From: rtrimana <rtrimana@uci.edu>
Date: Tue, 16 Aug 2016 22:37:32 +0000 (-0700)
Subject: Cleaning up proof part
X-Git-Url: http://plrg.eecs.uci.edu/git/?p=iotcloud.git;a=commitdiff_plain;h=b3268a5e142973ff90796c2afab456b5a22fce4c

Cleaning up proof part
---

diff --git a/doc/iotcloud.tex b/doc/iotcloud.tex
index 52ec3f0..a0e8abe 100644
--- a/doc/iotcloud.tex
+++ b/doc/iotcloud.tex
@@ -929,12 +929,12 @@ received the same message $\mathsf{t}$ with sequence number $\mathsf{s_t > s_n}$
 
 \begin{prop} 
 Every client $\mathsf{J}$ who sends a message $\mathsf{t}$ 
-has $\mathsf{\mathscr{P}_t}$ as its latest stored message, and 
-$\mathsf{s_t = s_{\mathscr{P}_t} + 1}$. 
+has parent $\mathsf{p_t}$ as its latest stored message, and 
+$\mathsf{s_t = s_{p_t} + 1}$. 
 \end{prop}
 \begin{proof} True by definition, because $J$ sets 
-$\mathsf{hmac_p(t) = hmac_c(\mathscr{P}_t)}$ and 
-$\mathsf{s_t = }$ $\mathsf{s_{\mathscr{P}_t + 1}}$ when a message 
+$\mathsf{hmac_p(t) = hmac_c(p_t)}$ and 
+$\mathsf{s_t = }$ $\mathsf{s_{p_t + 1}}$ when a message 
 is sent. 
 \end{proof}
 
@@ -969,10 +969,12 @@ call them $\mathsf{t}$ and $\mathsf{u}$ such that $\mathsf{s_t \le s_u}$.
 Then $\mathsf{t}$ is in the path of $\mathsf{u}$. 
 \end{lem}
 \begin{proof}
-Clearly $\mathsf{C}$ will throw an error if $\mathsf{s_t = s_u}$. So 
-$\mathsf{s_t < s_u}$. Additionally, if $\mathsf{C}$ receives $\mathsf{u}$ before 
-$\mathsf{t}$, this will cause it to throw an error, so $\mathsf{t}$ is received 
-before $\mathsf{u}$.
+By contradiction, we will prove that if $\mathsf{t}$ is not in the path of 
+$\mathsf{u}$, then it is impossible for client $\mathsf{C}$ to receive both
+messages without throwing any errors. Clearly $\mathsf{C}$ will throw an error 
+if $\mathsf{s_t = s_u}$. So $\mathsf{s_t < s_u}$. Additionally, if $\mathsf{C}$ 
+receives $\mathsf{u}$ before $\mathsf{t}$, this will cause it to throw an 
+error, so $\mathsf{t}$ is received before $\mathsf{u}$.
 
 Assume that $\mathsf{t}$ is not in the path of $\mathsf{u}$. Take $\mathsf{u}$ 
 to be the packet of smallest sequence number for which this occurs, and 
@@ -993,22 +995,28 @@ $\mathsf{{id_{self}}_J = GetMacID(r_1)}$, and $\mathsf{K}$ be the client who
 sent $\mathsf{l_1}$. Because no client can send two messages with the same sequence number, and 
 $\mathsf{s_{r_1} = s_{l_1} = s_q + 1}$, $\mathsf{J \neq K}$.
 
-We also know the following three facts: 
+We also know the following facts: 
 
 \begin{prop} No client sends both a message in 
-$\mathsf{(r_2,...,t)}$ and a message in $\mathsf{(l_2,...,u)}$. \end{prop}
+$\mathsf{(r_2,...,t)}$ and a message in $\mathsf{(l_2,...,u)}$. 
+\end{prop}
 
 \begin{proof}
 To send a message $\mathsf{p}$ that is the parent of some other 
 message, one must have received the parent of $\mathsf{p}$. Since 
 $\mathsf{u}$ is the message with smallest sequence number received by any 
 client that violates Lemma 1, no client receives both a message in $\mathsf{r}$ 
-and a message in $\mathsf{l}$. \end{proof}
+and a message in $\mathsf{l}$. 
+\end{proof}
 
-\begin{prop} $\mathsf{C}$ does not receive any message with 
-sequence number strictly between $\mathsf{s_t}$ and $\mathsf{s_u}$. \end{prop}
+\begin{prop} $\mathsf{C}$ does not receive any message with a
+sequence number strictly between $\mathsf{s_t}$ and $\mathsf{s_u}$. 
+\end{prop}
 
-\begin{proof} If there were such a message with sequence number smaller than $\mathsf{s_u}$, it would contradict the assumption that $\mathsf{u}$ is the message with least sequence number that violates Lemma 1. \end{proof}
+\begin{proof} If there were such a message with sequence number smaller than 
+$\mathsf{s_u}$, it would contradict the assumption that $\mathsf{u}$ is the 
+message with the least sequence number that violates Lemma 1. 
+\end{proof}
 
 There are two cases:
 \begin{itemize}