A message $\mathsf{t}$, is the tuple \r
\begin{center}\r
$\mathsf{t = \tuple{s, E(Dat_s)}}$ \\\r
-$\mathsf{Dat_s = \tuple{s,id,hmac_p, DE,hmac_c}}$\r
+$\mathsf{Dat_t = \tuple{s,id,hmac_p, DE,hmac_c}}$\r
\end{center}\r
-containing $\mathsf{s}$ as sequence number and $\mathsf{Dat_s}$ as its \r
-encrypted contents. $\mathsf{Dat_s}$ consists of $\mathsf{s}$, \r
+containing $\mathsf{s}$ as sequence number and $\mathsf{Dat_t}$ as its \r
+encrypted contents. $\mathsf{Dat_t}$ consists of $\mathsf{s}$, \r
$\mathsf{id}$ as machine ID of the sender, $\mathsf{hmac_p}$ as HMAC \r
from a previous message, $\mathsf{DE}$ as set of data entries, and \r
$\mathsf{hmac_c}$ as HMAC from message $\mathsf{t}$ respectively.\r
\r
\begin{defn}[Equality]\r
Two messages $\mathsf{t}$ and $\mathsf{u}$ are equal if their $\mathsf{s}$, \r
-and $\mathsf{Dat_s}$ are exactly the same.\r
+and $\mathsf{Dat_t}$ are exactly the same.\r
\end{defn}\r
\r
\begin{defn}[Parent]\r
-A parent of a message $\mathsf{t}$ is the message $\mathsf{A(t)}$, unique \r
-by the correctness of HMACs in $\mathsf{Dat_s}$, such that \r
-$\mathsf{hmac_p(t) = hmac_c(A(t))}$.\r
+A parent of a message $\mathsf{t}$ is the message $\mathsf{\mathscr{P}_t}$, \r
+unique by the correctness of HMACs in $\mathsf{Dat_t}$, such that \r
+$\mathsf{hmac_p(t) = hmac_c(\mathscr{P}_t)}$.\r
\end{defn}\r
\r
\begin{defn}[Chain]\r
A chain of messages with length $\mathsf{n \ge 1}$ is a message sequence \r
-$\mathsf{R = (r_i, r_{i+1}, ..., r_{i+n-1})}$ such that for every index \r
-$\mathsf{i < k \le i+n-1}$, $\mathsf{r_k}$ has sequence number $\mathsf{k}$ \r
-and is the parent of $\mathsf{r_{k-1}}$.\r
+$\mathsf{R = (r_s, r_{s+1}, ..., r_{s+n-1})}$ such that for every sequence \r
+number $\mathsf{s < k \le s+n-1}$, $\mathsf{r_k}$ has sequence number \r
+$\mathsf{k}$ and is the parent of $\mathsf{r_{k-1}}$.\r
\end{defn}\r
\r
\begin{defn}[Partial sequence]\r
\r
\begin{defn}[Total sequence]\r
A total sequence $\mathsf{T =}$ $\mathsf{(t_1, t_2, ..., t_n)}$ with \r
-length $\mathsf{n}$ is a chain of messages that starts at $\mathsf{i = 1}$.\r
+length $\mathsf{n}$ is a chain of messages that starts at $\mathsf{s = 1}$.\r
\end{defn}\r
\r
\begin{defn}[Path]\r
-The path of a message $\mathsf{t}$ is the chain that starts at $\mathsf{i = 1}$ \r
+The path of a message $\mathsf{t}$ is the chain that starts at $\mathsf{s = 1}$ \r
and whose last message is $\mathsf{t}$. The uniqueness of a path follows \r
from the uniqueness of a parent.\r
\end{defn}\r
\begin{defn}[Consistency]\r
A partial sequence $\mathsf{P}$ is consistent with a total sequence \r
$\mathsf{T}$ of length $\mathsf{n}$ if for every message $\mathsf{p \in P}$ \r
-with $\mathsf{i(p) \leq n}$, $\mathsf{t_{i(p)} = p}$. This implies that \r
-$\mathsf{\{p \in P | i(p) \le n\}}$ is a partial sequence of $\mathsf{T}$.\r
+with $\mathsf{s_p \leq n}$, $\mathsf{t_{s_p} = p}$. This implies that \r
+$\mathsf{\{p \in P | s_p \le n\}}$ is a partial sequence of $\mathsf{T}$.\r
\end{defn}\r
\r
\begin{defn}[Transitive closure]\r
-Transitive closure set at index $\mathsf{n}$ is a set $\mathscr{S}$ of \r
-clients comprising a connected component of an undirected graph, where two \r
-clients are connected by an edge if they both received the same message \r
-$\mathsf{t}$ with index $\mathsf{i(t) > n}$.\r
