diff --git a/TIforWildFJ.tex b/TIforWildFJ.tex
index 3157828..1598393 100644
--- a/TIforWildFJ.tex
+++ b/TIforWildFJ.tex
@@ -126,7 +126,7 @@ $\begin{array}{rcl}
 
 \input{tRules}
 
-\input{tiRules}
+%\input{tiRules}
 
 \input{constraints}
 
diff --git a/constraints.tex b/constraints.tex
index 15bacba..9ad78f2 100644
--- a/constraints.tex
+++ b/constraints.tex
@@ -1,14 +1,15 @@
 \section{Constraint generation}
-Our type inference algorithm is split into two parts.
-A constraint generation step \textbf{TYPE} and a \unify{} step.
+% Our type inference algorithm is split into two parts.
+% A constraint generation step \textbf{TYPE} and a \unify{} step.
 
-Method names are not unique.
-It is possible to define the same method in multiple classes.
-The \TYPE{} algorithm accounts for that by generating Or-Constraints.
-This can lead to multiple possible solutions.
+% Method names are not unique.
+% It is possible to define the same method in multiple classes.
+% The \TYPE{} algorithm accounts for that by generating Or-Constraints.
+% This can lead to multiple possible solutions.
 
 %\subsection{Well-Formedness}
 
+The \fjtype{} algorithm assumes capture conversions for every method parameter.
 
 %Why do we need a constraint generation step?
 %% The problem is NP-Hard
diff --git a/introduction.tex b/introduction.tex
index 82210e6..13d5872 100644
--- a/introduction.tex
+++ b/introduction.tex
@@ -17,11 +17,21 @@ In Java this is done implicitly by a process called capture conversion \cite{Jav
 The type system in \cite{WildcardsNeedWitnessProtection} makes this process explicit by using \texttt{let} statements.
 Our type inference algorithm will accept an input program without let statements and add them where necessary.
 
-The input to the type inference algorithm is a Featherweight Java program without let statements (see figure \ref{fig:syntax}).
+The input to the type inference algorithm is a Featherweight Java program (example in figure \ref{fig:nested-list-example-typeless}) conforming to the syntax shown in figure \ref{fig:syntax}.
 Type inference adds \texttt{let} statements in a fashion similar to the Java capture conversion \cite{WildFJ}.
+We wrap every parameter of a method invocation in \texttt{let} statement unknowing if a capture conversion is necessary (see figure \ref{fig:nested-list-example-let}).
+The \fjtype{} algorithm calculates constraints based on this intermediate representation,
+which are then solved by the \unify{} algorithm
+resulting in a correctly typed program (see figure \ref{fig:nested-list-example-typed}).
 
 We figured the \texttt{let} statements to be obsolete for our use case.
-Once the type inference algorithm found a correct type solution 
+Once the type inference algorithm found a correct type solution they can be inferred by the given type annotations.
+
+% 1. Constraint generation
+% 2. Insert typing
+
+% # Showing soundness
+% Every program in our calculus can be converted to a WildcardsNeedWitnessProtection program
 
 \begin{figure}[tp]
       \begin{subfigure}[t]{\linewidth}
@@ -48,6 +58,27 @@ class List<A> {
     List<A> add(A v) { ... }
 }
 
+class Example {
+    m(l, la, lb){
+        return let r2 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = {
+            let r1 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = l in {
+                let p1 : (*@$\exptype{List}{\type{Integer}}$@*) = {
+                    let xa = la in xa.add(1)
+                    } in x1.add(p1)    
+            } in {
+                let p2 = {
+                    let xb = lb in xb.add("str")
+                    } in x2.add(p2)
+            };
+    }
+}
+\end{lstlisting}
+
+\begin{lstlisting}[style=tfgj]
+class List<A> {
+    List<A> add(A v) { ... }
+}
+
 class Example {
     m((*@$\exptype{List}{\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}}$@*) l, List<Integer> la, List<String> lb){
         return l 
@@ -57,7 +88,7 @@ class Example {
 }
 \end{lstlisting}
         \caption{Featherweight Java Representation}
-        \label{fig:nested-list-example-typed}
+        \label{fig:nested-list-example-let}
       \end{subfigure}
       ~
       \begin{subfigure}[t]{\linewidth}
diff --git a/prolog.tex b/prolog.tex
index a185b1c..9504553 100755
--- a/prolog.tex
+++ b/prolog.tex
@@ -198,8 +198,8 @@
 \end{tcolorbox}
     }
 
