new system

new_system_beppe.tex

@@ -12,13 +12,22 @@
 
 \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]
+    \textbf{Expressions} &e &::=& 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]
+    \textbf{Values} & v  &::=& c\alt\lambda x.e\alt (v,v)
   \end{array}
 \end{equation}
 
+\subsubsection{Dynamic semantics}
+\begin{equation}
+  (\lambda x.e)v\reduces e\subst x{v}
+\end{equation}
+plus context rules to implement an leftmost outermost weak reduction semantics.
+
+
 \subsubsection{Typing rules}
-Rules of Definition 3.5 of Barbanera Dezani + pairs + negation
+Rules of Definition 3.5 of Barbanera Dezani + different union rule + pairs + negation
 
 \begin{mathpar}
     \Infer[Const]
@@ -62,20 +71,24 @@ Rules of Definition 3.5 of Barbanera Dezani + pairs + negation
   {\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 }
+  \Infer[Let]
+    {\Gamma \vdash e':t\and\Gamma, x:t'\vdash e:t}
     {
     \Gamma\vdash\letexp x {e'} e : t
     }
     { }
+  \Infer[$\vee$]
+  {\Gamma, x:t_1 \vdash e:t \and \Gamma,x:t_2 \vdash e:t}
+  {\Gamma x:t_1\vee t_2 \vdash e: t}
+  { }
+  \qquad 
     \\
-\Infer[$\neg_1$]
+ \Infer[$\wedge$]
+  {\Gamma \vdash e:t_1 \and \Gamma \vdash e:t_2}
+  {\Gamma \vdash e: {t_1}\wedge {t_2}}
+  { }
+  \qquad
+  \Infer[$\neg_1$]
   {
     \Gamma \vdash e:t \and \Gamma\vdash e_1: t_1
   }
@@ -95,6 +108,36 @@ Rules of Definition 3.5 of Barbanera Dezani + pairs + negation
     { }
 \end{mathpar}
 
+A different union rule, closer to the one by Barbanera Dezani is the
+following one, which is meant to replace both the union rule and the
+let rule above.
+\begin{mathpar}
+\Infer[$\vee$]
+    {     \Gamma \vdash e':t_1\vee t_2 \and
+      \Gamma, x:t_1\vdash e:t\and \Gamma, x:t_2\vdash e:t
+    }  
+    {
+    \Gamma\vdash\letexp x {e'} e : t
+    }
+    { }
+\end{mathpar}
+or better the following one that allows to deduce nested unions
+without reserting to nested let's.
+\begin{mathpar}
+\Infer[$\vee$]
+    {\Gamma \vdash e':\textstyle\bigvee_{i\in I} t_i
+        \and {\small(\forall i\in I)} \quad \Gamma, x:t_i\vdash e:t
+    }  
+    {
+    \Gamma\vdash\letexp x {e'} e : t
+    }
+    { }
+\end{mathpar}
+here however we can express less typings essentially because we use
+the let expression instead of the substitution meta-notation as in
+Barbanera-Dezani. However the idea is that on closed terms the two
+systems should be equivalent. 
+
 
 \subsection{Algorithmic type system}
 
@@ -127,6 +170,96 @@ just syntactic sugar for  $\letexp x {e}{\tcase {x} t {e_1}{e_2}}$ and
 
 So we  find exactly the typing rule for Cduce's typecase.
 
+Now we want to find a way to find a way to encode the derivations of
+the previous system so that every term has a unique type derivation
+that represent a family of type derivations in the previous system. To
+that end we add to the terms of the previous system explicit type
+annotations (ranged over by \anns) yielding the following
+definition.
+
+\subsubsection{Explicitly annotated terms}
+\begin{equation}
+  \begin{array}{lrclr}
+    \textbf{Annotations} & \anns &::=& \{\ann\Gamma t,\ldots{,}\ann\Gamma t\}\\ 
+    \textbf{Expressions} &e &::=& c\alt x\alt\lambda x{:}\anns.e
+    \alt e e\alt (e,e)\alt \pi_i e \alt  \letexp{x{:}\anns}{e}{e} \alt
+    \tcase{e}{t}{e}{e}\\[.3mm]
+    \textbf{Values} & v  &::=& c\alt\lambda x{:}\anns.e\alt (v,v)
+  \end{array}
+\end{equation}
+\subsubsection{Dynamic semantics}
+\begin{equation}
+  (\lambda x.e)v\reduces e\subst x{v}
+\end{equation}
+plus context rules to implement an leftmost outermost weak reduction semantics.
+
+\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 \vdash e_1: t' \\ \Gamma(x)\leq t
+  }
+  {\Gamma\vdash \tcase {x} t {e_1}{e_2}: t'}
+  {x\in\dom{\Gamma}}
+  \quad
+  \Infer[$\neg_2$]
+  {
+    \Gamma\vdash e_2: t' \\ \Gamma(x)\leq\neg t
+  }
+  {\Gamma\vdash \tcase {x} t {e_1}{e_2}: t'}
+  {x\in\dom{\Gamma}}
+  %% \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}s_i\\i=1..n \\ \Gamma, x:s_i\vdash e:t_i}
+    {
+    \Gamma\vdash\letexp {x{:}\{s_1,...,s_n\}} {e'} e :\textstyle \bigvee_{i=1..n}t_i
+    }
+    { }
+\end{mathpar}
+
 \subsubsection{Normal form terms with explicit annotations}
 
 \begin{equation}\label{expressions2}

setup.sty

@@ -84,6 +84,8 @@
 \newcommand{\Gp}[2]{\textsf{Env}^{#1}_{#2}}
 \newcommand{\ite}[4]{\ensuremath{\texttt{if}\;#1\in#2\;\texttt{then}\;#3\;\texttt{else}\;#4}}
 \newcommand{\Let}[3]{\ensuremath{\texttt{let}\;#1\;\texttt{=}\;#2\;\texttt{in}\;#3}}
+\newcommand{\anns}{\ensuremath{A}}
+\newcommand{\ann}[2]{#1{\triangleright}#2}
 \newcommand{\alt}{~|~}
 \newcommand{\arrow}[2]{#1\to #2}
 \newcommand{\pair}[2]{#1\times #2}
@@ -313,4 +315,4 @@
 
 \makeatletter % allow us to mention @-commands
 \def\arcr{\@arraycr}
-\makeatother
\ No newline at end of file
+\makeatother