diff --git a/chap-GS-background.tex b/chap-GS-background.tex
index ce184b5..a74117c 100644
--- a/chap-GS-background.tex
+++ b/chap-GS-background.tex
@@ -306,8 +306,7 @@ adversarially-controlled user.
   attacks} if, for any $\ppt$ adversary $\adv$ involved in Experiment~$\Exp{\textrm{mis-id}}{\adv}(\lambda)$
   described in Figure~\ref{exp:mis-id}, we have:
   \[\advantage{\adv}{\mathrm{mis}\textrm{-}\mathrm{id}}(\lambda) \triangleq
-    \Proba{\,\Exp{\mathrm{mis}\textrm{-}\mathrm{id}}{\adv}(\lambda)=1} =
-  \negl[\lambda].\]
+    \Proba{\,\Exp{\mathrm{mis}\textrm{-}\mathrm{id}}{\adv}(\lambda)=1} \leq \negl[\lambda].\]
 \end{definition}
 
 
@@ -364,7 +363,7 @@ The adversary eventually aims at framing an honest group member.
 %
   A dynamic group signature scheme is secure against \emph{framing attacks} if,
   for any $\ppt$ adversary $\adv$ involved in the experiment~$\Exp{\mathrm{fra}}{\adv}(\lambda)$ described Figure~\ref{exp:frame}), it holds that
-  \[ \advantage{\adv}{\mathrm{fra}}(\lambda)=\Proba{\Exp{\mathrm{fra}}{\adv}(\lambda)=1} \in \negl[\lambda]. \]
+  \[ \advantage{\adv}{\mathrm{fra}}(\lambda)=\Proba{\Exp{\mathrm{fra}}{\adv}(\lambda)=1} \leq \negl[\lambda]. \]
 %
 
 \end{definition}
diff --git a/chap-OT-LWE.tex b/chap-OT-LWE.tex
index 246e0b9..5344af4 100644
--- a/chap-OT-LWE.tex
+++ b/chap-OT-LWE.tex
@@ -103,7 +103,7 @@ The distribution of outputs of the environment in the different settings is deno
 
 \begin{definition}
   An AC-OT protocol is said to securely implement the functionality if for any real-world adversary $\adv$ and any real world environment $\mathcal E$, there exists an ideal-world simulator $\mathcal A'$ controlling the same parties in the ideal-world as $\adv$ does in the real-world, such that
-  \[ | \mathbf{Real}_{\mathcal E, \adv}(\lambda) - \mathbf{Ideal}_{\mathcal{E}, \adv}(\lambda) | = \negl(\lambda). \]
+  \[ | \mathbf{Real}_{\mathcal E, \adv}(\lambda) - \mathbf{Ideal}_{\mathcal{E}, \adv}(\lambda) | \leq \negl(\lambda). \]
 \end{definition}
 
 
diff --git a/chap-proofs.tex b/chap-proofs.tex
index 11d9a7e..dfb6298 100644
--- a/chap-proofs.tex
+++ b/chap-proofs.tex
@@ -123,7 +123,7 @@ an attack is successful if the probability that it succeed is noticeable.
   \index{Probability!Negligible} \index{Probability!Noticeable} \index{Probability!Overwhelming}
   Let $f : \NN \to [0,1]$ be a function. The function $f$ is said to be \emph{negligible} if $f(n) = n^{-\omega(1)}_{}$, and this is written $f(n) = \negl[n]$.\\
   Non-negligible functions are also called \emph{noticeable} functions.\\
-  Finally, if $f = 1- \negl[n]$, $f$ is said to be \emph{overwhelming}.
+  Finally, if $f = 1 - \negl[n]$, $f$ is said to be \emph{overwhelming}.
 \end{definition}
 
 Now, we have to define two more notions to be able to work on security proofs.