diff --git a/pdf/grandpa.pdf b/pdf/grandpa.pdf
index f760d1c..0b9836a 100644
Binary files a/pdf/grandpa.pdf and b/pdf/grandpa.pdf differ
diff --git a/pdf/grandpa.tex b/pdf/grandpa.tex
index ebb8789..7c69f4d 100644
--- a/pdf/grandpa.tex
+++ b/pdf/grandpa.tex
@@ -186,7 +186,7 @@ It is not clear that it is, nor is it clear how to prove liveness for such a rul
 
 \section{Preliminaries} \label{sec:prelims}
 
-{\bf Network model}: We will mostly be using a partially synchronous gossip network model, such as that described in \cite{Tendermint} II A.
+{\bf Network model}: We will mostly be using a partially synchronous gossip network model, such as that described in \cite{Tendermint} section II A.
 Participants communicate via a gossip network, where they are connected to a subset of other participants, and forward all messages they receive to all their connected peers.
 We assume that the network graph is such that any Byzantine participants are not able to cut off an honest participant and so any message sent or received by an honest participant reaches all honest participants.
 The partial synchrony we will use is the model where messages are received within time $T$, but possibly only after some Global Synchronisation Time $\GST$.
@@ -202,7 +202,7 @@ We need $n-f$ of voters to agree on finality. Whether or not block producers eve
 
 {\bf Rounds}: each participant has their own idea of what the current round number is. Every prevote and precommit has an associated round number. Honest voters only vote once (for each type of vote) in each round and don't vote in earlier rounds after later ones.
 
-Participants remember which block they see as currently being the latest finalised block and an estimate of which block could have been finalised in the last round.
+Participants need to keep track of which block they see as currently being the latest finalised block and an estimate of which block could have been finalised in the last round. 
 
 
 
@@ -218,7 +218,9 @@ Blocks are ordered as a tree with the genesis block as root. So any two blocks h
 
 A vote $v$ for a block $B$ by a validator $V$ is a message signed by $V$ containing the blockhash of $B$ and meta information like the round numbers and the type of vote. 
 
-We call a set $S$ of votes tolerant if the number of voters with more than one vote in $S$ is at most $f$. We say that $S$ has supermajority for a block $B$ if the set of voters with votes for blocks $\geq B$ has size at least $(n+f+1)/2$.
+
+
+A validator euivocates in a set of votes $S$ if they have more than one vote in $S$. We call a set $S$ of votes tolerant if the number of voters who equivocate in $S$ is at most $f$. We say that $S$ has supermajority for a block $B$ if the set of voters who either have a vote for blocks $\geq B$ or equivocate in $S$ has size at least $(n+f+1)/2$. (The reason to count equivocations like this is to retain monotonicity , that if $S \subset T$ then if $S$ has a supermajority for $B$ so does $T$, while being able to ignore yet more equivocating votes from an equivocating validator).
 
 The $2/3$-GHOST function $g(S)$ takes a set $S$ of votes and returns the block $B$ with highest block number such that $S$ has a supermajority for $B$.
 If there is no such block, then it returns `nil`. (if $f \neq \lfloor (n-1)/3 \rfloor$, then this is a misnomer and we may change the name accordingly.)
@@ -270,7 +272,7 @@ In round $r$ an honest participant $v$ does the following:
 \begin{enumerate}
 \item $v$ can start round $r > 1$ when round $r-1$ is completable and $v$ has cast votes in all previous rounds where they are a voter. Let $t_{r,v}$ be the time $v$ starts round $r$.
 
-\item At time $t_{r,v}$, if $v$ is the primary of this round and has not finalised $E_{r-1,v}$ then they broadcast $E_{r-1,v}$. If thy have finalised it, they can broadcast $E_{r-1,v}$ anyway (but do not need to).
+\item At time $t_{r,v}$, if $v$ is the primary of this round and has not finalised $E_{r-1,v}$ then they broadcast $E_{r-1,v}$. If they have finalised it, they can broadcast $E_{r-1,v}$ anyway (but do not need to).
 
 \item If $v$ is a voter for the prevote of round $r$, $v$ waits until either it is at least time $t_{r,v}+2T$ or round $r$ is completable, then broadcasts a prevote.
 They prevote for the head of the best chain containing $E_{r-1,v}$ unless we received a block $B$ from the primary and $g(V_{r-1,v}) \geq B > E_{r-1,v}$, in which case they use the best chain containing $B$ instead.
@@ -370,14 +372,15 @@ If there at most $f$ Byzantine voters in any vote, $B$ was finalised in round $r
 \subsection{Liveness }
 
 We show the protocol is deadlock free and also that it finalises new blocks quickly in a weakly synchronous model.
+For this section, we will assume that there are at most $f < n/3$ Byzantine voters for each vote, and so that the sets of prevotes and precommits for each round are tolerant.
 
 We define $V_{r,v,t}$ be the set $V_{r,v}$ at time $t$ and similarly for $C_{r,v,t}$ and the block $E_{r,v,t}$ .
 
+We first show that the completability of a round and the estimate for a completable round are monotone in the votes we see, in the latter case monotonically decreasing: 
 
 \begin{lemma} \label{lem:message-monotonicity-completed-estimate}
-Assume $3f < n-1$.
-Let $v,v'$ be (possibly identical) honest participants, $t,t'$ be times with $t \leq t'$, and $r$ be a round.
-Then if $V_{r,v,t} \subseteq V_{r,v',t'}$ and $C_{r,v,t} \subseteq C_{r,v',t'}$, all these sets are tolerant, and $v$ sees that $r$ is completable at time $t$, then $E_{r,v',t'} \leq E_{r,v,t}$ and $v'$ sees that $r$ is completable at time $t'$.
+Let $v,v'$ be (possibly identical) honest participants, $t,t'$ be times, and $r$ be a round.
+Then if $V_{r,v,t} \subseteq V_{r,v',t'}$ and $C_{r,v,t} \subseteq C_{r,v',t'}$and $v$ sees that $r$ is completable at time $t$, then $E_{r,v',t'} \leq E_{r,v,t}$ and $v'$ sees that $r$ is completable at time $t'$.
 \end{lemma}
 
 \begin{proof} 
@@ -385,7 +388,7 @@ Since $v$ sees that $r$ is completable at time $t$,
 either $E_{r,v} < g(V_{r,v})$ requiring $(n+f+1)/2 > 2f + 1$ votes, or else it is impossible for $C_{r,v}$ to have a supermajority for any children of $g(V_{r,v})$, requiring $2f + 1$ votes.
 In either case, both $V_{r,v,t}$ and $C_{r,v,t}$ contain votes from $2f + 1$ voters and so the same holds for $V_{r,v',t'}$ and $C_{r,v',t'}$. 
 By Lemma \ref{lem:ghost-monotonicity} (ii), $g(V_{r,v',t'}) \geq g(V_{r,v,t})$.
-As it is impossible for $C_{r,v,t}$ to have a supermajority for any children of $g(V_{r,v,t}$, it follows from Lemma \ref{lem:impossible} (i \& ii) that it is impossible for $C_{r,v',t'})$ as well, and so both $E_{r,v',t'} \leq g(V_{r,v,t})$ and $v'$ sees $r$ is completable at time $t'$.
+As it is impossible for $C_{r,v,t}$ to have a supermajority for any children of $g(V_{r,v,t})$, it follows from Lemma \ref{lem:impossible} (i \& ii) that it is impossible for $C_{r,v',t'})$ as well, and so both $E_{r,v',t'} \leq g(V_{r,v,t})$ and $v'$ sees $r$ is completable at time $t'$.
 But now $E_{r,v,t}$ and $E_{r,v',t'}$ are the last blocks on $\textrm{chain}(g(V_{r,v,t}))$ for which it is possible for $C_{r,v,t}$ and $C_{r,v',t'}$ respectively to have a supermajority, 
 As it is possible for $C_{r,v',t'}$ to have a supermajority for $E_{r,v',t'}$, then it is possible for $C_{r,v,t}$ to have a supermajority for $E_{r,v',t'}$ as well, by Lemma \ref{lem:impossible} (ii) and tolerance assumptions, so $E_{r,v',t'} \leq E_{r,v,t}$.
 \end{proof}