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