\subsection*{1.a}
\emph{Consider the train gate controller example of slide 9, lecture
\emph{Timed Automata}. Give an example of a series of timed transitions
(including intermediate states) of the composed system (so the product
construction of the three automata) showing a train approaching and finally
leaving the gate.}
\begin{figure}[h]
	\centering
	\includegraphics[width=.3\linewidth]{1a}
	\caption{Timed transitions of the composed system}
\end{figure}

\subsection*{1.b}
\emph{Consider the timed automaton in figure 1 of the paper ”Timed Automata” by
Rajeev Alur. Suppose initially we have a zone $(s0, [0\leq x\leq 4, 0\leq y
\leq 3])$. Give the zone after a sequence a.b and show the intermediate steps
in the derivation.}

We define:\\
$e_0=\langle s_0, a, [], [x:=0] s_1\rangle$ and\\
$e_1=\langle s_1, a, [], [y:=0] s_2\rangle$\\

And we derive:\\
\begin{align*}
	succ(e_1, succ(e_0, [0\leq x\leq 4, 0\leq y])) =&
		succ(e_1, ((([0\leq x\leq 4, 0\leq y]\wedge [])\Uparrow)\wedge []\wedge [x<1])[x:=0])\\
	=& succ(e_1, (([0\leq x\leq 4, 0\leq y]\Uparrow)\wedge []\wedge [x<1])[x:=0])\\
	=& succ(e_1, ([0\leq x<1, 0\leq y])[x:=0])\\
	=& succ(e_1, [x=1, 0\leq y][x:=0])\\
	=& ((([x=1, 0\leq y]\wedge [x<1])\Uparrow)\wedge [x<1]\wedge [x<1])[y:=0]\\
	=& (([x=1, 0\leq y]\Uparrow)\wedge [x<1]\wedge [x<1])[y:=0]\\
	=& [x<1, y=0]
\end{align*}

\subsection*{1.c}
\emph{Consider the timed automaton in figure 1 of the paper \emph{Timed
Automata} by Rajeev Alur. Give the zone automaton of the timed automaton, with
initial state $(s0, [x = 0, y = 0])$.}
