diff --git a/main.tex b/main.tex
index 5c66397e30fc5e62d186c5943c89b5bddfaac80c..5b9c7eef507453c9bc3fe2144022f9edaca27806 100644
--- a/main.tex
+++ b/main.tex
@@ -594,7 +594,7 @@ We consider the following generalization of the Markov Chain.
 
 Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. We define a Markov Chain $\mathcal{M}_G$ as the following process: initialize every vertex of $G$ independently to 0 with probability $p$ and 1 with probability $1-p$. Then at each step, select a uniformly random vertex that has value $0$ and resample it and its neighbourhood, all of them independently with the same probability $p$. The Markov Chain terminates when all vertices have value $1$. We use $\P^{G}$ to denote probabilities associated to this Markov Chain and $\E^G$ to denote expectation values.
 
-\begin{definition}[Events] \label{def:events}
+\begin{definition}[Events and notation] \label{def:events}
     Let $S\subseteq V$ be any subset of vertices.
     \begin{itemize}
         \item Define $\Z{S}$ as the event that \emph{all} vertices in $S$ become zero at any point in time before the Markov Chain terminates.
@@ -603,43 +603,33 @@ Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. We define a Marko
             \begin{align*}
                 \P^{G}_S(A) &= \P^{G}(A \mid \text{All vertices in $S$ get initialized to }1)
             \end{align*}
-        \item Boundary $\partial$ \todo{}
-        \item $d$-Neighbourhood $B(S;d)$ \todo{}
+            The condition does not apply to subsequent resamplings of vertices in $S$, it only specifies the initial assignment.
+        \item Define the $d$-neighbourhood $B(S;d)$ of $S$ as the set of vertices reachable from $S$ within $d$ steps.
+        \item Define the boundary $\partial S$ of $S$ as the set of vertices adjacent to $S$, excluding $S$ itself. In other words $\partial S = B(S;1) \setminus S$.
     \end{itemize}
 \end{definition}
 
-\begin{wrapfigure}[7]{r}{0.25\textwidth} % The first [] argument is number of lines that are narrowed
-    \centering
-    \includegraphics{diagram_groups.pdf}
-    \caption{\label{fig:separatedgroupsGen} Lemma \ref{lemma:eventindependenceGen}.}
-\end{wrapfigure}
-The following lemma considers two vertices $v,w$ that are never ``crossed'' so that two halves of the cycle become independent.
-\begin{lemma}[Conditional independence] \label{lemma:eventindependenceGen}
-    Let $b=b_1\land b_2\in\{0,1\}^n$ be a state with two separated groups of zeroes as in Figure \ref{fig:separatedgroupsGen}. Let $v$, $w$ be any indices inbetween the groups, such that $b_1$ lies on one side of them and $b_2$ on the other, as shown in the figure. Furthermore, let $A_1$ be any event that depends only on the sites ``on the $b_1$ side of $v,w$'', and similar for $A_2$ (for example $\mathrm{Z}^{(i)}$ for an $i$ on the correct side). Then we have
+The following Lemma says that if a set $S$ splits the graph in two, then those two parts become independent if the vertices in $S$ never become zero.
+\begin{lemma}[Splitting lemma] \label{lemma:splitting}
+    \todo{Picture of $S,X,Y$.}
+    Let $G=(V,E)$ be a graph. Let $S,X,Y\subseteq V$ be a partition of the vertices, such that $X$ and $Y$ are disconnected in the graph $G\setminus S$. Furthermore, let $A^X$ and $A^Y$ be any events that depends only on the sites in $X$ and $Y$ respectively. Then we have
     \begin{align*}
-        \P^{(n)}_b(\mathrm{NZ}^{(v,w)}, A_1, A_2)
+        \P^{G}_S(\NZ{S} \cap A^X \cap A^Y)
         &=
-        \P^{(n)}_{b_1}(\mathrm{NZ}^{(v,w)}, A_1)
-        \; \cdot \;
-        \P^{(n)}_{b_2}(\mathrm{NZ}^{(v,w)}, A_2) \\
-        \P^{(n)}_b(A_1, A_2 \mid \mathrm{NZ}^{(v,w)})
-        &=
-        \P^{(n)}_{b_1}(A_1 \mid \mathrm{NZ}^{(v,w)})
+        \P^{G\setminus Y}_S(\NZ{S} \cap A^X)
         \; \cdot \;
-        \P^{(n)}_{b_2}(A_2 \mid \mathrm{NZ}^{(v,w)}) .%\\
-        %R_{b,\mathrm{NZ}^{(v,w)},A_1,A_2}
-        %&=
-        %R_{b_1,\mathrm{NZ}^{(v,w)},A_1}
-        %\; + \;
-        %R_{b_2,\mathrm{NZ}^{(v,w)},A_2}
+        \P^{G\setminus X}_S(\NZ{S} \cap A^Y)
     \end{align*}
-    %up to any order in $p$.
 \end{lemma}
 
