Laboratoire de l'Informatique et du Parallélisme » Linear Arithmetic

\begin{figure}

\[

\small

\begin{array}{l}

\begin{array}{cccc}

%\vlinf{\lefrul{\bot}}{}{p, \lnot{p} \seqar }{}

%& \vlinf{\id}{}{p \seqar p}{}

%& \vlinf{\rigrul{\bot}}{}{\seqar p, \lnot{p}}{}

%& \vliinf{\cut}{}{\Gamma, \Sigma \seqar \Delta , \Pi}{ \Gamma \seqar \Delta, A }{\Sigma, A \seqar \Pi}

 \vlinf{id}{}{p \seqar p}{}

& \vliinf{cut}{}{\Gamma \seqar \Delta }{ \Gamma \seqar \Delta, A }{\Gamma, A \seqar \Delta}

&&

\\

\noalign{\bigskip}

%\noalign{\bigskip}

\vliinf{\lefrul{\cor}}{}{\Gamma, A \cor B \seqar \Delta}{\Gamma , A \seqar \Delta}{\Gamma, B \seqar \Delta}

&

\vlinf{\lefrul{\cand}}{}{\Gamma, A\cand B \seqar \Delta}{\Gamma, A , B \seqar \Delta}

&

%\vlinf{\lefrul{\laand}}{}{\Gamma, A\laand B \seqar \Delta}{\Gamma, B \seqar \Delta}

%\quad

\vlinf{\rigrul{\cor}}{}{\Gamma \seqar \Delta, A \cor B}{\Gamma \seqar \Delta, A, B}

&

%\vlinf{\rigrul{\laor}}{}{\Gamma \seqar \Delta, A\laor B}{\Gamma \seqar \Delta, B}

%\quad

\vliinf{\rigrul{\cand}}{}{\Gamma \seqar \Delta, A \cand B }{\Gamma \seqar \Delta, A}{\Gamma \seqar \Delta, B}

\\

\noalign{\bigskip}

\vliinf{\lefrul{\cimp}}{}{\Gamma, A \cimp B \seqar \Delta}{\Gamma \seqar A, \Delta}{\Gamma, B \seqar \Delta}

&

\vlinf{\lefrul{\neg}}{}{\Gamma, \neg A \seqar \Delta}{\Gamma \seqar A, \Delta}

&

\vlinf{\rigrul{\cimp}}{}{\Gamma \seqar \Delta, A \cimp B}{\Gamma, A \seqar \Delta,  B}

&

\vlinf{\lefrul{\neg}}{}{\Gamma, \seqar \neg A, \Delta}{\Gamma, A \seqar  \Delta}

\\

\noalign{\bigskip}

%\text{Structural:} & & & \\

%\noalign{\bigskip}

\vlinf{\lefrul{\wk}}{}{\Gamma, A \seqar \Delta}{\Gamma \seqar \Delta}

&

\vlinf{\lefrul{\cntr}}{}{\Gamma, A \seqar \Delta}{\Gamma, A, A \seqar \Delta}

&

\vlinf{\rigrul{\wk}}{}{\Gamma \seqar \Delta, A }{\Gamma \seqar \Delta}

&

\vlinf{\rigrul{\cntr}}{}{\Gamma \seqar \Delta, A}{\Gamma \seqar \Delta, A, A}

\\

\noalign{\bigskip}

\vlinf{\lefrul{\exists}}{}{\Gamma, \exists x . A(x) \seqar \Delta}{\Gamma, A(a) \seqar \Delta}

&

\vlinf{\lefrul{\forall}}{}{\Gamma, \forall x. A(x) \seqar \Delta}{\Gamma, A(t) \seqar \Delta}

&

\vlinf{\rigrul{\exists}}{}{\Gamma \seqar \Delta, \exists x . A(x)}{ \Gamma \seqar \Delta, A(t)}

&

\vlinf{\rigrul{\forall}}{}{\Gamma \seqar \Delta, \forall x . A(x)}{ \Gamma \seqar \Delta, A(a) } \\

%\noalign{\bigskip}

% \vliinf{mix}{}{\Gamma, \Sigma \seqar \Delta , \Pi}{ \Gamma \seqar \Delta}{\Sigma \seqar \Pi} &&&

\end{array}

\end{array}

\]

\caption{Sequent calculus rules}\label{fig:sequentcalculus}

\end{figure}

 We denote sequence as $\Gamma \seqar \Delta$ where $\Gamma$, $\Delta$ are multi sets of formulas. The sequent calculus rules are displayed on Fig. \ref{fig:sequentcalculus},  where $p$ is atomic, $i \in \{ 1,2 \}$, $t$ is a term and the eigenvariable $a$ does not occur free in $\Gamma$ or $\Delta$.