-\newcommand{\wcNtype}[2]{#1 .\ntype{#2}}
-\newcommand{\wctype}[3]{#1 .\exptype{#2}{#3}}
+\newcommand{\wcNtype}[2]{\exists #1 .\ntype{#2}}
+\newcommand{\wctype}[3]{\exists #1 .\exptype{#2}{#3}}
 \newcommand{\wtype}[1]{\mathit{#1}}
 \newcommand{\ntype}[1]{\mathtt{#1}}
 
diff --git a/soundness.tex b/soundness.tex
index 7ffbe99..eebaaca 100644
--- a/soundness.tex
+++ b/soundness.tex
@@ -420,3 +420,152 @@ $\sigma(\wildcardEnv) = \sigma([\type{T}/\tv{a}]\wildcardEnv)$
 %Proof by Lemma 5 \emph{Type substitution preserves subtyping} from \cite{WildcardsNeedWitnessProtection}.
 Same as Subst
 \end{description}
+
+
+\subsection{Converting to Wild FJ}
+Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
+They have been merged with let statements and simplified.
+
+The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
+are necessary for the soundness proof.
+
+%TODO: Show that well-formed implies witnessed!
+We change the type rules to require the well-formedness instead of the witnessed property.
+See figure \ref{fig:well-formedness}.
+
+Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
+\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
+With \texttt{witnessed} being the stronger one.
+We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
+$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
+Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
+Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
+
+The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
+The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
+Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
+$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
+
+A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
+$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
+$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
+and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
+
+\begin{figure}[tp]
+$\begin{array}{l}
+    \typerule{T-Var}\\
+    \begin{array}{@{}c}
+        x : \type{T} \in \Gamma
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta | \Gamma \vdash x : \type{T}
+    \end{array}
+\end{array}$
+\\[1em]
+$\begin{array}{l}
+    \typerule{T-New}\\
+    \begin{array}{@{}c}
+        \Delta, \overline{\Delta} \vdash  \exptype{C}{\ol{T}} \ \ok \quad \quad
+        \text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
+        \Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
+        \Delta \vdash  \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
+        \Delta, \overline{\Delta} \vdash  \overline{\type{N}} <: \overline{\type{U}} \quad \quad
+        \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
+        \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
+        \Delta \vdash \type{T},  \overline{\wcNtype{\Delta}{N}} \ \ok
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+    \Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T}
+    \end{array}
+\end{array}$
+\\[1em]
+$\begin{array}{l}
+    \typerule{T-Field}\\
+    \begin{array}{@{}c}
+        \Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad 
+        \Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
+        \textit{fields}(\type{N}) = \ol{U\ f} \\
+\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
+\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
+\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
+    \end{array}
+\end{array}$
+\\[1em]
+$\begin{array}{l}
+    \typerule{T-Call}\\
+    \begin{array}{@{}c}
+        \Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
+        \textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad
+        \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
+        \\
+        \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
+        \Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad
+        \Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad
+        \Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
+        \Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
+        \\
+        \Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
+        \Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
+        \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
+        \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}
+        {\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
+    \end{array}
+\end{array}$
+\\[1em]
+$\begin{array}{l}
+    \typerule{T-Elvis}\\
+    \begin{array}{@{}c}
+        \Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad 
+        \Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad
+        \Delta \vdash \type{T}_1 <: \type{T} \quad \quad
+        \Delta \vdash \type{T}_2 <: \type{T}
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T}
+    \end{array}
+\end{array}$
+\\[1em]
+$\begin{array}{l}
+        \typerule{T-Method}\\
+    \begin{array}{@{}c}
+        \text{dom}(\Delta)=\ol{X} \quad \quad
+        \Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad
+        \Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad
+        \texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\
+        \Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad
+        \Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad 
+        \text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C}
+    \end{array}
+\end{array}$
+\\[1em]
+$\begin{array}{l}
+        \typerule{T-Class}\\
+    \begin{array}{@{}c}
+        \Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad 
+        \Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad
+        \Delta \vdash \type{N} \ \ok \quad \quad
+        \Delta \vdash \ol{M} \ \ok \texttt{ in C}
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok
+    \end{array}
+\end{array}$
+\caption{T-Call and T-Field} \label{fig:tletexpr}
+\end{figure}
+
diff --git a/tRules.tex b/tRules.tex
index c25eb1b..c19abf5 100644
--- a/tRules.tex
+++ b/tRules.tex
@@ -1,4 +1,15 @@
-\section{Syntax}
+\section{Syntax and Typing}
+
+The input syntax for our algorithm is shown in figure \ref{fig:syntax}
+and the respective type rules in figure \ref{fig:expressionTyping} and \ref{fig:typing}.
+
+The type system in \cite{WildcardsNeedWitnessProtection} allows a method to \textit{override} an existing method declaration in one of its super classes,
+but only by a method with the exact same type.
+The type system presented here does not allow the \textit{overriding} of methods.
+Our type inference algorithm consumes the input classes in succession and could only do a type check instead of type inference
+on overriding methods, because their type is already determined.
+Allowing overriding therefore has no implication on our type inference algorithm.
+
 \begin{figure}
 $
 \begin{array}{lrcl}
@@ -21,13 +32,12 @@ $
 \caption{Input Syntax}\label{fig:syntax}
 \end{figure}
 
