Add Fixpoint example and fix some errors in table one.

code_table.tex

@@ -53,7 +53,7 @@ let or_ = fun (x : Any) -> fun (y: Any) ->
  else if y is True then true else false
 \end{lstlisting}
  &\vfill
-   $(\True\to\textsf{Any}\to\True)\land(\textsf{Any}\to\True\to\True)\;\land$\newline
+   $(\True\to\textsf{Any}\to\True)\land(\lnot\Keyw{True}\to\True\to\True)\;\land$\newline
    $(\lnot\True\to\lnot\True\to\False)$
 \\\hline
     6 &
@@ -137,10 +137,9 @@ $(\lnot\True\to((\True\to\True) \land (\lnot\True\to\False))$
     \\\hline
  10 & \begin{lstlisting}
 (* f, g have type: (Int->Int) & (Any->Bool) *)
-
 let example10 = fun (x : Any) ->
   if (f x, g x) is (Int, Bool) then 1 else 2
-\end{lstlisting} &\vfill\medskip\smallskip
+\end{lstlisting} &\vfill
 $(\Int\to\textsf{Empty})\land(\neg\Int\to{}2)$\newline
 \texttt{Warning: line 4, 39-40: unreachable expression}
     \\\hline
@@ -172,22 +171,17 @@ $((\Keyw{Nil}\,|\,\orecord{\texttt{l}\,=\,?\Keyw{Empty},\, \texttt{prototype}\,=
 $\Keyw{Object}\to\Keyw{Any}$
     \\\hline
   12 & \begin{lstlisting}
-atom character
-atom boolean
-atom undefined
-type String = Number | Character | Boolean | Undefined
-
-let typeof = fun (x:Any) ->
-    if x is Int then number
+(* number, character,... are atoms that represent
+   the strings "number", "string",... from JS. *)
+let typeof = fun (x:Any) -> if x is Int then number
     else if x is Char then character
-    else if x is Bool then boolean
-    else undefined
+    else if x is Bool then boolean else object
 
 let test = fun (x:Any) ->
     if typeof x is Number then incr x
     else if typeof x is Character then charcode x
     else if typeof x is Boolean then int_of_bool x
-    else 0\end{lstlisting} &\vfill\bigskip\medskip\smallskip
+    else 0\end{lstlisting} &\vfill\smallskip
 $(\Int\to\textsf{Number}) \wedge$\newline
 $(\Char\to\textsf{Character})\wedge$\newline
 $(\Bool\to\textsf{Boolean})\wedge$\newline
@@ -199,6 +193,17 @@ $(\Int \to \Int) \wedge (\Char \to \Int)  \wedge
 $(\lnot(\Bool{\vee} \Int {\vee} \Char) \to 0)\wedge  \ldots$\newline
     (two other redundant cases omitted) 
     \\\hline
+    13 & \begin{lstlisting}
+type X = X -> Empty -> Any
+let z = fun (((Empty -> Any) -> Empty -> Any ) ->
+   (Empty -> Any)) f ->
+      let delta = fun ( X -> (Empty -> Any) ) x ->
+          f ( fun (Empty -> Any) v -> ( x x v ))
+        in delta delta    
+    \end{lstlisting} &\vfill
+    $(\Empty\to\Any)\to\Empty\to\Any$
+    \\\hline
+
 \end{tabular}
 }
 \caption{Types inferred by the implementation}

practical.tex

@@ -212,7 +212,7 @@ define a well-typed fix-point combinator (for the reader convinience
 we show its definition both in \Appendix\ref{sec:fixpoint} and in our
 on-line prototype).
 
-Finally, Code~12 implements the typical type-switching pattern used in
+Code~12 implements the typical type-switching pattern used in
 JavaScript. While languages such as Scheme and Racket provides specific
 type predicates for each type---predicates that in our system must not
 be provided since they can be
@@ -228,6 +228,13 @@ system. Since our prototype has no type for strings, then we
   Table~\ref{tab:implem} is
   equivalent to $(\textsf{Any}\to\Int)\wedge(\lnot(\Bool{\vee} \Int {\vee} \Char) \to 0)$).
 
+Finally, Code~13 shows that, thanks to recursive function types, one can easily
+define and typecheck a fixpoint combinator. We demonstrate it by first defining a
+recursive function type $X=(X\to \Empty\to\Any)$ and then using it to annotate
+the functions occuring in the definition of $Z$ (the eta-expanded version of
+Curry's $Y$ combinator).
+
+
 \input{code_table2}
 In Table~\ref{tab:implem2}, we reproduce in our syntax the 14
 archetypal examples of