trying some new rules... (wip)

main.tex

@@ -350,4 +350,9 @@ The authors thank Paul-André Melliès for his help on type ranking.
 \section{Full new system with normal form}
 \input{new_system}
 
+\newpage
+
+\section{Full new system with normal form (experimentation)}
+\input{new_system2}
+
 \end{document}

new_system.tex

@@ -184,7 +184,7 @@ TODO: Algorithmic rules
   let y = x in
   let z = if y is True then (fun (z:True) -> true) else (fun (z:False) -> false)
   in z x
-  \end{lstlisting}
+  \end{lstlisting}\vspace{1em}
 
   The typing rules should return a list of pair (type, environments) in order to be
   as precise as possible (= remember correlations). When typing a let, the candidates of $e_2$

new_system2.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 (a,a)\\[.3mm]
+    \textbf{U-Expr} &u &::=& a a\alt \tcase{x}{t}{e}{e}\alt \pi_i a\\[.3mm]
+    \textbf{P-Expr} &p &::=& u\alt a\\[.3mm]
+    \textbf{Expressions} &e &::=& \letexp{x}{p}{e}\alt p \\[.3mm]
+    \textbf{Values} &v &::=& c\alt\lambda x:t.e\alt (v,v)\\
+  \end{array}
+\end{equation}
+
+TODO: the then and else expressions of typecases have no associated variable,
+altough the env refinement rules could refine their types...
+Is this an issue? Should we require an atomic in the then and else branchs
+(seems difficult: there would be no equivalent normal form)?
+
+TODO: Which restrictions on types? (cannot test a precise arrow type, etc)
+
+\subsection{Typing rules}
+
+\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 }
+% { }
+\qquad
+\Infer[Subs]
+{ \Gamma \vdash e:t\\t\leq t' }
+{ \Gamma \vdash e: t' }
+{ }
+\\
+\Infer[Efq]
+{ \exists x\in\dom{\Gamma}.\ \Gamma(x)=\Empty }
+{ \Gamma \vdash e: t }
+{ }
+\qquad
+\Infer[Proj]
+{\Gamma \vdash a:\pair{t_1}{t_2}}
+{\Gamma \vdash \pi_i a:t_i}
+{ }
+\qquad
+\Infer[Pair]
+{\Gamma \vdash a_1:t_1 \and \Gamma \vdash a_2:t_2}
+{\Gamma \vdash (a_1,a_2):\pair {t_1} {t_2}}
+{ }
+\\
+\Infer[App]
+{
+  \Gamma \vdash a_1: \arrow {t_1}{t_2}\quad
+  \Gamma \vdash a_2: t_1
+}
+{ \Gamma \vdash {a_1}{a_2}: t_2 }
+{ }
+\qquad
+\Infer[Split]
+{
+  \Gamma,(x:t_1) \vdash e: t\quad
+  \Gamma,(x:t_2) \vdash e: t
+}
+{ \Gamma,(x:t_1\lor t_2) \vdash e: t }
+{ }
+\\
+\Infer[Case]
+{
+x\in\dom\Gamma\\  
+\Gamma\subst{x}{\Gamma(x)\land t} \vdash e_1:t'\\
+\Gamma\subst{x}{\Gamma(x)\land \neg t} \vdash e_2:t'}
+{\Gamma\vdash \tcase {x} t {e_1}{e_2}: t'}
+{ }
+\\
+\Infer[Abs]
+  {\Gamma,(x:s)\vdash e\triangleright_x \dt\\
+  \textstyle {\dt'=\dt\cup\left\{s\setminus\bigcup_{s'\in\dt}s'\right\}}\\
+  T=\{(s',t')\,|\,s'\in\dt' \text{ and } \Gamma,(x:s')\vdash e:t'\}}
+  {
+  \Gamma\vdash\lambda x:s.e:\textstyle \bigwedge_{(s',t')\in T}\arrow {s'} {t'}
+  }
+  { }
+\\
+\Infer[Let]
+  {
+  \Gamma\bvdash{p}{u} (\Gamma_u^i)_{i\in I_u}\\
+  \forall i\in I_u.\ \Gamma_u^i,(x:u)\vdash e:t}
+  {
+  \Gamma\vdash_{u}\letexp x p e : t
+  }
+  { }
+  \\
+  \Infer[LetSplit]
+  {
+  \Gamma,(x:t')\vdash e\triangleright_x\dt\\
+  \forall u\in \dt.\ \Gamma\vdash_{u}\letexp x p e : t}
+  {
+  \Gamma\vdash_{t'}\letexp x p e : t
+  }
+  { }
+  \\
+\Infer[Let]
+  {\Gamma\vdash p:t'\\