-Each class type has a set of wildcard types $\overline{\Delta}$ attached to it.
-The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$,
-which can be used inside the type parameters $\ol{T}$.
-
-\section{Type rules}
+% Each class type has a set of wildcard types $\overline{\Delta}$ attached to it.
+% The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$,
+% which can be used inside the type parameters $\ol{T}$.
 
 \begin{figure}[tp]
+\begin{center}
 $\begin{array}{l}
     \typerule{S-Refl}\\
     \begin{array}{@{}c}
@@ -74,7 +84,7 @@ $\begin{array}{l}
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta \vdash \wctype{\Delta'}{C}{\ol{T}} <: \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{S}}
+        \Delta \vdash \wcNtype{\Delta'}{\type{N}} <: \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}}
     \end{array}
 \end{array}$
 $\begin{array}{l}
@@ -82,15 +92,16 @@ $\begin{array}{l}
     \begin{array}{@{}c}
         \Delta', \Delta \vdash [\ol{T}/\ol{\type{X}}]\ol{L} <: \ol{T} \quad \quad
         \Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\
-        \text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad \quad
+        \text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad
         \text{dom}(\Delta') \cap \text{fv}(\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}) = \emptyset
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta \vdash \wctype{\Delta'}{C}{[\ol{T}/\ol{\type{X}}]\ol{S}} <:
-        \wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}
+        \Delta \vdash \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}} <:
+        \wcNtype{\ol{\wildcard{X}{U}{L}}}{N}
     \end{array}
 \end{array}$
+\end{center}
 \caption{Subtyping}\label{fig:subtyping}
 \end{figure}
 
@@ -141,69 +152,32 @@ $\begin{array}{l}
 \end{center}
 \caption{Well-formedness}\label{fig:well-formedness}
 \end{figure}
-%TODO: Proof that well-formed (ok) implies that a type is witnessed
-We change the type rules to require the well-formedness instead of the witnessed property.
-See figure \ref{fig:well-formedness}.
-
-Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
-\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
-With \texttt{witnessed} being the stronger one.
-We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
-$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
-Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
-Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
-
-The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
-The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
-Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
-$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
-
-A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
-$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
-$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
-and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
-
-%Not necessary!
-% Additionally we do not allow nested wildcards. %TODO: Unify cannot create them (or does it?) What is when the input contains them
-
-% The \unify{} algorithm is not capable of generating types like $\wctype{\rwildcard{X}, \wildcard{Y}{\bot}{\rwildcard{X}}}{Pair}{\rwildcard{X}, \rwildcard{Y}}$.
-% %But they could be created in intermidiate types (\ol{S}) in method calls by capture conversion:
-% class WList<A> extends List<List<? extends A>>
-% m(List<X> l) % TODO
-% m(X.Wlist<X>) % S = Y^X.List<Y>
-
-% X.WList<X> <. List<x?>
-% X.List<Y^X.List<Y>> <. List<x?>
-% x =. Y^X.List<Y>
-
-% %TODO: do we need \Delta' \Delta \vdash T, L ,U ok ? or is \Delta \vdashh ... sufficient?
-% %<A> m(List<? extends A> l)
-% %This is important because it means different things maybe for the proof of our OK being more restrictive than "witnessed"
-
-% % where can nested wildcards occur?
-% WBox<X> extends Box<Box<? extends X>>
-% WBox<?> => Y.WBox<Y> <: Y.Box<X^Y.Box<X>>
-
-% Everything else regarding subtyping stays the same as in \cite{WildcardsNeedWitnessProtection}.
-% \begin{lemma}
-%     If $\Delta \vdash $
-% \end{lemma}
-
-Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
-They have been merged with let statements and simplified.
-
-The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
-are necessary for the soundness proof.
 
