From 29be8ea7ee3c4747c41cf6b5400e09ef4f858d7c 2017-09-07 02:44:49
From: András Gilyén <gilyen@cwi.nl>
Date: 2017-09-07 02:44:49
Subject: [PATCH] simplified proof

---

diff --git a/main.tex b/main.tex
index 6f11925f9f61fdb02bf30f484e20b88e21281bb3..23e4e6342b921b79bae49f773187e43289ab488e 100644
--- a/main.tex
+++ b/main.tex
@@ -666,7 +666,7 @@ The intuition of the following lemma is that the far right can only affect the z
  		$$\P^{[k]}(\Z{1})=\sum_{P\text{ patch}\,:\,1\in P}\P^{[k]}(P\in\mathcal{P})$$
  		$$\P^{[k]}(\Z{k})=\sum_{P\text{ patch}\,:\,k\in P}\P^{[k]}(P\in\mathcal{P})$$
  		
- 		Suppose we proved the statement up to $k-$, then we proceed using induction similarly to the above
+ 		Suppose we proved the statement up to $k-1$, then we proceed using induction similarly to the above
  		\begin{align*}
  		&\P^{[k]}(\Z{1}\cap \Z{k})=\\
  		&=\sum_{\ell, r\in [k]: \ell<r-1}\P^{[k]}([\ell],[r,k]\in\mathcal{P})