\section{An arithmetic for the polynomial hierarchy}\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 $\succ{}(2\cdot x)$,
 
 
 \[
 \begin{array}{l}
 \safe (0) \\
 \forall x^\safe . \safe (\succ{} x) \\
 \forall x^\safe . 0 \neq \succ{} (x) \\
 \forall x^\safe , y^\safe . (\succ{} x = \succ{} y \cimp x = y) \\
 \forall x^\safe . (x = 0 \cor \exists y^\safe.\  x = \succ{} y   )\\
 \forall x^\safe, y^\safe . \safe(x+y)\\
 \forall u^\normal, x^\safe . \safe(u\times x) \\
 \forall u^\normal , v^\normal . \safe (u \smsh v)\\
 \forall u^\normal \safe(u) \\
 \forall u^\safe \safe(\hlf{u})\\
 \forall u^\normal \safe(|x|)  
 \end{array}
 \]
 
$$
%\begin{equation}
\begin{array}{l}
 \forall x^{\safe}, y^{\safe}.  (y\leq x) \cimp (y \leq \succ{} x) \\
\forall x^{\safe}. x \neq \succ{} x\\
 \forall x^{\safe}.0 \leq x\\
 \forall x^{\safe}, y^{\safe}.(x\leq y \cand x \neq y) \leftrightarrow \succ{} x \leq y\\
\forall x^{\safe}. x\neq 0 \cimp \succ{0}x \neq 0\\
\forall x^{\safe}, y^{\safe}. y\leq x \cor x \leq y\\
\forall x^{\safe}, y^{\safe}. x\leq y \cand y\leq x \cimp x=y\\
\forall x^{\safe}, y^{\safe}, z^{\safe}. x\leq y \cand y\leq z \cimp x\leq z\\
  |0|=0\\
  \forall x^{\safe}, y^{\safe}.x\neq 0 \cimp  |\succ{0}x|=\succ{}( |x|) \cand |\succ{0}x|= \succ{}(|x|) \\
   |\succ{}0|=\succ{} 0\\
\forall x^{\safe}, y^{\safe}.   x\leq y \cimp   |x|\leq  |y|\\
\forall x^{\safe}, y^{\normal}.    |x\smsh y|=\succ{}( |x|\cdot  |y|)\\
\forall y^{\normal}.    0 \smsh y=\succ{} 0\\
\forall x^{\safe}.    x\neq 0 \cimp 1 \smsh(\succ{0}x)=\succ{0}(1\smsh x) \cand 1 \smsh(\succ{1}x)=\succ{0}(1\smsh x)\\
\forall x^{\normal}, y^{\normal}.    x \smsh y = y \smsh x\\
  \forall x^{\safe}, y^{\safe}, z^{\normal}.    |x|= |y| \cimp x\smsh z = y\smsh z\\
  \forall x^{\safe}, u^{\safe}, v^{\safe}, y^{\normal}.      |x|= |u|+  |v| \cimp x\smsh y=(u\smsh y)\cdot (v\smsh y)\\
  \forall x^{\safe}, y^{\safe}.      x\leq x+y\\
 \forall x^{\safe}, y^{\safe}.       x\leq y \cand x\neq y \cimp \succ{}(\succ{0}x) \leq \succ{0}y \cand  \succ{}(\succ{0}x) \neq \succ{0}y\\
   \forall x^{\safe}, y^{\safe}.     x+y=y+x\\
 \forall x^{\safe}.       x+0=x\\
 \forall x^{\safe}, y^{\safe}.       x+\succ{}y=\succ{}(x+y)\\
  \forall x^{\safe}, y^{\safe}, z^{\safe}.      (x+y)+z=x+(y+z)\\
   \forall x^{\safe}, y^{\safe}, z^{\safe}.     x+y \leq x+z \leftrightarrow y\leq z\\
 \forall x^{\normal}       x\cdot 0=0\\
      \forall x^{\normal}, y^{\safe}.  x\cdot(\succ{}y)=(x\cdot y)+x\\
       \forall x^{\normal}, y^{\normal}. x\cdot y=y\cdot x\\
      \forall x^{\normal}, y^{\safe}, z^{\safe}.  x\cdot(y+z)=(x\cdot y)+(x\cdot z)\\
  \forall x^{\normal}, y^{\safe}, z^{\safe}.      x\geq \succ{} 0 \cimp (x\cdot y \leq x\cdot z \leftrightarrow y\leq z)\\
 \forall x^{\normal}       x\neq 0 \cimp  |x|=\succ{}(\hlf{x})\\
   \forall x^{\safe}, y^{\normal}.     x= \hlf{y} \leftrightarrow (\succ{0}x=y \cor \succ{}(\succ{0}x)=y)
 \end{array}
%\end{equation}
$$