 \begin{figure}[tp]
+\begin{center}
 $\begin{array}{l}
     \typerule{T-Var}\\
     \begin{array}{@{}c}
-        x : \type{T} \in \Gamma
+        \texttt{x} : \type{T} \in \Gamma
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta | \Gamma \vdash x : \type{T}
+        \triangle | \Gamma \vdash \texttt{x} : \type{T}
+    \end{array}
+\end{array}$ \hfill
+$\begin{array}{l}
+    \typerule{T-Field}\\
+    \begin{array}{@{}c}
+        \Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad 
+        \Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
+        \textit{fields}(\type{N}) = \ol{U\ f} \\
+\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
+\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
+\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S}
     \end{array}
 \end{array}$
 \\[1em]
@@ -221,23 +195,7 @@ $\begin{array}{l}
         \\
         \hline
     \vspace*{-0.3cm}\\
-    \Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T}
-    \end{array}
-\end{array}$
-\\[1em]
-$\begin{array}{l}
-    \typerule{T-Field}\\
-    \begin{array}{@{}c}
-        \Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad 
-        \Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
-        \textit{fields}(\type{N}) = \ol{U\ f} \\
-\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
-\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
-\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
-        \\
-        \hline
-    \vspace*{-0.3cm}\\
-        \Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
+        \triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}}
     \end{array}
 \end{array}$
 \\[1em]
@@ -245,72 +203,109 @@ $\begin{array}{l}
     \typerule{T-Call}\\
     \begin{array}{@{}c}
         \Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
-        \textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad
+        \generics{\ol{X \triangleleft U'}} \type{N} \to \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
         \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
         \\
         \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
-        \Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad
-        \Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad
+        \Delta | \Gamma \vdash \texttt{e} : \type{T}_r \quad \quad
+        \Delta | \Gamma \vdash \ol{e} : \ol{T} \quad \quad
         \Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
-        \Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
+        \Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}}
         \\
         \Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
-        \Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
+        \Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
         \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
         \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}
-        {\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
+        \Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T}
     \end{array}
 \end{array}$
 \\[1em]
 $\begin{array}{l}
     \typerule{T-Elvis}\\
     \begin{array}{@{}c}
-        \Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad 
-        \Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad
-        \Delta \vdash \type{T}_1 <: \type{T} \quad \quad
-        \Delta \vdash \type{T}_2 <: \type{T}
+        \triangle | \Gamma \vdash \texttt{t} : \type{T}_1 \quad \quad
+        \triangle | \Gamma \vdash \texttt{t}_2 : \type{T}_2 \quad \quad
+        \triangle \vdash \type{T}_1 <: \type{T} \quad \quad
+        \triangle \vdash \type{T}_2 <: \type{T} \quad \quad
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T}
+        \triangle | \Gamma \vdash \texttt{t}_1 \ \texttt{?:} \ \texttt{t}_2 : \type{T}
+    \end{array}
+\end{array}$
+\end{center}
+\caption{Expression Typing}\label{fig:expressionTyping}
+\end{figure}
+
+\begin{figure}
+\begin{center}
+$\begin{array}{l}
+    \typerule{T-Method}\\
+    \begin{array}{@{}c}
+        \exptype{C}{\ol{X}} \to \ol{T} \to \type{T} \in \mathtt{\Pi}(\texttt{m})\quad \quad
+        %\text{dom}(\triangle) = \ol{X} \quad \quad
+        \triangle' =  \overline{\type{Y} : \bot .. \type{P}} \quad \quad
+        \triangle, \triangle' \vdash \ol{U}, \type{T}, \ol{T} \ \ok \\
+        %\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \ \{ \ldots \} \\
+        \mathtt{\Pi} | \triangle, \triangle' | \ol{x : T}, \texttt{this} : \exptype{C}{\ol{X}} \vdash \texttt{e} : \type{S} \quad \quad 
+        \triangle \vdash \type{S} <: \type{T}
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \triangle \vdash \texttt{m}(\ol{x}) = \texttt{e} \ \ok \text{ in C with } \generics{\ol{Y \triangleleft P}}
     \end{array}
 \end{array}$
 \\[1em]
 $\begin{array}{l}
-        \typerule{T-Method}\\
+    \typerule{T-Class}\\
     \begin{array}{@{}c}
-        \text{dom}(\Delta)=\ol{X} \quad \quad
-        \Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad
-        \Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad
-        \texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\
-        \Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad
-        \Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad 
-        \text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
+        \mathtt{\Pi}' = \mathtt{\Pi} \cup \set{ \texttt{m} \mapsto (\exptype{C}{\ol{X}} \to \ol{T}_\texttt{m} \to \type{T}_\texttt{m}) \mid \texttt{m} \in \ol{M}} \\
+        \mathtt{\Pi}'' = \mathtt{\Pi} \cup \set{ \texttt{m} \mapsto
+            \generics{\ol{X \triangleleft \type{N}}, \ol{Y \triangleleft P}}(\exptype{C}{\ol{X}} \to \ol{T}_\texttt{m} \to \type{T}_\texttt{m}) \mid \texttt{m} \in \ol{M} } \\
+        \triangle = \overline{\type{X} : \bot .. \type{U}} \quad \quad
+        \triangle \vdash \ol{U}, \ol{T}, \type{N} \ \ok \quad \quad
+        \triangle \vdash \ol{M} \ \ok \text{ in C with} \ \generics{\ol{Y \triangleleft P}}
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C}
+        \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \{ \ol{T\ f}; \ol{M} \}  : \mathtt{\Pi}''
     \end{array}
 \end{array}$
 \\[1em]
 $\begin{array}{l}
-        \typerule{T-Class}\\
+    \typerule{T-Program}\\
     \begin{array}{@{}c}
-        \Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad 
-        \Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad
-        \Delta \vdash \type{N} \ \ok \quad \quad
-        \Delta \vdash \ol{M} \ \ok \texttt{ in C}
+        \emptyset \vdash \texttt{L}_1 : \mathtt{\Pi}_1 \quad \quad 
+        \mathtt{\Pi}_1 \vdash \texttt{L}_2 : \mathtt{\Pi}_1 \quad \quad 
+        \ldots \quad \quad 
+        \mathtt{\Pi}_{n-1} \vdash \texttt{L}_n : \mathtt{\Pi}_n \quad \quad 
         \\
         \hline
     \vspace*{-0.3cm}\\
-        \texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok
+        \vdash \ol{L} : \mathtt{\Pi}_n
     \end{array}
 \end{array}$
-\caption{T-Call and T-Field} \label{fig:tletexpr}
+\end{center}
+\caption{Class and Method Typing rules}\label{fig:typing}
 \end{figure}
 
