author rtrimana Tue, 16 Aug 2016 22:38:56 +0000 (15:38 -0700) committer rtrimana Tue, 16 Aug 2016 22:38:56 +0000 (15:38 -0700)
1  2
doc/iotcloud.tex

@@@ -969,17 -951,12 +951,14 @@@ call them $\mathsf{t}$ and $\mathsf{u} Then$\mathsf{t}$is in the path of$\mathsf{u}$. \r \end{lem}\r \begin{proof}\r - By contradiction, we will prove that if$\mathsf{t}$is not in the path of \r -$\mathsf{u}$, then it is impossible for client$\mathsf{C}$to receive both\r - messages without throwing any errors. Clearly$\mathsf{C}$will throw an error \r - if$\mathsf{s_t = s_u}$. So$\mathsf{s_t < s_u}$. Additionally, if$\mathsf{C}$\r - receives$\mathsf{u}$before$\mathsf{t}$, this will cause it to throw an \r - error, so$\mathsf{t}$is received before$\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}$.\r -Assume otherwise. Then there are some pairs$\mathsf{(t,u)}$that violate this lemma. Take a specific$\mathsf{(t,u)}$such that$\mathsf{s_u}$is minimized and$\mathsf{s_t}$is maximized for this choice of$\mathsf{s_u}$.\r ++Assume otherwise. Then there are some pairs$\mathsf{(t,u)}$that violate this lemma. \r ++Take a specific$\mathsf{(t,u)}$such that$\mathsf{s_u}$is minimized and \r ++$\mathsf{s_t}$is maximized for this choice of$\mathsf{s_u}$.\r \r + Clearly$\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}$. We will prove that an error occurs upon receipt of$\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