From b554f1fbc4435df678b9f5eeaf284ec6f451ce49 2017-09-08 17:09:07
From: Andras Gilyen <gilyen@clayoquot.swat.cwi.nl>
Date: 2017-09-08 17:09:07
Subject: [PATCH] Incr.

---

diff --git a/main.tex b/main.tex
index 6240f3d608ec1b934801be6c261eabcb61569b99..ff16acc0fb35a77cad5a0a42b7d3e5b92c9f6723 100644
--- a/main.tex
+++ b/main.tex
@@ -62,6 +62,8 @@
 \newcommand{\paths}[1]{\mathcal{P}\left(#1\to\mathbf{1}\right)}
 \newcommand{\start}[1]{\textsc{start}\left(#1\right)}
 \newcommand{\initone}[1]{\textsc{InitOne}\left(#1\right)}
+\newcommand{\patch}[1]{\textsc{Patch}\left(#1\right)}
+\newcommand{\patches}[1]{\textsc{Patches}\left(#1\right)}
 \newcommand{\maxgap}[1]{\mathrm{maxgap}\left(#1\right)}
 \newcommand{\gaps}[1]{#1_{\mathrm{gaps}}}
 \renewcommand{\P}{\mathbb{P}}
@@ -794,12 +796,12 @@ The following lemma considers two vertices $v,w$ that are never ``crossed'' so t
 \end{proof}
 
 	\begin{definition}[Connected patches]
-		Let $P\subseteq V$ be a connected component of $G$. We say that $P$ is a patch of a particular run of the process if $P$ is a maximal connected component of the vertices that have ever become $0$ before termination. We denote the set of patches of a run by $\mathcal{P}$. For a patch $P$ let $P\in \mathcal{P}$ denote the event that one of the patches is equal to $P$. 
+		Let $P\subseteq V$ be a connected component of $G$. We say that $P$ is a patch of a particular run of the process if $P$ is a maximal connected component of the vertices that have ever become $0$ before termination. We denote the set of patches of a run by $\mathcal{P}$. For a patch $P$ let $\patch{P}$ denote the event that one of the patches is equal to $P$. 
 		In other words
 		\begin{align*}
-		P\in\mathcal{P} := \NZ{\overline{\partial}P} \cap \Z{P}.
+		\patch{P} := \NZ{\overline{\partial}P} \cap \Z{P}.
 		\end{align*}
-		For $\mathcal{I}'\subseteq 2^{2^V}$ a set of patches we denote by $\mathcal{P}'\in \mathcal{P}$ the event that $\mathcal{P}'$ is a subset of the patches, i.e.,
+		For a set of patches $\mathcal{P}$ 	
 		\begin{align*}
 			\mathcal{P}'\in \mathcal{P} := \bigcup_{P\in \mathcal{P}'}\NZ{\overline{\partial}P} \cap \Z{P}.
 		\end{align*}