comment previous 'new systems' and add rules for the declarative system

main.tex

@@ -335,24 +335,29 @@ The authors thank Paul-André Melliès for his help on type ranking.
 \section{A crack to Flow Analysis}\label{app:flowanalysis}
 \input{flowanalysis}
 
-\newpage
+%\newpage
 
-\section{New typecase system}
-\input{typecase}
+%\section{New typecase system}
+%\input{typecase}
 
-\newpage
+%\newpage
 
-\section{New function refinement rules}
-\input{refining2}
+%\section{New function refinement rules}
+%\input{refining2}
 
-\newpage
+%\newpage
+
+%\section{Full new system with normal form}
+%\input{new_system}
+
+%\newpage
 
-\section{Full new system with normal form}
-\input{new_system}
+%\section{Full new system with normal form (experimentation)}
+%\input{new_system2}
 
 \newpage
 
-\section{Full new system with normal form (experimentation)}
-\input{new_system2}
+\section{New system with normal form}
+\input{new_system3}
 
 \end{document}

new_system3.tex

+\subsection{Syntax}
+
+\[
+\begin{array}{lrcl}
+\textbf{Types} & t & ::= & b\alt t\to t\alt t\times t\alt t\vee t \alt \neg t \alt \Empty 
+\end{array}
+\]
+
+\begin{equation}\label{expressions2}
+  \begin{array}{lrclr}
+    \textbf{Atomic expr} &a &::=& c\alt x\alt\lambda x:t.e\alt (x,x)
+    \alt x x\alt \tcase{x}{t}{e}{e}\alt \pi_i x\\[.3mm]
+    \textbf{Expressions} &e &::=& \letexp{x}{a}{e}\alt a \\[.3mm]
+    \textbf{Values} &v &::=& c\alt\lambda x:t.e\alt (v,v)\\
+  \end{array}
+\end{equation}
+
+TODO: Which restrictions on types? (cannot test a precise arrow type, etc)
+
+\subsection{Declarative type system}
+
+\begin{mathpar}
+    \Infer[Const]
+    { }
+    {\Gamma\vdash c:\basic{c}}
+    { }
+    \quad
+    \Infer[Var]
+  { }
+  { \Gamma \vdash x: \Gamma(x) }
+  { x\in\dom\Gamma }
+  % \qquad
+  % \Infer[Inter]
+  % { \Gamma \vdash e:t_1\\\Gamma \vdash e:t_2 }
+  % { \Gamma \vdash e: t_1 \wedge t_2 }
+  % { }
+  \\
+  \Infer[Proj]
+  {\Gamma \vdash x:\pair{t_1}{t_2}}
+  {\Gamma \vdash \pi_i x:t_i}
+  { }
+  \qquad
+  \Infer[Pair]
+  {\Gamma \vdash x_1:t_1 \and \Gamma \vdash x_2:t_2}
+  {\Gamma \vdash (x_1,x_2):\pair {t_1} {t_2}}
+  { }
+  \\
+  \Infer[App]
+  {
+    \Gamma \vdash x_1: \arrow {t_1}{t_2}\quad
+    \Gamma \vdash x_2: t_1
+  }
+  { \Gamma \vdash {x_1}{x_2}: t_2 }
+  { }
+  \\
+  \Infer[Case]
+  {
+  \Gamma\vdash x:t_\circ\\  
+  \Gamma\subst{x}{t_\circ\land t} \vdash e_1:t'\\
+  \Gamma\subst{x}{t_\circ\land \neg t} \vdash e_2:t'}
+  {\Gamma\vdash \tcase {x} t {e_1}{e_2}: t'}
+  { }
+  \\
+  \Infer[Abs]
+    {\Gamma,(x:s)\vdash e:t}
+    {\Gamma\vdash\lambda x:s.e: t}
+    { }
+  \qquad
+  \Infer[Let]
+    {\Gamma\vdash a:t_\circ\\
+    \Gamma,(x:t_\circ)\vdash e:t}
+    {
+    \Gamma\vdash\letexp x a e : t
+    }
+    { }
+  \end{mathpar}
+
+\begin{mathpar}
+    \Infer[Subs]
+    { \Gamma \vdash e:t\\t\leq t' }
+    { \Gamma \vdash e: t' }
+    { }
+    \qquad
+    \Infer[Efq]
+    { \exists x\in\dom{\Gamma}.\ \Gamma(x)=\Empty }
+    { \Gamma \vdash e: t }
+    { }
+\\
+  \Infer[Split]
+  {
+    \Gamma, (x:t_1)\vdash e: t\\
+    \Gamma, (x:t_2)\vdash e: t\\
+  }
+  {
+    \Gamma, (x:t_1\vee t_2)\vdash e: t
+  }
+  { }
+  \Infer[AbsSplit]
+  {
+    \Gamma\vdash \lambda x:s_1.\ e: t_1\\
+    \Gamma\vdash \lambda x:s_2.\ e: t_2\\
+  }
+  {
+    \Gamma\vdash \lambda x:s_1\vee s_2.\ e: t_1\wedge t_2
+  }
+  { }
+\end{mathpar}
+
+TODO: Inter rule needed?
+\newpage
+
+\subsection{Algorithmic type system}