+Transitive closure set at sequence number $\mathsf{s_n}$ is a set \r
+$\mathsf{\mathscr{S}}$ of clients comprising a connected component of an \r
+undirected graph, where two clients are connected by an edge if they both \r
+received the same message $\mathsf{t}$ with sequence number $\mathsf{s_t > n}$.\r
\end{defn}\r
\r
-%\begin{enumerate}\r
-%\item Equality: Two messages $t$ and $u$ are equal if their sequence numbers, senders, and contents are exactly the same.\r
-%\item Message: A message $t$, is the tuple $t = (i(t), s(t), contents(t))$ containing the sequence number, machine ID of the sender, and contents of $t$ respectively.\r
-%\item Parent: A parent of a message $t$ is the message $A(t)$, unique by the correctness of HMACs, such that $HMAC_C(t) = HMAC_P(A(t))$.\r
-%\item Chain: A chain of messages with length $n \ge 1$ is a message sequence $R = (R_i, R_{i+1}, ..., R_{i+n-1})$ such that for every index $i < k \le i+n-1$, $R_k$ has sequence number $k$ and is the parent of $R_{k-1}$.\r
-%\item Partial message sequence: A partial message sequence is a sequence of messages, no two with the same sequence number, that can be divided into disjoint chains.\r
-%\item Total message sequence: A total message sequence $T$ with length $n$ is a chain of messages that starts at $i = 1$.\r
-%\item Path: The path of a message $t$ is the chain that starts at $i = 1$ and whose last message is $t$. The uniqueness of a path follows from the uniqueness of a parent.\r
-%\item Consistency: A partial message sequence $P$ is consistent with a total message sequence $T$ of length $n$ if for every message $p \in P$ with $i(p) < n$, $T_{i(p)} = p$. This implies that $\{p \in P | i(p) \le n\}$ is a subsequence of T.\r
-%\item Transitive closure set at index $n$: A set $\mathscr{S}$ of clients comprising a connected component of an undirected graph, where two clients are connected by an edge if they both received the same message $t$ with index $i(t) > n$.\r
-%\end{enumerate}\r
-\r
\subsubsection{Lemmas and Proofs}\r
\r
-\begin{prop} Every client $J$ who sends a message $t$ has $A(t)$ as its latest stored message, and $i(t) = i(A(t)) + 1$. \end{prop}\r
-\begin{proof} True by definition, because $J$ sets $HMAC_P(t) = HMAC_C(A(t))$ and $i(t) = i(A(t)) + 1$ when a message is sent. \end{proof}\r
+\begin{prop} \r
+Every client $\mathsf{J}$ who sends a message $\mathsf{t}$ \r
+has $\mathsf{\mathscr{P}_t}$ as its latest stored message, and \r
+$\mathsf{s_t = s_{\mathscr{P}_t} + 1}$. \r
+\end{prop}\r
+\begin{proof} True by definition, because $J$ sets \r
+$\mathsf{hmac_p(t) = hmac_c(\mathscr{P}_t)}$ and \r
+$\mathsf{s_t = }$ $\mathsf{s_{\mathscr{P}_t + 1}}$ when a message \r
+is sent. \r
+\end{proof}\r
\r
-\begin{prop} If a rejected message entry is added to the RML at index $i$, the message will remain in the RML until every client has seen it. \end{prop}\r
-\begin{proof} Every RML entry $e$ remains in the queue until it reaches the tail, and is refreshed by the next sender $J$ at that time if $min(MS) > i(e)$; that is, until every client has sent a message with sequence number greater than $i(e)$. Because every client who sends a message with index $i$ has the state of the queue at $i - 1$, this client will have seen the message at $i(e)$. \end{proof}\r
+\begin{prop} If a rejected message entry is added to the set $\mathsf{CR}$ \r
+at sequence number $\mathsf{s}$, the message will remain in $\mathsf{CR}$ \r
+until every client has seen it. \r
+\end{prop}\r
+\begin{proof} Every $\mathsf{CR}$ entry $\mathsf{cr}$ remains in the queue until it \r
+reaches the tail, and is refreshed by the next sender $\mathsf{J}$ at that time if \r
+$\mathsf{min(MS) > s_{cr}}$; that is, until every client has sent a message with \r
+sequence number greater than $\mathsf{s_{cr}}$. Because every client who sends a \r
+message with sequence number $\mathsf{s}$ has the state of the set $\mathsf{SL}$ at \r
+$\mathsf{s - 1}$, this client will have seen the message at $\mathsf{s_{cr}}$. \r
+\end{proof}\r
\r
\begin{figure}[h]\r
\centering\r
- \xymatrix{ & & S \\\r
-\dots \ar[r] & q \ar[dr]_{J} \ar[r]^{K} & S_1 \ar[r] & S_2 \ar[r] & \dots \ar[r] & p \ar[r] & \dots \ar[r] & S_n = u \\\r
-& & R_1 \ar[r] & R_2 \ar[r] & \dots \ar[r] & R_m = t \\\r
+ \xymatrix{ & & L \\\r
+\dots \ar[r] & q \ar[dr]_{J} \ar[r]^{K} & l_1 \ar[r] & l_2 \ar[r] & \dots \ar[r] & p \ar[r] & \dots \ar[r] & l_n = u \\\r
+& & r_1 \ar[r] & r_2 \ar[r] & \dots \ar[r] & r_m = t \\\r
& & R\r
\save "2,3"."2,8"*+\frm{^\}}\r
\save "3,3"."3,6"*+\frm{_\}}\r
\restore\r
\restore\r
}\r
-\caption{By Lemma 1, receiving $u$ after $t$ is impossible.}\r
+\caption{By Lemma 1, receiving $t$ after $u$ is impossible.}\r
\end{figure}\r
\r
-\begin{lem} Two messages are received without errors by a client $C$; call them $t$ and $u$ such that $i(t) \le i(u)$. Then $t$ is in the path of $u$. \end{lem}\r
+\begin{lem} Two messages are received without errors by a client $\mathsf{C}$; \r
+call them $\mathsf{t}$ and $\mathsf{u}$ such that $\mathsf{s_t \le s_u}$. \r
+Then $\mathsf{t}$ is in the path of $\mathsf{u}$. \r
+\end{lem}\r
\begin{proof}\r
-Clearly $C$ will throw an error if $i(t) = i(u)$. So $i(t) < i(u)$. Additionally, if $C$ receives $u$ before $t$, this will cause it to throw an error, so $t$ is received before $u$.\r
-\r
-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 greatest index for this $u$. We will prove that an error occurs upon receipt of $u$.\r
-\r
-Let $R_1$ be the earliest member of the path of $t$ that is not in the path of $u$, and $q$ be its parent. $q$, the last common ancestor of $t$ and $u$, must exist, since all clients and the server were initialized with the same state. Let $S_1$ be the successor of $q$ that is in the path of $u$; we know $S_1 \neq R_1$. Let $R = (R_1, R_2, \dots, R_m = t)$ be the distinct portion of the path of $t$, and similarly let $S$ be the distinct portion of the path of $S_n = u$.\r
-\r
-Let $J$ be the client who sent $R_1$; that is, such that ${id_self}_J = GetMacID(R_1)$, and $K$ be the client who sent $S_1$.\r
+Clearly $\mathsf{C}$ will throw an error if $\mathsf{s_t = s_u}$. So \r
+$\mathsf{s_t < s_u}$. Additionally, if $\mathsf{C}$ receives $\mathsf{u}$ before \r
+$\mathsf{t}$, this will cause it to throw an error, so $\mathsf{t}$ is received \r
+before $\mathsf{u}$.\r
+\r
+Assume that $\mathsf{t}$ is not in the path of $\mathsf{u}$. Take $\mathsf{u}$ \r
+to be the packet of smallest sequence number for which this occurs, and \r
+$\mathsf{t}$ be the packet with greatest sequence number for this $\mathsf{u}$. \r
+We will prove that an error occurs upon receipt of $\mathsf{u}$.\r
