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-Hello world
+\section{Object centered approach}
+We define:
+All boxes $B=\{b^0, b^1, \ldots, b^k\}$ and all targets $T=\{t^0, t^1, \ldots,
+t^k\}$.
+\subsection{Objects}
+We define the set $lex$ as being the set containing all legal $x$ values and
+the function $ley(x)$ that for a given $x$ returns the set of all legal $y$
+values for that $x$.
+
+To reduce size of the encoding we introduce a module called \texttt{ob}
+representing an object in the screen. The module contains two integer variables
+$x$ and $y$ that contains the position of the object in the field. The $x$ and
+$y$ values are bounded by the level size through their domain and restricted to
+the non-wall positions with the following invariant:
+$$\bigvee_{i\in lex}\left(x=i\wedge\bigvee_{j\in ley(x)}y=j\right)$$
+
+The initial values are determined by the values passed to it in the
+\texttt{main} module.
+
+In the \texttt{main} module we define for all boxes and the agent such an
+\texttt{ob} that stores the position of that object in that particular state.
+
+\subsection{Movement}
+We define a variable $movement=\{left,up,right,down,finished\}$ which describes
+the move in that state. From the movement variable new positions of objects are
+calculated using delta variables.
+$$\Delta_x=\left\{\begin{array}{ll}
+	-1 & movement=left\\
+	1 & movement=right\\
+	0 & \text{otherwise}
+\end{array}\right.
+\Delta_y=\left\{\begin{array}{ll}
+	-1 & movement=up\\
+	1 & movement=down\\
+	0 & \text{otherwise}
+\end{array}\right.$$
+
+The next position of the agent is defined by the following invariant:
+$$next(agent_x)=agent_x+\Delta_x\wedge
+next(agent_y)=agent_y+\Delta_x$$
+
+Determining the next position of the boxes depends on the position of the
+agent. This is described for the $x$ coordinate and analogous for the $y$
+value.
+$$next(b^i_x)=\left\{\begin{array}{ll}
+	b^i_x+\Delta_x & next(agent_x)=b^i_x \wedge next(agent_y)=b^i_y\\
+	b^i_x & \text{otherwise}
+\end{array}\right.$$
+
+To restrict the boxes to being on top of each other we define the following
+invariant:
+$$\bigwedge_{i=0}^{k-1}\bigwedge_{j=i+1}^k\neg(b^i_x=b^j_x \wedge b^i_y=b^j_y)
+$$
+
+\subsection{Goal}
+When the goal state is reached the $movement$ variable should be set to
+$finished$ to be able to specify the CTL specification more easily. Via the
+following invariant the goal state is tied to the $movement$ variable:
+$$\left(\bigvee_{B`\in perms(B)}
+	\bigwedge_{b\in B`, i\in 0\ldots k}\left(
+		b_x=t^i_x \wedge b_y=t^i_y
+\right)\right)\leftrightarrow movement=finished$$
+
+And a counterexample for the following CTL specification can then be seen as an
+example solution:
+$$\nexists\lozenge(movement=finished)$$