Laboratoire de l'Informatique et du Parallélisme » Linear Arithmetic

\subsection{Graphs of some basic functions}\label{sect:graphsbasicfunctions}

We say that a function $f$ is \emph{represented} by a formula $A_f$ in a theory if we can prove a formula of the form $\forall \vec u^{\normal} , \forall  x^{\safe}, \exists! y^{\safe}. A_f$. The variables $\vec u$ and $\vec x$ can respectively be thought of as normal and safe arguments of $f$, and $y$ is the output of $f(\vec u; \vec x)$.

Let us give a few examples for basic functions representable in $\arith^1$, wherein proofs of totality are routine:

\begin{itemize}

%\item Successor $\succ{}$: $\forall x^{\safe} , \exists^{\safe} y. y=x+1.$. The formulas for the binary successors $\succ{0}$, $\succ{1}$ and the constant functions $\epsilon^k$ are defined in a similar way.

%\item Predecessor $p$:   $\forall^{\safe} x, \exists^{\safe} y. (x=\succ{0} y \cor x=\succ{1} y \cor (x=\epsilon \cand y= \epsilon)) .$

\item Conditional $C$: 

$$\begin{array}{ll}

\end{array}

$$

%\item Addition:

  This is a predicate that we will need for technical reasons, in the completeness proof. The predicate $\pref(k,x,y)$ holds if the prefix of string $x$

  of length $k$ is $y$. It is defined as:

  $$\begin{array}{ll}

                                                                                                   & \cand \; |z|=l\\

                                                                                                   & \cand \; x=y\smsh(2,l)+z)

                                                                                                   \end{array}

$\zerobit(x,k)$ (resp. $\onebit(x,k)$) holds iff the $k$th bit of $x$ is 0 (resp. 1). The predicate $\zerobit(x,k)$  for instance can be

defined by:

\end{itemize}

\subsection{A sequent calculus presentation}