+\r
+Let $\mathsf{r_1}$ be the earliest member of the path of $\mathsf{t}$ that is \r
+not in the path of $\mathsf{u}$, and $\mathsf{q}$ be its parent. Message \r
+$\mathsf{q}$, the last common ancestor of $\mathsf{t}$ and $\mathsf{u}$, must exist, \r
+since all clients and the server were initialized with the same state. Let \r
+$\mathsf{l_1}$ be the successor of $\mathsf{q}$ that is in the path of $\mathsf{u}$; \r
+we know $\mathsf{l_1 \neq r_1}$. Let $\mathsf{R = (r_1, r_2, \dots, r_m = t)}$ be \r
+the distinct portion of the path of $\mathsf{t}$, and similarly let $\mathsf{L}$ \r
+be the distinct portion of the path of $\mathsf{l_n = u}$.\r
+\r
+Let $\mathsf{J}$ be the client who sent $\mathsf{r_1}$; that is, such that \r
+$\mathsf{{id_{self}}_J = GetMacID(r_1)}$, and $\mathsf{K}$ be the client who \r
+sent $\mathsf{l_1}$.\r
\r
We know the following three facts: \r
\r
\begin{enumerate}\r
\r
-\item Because no client can send two messages with the same index, and $i(R_1) = i(S_1) = i(q) + 1$, $J \neq K$.\r
+\item Because no client can send two messages with the same sequence number, and \r
+$\mathsf{s_{r_1} = s_{l_1} = s_q + 1}$, $\mathsf{J \neq K}$.\r
\r
-\item To send a message $p$ that is the parent of some other message, one must have the received the parent of $p$. Since $u$ is the message with smallest sequence number received by any client that violate this lemma, no client receives both a message in $R$ and a message in $S$; therefore, no client sends both a message in $(R_2,...,t)$ and a message in $(S_2,...,u)$.\r
+\item To send a message $\mathsf{p}$ that is the parent of some other \r
+message, one must have received the parent of $\mathsf{p}$. Since \r
+$\mathsf{u}$ is the message with smallest sequence number received by any \r
+client that violates this lemma, no client receives both a message in $\mathsf{r}$ \r
+and a message in $\mathsf{l}$; therefore, no client sends both a message in \r
+$\mathsf{(r_2,...,t)}$ and a message in $\mathsf{(l_2,...,u)}$.\r
\r
-\item Since $u$ are the greatest- and least- sequence number messages that violate this lemma, $C$ does not receive any message with sequence number strictly between $i(t)$ and $i(u)$. Because the $s_{last}$ that $C$ stores increases at every message receipt event, $C$ also does not receive any message after $t$ and before $u$ in real time.\r
+\item Since $\mathsf{u}$ is the message with the greatest- and least- sequence \r
+number that violates this lemma, $\mathsf{C}$ does not receive any message with \r
+sequence number strictly between $\mathsf{s_t}$ and $\mathsf{s_u}$. Because the \r
+$\mathsf{s_{last}}$ that $\mathsf{C}$ stores increases at every message receipt \r
+event, $\mathsf{C}$ also does not receive any message after $\mathsf{t}$ and before \r
+$\mathsf{u}$ in real time.\r
\r
\end{enumerate}\r
-\r
There are two cases:\r
-\r
\begin{itemize}\r
-\item Case 1: $J$ did not send a message in $S$. Then $v_J(t) > v_J(q) = v_J(u)$.\r
+\item Case 1: $\mathsf{J}$ did not send a message in $\mathsf{L}$. Then \r
+%$\mathsf{v_J(t) > v_J(q) = v_J(u)}$.\r
+$\mathsf{s_{t_J} > s_{q_J} = s_{u_J}}$.\r
\begin{itemize}\r
-\item Case 1.1: $C$ never updates its slot sequence list between receiving $t$ and receiving $u$; this can only happen if $i(t) = i(u) - 1$. Since $t$ is not the parent of $u$, $HMAC_p(u) \neq HMAC_c(t)$, causing $C$ to error.\r
