Fix doc after merge

Author: Velnbur

docs/paper/nero.pdf

@@ -314,60 +314,6 @@ \subsection{Implementation on Bitcoin}
   \Return{$z_1,\dots,z_n$}
 \end{algorithm}\label{alg:intermediate_steps}
 
-\begin{example}
-  Consider a fairly simple program $f$:
-  \begin{equation*}
-    f(a,b) = 25a^2b^2{(a+b)}^2
-  \end{equation*}
-
-  Additionally, assume that we can abstract the multiplication operation
-  via the opcode \texttt{OP\_MUL} (which, in fact, is already natively
-    implemented in Bitcoin Script, although the size of such script
-  for two 254-bit numbers exceeds 60kB).
-  Then, the implementation of $f$ in Bitcoin Script could be:
-  \begin{empheqboxed}
-    \begin{align*}
-      &\elem{a} \elem{b} \opcode{\texttt{OP\_2DUP}}
-      \opcode{\texttt{OP\_ADD}} \opcode{\texttt{OP\_MUL}}
-      \opcode{\texttt{OP\_MUL}}
-      \opcode{\texttt{OP\_DUP}} \opcode{\texttt{OP\_DUP}} \\&
-      \opcode{\texttt{OP\_ADD}} \opcode{\texttt{OP\_DUP}}
-      \opcode{\texttt{OP\_ADD}} \opcode{\texttt{OP\_ADD}}
-      \opcode{\texttt{OP\_DUP}} \opcode{\texttt{OP\_MUL}}
-    \end{align*}
-  \end{empheqboxed}
-
-  Fairly complex, right? Let us split the function into three shards
-  $\textcolor{red!80!white}{f_1}$, $\textcolor{blue}{f_2}$, and
-  $\textcolor{green!60!black}{f_3}$:
-  \begin{align*}
-    \textcolor{red!80!white}{f_1}(x,y) = xy(x+y), \quad
-    \textcolor{blue}{f_2}(z) = 5z, \quad
-    \textcolor{green!60!black}{f_3}(w) = w^2
-  \end{align*}
-
-  This way, it is fairly easy to see that $f(a,b) =
-  \textcolor{green!60!black}{f_3} \circ \textcolor{blue}{f_2} \circ
-  \textcolor{red!80!white}{f_1}(a,b)$. In turn, in Bitcoin
-  script we can represent $f$ as $\textcolor{red!80!white}{f_1}
-  \parallel \textcolor{blue}{f_2} \parallel \textcolor{green!60!black}{f_3}$:
-  \begin{empheqboxed}
-    \begin{align*}
-      &\elem{a} \elem{b}
-      \textcolor{red!80!white}{\opcode{\texttt{OP\_2DUP}}
-        \opcode{\texttt{OP\_ADD}} \opcode{\texttt{OP\_MUL}}
-      \opcode{\texttt{OP\_MUL}}} &&
-      \textcolor{gray!80!black}{\text{// $xy(x+y)$}} \\
-      &\textcolor{blue}{\opcode{\texttt{OP\_DUP}}
-        \opcode{\texttt{OP\_DUP}} \opcode{\texttt{OP\_ADD}}
-        \opcode{\texttt{OP\_DUP}} \opcode{\texttt{OP\_ADD}}
-      \opcode{\texttt{OP\_ADD}}} && \textcolor{gray!80!black}{\text{// $5z$}} \\
-      &\textcolor{green!60!black}{\opcode{\texttt{OP\_DUP}}
-      \opcode{\texttt{OP\_MUL}}}&&\textcolor{gray!80!black}{\text{// $w^2$}}
-    \end{align*}
-  \end{empheqboxed}
-\end{example}
-
 \begin{example}
   Consider a fairly simple program $f$:
   \begin{equation*}