%\section{Appendix: remaining axioms of $\basic$}\label{appendix:arithmetic}
%
%We give here the  list of remaining  axioms of $\basic$, which are directly inspired by the $\basic$ theory of Buss's bounded arithmetic \cite{Buss86book}:
%%$\succ{0}(x)$ stand for $2\cdot x$ and $\succ{1}(x)$ stand for $\succ{}(2\cdot x)$,
% 
% 