-\item Case 1.2: Case 1.1 does not occur; therefore, $C$ must update its slot sequence list at some point between receiving $t$ and $u$. The latest index of $J$ decreases during this time, which means it must decrease when some message is received, which means C throws an error in the $CheckLastSeqN$ subroutine.\r
+\item Case 1.1: $\mathsf{C}$ never updates its slot sequence list $\mathsf{SS}$ \r
+between receiving $\mathsf{t}$ and receiving $\mathsf{u}$; this can only happen if \r
+$\mathsf{s_t = s_u - 1}$. Since $\mathsf{t}$ is not the parent of $\mathsf{u}$, \r
+$\mathsf{hmac_p(u) \neq hmac_c(t)}$, causing $\mathsf{C}$ to error.\r
+\item Case 1.2: Case 1.1 does not occur; therefore, $\mathsf{C}$ must update \r
+its slot sequence list $\mathsf{SS}$ at some point between receiving $\mathsf{t}$ \r
+and $\mathsf{u}$. \r
+The latest sequence number of $\mathsf{J}$ decreases during this time, which \r
+means it must decrease when some message is received, which means $\mathsf{C}$ \r
+throws an error in the $\mathsf{CheckLastSeqN}$ subroutine.\r
\end{itemize}\r
\r
-\item Case 2: $J$ sent at least one message in $S$. Call the first one $p$. We know that $i(p) > i(R_1)$, since $J \neq K$ and $p \neq S_1$. $R_1$ must be sent either before or after $p$.\r
+\item Case 2: $\mathsf{J}$ sent at least one message in $\mathsf{L}$. Call the \r
+first one $\mathsf{p}$. We know that $\mathsf{s_p > s_{r_1}}$, since \r
+$\mathsf{J \neq K}$ and $\mathsf{p \neq l_1}$. Message $\mathsf{r_1}$ must be sent \r
+either before or after $\mathsf{p}$.\r
\begin{itemize}\r
-\item Case 2.1: Client $J$ sends $p$, and then $R_1$. Before sending $p$, the greatest sequence number of a message that $J$ has received, ${s_{last}}_j$, must be equal to $i(p) - 1 \ge i(R_1)$. Since ${s_{last}}_j$ never decreases, Client $J$ cannot then send a message with sequence number $i(R_1)$, a contradiction.\r
-\item Case 2.2: Client $J$ sends $R_1$, and then $p$. Let $X = (R_1, \dots )$ be the list of messages $J$ sends starting before $R_1$ and ending before $p$.\r
+\item Case 2.1: Client $\mathsf{J}$ sends $\mathsf{p}$, and then $\mathsf{r_1}$. \r
+Before sending $\mathsf{p}$, the greatest sequence number of a message that \r
+$\mathsf{J}$ has received, $\mathsf{{s_{last}}_J}$, must be equal to \r
+$\mathsf{s_p - 1 \ge s_{r_1}}$. Since $\mathsf{{s_{last}}_J}$ never decreases, \r
+client $\mathsf{J}$ cannot then send a message with sequence number \r
+$\mathsf{s_{r_1}}$, a contradiction.\r
+\item Case 2.2: Client $\mathsf{J}$ sends $\mathsf{r_1}$, and then $\mathsf{p}$. \r
+Let $\mathsf{X = (r_1, \dots )}$ be the list of messages $\mathsf{J}$ sends \r
+starting before $\mathsf{r_1}$ and ending before $p$.\r
\begin{itemize}\r
-\item Case 2.2.1: Some message in $X$ was accepted. Let $v_J(w)$ be the greatest sequence number of the messages that client $J$ sent in the path of message $w$. In this case, before sending $p$, $J$'s value $SM_J[J]$ for its own latest index would be strictly greater than $v_J(q)$. If there is a sequence of messages with contiguous sequence numbers that $J$ receives between $R_1$ and $p$, J throws an error for a similar reason as Case 1.1. Otherwise, when preparing to send $p$, $J$ would have received an update of its own latest index as at most $v_J(q)$. $J$ throws an error before sending $p$, because its own latest index decreases.\r
