Remove comments (cleanup). Add Clear and Exclude rules. Change Unify Soundness premise

constraints.tex

@@ -144,11 +144,11 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
     \typeExpr{} &({\mtypeEnvironment} , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2) = \\
                 & \begin{array}{ll}
                     \textbf{let} 
-                    & \tv{e} \ \text{fresh} \\
+                    & \tv{e}, \tv{x} \ \text{fresh} \\
                     & \consSet_R = \typeExpr({\mtypeEnvironment}, \expr{e}_1, \tv{e})\\
                     & \constraint =
                       \set{
-                      \tv{e} \lessdotCC \tv{x} 
+                      \tv{e} \lessdot \tv{x} 
                       }\\
                     {\mathbf{in}} & {
                     \consSet_R \cup \set{\constraint}} \cup \typeExpr(\mtypeEnvironment \cup \set{\expr{x} : \tv{x}})
@@ -190,7 +190,7 @@ The ones to already typed methods and calls to untyped methods.
       \textbf{let}
       & \tv{r}, \ol{\tv{r}} \text{ fresh} \\
       & \constraint = [\overline{\wtv{b}}/\ol{Y}]\set{
-                    \ol{S} \lessdot \ol{T}, \type{T} \lessdot \tv{a}, 
+                    \ol{S} \lessdotCC \ol{T}, \type{T} \lessdot \tv{a}, 
                     \ol{Y} \lessdot \ol{N} }\\
       \mathbf{in} & (\consSet_R \cup \overline{\consSet} \cup \constraint, \type{T}) \\
       & \mathbf{where}\ \begin{array}[t]{l}

soundness.tex

@@ -329,18 +329,17 @@ If there is a solution for a constraint set $C$, then there is also a solution f
 % does this really work. We have to show that the algorithm never gets stuck as long as there is a solution
 % maybe there are substitutions with types containing free variables that make additional solutions possible. Check!
 
-\begin{lemma}\label{lemma:unifySoundness}
+\begin{lemma}{\unify{} Soundness:}\label{lemma:unifySoundness}
-The \unify{} algorithm only produces correct output.
+\unify{}'s type solutions are correct respective to the subtyping rules defined in figure \ref{fig:subtyping}.
 \begin{description}
 \item[If] $(\sigma, \Delta) = \unify{}( \Delta', \, \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdotCC \type{T'} } )$ %\cup \overline{ \type{S} \doteq \type{S'} })$
-%\item[and] $fv(\overline{ \type{S} }) = \emptyset$, $fv(\overline{ \type{T} }) = \emptyset$
+\item[Then] there exists a substitution $\sigma'$ and a set of types $\overline{\wcNtype{\Delta}{N}}$ with:
-\item[Then] there exists a $\sigma'$ with:
+\begin{itemize}
-$\sigma \subseteq \sigma'$ and
+\item $\sigma \subseteq \sigma'$
-$\Delta, \Delta' \vdash \overline{\sigma'(\type{S}) <: \sigma'(\type{T})}$
+\item $\Delta, \Delta' \vdash \overline{\sigma'(\type{S}) <: \sigma'(\type{T})}$
-and $\Delta, \Delta', \overline{\Delta} \vdash \overline{\type{N} <: \sigma'(\type{T'})}$ where $\overline{\sigma(\type{S'}) = \wcNtype{\Delta}{N}}$,
+\item $\Delta, \Delta' \vdash \overline{\sigma'(\type{S'}) <: \wcNtype{\Delta}{N}}$ 
-otherwise $\Delta, \Delta' \vdash \overline{\sigma'(\type{S'}) <: \sigma'(\type{T'})}$
+\item $\Delta, \Delta', \overline{\Delta} \vdash \overline{\type{N} <: \sigma'(\type{T'})}$
-% and $\sigma(\type{T'}) = \sigma(\type{T'})$.
+\end{itemize}
-
 \end{description}
 \end{lemma}
 

unify.tex

@@ -259,11 +259,8 @@ $
 \end{figure}
 
 
-The capture constraints are preserved when applying the \rulename{Upper} rule.
+%The capture constraints are preserved when applying the \rulename{Upper} rule.
-%because \texttt{let} statements like
+% 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
-%let x : X = v in
-%can be transformed to 
-%let x : U = v in 
 
