Let $S$ be the parallel composition of $S_1$ and $S_2$.
Prove that if $P$ is an invariant of $S_1$ and $S_3$, then $P$ is an muariant of $S$. Prove that if $P$ is a $Q$-derivate of $S_1$ and $S_2$, then $P$ is a $Q$-dervate of $S$. Give an example where $P$ is always true in both $S_1$ and $S_2$, but not in $S$.
The next exercise concerns well-founded sets. A set can be well-founded, even when it has an element smaller than which there exist infinitely many
elements in the set. The lericographic order on vectors $a=\left(a_1, a_2, \ldots, a_n\right)$ and $b=\left(b_1, b_2, \ldots, b_n\right)$ is defined by the relation
$$
a<1 b \Longleftrightarrow a_1<b_1 \vee\left(a_1=b_1 \wedge\left(a_2, \ldots, a_n\right)<1\left(b_2, \ldots, b_n\right)\right) .
$$