-\item Case 2.2.2: All messages in $X$ were rejected, making $p$ the first message of $J$ that is accepted after $R_1$. Note that $u$ is the message with least sequence number that violates this lemma, and therefore the first one that $C$ receives after $t$. Therefore, $C$ could not have seen a message after $t$ with index less than $i(p)$. In the $PutDataEntries$ subroutine, $J$ adds every rejected message to $CR(P)$; so for every $i$, $1 \leq i < |X|$, the $i$th element of $X$ will be recorded in the RML of all further elements of $X$; and every element of $X$ will be recorded in $CR_C(p)$. Since every rejected message in $CR(p)$ will be in $CR_C(u)$, $R_1$ will be recorded in $CR_C(p)$. When $C$ receives $u$, $C$ will throw an error from the match $(J, i(q)+1)$ in $CR_C(p)$.\r
+\item Case 2.2.1: Some message in $\mathsf{X}$ was accepted. Let $\mathsf{s_{w_J}}$ \r
+be the greatest sequence number of the messages that client $\mathsf{J}$ sent in \r
+the path of message $\mathsf{w}$. In this case, before sending $\mathsf{p}$, \r
+$\mathsf{J}$'s value in $\mathsf{SM_J}$ for its own latest sequence number would \r
+be strictly greater than $\mathsf{s_{q_J}}$. If there is a sequence of messages with \r
+contiguous sequence numbers that $\mathsf{J}$ receives between $\mathsf{r_1}$ and \r
+$\mathsf{p}$, $\mathsf{J}$ throws an error for a similar reason as Case 1.1. Otherwise, \r
+when preparing to send $\mathsf{p}$, $\mathsf{J}$ would have received an update of its \r
+own latest sequence number as at most $\mathsf{s_{q_J}}$. $J$ throws an error before \r
+sending $\mathsf{p}$, because its own latest sequence number decreases.\r
+\item Case 2.2.2: All messages in $\mathsf{X}$ were rejected, making $\mathsf{p}$ \r
+the first message of $\mathsf{J}$ that is accepted after $\mathsf{r_1}$. Note that \r
+$\mathsf{u}$ is the message with least sequence number that violates this lemma, and \r
+therefore the first one that $\mathsf{C}$ receives after $\mathsf{t}$. Therefore, \r
+$\mathsf{C}$ could not have seen a message after $\mathsf{t}$ with sequence number less \r
+than $\mathsf{s_p}$. In the $\mathsf{PutDataEntries}$ subroutine, $\mathsf{J}$ adds every \r
+%TODO: Check with Thomas about this part\r
+rejected message to $\mathsf{CR(p)}$; so for every $\mathsf{s}$, $\mathsf{1 \leq s < |X|}$, \r
+the $\mathsf{s}$-th element of $\mathsf{X}$ will be recorded in the rejected message list\r
+$\mathsf{CR}$ of all further elements of $\mathsf{X}$; and every element of \r
+$\mathsf{X}$ will be recorded in $\mathsf{CR_C(p)}$. Since every rejected message in \r
+$\mathsf{CR(p)}$ will be in $\mathsf{CR_C(u)}$, ${r_1}$ will be recorded in \r
+$\mathsf{CR_C(p)}$. When $\mathsf{C}$ receives $\mathsf{u}$, $\mathsf{C}$ will throw \r
+an error from the match $\mathsf{\tuple{s_q+1, id_J}}$ in $\mathsf{CR_C(p)}$.\r
\item Case 2.2.2.1: \r
\end{itemize}\r
\end{itemize}\r
\end{itemize}\r
\end{proof}\r
\r
-\begin{lem} If there are two packets $t$ and $u$, with $i(t) <= i(u)$, such that $t$ is in the path of $u$, then for any message $p$ with $i(p) <= i(t)$, iff $p$ is in the path of $t$, it is in the path of $u$. \end{lem}\r