 \begin{figure}
   \begin{center}
@@ -439,6 +436,11 @@ The capture constraints are preserved when applying the \rulename{Upper} rule.
         \wildcardEnv \vdash C \cup \set{\type{N} \doteq \rwildcard{A}}\\
         \hline
         \wildcardEnv \vdash C \cup \set{\rwildcard{A} \doteq \type{N}}
+        \end{array} \quad
+        \begin{array}[c]{l}
+        \wildcardEnv \vdash C \cup \set{\ntv{a} \doteq \rwildcard{A}}\\
+        \hline
+        \wildcardEnv \vdash C \cup \set{\rwildcard{A} \doteq \ntv{a}}
         \end{array}$
 \end{tabular}}
 \end{center}
@@ -583,6 +585,46 @@ Their upper and lower bounds are fresh type variables.
     \end{array}
     $
   \\\\
+  \rulename{Clear}
+  & 
+  $
+  \begin{array}[c]{@{}ll}
+  \begin{array}[c]{l}
+      \wildcardEnv \cup \set{\wildcard{A}{\type{U}}{\type{L}}} \vdash 
+      C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\ 
+    \hline
+    \vspace*{-0.4cm}\\
+    \subst{\type{U}}{\rwildcard{A}}\wildcardEnv \vdash 
+      [\type{U}/\rwildcard{A}]C \cup \, [\type{U}/\rwildcard{A}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T}, \type{U} \doteq \type{L} } \\  
+  \end{array}
+   %\quad \ol{Y} = \textit{fresh}(\ol{X})
+   \quad \begin{array}[c]{l}
+    \Delta \neq \emptyset\\
+    \rwildcard{A} \in \text{fv}(\type{T})
+    \end{array}
+  \end{array}
+  $
+\\\\
+\rulename{Exclude}
+& 
+$
+\begin{array}[c]{@{}ll}
+\begin{array}[c]{l}
+    \wildcardEnv \vdash 
+    C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\ 
+  \hline
+  \vspace*{-0.4cm}\\
+  \subst{\tv{a}}{\wtv{a}}\wildcardEnv \vdash 
+    [\tv{a}/\wtv{a}]C \cup \, [\tv{a}/\wtv{a}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T}, \type{U} \doteq \type{L} } \\  
+\end{array}
+ %\quad \ol{Y} = \textit{fresh}(\ol{X})
+ \quad \begin{array}[c]{l}
+  \Delta \neq \emptyset\\
+  \wtv{a} \in \text{fv}(\type{T}), \tv{a} \ \text{fresh}
+  \end{array}
+\end{array}
+$
+\\\\
         \rulename{Adopt}
         & $
         \begin{array}[c]{@{}ll}
@@ -1002,137 +1044,6 @@ This builds a search tree over multiple possible solutions.
   \label{fig:step2-rules}
   \end{figure}
 