 
+
+\begin{figure}
+    $\text{fields}(\exptype{Object}{}) = \emptyset$
+    \quad \quad
+    $\begin{array}{l}
+    \typerule{F-Class}\\
+    \begin{array}{@{}c}
+        \texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ol{S\ f}; \ol{M}} \quad \quad 
+        \text{fields}([\ol{T}/\ol{X}]\type{N}) = \ol{U\ g}
+        \\
+        \hline
+    \vspace*{-0.3cm}\\
+        \text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f}
+    \end{array}
+\end{array}$
+\end{figure}
\ No newline at end of file
diff --git a/tiRules.tex b/tiRules.tex
index 9ef32d5..4b3f17e 100644
--- a/tiRules.tex
+++ b/tiRules.tex
@@ -1,24 +1,5 @@
 \section{Typeless FJ}
 
-\subsection{Syntax}
-We use the TameFJ syntax for the most part (\ref{fig:syntax-tamingfj}).
-
-\begin{figure}[tp]
-  \begin{align*}
-    \mv N &::= \exptype{C}{\ol{W}}\\
-    \mv T &::= \mv X \mid  \wctype{\ol{W}}{C}{\ol{W}}\\
-    \mv W &::= \mv T \mid \wildcard{Z}{T}{T}\\
-    \mv L &::= \mathtt{class} \ \exptype{C}{\ol{X} \triangleleft \ol{T}} \triangleleft \ \mv N\ \{ \ol{T} \ \ol{f}; \,\mv K \, \ol{M} \} \\
-    \mv K &::= \mv C(\ol{f})\ \{\mathtt{super}(\ol{f}); \ \mathtt{this}.\ol{f}=\ol{f};\} \\
-    \mv M &::= \mathtt{m}(\ol{x})\ = \mv e \\
-    \mv e &::= \mv x \mid \mv e.\mv f \mid
-               \mv e.\mathtt{m}(\ol{e}) \mid \mathtt{new}\ \mathtt{C}(\ol{e})
-               \mid \mv e \elvis{} \mv e
-  \end{align*}
-    \caption{Syntax of TameFJ + Type Inference}
-    \label{fig:syntax-tamingfj}
-  \end{figure}
-
 \subsection{Type System}
 
 The type system is similar to the ones used in \emph{Wildcards need Witness Protection}.