+\begin{lem} If there are two packets $\mathsf{t}$ and $\mathsf{u}$, with \r
+$\mathsf{s_t \leq s_u}$, such that $\mathsf{t}$ is in the path of $\mathsf{u}$, \r
+then for any message $\mathsf{p}$ with $\mathsf{s_p \leq s_t}$, iff $\mathsf{p}$ is in \r
+the path of $\mathsf{t}$, it is in the path of $\mathsf{u}$. \r
+\end{lem}\r
\r
\begin{proof}\r
-If $i(t) = i(u)$ or $i(p) = i(t)$, then we are done, because the two relevant messages are the same.\r
-\r
-Reverse direction: The definition of $t$ being in the path of $u$ is the existence of a message sequence $(..., t, ..., u)$ such that each message except $u$ is the parent of the succeeding message. The path of $u$ must contain some message with index $i(p)$; because $p$ is in the path of $u$, this message is $p$ itself. The path of $t$ is then the prefix of this path ending at $t$, which clearly contains $p$.\r
-\r
-Forward direction: The path of $t$ is a substring of the path of $u$, so if the path of $t$ contains $p$, so does the path of $u$.\r
+If $\mathsf{s_t = s_u}$ or $\mathsf{s_p = s_t}$, then we are done, because the two \r
+relevant messages are the same. If they are different messages, then:\r
+\begin{itemize}\r
+\item Reverse direction: The definition of $\mathsf{t}$ being in the path of \r
+$\mathsf{u}$ is the existence of a message sequence $\mathsf{(\dots, t, \dots, u)}$ \r
+such that each message except $\mathsf{u}$ is the parent of the succeeding message. \r
+The path of $\mathsf{u}$ must contain some message with sequence number $\mathsf{s_p}$; \r
+because $\mathsf{p}$ is in the path of $\mathsf{u}$, this message is $\mathsf{p}$ \r
+itself. The path of $\mathsf{t}$ is then the prefix of this path ending at $\mathsf{t}$, \r
+which clearly contains $\mathsf{p}$.\r
+\r
+\item Forward direction: The path of $\mathsf{t}$ is a substring of the path of \r
+$\mathsf{u}$, so if the path of $\mathsf{t}$ contains $\mathsf{p}$, so does the path \r
+of $\mathsf{u}$.\r
+\end{itemize}\r
\end{proof}\r
\r
\begin{theorem}\r
-Suppose that there is a transitive closure set $\mathscr{S}$ of clients, at index $n$. Then there is some total sequence $T$ of length $n$ such that every client $C$ in $\mathscr{S}$ sees a partial sequence $P_C$ consistent with $T$. \end{theorem}\r
+Suppose that there is a transitive closure set $\mathsf{\mathscr{S}}$ of clients, \r
+at sequence number $\mathsf{s_n}$. Then there is some total sequence $\mathsf{T}$ of \r
+length $\mathsf{n}$ such that every client $\mathsf{C}$ in $\mathsf{\mathscr{S}}$ \r
+sees a partial sequence $\mathsf{P_C}$ consistent with $\mathsf{T}$. \r
+\end{theorem}\r
\r
\begin{proof}\r
\r
-The definition of consistency of $P_C$ with $T$ is that every message $p \in P_C$ with index $i(p) \le n$ is equal to the message in that slot in $T$. Let $C_1$ be some client in the transitive closure set, with partial sequence $P_{C_1}$, and let $u$ be some message with $i(u) > n$ that $C_1$ shares with another client. Then let $T$ be the portion of the path of $u$ ending at index $n$ and $t$ be the message at that index. Clearly, by Lemma 1, $P_{C_1}$ is consistent with $T$. We will show that, for every other client $D$ with partial sequence $P_D$, $P_D$ has some message whose path includes $t$. Because $D$ is in the transitive closure, there is a sequence of clients $\mathscr{C} = (C_1, C_2, ..., D)$ from $C_1$ to $D$, where each shares an edge with the next.\r
