Explain let scoping in unify

unify.tex

@@ -115,14 +115,47 @@ The vital part are the \rulename{Subst} and \rulename{Normalize} rules.
 They assert that a normal type placeholder is never replaced by a type containing free variables.
 \rulename{Normalize} replaces Wildcard placeholders with normal placeholders right before they get substituted by \rulename{Subst}.
 The idea is to keep the possibility of replacing a wildcard placeholder with a free variable as long as possible.
-As soon as they appear in a $\ntv{a} \doteq \type{T}$ constraint they can no longer be replaced by free variables.
+As soon as they appear in a $\ntv{a} \doteq \type{T}$ constraint they have to be replaced with normal placeholders.
 
 A type solution for a normal type placeholder will never contain free variables.
 This is needed to guarantee well-formed type solutions and also keep free variables inside their scope.
 
+\begin{example}{Free variables must not leave the scope of the surrounding \texttt{let} statement}
+
+\noindent
+\begin{minipage}{0.40\textwidth}
 \begin{lstlisting}
-let y = { let x = v in v.get() } in y.get()
-\end{lstlisting} %TODO: explain: here y has to be a type without free variables.
+m(l) = let v = l in v.get()
+\end{lstlisting} 
+\end{minipage}%
+\hfill
+\begin{minipage}{0.59\textwidth}
+\begin{constraintset}
+$\tv{l} \lessdot \tv{v}, \tv{v} \lessdotCC \exptype{List}{\wtv{x}},
+\wtv{x} \lessdot \tv{r}$
+\end{constraintset}
+\end{minipage}
+Lets assume the variables \texttt{l} and \texttt{v}
+get the type $\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}}$ assigned to them.
+After application of the \rulename{Capture} and \rulename{SubstWC} rules the constraint set looks like this:
+
+$\begin{array}[c]{l}
+\wctype{\wildcard{X}{\type{Object}}{\type{String}}}{List}{\rwildcard{X}} \lessdotCC \exptype{List}{\wtv{x}}, \wtv{x} \lessdot \ntv{r}
+  \\
+\hline
+\vspace*{-0.4cm}\\
+\wildcard{X}{\type{Object}}{\type{String}} \vdash \wtv{x} \doteq \rwildcard{X}, \rwildcard{X} \lessdot \ntv{r}
+\end{array}
+$
+
+Replacing $\tv{r}$ with $\rwildcard{X}$ would solve the constraint set but lead to the method type
+
+\texttt{X m(List<? super String> l)} which makes no sense.
+The normal type placeholder $\ntv{r}$ has to be replaced by a type without free type variables
+($\ntv{r} \doteq \type{Object}$) leading to
+
+\texttt{Object m(List<? super String> l)}.
+\end{example}
 
 \subsection{Algorithm}
 
@@ -438,18 +471,14 @@ C \cup [\type{U}/\type{A}]\set{\ntv{a} \doteq \type{T}, \type{L} \doteq \type{U}
 There are n different rules to deal with $\type{N} \lessdot \type{N}$ constraints.
 Prepare, Capture, Reduce, Trim, Clear, Exclude, Adapt
 
+% % TODO: 
+% a <c C<X> 
+% -------------
+% a <. X.C<X>, X.C<X> <c C<X>
 
-a <c C<x?>
-x? =. X
-a <c C<X>
-
-a <c C<X> 
--------------
-a <. X.C<X>, X.C<X> <c C<X>
-
-a <. C<X>
----------
-a <. C<U>, U = L
+% a <. C<X>
+% ---------
+% a <. C<U>, U = L
 
 %The capture constraints are preserved when applying the \rulename{Upper} rule.
 % This is legal: a T <c S constraint indicates a let-statement can be inserted. Therefore there must exist a type T' with T <. T' <c S