$ j $=0$ i $$ j $=0\}$

i $=$ j $-1$;

k $=$ i $*$ j ;

j $=$ i $+1$;

$\{$ i $=0$ i $=-$j $$ k $=$ i $$ j $\}$

$(4)\{$ false $\}$

if $($ n $<$ m $)$

x $=$ n ;

else

x $=$ m ;

$\{$ x $=$ max$($n$,$ m$)\}$