From: tkwa <kwathomas0@gmail.com>
Date: Mon, 1 Aug 2016 20:38:16 +0000 (-0700)
Subject: Added simple diagram for lemma in proof
X-Git-Url: http://plrg.eecs.uci.edu/git/?p=iotcloud.git;a=commitdiff_plain;h=fc4b68411ca2e501708166c061b1bb4f896ad4a0;hp=6f2c3c4e73a56ce088dcbf9cac647ec4f827bf89;ds=sidebyside

Added simple diagram for lemma in proof
---

diff --git a/doc/iotcloud.tex b/doc/iotcloud.tex
index 143791d..250ddcf 100644
--- a/doc/iotcloud.tex
+++ b/doc/iotcloud.tex
@@ -6,6 +6,7 @@
 \usepackage{graphicx}
 \usepackage{mathrsfs}
 \usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx
+\usepackage[all]{xy}
 \newtheorem{theorem}{Theorem}
 \newtheorem{prop}{Proposition}
 \newtheorem{lem}{Lemma}
@@ -685,6 +686,14 @@ $\tuple{ck,\tuple{k, v}} \in KV_s \wedge
 
 \end{lem}
 
+\begin{figure}[h]
+  \centering
+      \xymatrix{
+\dots \ar[r] & q \ar[dr]_{J} \ar[r]^{K} & S_1 \ar[r] & S_2 \ar[rr] & & \dots \ar[r] & S_n = u \\
+& & R_1 \ar[r] & R_2 \ar[r] & \dots \ar[r] & R_m = t}
+\caption{By Lemma 2, receiving $t$ before $u$ is impossible.}
+\end{figure}
+
 \begin{lem} If two packets $t$ and $u$, with $i(t) \le i(u)$, are received without errors by a client $C$, then $t$ is in the path of $u$. \end{lem}
 \begin{proof}
 Assume that $t$ is not in the path of $u$. Take $u$ to be the packet of smallest index for which this occurs, and $t$ be the packet with largest index for this $u$. We will prove that an error occurs upon receipt of $u$.