+%\newcommand{\initone}[1]{\textsc{InitOne}_#1}
 \begin{proof}
-    From any path $\xi\in\start{b} \cap \mathrm{NZ}^{(v,w)}$ we can construct paths $\xi_1\in\start{b_1}\cap \mathrm{NZ}^{(v,w)}$ and $\xi_2\in\start{b_2}\cap\mathrm{NZ}^{(v,w)}$ as follows. Let us write the path $\xi$ as
-    $$\xi=\left( (\text{initialize }b), (z_1, s_1, r_1), (z_2, s_2, r_2), ..., (z_{|\xi|}, s_{|\xi|}, r_{|\xi|}) \right)$$
-    where $z_i\in[n]$ denotes the number of zeroes in the state before the $i$th step, $s_i\in [n]$ denotes the site that was resampled and $r_i\in \{0,1\}^3$ is the result of the three resampled bits. We have
+    We are considering three different Markov Chains and we will consider paths (i.e. resampling sequences) of them. We will use a superscript to denote to which Markov Chain a path belongs. Let $\xi^G \in \NZ{S}$ be a path of the Markov Chain associated to the resample process on the graph $G$, that satisfies the event $\NZ{S}$. 
+    From $\xi^G$ we will now construct paths $\xi^{G\setminus Y} \in \NZ{S}$ and $\xi^{G \setminus X} \in \NZ{S}$ of the other Markov Chains satisfying the corresponding events on those Markov Chains.
+    Let us write the path $\xi^G$ as an initialization and a sequence of resamplings:
+    $$\xi^G=\left( (\text{initialize to }b), (z_1, v_1, r_1), (z_2, v_2, r_2), ..., (z_{|\xi^G|}, v_{|\xi^G|}, r_{|\xi^G|}) \right)$$
+    where $1 \leq z_i \leq |V|$ denotes the number of zeroes in the state before the $i$th step, $v_i\in V$ denotes the site that was resampled and $r_i\in \{0,1\}^{d(v_i)+1}$ is the result of the resampled bits. Here $d(v_i)$ is the degree of vertex $v_i$. We have
+    \todo{from here}
     \begin{align*}
         \P^{(n)}_b[\xi] &= \P(\text{pick }s_1 | z_1) \P(r_1) \P(\text{pick }s_2 | z_2) \P(r_2) \cdots \P(\text{pick }s_{|\xi|} | z_{|\xi|}) \P(r_{|\xi|}) \\
                 &= \frac{1}{z_1} \P(r_1) \frac{1}{z_2} \P(r_2) \cdots \frac{1}{z_{|\xi|}} \P(r_{|\xi|}) .
@@ -738,7 +728,7 @@ The following lemma considers two vertices $v,w$ that are never ``crossed'' so t
         \P^{(n)}_{b} (\NZ{v,w} \cap A) = \P^{[v,w]}_b (\NZ{v,w}\cap A) .
     \end{align*}
     If vertex $v$ and $w$ never become zero, then the zeroes never get outside of the interval $[v,w]$ and we can ignore the entire circle and only focus on the process within $[v,w]$.
-    We can apply this to the result of Lemma \ref{lemma:eventindependenceGen}, to get
+    We can apply this to the result of Lemma \ref{lemma:splitting}, to get
     \begin{align*}
         \P^{(n)}_b(\mathrm{NZ}^{(v,w)} \cap A \cap B)
         &=
