Added declarative rules

main.tex

@@ -356,7 +356,10 @@ The authors thank Paul-André Melliès for his help on type ranking.
 %\input{new_system2}
 
 \newpage
+\section{New Formalization (Beppe)}
+\input{new_system_beppe}
 
+\newpage
 \section{New system with normal form}
 \input{new_system3}
 

new_system_beppe.tex

+\subsection{Declarative system}
+
+\subsubsection{Types}
+
+\[
+\begin{array}{lrcl}
+\textbf{Types} & t & ::= & b\alt t\to t\alt t\times t\alt t\vee t \alt \neg t \alt \Empty 
+\end{array}
+\]
+
+\subsubsection{Terms}
+
+\begin{equation}
+  \begin{array}{lrclr}
+    \textbf{Expressions} &a &::=& c\alt x\alt\lambda x.e
+    \alt e e\alt (e,e)\alt \pi_i e \alt  \letexp{x}{e}{e} \alt \tcase{e}{t}{e}{e}\\[.3mm]
+  \end{array}
+\end{equation}
+
+\subsubsection{Typing rules}
+Rules of Definition 3.5 of Barbanera Dezani + pairs + negation
+
+\begin{mathpar}
+    \Infer[Const]
+    { }
+    {\Gamma\vdash c:\basic{c}}
+    { }
+    \quad
+    \Infer[Ax]
+  { }
+  { \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[$\to$I]
+    {\Gamma,x:t_1\vdash e:t_2}
+    {\Gamma\vdash\lambda x.e: \arrow{t_1}{t_2}}
+    { }
+  \qquad  \Infer[$\to$E]
+  {
+    \Gamma \vdash e_1: \arrow {t_1}{t_2}\quad
+    \Gamma \vdash e_2: t_1
+  }
+  { \Gamma \vdash {e_1}{e_2}: t_2 }
+  { }
+  \\
+  \Infer[$\times$I]
+  {\Gamma \vdash e_1:t_1 \and \Gamma \vdash e_2:t_2}
+  {\Gamma \vdash (e_1,e_2):\pair {t_1} {t_2}}
+  { }
+ \qquad
+  \Infer[$\times$E$_1$]
+  {\Gamma \vdash e:\pair{t_1}{t_2}}
+  {\Gamma \vdash \pi_1 e:t_1}
+  { } \quad
+  \Infer[$\times$E$_2$]
+  {\Gamma \vdash e:\pair{t_1}{t_2}}
+  {\Gamma \vdash \pi_2 e:t_2}
+  { }
+  \\  
+  \Infer[$\wedge$]
+  {\Gamma \vdash e:t_1 \and \Gamma \vdash e:t_2}
+  {\Gamma \vdash e: {t_1}\wedge {t_2}}
+  { }
+  \qquad
+  \Infer[$\vee$]
+    {\Gamma, x:t_1\vdash e:t\and \Gamma, x:t_2\vdash e:t\and
+      \Gamma \vdash e':t_1\vee t_2 }
+    {
+    \Gamma\vdash\letexp x {e'} e : t
+    }
+    { }
+    \\
+\Infer[$\neg_1$]
+  {
+    \Gamma \vdash e:t \and \Gamma\vdash e_1: t_1
+  }
+  {\Gamma\vdash \tcase {e} t {e_1}{e_2}: t_1}
+  { }
+  \qquad
+  \Infer[$\neg_2$]
+  {
+    \Gamma \vdash e:\neg t \and \Gamma\vdash e_2: t_2
+  }
+  {\Gamma\vdash \tcase {e} t {e_1}{e_2}: t_2}
+  { }
+  \\
+    \Infer[Subsumption]
+    { \Gamma \vdash e:t\\t\leq t' }
+    { \Gamma \vdash e: t' }
+    { }
+\end{mathpar}
+
+
+\subsection{Algorithmic type system}
+
+The idea is to translate terms above in to explicitly annotated
+notrmal form and prove that a term above is well typed if and only if
+its translation is well typed for a suitable annotation.
+
+In particular the typing rule for cases will be obtained by combining
+the union rule with the case rule and using two subsumption rules. Intuitively
+\begin{mathpar}
+  \inferrule*{
+    \inferrule*{
+         \inferrule*{\Gamma, x: t{\wedge} t_\circ\vdash x: t{\wedge}
