From 3591b2e40b1d119dc80a85c5e28d31dc88ac7f7c Mon Sep 17 00:00:00 2001 From: Jeff Burdges Date: Sun, 11 Nov 2018 01:53:51 +0100 Subject: [PATCH] Oxford comma --- pdf/grandpa.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/pdf/grandpa.tex b/pdf/grandpa.tex index 124c7d5..95f4c8a 100644 --- a/pdf/grandpa.tex +++ b/pdf/grandpa.tex @@ -343,10 +343,10 @@ By Lemma \ref{lem:impossible} (i), this means $C_{r-1,v}$ did not have a superma Thus we have that, at the time of the vote, for one of $V_{r-1,v}$, $C_{r-1,v}$, it was impossible to have a supermajority for $B$. The current sets $V_{r-1,v}$ and $C_{r-1,v}$ are supersets of those at the time of the vote, and so by Lemma \ref{lem:impossible} (ii), it is still impossible. Thus $v$ can respond validly. -This is enough to show Theorem \ref{thm:accountable} . Not that if $v$ sees a commit message for a block $B$ in round $r$ and has that $E_{r',v} \not\geq B$, for some completable round $r' \geq r$, then they should also be able to start a challenge procedure that successfully identifies at least $f+1$ Byzantine voters in some round. Thus we have that: +This is enough to show Theorem \ref{thm:accountable}. Note that if $v$ sees a commit message for a block $B$ in round $r$ and has that $E_{r',v} \not\geq B$, for some completable round $r' \geq r$, then they should also be able to start a challenge procedure that successfully identifies at least $f+1$ Byzantine voters in some round. Thus we have that: \begin{lemma} \label{lem:overestimate-final} -If there at most $f$ Byzantine voters in any vote, $B$ was finalised in round $r$ and an honest participant $v$ sees that round $r' \geq r$ is completable, then $E_{r',v} \geq B$. +If there at most $f$ Byzantine voters in any vote, $B$ was finalised in round $r$, and an honest participant $v$ sees that round $r' \geq r$ is completable, then $E_{r',v} \geq B$. \end{lemma} \subsection{Liveness }