-\r
-We prove by induction that $P_D$ has a message whose path includes $t$.\r
+The definition of consistency of $\mathsf{P_C}$ with $\mathsf{T}$ is that every message \r
+$\mathsf{p \in P_C}$ with sequence number $\mathsf{s_p \le s_n}$ is equal to the message \r
+in that slot in $\mathsf{T}$. Let $\mathsf{C_1}$ be some client in the transitive closure \r
+set, with partial sequence $\mathsf{P_{C_1}}$, and let $\mathsf{u}$ be some message with \r
+$\mathsf{s_u > s_n}$ that $\mathsf{C_1}$ shares with another client. Then let $\mathsf{T}$ \r
+be the portion of the path of $\mathsf{u}$ ending at sequence number $\mathsf{s_n}$ and \r
+$\mathsf{t}$ be the message at that sequence number. Clearly, by Lemma 1, $\mathsf{P_{C_1}}$ \r
+is consistent with $\mathsf{T}$. We will show that, for every other client $\mathsf{D}$ \r
+with partial sequence $\mathsf{P_D}$, $\mathsf{P_D}$ has some message whose path includes \r
+$\mathsf{t}$. Because $\mathsf{D}$ is in the transitive closure, there is a sequence of \r
+clients $\mathsf{\mathscr{C} = (C_1, C_2, ..., D)}$ from $\mathsf{C_1}$ to $\mathsf{D}$, \r
+where each shares an edge with the next.\r
+We prove by induction that $\mathsf{P_D}$ has a message whose path includes $\mathsf{t}$.\r
\begin{itemize}\r
-\item Base case: $P_{C_1}$ includes $u$, whose path includes $t$.\r
-\r
-\item Inductive step: Each client in $\mathscr{C}$ has a partial sequence with a message that includes $t$ if the previous client does. Suppose $P_{C_k}$ has a message $w$ with a path that includes $t$, and shares message $x$ with $P_{C_{k+1}}$ such that $i(x) > n$. By Lemma 1, $w$ or $x$, whichever has the least sequence number, is in the path of the other, and therefore by Lemma 2, $t$ is in the path of $x$.\r
-\r
-\item Let $z$ be the message of $D$ whose path includes $t$. By Lemma 1, every message in $P_D$ with index smaller than $i(w)$ is in the path of $z$. Since $t$ is in the path of $z$, every message in $P_D$ with smaller index than $i(t)$ is in $T$. Therefore, $P_D$ is consistent with $T$.\r
+\item Base case: $\mathsf{P_{C_1}}$ includes $\mathsf{u}$, whose path includes $\mathsf{t}$.\r
+\r
+\item Inductive step: Each client in $\mathsf{\mathscr{C}}$ has a partial sequence with a message \r
+that includes $\mathsf{t}$ if the previous client does. Suppose $\mathsf{P_{C_k}}$ has \r
+a message $\mathsf{w}$ with a path that includes $\mathsf{t}$, and shares message $\mathsf{x}$ \r
+with $\mathsf{P_{C_{k+1}}}$ such that $\mathsf{s_x > s_n}$. By Lemma 1, $\mathsf{w}$ or \r
+$\mathsf{x}$, whichever has the least sequence number, is in the path of the other, and therefore \r
+by Lemma 2, $\mathsf{t}$ is in the path of $\mathsf{x}$.\r
+\r
+\item Let $\mathsf{z}$ be the message of $\mathsf{D}$ whose path includes $\mathsf{t}$. \r
+By Lemma 1, every message in $\mathsf{P_D}$ with sequence number smaller than $\mathsf{s_w}$ \r
+is in the path of $\mathsf{z}$. Since $\mathsf{t}$ is in the path of $\mathsf{z}$, every \r
+message in $\mathsf{P_D}$ with smaller sequence number than $\mathsf{s_t = s_n}$ is in $\mathsf{T}$. \r
+Therefore, $\mathsf{P_D}$ is consistent with $\mathsf{T}$.\r
\r
\end{itemize}\r
\end{proof}\r
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