+           t_\circ\quad t\wedge t_\circ\leq t}{\Gamma, x: t\wedge t_\circ\vdash x: t}
+          \Gamma, x: t{\wedge} t_\circ\vdash e_1: t'}
+               {\Gamma, x: t\wedge t_\circ\vdash \tcase {x} t {e_1}{e_2}:t'}
+        \inferrule{
+         \inferrule*{...}{\Gamma, x:
+           \neg t\wedge t_\circ\vdash x: \neg t}
+          \Gamma, x: \neg t{\wedge} t_\circ\vdash e_2: t}       
+               {\Gamma, x:\neg t\wedge t_\circ\vdash \tcase {x} t {e_1}{e_2}:t'}
+    \Gamma\vdash e:t_\circ
+  }
+  {\Gamma\vdash \letexp x {e}{\tcase {x} t {e_1}{e_2}:t'}}
+\end{mathpar}
+Note: $t_\circ =(t\wedge t_\circ)\vee(\neg t\wedge t_\circ$)
+
+
+\subsubsection{Normal form terms with explicit annotations}
+
+\begin{equation}\label{expressions2}
+  \begin{array}{lrclr}
+    \textbf{Atomic expr} &a &::=& c\alt x\alt\lambda x{:}\{t,...,t\}.e\alt x x\alt \pi_i x\alt (x,x)
+    \alt \tcase{x}{t}{e}{e}\\[.3mm]
+    \textbf{Expressions} &e &::=& \letexp{x{:}\{t,...,t\}}{a}{e}\alt a \\[.3mm]
+    \textbf{Values} &v &::=& c\alt\lambda x{:}\{t,...,t\}.e\alt (v,v)\\
+  \end{array}
+\end{equation}
+\subsubsection{Algorithmic typing rules}
+
+\begin{mathpar}
+    \Infer[Const]
+    { }
+    {\Gamma\vdash c:\basic{c}}
+    { }
+    \quad
+    \Infer[Ax]
+  { }
+  { \Gamma \vdash x: \Gamma(x) }
+  { x\in\dom\Gamma }
+ \\
+  \Infer[$\to$I]
+    {i=1..n\\\Gamma,x:t_i\vdash e:s_i}
+    {\Gamma\vdash\lambda x{:}\{t_1,...,t_n\}.e: \textstyle\bigwedge_{i=1..n}\arrow{t_i}{s_i}}
+    { }
+  \qquad  \Infer[$\to$E]
+  {
+    \Gamma \vdash e_1: \arrow {t_1}\quad
+    \Gamma \vdash e_2: t_2
+  }
+  { \Gamma \vdash {e_1}{e_2}: t_1\circ t_2 }
+  {t_1\leq\Empty\to\Any,t_2\leq\dom{t_1} }
+  \\
+  \Infer[$\times$I]
+  {\Gamma \vdash x_1:t_1 \and \Gamma \vdash x_2:t_2}
+  {\Gamma \vdash (x_1,x_2):\pair {t_1} {t_2}}
+  { }
+ \qquad
+  \Infer[$\times$E$_1$]
+  {\Gamma \vdash x:t\leq(\Any\times\Any)}
+  {\Gamma \vdash \pi_1 x:\pi_1(t_1)}
+  { } \quad
+  \Infer[$\times$E$_2$]
+  {\Gamma \vdash  x:t\leq(\Any\times\Any)}
+  {\Gamma \vdash \pi_2 x:\pi_2(t)}
+  { }
+  \\
+  \Infer[$\neg_1$]
+  {
+    \Gamma, x: t_0{\wedge} t\vdash e_1: t'
+  }
+  {\Gamma,x:t_0\vdash \tcase {x} t {e_1}{e_2}: t'}
+  {t_0\wedge\neg t = \Empty}
+  \quad
+  \Infer[$\neg_2$]
+  {
+    \Gamma, x: t_0{\wedge}
+    \neg t\vdash e_2: t'
+  }
+  {\Gamma,x:t_0\vdash \tcase {x} t {e_1}{e_2}: t'}
+  {t_0\wedge t = \Empty}
+  \and
+    \Infer[$\neg$]
+  {
+    \Gamma, x: t_0{\wedge} t\vdash e_1: t' \\ \Gamma, x: t_0{\wedge}
+    \neg t\vdash e_2: t'
+  }
+  {\Gamma,x:t_0\vdash \tcase {x} t {e_1}{e_2}: t'}
+  {\text{(otherwise)}}
+  \\
+  \Infer[$\vee$]
+    {\Gamma\vdash e':\textstyle\bigvee_{i=1..n}t_i\\i=1..n \\ \Gamma, x:t_i\vdash e:t}
+    {
+    \Gamma\vdash\letexp {x{:}\{t_1,...,t_n\}} {e'} e : t
+    }
+    { }
+\end{mathpar}