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\section{An arithmetic for the polynomial hierarchy}\label{appendix:arithmetic}
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\section{Appendix: remaining axioms of $\basic$}\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{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}, 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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\forall x^{\safe}, y^{\safe}. ((x\leq y \cand x \neq y) \ciff \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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\forall x^{\safe}, y^{\safe}.( x\neq 0 \cimp (|\succ{0}x|=\succ{}( |x|) \cand |\succ{1}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 x^{\safe}, y^{\safe}. (x\leq y \cimp |x|\leq |y|)\\
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\forall x^{\normal}, 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}. (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^{\normal}, y^{\normal}, z^{\normal}. ( |x|= |y| \cimp x\smsh z = y\smsh z)\\
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\forall x^{\normal}, u^{\normal}, v^{\normal}, 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\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^{\safe}, y^{\safe}, z^{\safe}. ( x+y \leq x+z \ciff y\leq z)\\
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\forall x^{\safe} 0\cdot x =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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\forall x^{\normal}, y^{\safe}, z^{\safe}. (x\geq \succ{} 0 \cimp (x\cdot y \leq x\cdot z \ciff 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^{\safe}. ( x= \hlf{y} \ciff (\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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$$ |
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It is often useful for us to work with \emph{length-induction}, which is equivalent to polynomial induction and well known from bounded arithmetic:
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\begin{proposition}
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[Length induction] |
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The axiom schema of formulae, |
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\begin{equation}
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\label{eqn:lind}
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( A(0) \cand \forall x^\normal . (A(x) \cimp A(\succ{} x)) ) \cimp \forall x^\safe. A(|x|)
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\end{equation}
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for formulae $A \in \Sigma^\safe_i$ |
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is equivalent to $\cpind{\Sigma^\safe_i}$.
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\end{proposition}
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\begin{proof}
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Suppose we have $A(0)$ and $A(a) \cimp A(\succ{} a)$ for each $a \in \normal$.
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Then, by $\basic$, we have that $A(|a|) \cimp A(|2a|)$ and $A(|a|) \cimp A(|2a+1|)$ for each $a \in \normal$, whence we may conclude $\forall x. A(|x|)$ by polynomial induction on $A(|x|)$. |
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\end{proof}
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Let us refer to the axiom schema in \eqref{eqn:lind} as $\clind{\mathcal C}$, when $A \in \mathcal C$.
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We will freely use this in place of polynomial induction whenever it is convenient. |
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% |
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%It is often useful for us to work with \emph{length-induction}, which is equivalent to polynomial induction and well known from bounded arithmetic:
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%\begin{proposition}
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% [Length induction] |
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% The axiom schema of formulae, |
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% \begin{equation}
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% \label{eqn:lind}
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% ( A(0) \cand \forall x^\normal . (A(x) \cimp A(\succ{} x)) ) \cimp \forall x^\safe. A(|x|)
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% \end{equation}
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% for formulae $A \in \Sigma^\safe_i$ |
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% is equivalent to $\cpind{\Sigma^\safe_i}$.
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%\end{proposition}
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%\begin{proof}
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% Suppose we have $A(0)$ and $A(a) \cimp A(\succ{} a)$ for each $a \in \normal$.
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% Then, by $\basic$, we have that $A(|a|) \cimp A(|2a|)$ and $A(|a|) \cimp A(|2a+1|)$ for each $a \in \normal$, whence we may conclude $\forall x. A(|x|)$ by polynomial induction on $A(|x|)$. |
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%\end{proof}
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% |
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%Let us refer to the axiom schema in \eqref{eqn:lind} as $\clind{\mathcal C}$, when $A \in \mathcal C$.
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%We will freely use this in place of polynomial induction whenever it is convenient. |
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