update code'

git pus"g

diff --git a/a2/1.tex b/a2/1.tex
index b294ecc..94a01cb 100644 (file)
--- a/a2/1.tex
+++ b/a2/1.tex
@@ -1,5 +1,5 @@
 \section{Problem 1}
 \section{Problem 1}

-{\em Three non-self-supporting villages $A$, $B$ and $C$ in the middle of
+\emph{Three non-self-supporting villages $A$, $B$ and $C$ in the middle of

 nowhere consume one food package each per time unit. The required food packages
 are delivered by a truck, having a capacity of $300$ food packages. The
 locations of the villages are given in the following picture, in which the
 nowhere consume one food package each per time unit. The required food packages
 are delivered by a truck, having a capacity of $300$ food packages. The
 locations of the villages are given in the following picture, in which the


@@ -19,28 +19,30 @@ city $B$, city $C$, the truck, the trucks location and a help variable to
 record how much food is dumped in the city. The distance between two cities is
 described with the function $dist(x, y)$. For all cities except $S$ a variable
 is defined that states the maximum capacity. 
 record how much food is dumped in the city. The distance between two cities is
 described with the function $dist(x, y)$. For all cities except $S$ a variable
 is defined that states the maximum capacity. 

-$$a_{max}=120\wedge b_{max}=120\wedge c_{max}=200$$
+$$a_{\max}=120\wedge b_{\max}=120\wedge c_{\max}=200$$

 Iteration $0$ depicts the initial state and can be described with a simple
 conjuction:
 Iteration $0$ depicts the initial state and can be described with a simple
 conjuction:

-$$a_0=40\wedge b_0=30\wedge c_0=145\wedge t_0=300\wedge l_0=S\wedge d_0=0$$.
+$$a_0=40\wedge b_0=30\wedge c_0=145\wedge t_0=300\wedge l_0=S\wedge d_0=0$$

 
 After that all iteration can be formalized as
 
 After that all iteration can be formalized as

-\begin{align}
+$$\begin{array}{lrl}

        \bigwedge_{i\in I} \Bigg(
        % Bent in S
        \bigwedge_{i\in I} \Bigg(
        % Bent in S

-               & \Bigg[l_i=S\wedge d_i=0\wedge t_i=300\wedge
-               \bigvee_{k\in\{A,B\}}\bigg(l_{i-1}=k\wedge
-               \bigwedge_{s\in\{A,B,C\}}s_i=s_{i-1}-dist(S, k)\bigg)\Bigg]\vee\\
+       & \Bigg[ & l_i=S\wedge d_i=0\wedge t_i=300\wedge\\
+       & & \bigvee_{k\in\{A,B\}}
+               \bigg(l_{i-1}=k\wedge
+               \bigwedge_{s\in\{A,B,C\}}
+                       s_i=s_{i-1}-dist(S, k)\bigg)\Bigg]\vee\\

        % Bent in A,B,C
        % Bent in A,B,C

-               & \bigvee_{k\in\{a,b,c\}}
-                       \Bigg[l_i=k\wedge d_i>0\wedge d_i<t_{i-1}\wedge d_i<k_{max}-k_{i-1}
-                       t_i=t_i-d_i\wedge k_i=k_i+d_i\wedge\\
-               & \qquad\qquad\bigwedge_{k'\in\{k\}\setminus\{S,A,B,C\}}\bigg(
-               l_{i-1}=k'\wedge
+       & \bigvee_{k\in\{a,b,c\}}\Bigg[ & l_i=k\wedge\\
+       & & d_i>0\wedge d_i<t_{i-1}\wedge d_i<k_{\max}-k_{i-1}\wedge\\
+       & & t_i=t_{i-1}-d_i\wedge k_i=k_{i-1}+d_i\wedge\\
+       & & \bigwedge_{k'\in\{k\}\setminus\{S,A,B,C\}}
+               \bigg(l_{i-1}=k'\wedge

                \bigwedge_{s\in\{k\}\setminus\{A,B,C\}}
                \bigwedge_{s\in\{k\}\setminus\{A,B,C\}}

-                       s_i=s_{i-1}-dist(k, s)\bigg)\Bigg]\\
-       \nonumber\Bigg)
-\end{align}
+                       s_i=s_{i-1}-dist(k, k')\bigg)\Bigg]\\
+       & \Bigg)
+\end{array}$$

 
 \begin{enumerate}
        \item This part of the formula describes what happens if the truck is in
 
 \begin{enumerate}
        \item This part of the formula describes what happens if the truck is in


@@ -52,14 +54,14 @@ After that all iteration can be formalized as
 \end{enumerate}
 
 \begin{enumerate}[a.]
 \end{enumerate}
 
 \begin{enumerate}[a.]

-       \item {\em Show that it is impossible to deliver food packages in such a
+       \item \emph{Show that it is impossible to deliver food packages in such a

                way that each of the villages consumes one food package per time unit
                forever.}
                way that each of the villages consumes one food package per time unit
                forever.}

-       \item {\em Show that this is possible if the capacity of the truck is
+       \item \emph{Show that this is possible if the capacity of the truck is

                increased to 320 food packages. (Note that a finite graph contains an
                infinite path starting in a node $v$ if and only if there is a path
                from $v$ to a node $w$ for which there is a non-empty path from $w$ to
                itself.)}
                increased to 320 food packages. (Note that a finite graph contains an
                infinite path starting in a node $v$ if and only if there is a path
                from $v$ to a node $w$ for which there is a non-empty path from $w$ to
                itself.)}

-       \item {\em Figure out whether it is possible if the capacity of the truck
+       \item \emph{Figure out whether it is possible if the capacity of the truck

                is set to $318$.}
 \end{enumerate}
                is set to $318$.}
 \end{enumerate}