-% \begin{figure}
-% \begin{center}
-%         \fbox{
-%         \begin{tabular}[t]{l@{~}l}
-%         \rulename{Same}
-%         & $
-%         \begin{array}[c]{l}
-%           \wildcardEnv \vdash
-%           C \cup \type{G} \lessdot \tv{a}\\
-%           \hline
-%           \wildcardEnv \vdash
-%           C \cup \set{
-%             \tv{a} \doteq \type{G}
-%         }
-%         \end{array} \quad \begin{array}[c]{l}
-%           \text{fv}(\type{G}) \in \Delta_{in}
-%         \end{array}
-%           $
-%       \\\\ 
-%           \rulename{\generalizeRule}
-%           & $
-%           \begin{array}[c]{l}
-%             \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
-%             \hline
-%             \wildcardEnv \vdash C \cup \set{\wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a},
-%           \tv{a} \doteq \wctype{\overline{\wildcard{X}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{X}}},
-%           %\overline{\tv{l} \lessdot \tv{u}}, % not needed, due to subst and reduce rule which are used afterwards
-%           \overline{\tv{u} \lessdot \type{S}}
-%           }
-%           \end{array} \quad \begin{array}[c]{l}
-%             \texttt{class} \ \exptype{C}{\ol{X \triangleleft \type{S}}} \triangleleft \exptype{D}{\ol{N}} \\
-%             \text{fresh}\ \overline{\wildcard{X}{\tv{u}}{\tv{l}}}
-%           \end{array}
-%             $
-%         \\\\ 
-%         \rulename{Subst-X}
-%         & $
-%         \begin{array}[c]{l}
-%           \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
-%           C \cup \rwildcard{X} \lessdot \tv{a}\\
-%           \hline
-%           \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
-%           C \cup \set{
-%             \tv{a} \doteq \rwildcard{X}
-%         }
-%         \end{array} \quad \begin{array}[c]{l}
-%           \rwildcard{X} \in \Delta_{in}
-%         \end{array}
-%           $
-%       \\\\ 
-%       \rulename{Gen-X}
-%       & $
-%       \begin{array}[c]{l}
-%         \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
-%         C \cup \rwildcard{X} \lessdot \tv{a}\\
-%         \hline
-%         \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
-%         C \cup \set{
-%           \type{U} \lessdot \tv{a}
-%       }
-%       \end{array}
-%         $
-%     \\\\ 
-%         \rulename{Super}
-%         & $
-%         \begin{array}[c]{l}
-%           \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
-%           \hline
-%           \wildcardEnv \vdash C \cup \set{ \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{N}} \lessdot \tv{a}   }
-%         %\set{\wctype{\ol{\wtype{W}}}{D}{[\ol{X}/\ol{Y}]\ol{Z}} \lessdot \tv{a}}
-%         \end{array} \quad
-%         \begin{array}{l}
-%           \texttt{class} \ \exptype{C}{\ol{X}} \triangleleft \exptype{D}{\ol{N}} \\
-%           \ol{X} \notin \wildcardEnv \cup \Delta,\, \Delta' = \Delta \cap \text{fv}([\ol{T}/\ol{X}]\ol{N})
-%         \end{array}
-%         $
-%       \\\\ 
-%       \rulename{Settle}
-%       & $
-%       \begin{array}[c]{l}
-%         \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
-%         \tv{a} \lessdot \tv{b}}
-%         \\
-%         \hline
-%         \wildcardEnv \vdash C \cup \set{  \tv{a} \lessdot \tv{b}, \tv{b} \lessdot \type{N}  }
-%       \end{array}
-%       $
-%     \\\\ 
-%     \rulename{Raise}
-%     & $
-%     \begin{array}[c]{l}
-%       \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
-%       \tv{a} \lessdot \tv{b}}
-%       \\
-%       \hline
-%       \wildcardEnv \vdash C \cup \set{\tv{a} \lessdot \type{N}, \type{N} \lessdot \tv{b}  }
-%     \end{array}
-%     $
-%   \\\\ 
-%     \end{tabular}}
-%   \end{center}
-% \caption{Step 2 branching: Multiple rules can be applied to the same constraint}
-% \label{fig:step2-rules}
-% \end{figure}
-
-
-%For every $\type{T} \lessdot \tv{a}$ constraint, the unify algorithm has to consider every possible supertype of $\type{T}$.
-%For every $\type{N}$ with $\type{T} \leq \type{N}$: ($\texttt{class} \ \exptype{T}{\ol{Y} \triangleleft \ol{N}} \triangleleft \type{N}$) 
-%There are two different ways of handling a $\type{T} \lessdot \tv{a}$ constraint:
-%TODO: why the \generalizeRule is basically the Same rule for regular type placeholders
-
-%where is the mistake in the old unify algorithm? 
-%when working with equality the problems arise! Free variables should not escape their scope
-% Replacing regular type placeholders causes problems related to method calls and capture conversion.
-
-% <X> List<X> same(List<X> a, List<X> b){}
-
-% This program has no correct type. the same method requires 
-% \begin{lstlisting}
-% List<?> f;
-% List<?> problem(){
-%   return same(problem(), problem()) ?: f;
-% }
-% \end{lstlisting}
-% \begin{constraints}
-% r <. List<x?>
-% r <. List<x?>
-% X.List<X> <. r
-% \end{constraints}
-% %TODO
-
 \unify{} generates wildcard types for constraints of the form $\type{N} \lessdot \tv{a}$ with the \rulename{\generalizeRule} rule.
 Otherwise only the wildcards already defined in the input constraints will be included in the result.
 \rulename{\generalizeRule} attempts to give $\tv{a}$ a more general type by replacing only the type parameters with fresh wildcards.
@@ -1145,15 +1056,6 @@ need to be handled in a similiar fashion.
 The type variable $\tv{b}$ could either be a sub or a supertype of the type $\type{N}$.
 We have to consider both possibilities.
 \\[1em]
-% The specification of the \unify{} algorithm has only two rules generating $\doteq$-Constraints
-% , \rulename{Reduce} and \rulename{Ground}.
-% $\doteq$-Constraints and the accompaning substitutions are dangerous respective to the soundness of the algorithm.
-% For the soundness proof of the \unify{} algorithm we have to show every generation of equals constraints
-% and the subsequent application of the \rulename{subst} rule is correct.
-% We try to use them as sparsely as possible to simplify the soundness proof.
-% You can notice this at \rulename{Equals} or \rulename{Force}:
-% Instead of setting $\type{U} \doteq \type{L}$, we say
-% $\type{U} \lessdot \type{L}, \type{L} \lessdot \type{U}$.
 