We consider \emph{systems} of `nonlogical' rules extending this sequent calculus, which we write as follows,

 \[

 \begin{array}{cc}

    \vlinf{(R)}{}{ \Gamma , \Sigma' \seqar \Delta' , \Pi  }{ \{\Gamma , \Sigma_i \seqar \Delta_i , \Pi \}_{i \in I} }

\end{array}

\]

 where, in each rule $(R)$, $I$ is a finite possibly empty set (indicating the number of premises) and we assume the following conditions and terminology:

 \begin{enumerate}

 \item In $(R)$ the formulas of $\Sigma', \Delta'$  are called \textit{principal}, those of $\Sigma_i, \Delta_i$ are called \textit{active}, and those of   

$ \Gamma,  \Pi$ are called \textit{context formulas}. 

\item Each rule $(R)$ comes with a list $a_1$, \dots, $a_k$ of eigenvariables such that each $a_j$ appears in exactly one $\Sigma_i, \Delta_i$ (so in some active formulas of exactly one premise)  and does not appear in  $\Sigma', \Delta'$ or $ \Gamma,  \Pi$.

    \item A system $\mathcal{S}$ of rules must be closed under substitutions of free variables by terms (where these substitutions do not contain the eigenvariables $a_j$ in their domain or codomain).  

   \end{enumerate}

%The distinction between modal and nonmodal formulae in $(R)$ induces condition 1

 Conditions 2 and 3 are standard requirements for nonlogical rules, independently of the logical setting, cf.\ \cite{Beckmann11}. Condition 2 reflects the intuitive idea that, in our nonlogical rules, we often need a notion of \textit{bound} variables in the active formulas (typically for induction rules), for which we rely on eigenvariables. Condition 3 is needed for our proof system to admit elimination of cuts on quantified formulas.

\todo{State theorem, with references (Takeuti, Cook-Nguyen) and present the important corollaries for this work.}

Since our nonlogical rules may have many principal formulae on which cuts may be anchored, we need a slightly more general notion of principality.

    \begin{definition}\label{def:anchoredcut}

  We define the notions of \textit{hereditarily principal formula} and \textit{anchored cut} in a $\system$-proof, for a system $\system$, by mutual induction as follows:

  \begin{itemize}

  \item A formula $A$ in a sequent $\Gamma \seqar \Delta$ is \textit{hereditarily principal} for a rule instance (S) if either (i) the sequent is in the conclusion of (S) and $A$ is principal in it, or 

(ii)  the sequent is in the conclusion of an anchored cut, the direct ancestor of $A$ in the corresponding premise is hereditarily principal for the rule instance (S), and the rule (S) is nonlogical.

  \item A cut-step is an \textit{anchored cut} if the two occurrences of its cut-formula $A$ in each premise are hereditarily principal for nonlogical steps, or one is hereditarily principal for a nonlogical step and the other one is principal for a logical step.

  \end{itemize}

     A cut which is not anchored will also be called a \textit{free-cut}.

  \end{definition}

  As a consequence of this definition, an anchored cut on a formula $A$ has the following properties:

  \begin{itemize}

  \item At least one of the two premises of the cut has above it a sub-branch of the proof which starts (top-down) with a nonlogical step (R) with $A$ as one of its principal formulas, and then a sequence of anchored cuts in which $A$ is part of the context.

  \item The other premise is either of the same form or is a logical step with principal formula $A$. 

  \end{itemize}

   Now we have (see \cite{Takeuti87}): 

   \begin{theorem}

   [Free-cut elimination]

   \label{thm:free-cut-elim}

    Given a system  $\mathcal{S}$, any  $\mathcal{S}$-proof $\pi$ can be transformed into a $\system$-proof $\pi'$ with same end sequent and without any free-cut.

   \end{theorem}