\appendix

\section{Appendix}

\begin{figure*}
  \centering
  \begin{minipage}{1.0\linewidth}
    \begin{gather*}
      \label{eq:1}
      \inferrule[Skip]{ }{\textit{srun}\ \sigma\ \epsilon\ \sigma}\\
      %
      \inferrule[Seq]{\textit{srun}\ \sigma_{0}\ \textit{s}_{1}\ \sigma_{1} \\ \textit{srun}\ \sigma_{1}\ \textit{s}_{2}\ \sigma_{2}}{\textit{srun}\ \sigma_{0}\ (\textit{s}_{1}\ \textit{s}_{2})\ \sigma_{2}}\\
      %
      \inferrule[CondTrue]{\textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ c\ v_{c} \\ \yhfunction{valToB}\ v_{c} = \yhconstant{true} \\ \textit{srun}\ \sigma_{0}\ \textit{st}\ \sigma_{1}}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{if(} c \yhkeyword{) } \textit{st} \yhkeywordsp{else } \textit{sf})\ \sigma_{1}}\\
      %
      \inferrule[CondFalse]{\textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ c\ v_{\rm c} \\ \yhfunction{valToB}\ v_{\rm c} = \yhconstant{false} \\ \textit{srun}\ \sigma_{0}\ \textit{sf}\ \sigma_{1}}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{if(} c \yhkeyword{) } \textit{st} \yhkeywordsp{else } \textit{sf})\ \sigma_{1}}\\
      %
      \inferrule[CaseNoMatch]{\textit{srun}\ \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } cs\ \textit{def} \yhkeywordsp{endcase})\ \sigma_{1} \\ \textit{erun}\  \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ me\ mve \\ \textit{erun}\  \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ e\ ve \\ mve \neq ve}{\textit{srun}\  \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } ((me : sc) :: cs)\ \textit{def} \yhkeywordsp{endcase})\ \sigma_{1}}\\
      %
      \inferrule[CaseMatch]{\textit{srun}\  \sigma_{0}\ sc\ \sigma_{1} \\ \textit{erun}\  \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ e\ ve \\ \textit{erun}\  \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ me\ mve \\ mve = ve}{\textit{srun}\  \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } ((me : sc) :: cs)\ \textit{def} \yhkeywordsp{endcase})\ \sigma_{1}}\\
      %
      \inferrule[CaseDefault]{\textit{srun}\  \sigma_{0}\ s\ \sigma_{1}}{\textit{srun}\  \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } []\ (\yhconstant{Some}\ s) \yhkeywordsp{endcase})\ \sigma_{1}}\\
      %
      \inferrule[Blocking Reg]{\yhfunction{name}\ \textit{lhs} = \yhconstant{OK}\ n \\ \textit{erun}\  \Gamma_{\rm r}\ \Gamma_{\rm a}\ \textit{rhs}\ v_{\rm rhs}}{\textit{srun}\  (\Gamma_{\rm r},\Gamma_{\rm a},\Delta_{\rm r},\Delta_{\rm a})\ (\textit{lhs} = \textit{rhs})\ (\Gamma_{\rm r} [n \mapsto v_{\rm rhs}], \Gamma_{\rm a}, \Delta_{\rm r}, \Delta_{\rm a})}\\
      %
      \inferrule[Nonblocking Reg]{\yhfunction{name}\ \textit{lhs} = \yhconstant{OK}\ n \\ \textit{erun}\  \Gamma_{\rm r}\ \Gamma_{\rm a}\ \textit{rhs}\ v_{\rm rhs}}{\textit{srun}\  (\Gamma_{\rm r}, \Gamma_{\rm a}, \Delta_{\rm r}, \Delta_{\rm a})\ (\textit{lhs} \Leftarrow \textit{rhs})\ (\Gamma_{\rm r}, \Gamma_{\rm a}, \Delta_{\rm r} [n \mapsto v_{\rm rhs}], \Delta_{\rm a})}\\
      \inferrule[Blocking Array]{\yhkeyword{name}\ \textit{lhs} = \yhkeyword{OK}\ n \\ \textit{erun}\  \Gamma_{r}\ \Gamma_{a}\ \textit{rhs}\ v_{\textit{rhs}}}{\textit{srun}\  (\Gamma_{r},\Gamma_{a},\Delta_{r},\Delta_{a})\ (\textit{lhs} = \textit{rhs})\ (\Gamma_{r} // \{n \rightarrow v_{\textit{rhs}}\}, \Gamma_{a}, \Delta_{r}, \Delta_{a})}\\
      %
      \inferrule[Nonblocking Array]{\yhkeyword{name}\ \textit{lhs} = \yhkeyword{OK}\ n \\ \textit{erun}\  \Gamma\ \textit{rhs}\ v_{\textit{rhs}}}{\textit{srun}\  (\Gamma_{r}, \Gamma_{a}, \Delta_{r}, \Delta_{a})\ (\textit{lhs} \Leftarrow \textit{rhs})\ (\Gamma_{r}, \Gamma_{a}, \Delta_{r} // \{n \rightarrow v_{\textit{rhs}}\}, \Delta_{a})}
    \end{gather*}
  \end{minipage}
  \caption{Inferrence rules for statements.}\label{fig:inferrence_statements}
\end{figure*}

\begin{figure*}
  \centering
  \begin{minipage}{1.0\linewidth}
    \begin{gather*}
      \inferrule[Module]{\Gamma_{r} ! s_{t} = \texttt{Some } v \\ (m_{i}, \Gamma_{r}^{0}, \Gamma_{a}^{0}, \epsilon, \epsilon\ l)\ \longrightarrow_{\vec{m}} (m_{i}, \Gamma_{r}^{1}, \Gamma_{a}^{1}, \Delta_{r}^{1}, \Delta_{a}^{1}) \\ (\Gamma_{r}^{1} // \Delta_{r}^{1}) ! s_{t} = \texttt{Some } v'}{\texttt{State } \textit{sf }\ m\ v\ \Gamma_{r}^{0}\ \Gamma_{a}^{0} \longrightarrow \texttt{State } \textit{sf }\ m\ v'\ (\Gamma_{r}^{1} // \Delta_{r}^{1})\ (\Gamma_{a}^{1} // \Delta_{a}^{1})}\\
      %
      \inferrule[Finish]{\Gamma_{r}!\textit{fin} = \texttt{Some } 1 \\ \Gamma_{r}!\textit{ret} = \texttt{Some } r}{\texttt{State } \textit{sf }\ m\ s_{t}\ \Gamma_{r}\ \Gamma_{a} \longrightarrow \texttt{Returnstate } \textit{sf }\ r}\\
      %
      \inferrule[Call]{ }{\texttt{Callstate } \textit{sf }\ m\ \vec{r} \longrightarrow \texttt{State } \textit{sf }\ m\ n\ (\textit{init\_params }\ \vec{r}\ a // \{s_{t} \rightarrow n\})}\\
      %
      \inferrule[Return]{ }{\texttt{Returnstate } (\texttt{Stackframe } r\ m\ \textit{pc }\ \Gamma_{r}\ \Gamma_{a} :: \textit{sf}) \longrightarrow \texttt{State } \textit{sf }\ m\ \textit{pc }\ (\Gamma_{r} // \{ \textit{st} \rightarrow \textit{pc}, r \rightarrow i \})\ \epsilon}
    \end{gather*}
  \end{minipage}
  \caption{Inferrence rules for modules}%
  \label{fig:inferrence_module}
\end{figure*}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% TeX-command-extra-options: "-shell-escape"
%%% End: