From: tkwa 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 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\$.