+  \Gamma\vdash_{t'}\letexp x p e : t}
+  {
+  \Gamma\vdash\letexp x p e : t
+  }
+  { }
+\end{mathpar}
+
+TODO: Inter rule needed?
+
+TODO: Algorithmic rules
+
+NOTE: In the algorithmic system, the Split rule can be guaranteed by stronger
+typing rules for the application and the typecase: it should return a list of
+pair (typ, env). It allows us to consider the two possibilities of a typecase
+without making any approximation (and similarly for the application of a union of arrows).
+
+\subsection{Candidates generation rules}
+
+\begin{mathpar}
+  \Infer[Var]
+      {
+      }
+      { \Gamma \vdash y \triangleright_x \varnothing }
+      {}
+  \hfill
+  \Infer[Const]
+      {
+      }
+      { \Gamma \vdash c \triangleright_x \varnothing }
+      {}
+  \hfill
+  \Infer[Abs]
+      {\Gamma,(y:s)\vdash e\triangleright_x\dt}
+      {
+      \Gamma\vdash\lambda y:s.\ e\triangleright_x\dt
+      }
+      {x\neq y}
+    \vspace{-2.3mm}\\ 
+    \Infer[Pair]
+          {\Gamma \vdash a_1\triangleright_x\dt_1 \and \Gamma \vdash a_2\triangleright_x\dt_2}
+          {\Gamma \vdash (a_1,a_2)\triangleright_x\dt_1\cup\dt_2}
+          {}
+  \hfill
+   \Infer[Proj]
+          {\Gamma \vdash a\triangleright_x\dt}
+          {\Gamma \vdash \pi_i a\triangleright_x\dt}
+          {}
+  \vspace{-2.3mm}\\        
+  \Infer[Case]
+        {\Gamma\subst{y}{\Gamma(y)\land t}\vdash e_1\triangleright_x\dt_1\\
+        \Gamma\subst{y}{\Gamma(y)\land \neg t}\vdash e_2\triangleright_x\dt_2}
+        {\Gamma\vdash \tcase{y}{t}{e_1}{e_2}\triangleright_x\dt_1\cup\dt_2\cup
+            \left\{
+              \begin{array}{l}
+                \{\Gamma(x)\land t,\Gamma(x)\land\neg t\} \text{ if } y=x\arcr
+                \varnothing\text{ otherwise}
+              \end{array}
+            \right.}
+        {}
+  \vspace{-2.3mm}\\
+  \Infer[App]
+      {
+        \Gamma \vdash a_1 : \textstyle{\bigvee \bigwedge_{i \in I}s_i\to t_i}\\
+        \Gamma \vdash a_1\triangleright_x\dt_1\\
+        \forall i\in I.\ \Gamma\fvdash{a_2}{s_i}\{\Gamma_i^j\}_{j\in J_i}
+      }
+      { \Gamma \vdash a_1 a_2\triangleright_x\dt_1\cup\textstyle{\bigcup_{i\in I}
+        \bigcup_{j\in J_i}\{\Gamma_i^j(x)\}} }
+      {}
+  \vspace{-2.3mm}\\
+  \Infer[Let]
+      {
+        \Gamma \vdash p: t\\
+        \Gamma, (y:t) \vdash e\triangleright_x\dt_x\\
+        \Gamma, (y:t) \vdash e\triangleright_y\dt_y\\
+        \forall u\in \dt_y.\ \Gamma\fvdash{p}{u}\{\Gamma_u^i\}_{i\in I_u}
+      }
+      { \Gamma \vdash\letexp{y}{p}{e}\triangleright_x\dt_x\cup\textstyle{
+        \bigcup_{u\in \dt_y}\bigcup_{i\in I_u}\{\Gamma_u^i(x)\}
+      }}
+      {}
+  \end{mathpar}
+
+  TODO: example: application $x y$ with $x$ an union of arrow. Candidates
+  for $x$ should be all the different union of arrows of its type?
+
+  TODO: Candidates for a variable should be a special "all splits".
+  Otherwise, $\letexp{x}{\cdots}{x}$ will never be splitted.
+
+  TODO: Candidates should always be a partition of the initial type of $x$
+
+\subsection{Forward refinement rules}
+
+\subsection{Backward refinement rules}