diff --git a/main.tex b/main.tex
index 946c01b4ba2b434b6849bbb8d8e902c626f48cd9..04ed76fecfb3254a9a087855fd177dce06c081b1 100644
--- a/main.tex
+++ b/main.tex
@@ -62,6 +62,7 @@
 \newcommand{\pop}{\mathsf{pop}}
 \newcommand{\auta}{\mathcal A}
 \newcommand{\autb}{\mathcal B}
+\newcommand{\astb}{{\star}}
 
 \newtheorem{theorem}{Theorem}
 \newtheorem{corollary}{Corollary}
@@ -103,7 +104,7 @@
 \paragraph{Signature}
 A \emph{signature} $\ssign$ is a set $S$ of \emph{symbols}\footnote{We only consider relational signatures.}, together with an \emph{arity function}, which we denote by $\ar:S\rightarrow \nat$. Abusing notations, we will often write $R\in \ssign$ instead of $R\in S$.
 
-\paragraph{Structures} A \emph{structure} (sometimes \emph{model}) $A$ over a signature $\ssign$ is given as a \emph{domain} $\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)\subseteq \dom(A)^r$, with $r=\ar(R)$.
+\paragraph{Structures} A \emph{structure} (sometimes \emph{model}) $A$ over a signature $\ssign$ is given as a \emph{domain} $\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)\subseteq \dom_A^r$, with $r=\ar(R)$.
 
 \paragraph{Operations}
 Any structure over a signature can be seen as a structure over a larger signature where additional symbols are interpreted as empty sets.
@@ -219,7 +220,7 @@ From the backward translation theorem.
 \subsubsection{Exponential growth}
 
 \begin{remark}
-An \msomi of dimension $d$ has growth $O((2^n)^d)=O(2^{dn})=2^{O(n)}$.
+An \msomi of dimension $d$ has growth $\leq (2^n)^d=2^{dn}=2^{O(n)}$.
 
 \end{remark}
 \begin{corollary}
@@ -358,6 +359,26 @@ For rational turing machines, the following hold:
 \end{enumerate}
 \end{theorem}
 
+\section{List functions}
+
+Types:
+
+\begin{itemize}
+\item $\tau^*$
+\item $\tau^{\underline *}$
+\item $\tau + \sigma$
+\item $\tau \times \sigma$
+\item $\tau \rightarrow \sigma$
+\end{itemize}
+
+Atomic programs: same as for polyregular list functions, except for \texttt{split}. Moreover each program involving $*$ has an equivalent version using $\astb$. Moreover we introduce a new basic program:
+
+\begin{itemize}
+\item $\texttt{subsets}^\tau: \tau^*\rightarrow ((\tau+\tau)^*)^\astb$
+
+\end{itemize}
+
+
 
 \section{The case of the successor}