 \noindent
 \textbf{Step 3} 
@@ -1164,23 +1066,6 @@ If $C$ does not contain any wildcard variables the algorithm proceeds with step
 \begin{center}
   \fbox{
   \begin{tabular}[t]{l@{~}l}
-%     \rulename{Remove} % kann beim Subst step gemacht werden und ist nur nötig wenn ein a =. T constraint mit wtv(T) > 1 entsteht
-%     & $
-%     \begin{array}[c]{@{}ll}
-%     \begin{array}[c]{l}
-%        \wildcardEnv \vdash C \\
-%         % \cup \, \set{ \wtv{a} \lessdot \type{T} }\\ 
-%       \hline
-%       \vspace*{-0.4cm}\\
-%       \subst{\tv{a}}{\wtv{a}}\wildcardEnv \vdash [\tv{a}/\wtv{a}]C 
-%     \end{array}
-%     &\begin{array}[c]{l}
-%       \wtv{a} \in C \\
-%       \tv{a} \ \text{fresh}
-%    \end{array}
-%     \end{array}
-% $
-% \\\\
 \rulename{Remove-Cons} & $
     \begin{array}[c]{l}
        \wildcardEnv \vdash C \cup \set{\type{S} \lessdotCC \type{T} } \\
@@ -1192,23 +1077,6 @@ If $C$ does not contain any wildcard variables the algorithm proceeds with step
 \end{tabular}}
 \end{center}
 
-% \textbf{Step 4:}
-% If there are constraints of the form $(\tv{a} \lessdot \tv{b})$ remaining in the constraint set then
-% apply the \rulename{Sub-Elim} rule and start over with step 1.
-% Otherwise continue to step 5.
-% \begin{center}
-%   \fbox{
-%   \begin{tabular}[t]{l@{~}l}
-%     \rulename{SubElim} 
-%     & $\begin{array}[c]{l}
-%       \wildcardEnv \vdash C \cup \set{\tv{a} \lessdot \tv{b}}\\
-%       \hline
-%       [\tv{a}/\tv{b}]\wildcardEnv \vdash [\tv{a}/\tv{b}]C \cup \set{ \tv{b} \doteq \tv{a} }
-%     \end{array}
-%     $
-% \end{tabular}}
-% \end{center}
-
 \noindent
 \textbf{Step 4:}
 We apply the rules in figure \ref{fig:cleanup-rules} exhaustively and proceed with step 6.
@@ -1242,43 +1110,6 @@ C \cup \set{ \tv{a} \doteq \bot }
 %\text{length}( \overline{\tv{a} \lessdot \type{T}} ) > 1
 \end{array}
 \end{array}$
-% \\\\  % The force rule is unnecessary because every type placeholder has an upper bound Object (a <. Object) The match rule eliminates those wildcards
-% \rulename{Force} &$
-%   \begin{array}[c]{@{}ll}
-%   \begin{array}[c]{l}
-%      \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
-% \vdash C \cup \, \set{
-%         \tv{a} \lessdot \type{N} } \\ 
-%     \hline
-%     \vspace*{-0.4cm}\\
-%     \wildcardEnv 
-% \vdash
-%     C \cup \, \set{ \tv{a} \lessdot [\type{U}/\rwildcard{X}]\type{N}, 
-%      \type{U} \doteq \type{L} }
-%   \end{array}
-%   &\begin{array}[c]{l}
-%     \type{X} \in \text{fv}(\type{N}) \\
-%     %\Delta' = \Delta \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
-%  \end{array}
-%   \end{array}$
-% \\\\
-% \rulename{FlatOut} &$
-%   \begin{array}[c]{@{}ll}
-%   \begin{array}[c]{l}
-%      \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
-% \vdash C \cup \, \set{
-%         \tv{a} \lessdot \wcNtype{\Delta}{N} } \\ 
-%     \hline
-%     \vspace*{-0.4cm}\\
-%     \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
-% \vdash
-%     C
-%   \end{array}
-%   &\begin{array}[c]{l}
-%     \type{X} \in \text{fv}(\wcNtype{\Delta}{N}) \\
-%     \tv{a} \notin C , \, \tv{a} \notin \wildcardEnv, \tv{a} \notin \sigma
-%  \end{array}
-%   \end{array}$
   \end{tabular}}
 \end{center}
 \caption{Cleanup rules}\label{fig:cleanup-rules}