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fjtyperules.tex
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We start with some notation.
An environment $\mv\Gamma$ is a finite mapping from variables to types,
written $\ol x:\ol T$; a type environment $\mv\Delta$ is a finite mapping
from type variables to nonvariable types, written $\ol X\subeq\ol
N$, which takes each type variable to its bound. As in FGJ, we do not
impose an ordering on environment entries to enable F-bounded
polymorphism.
There is a new method environment $\mv\Pi$ which maps pairs of a class header
$\exptype{C}{\ol X}$ and a method name $\mv m$ to a set of method
types of the form $\exptype{}{\ol{Y} \triangleleft \ol{P}} \ol{T} \to \mv{T}$.
It supports the \textit{mtype} function that relates a nonvariable type
$\mv N$ and a method name $\mv m$ to a method type.
The judgments for subtyping $\mathtt{\Delta \vdash S \subeq T}$ and
well-formedness of types $\mathtt{\Delta \vdash T\ \mathtt{ok}}$
(Figure~\ref{fig:well-formedness-and-subtyping}) stay the same as in
FGJ.
The overall approach to typing changes with respect to FGJ. In FGJ,
classes can be checked in any order as the method typings of all other
classes are available in the syntax. \TFGJ processes classes in
order such that early classes do not invoke methods in late classes.
The new program typing rule \rulename{GT-PROGRAM} for the judgment
$\vdash \ol{L} : \mv\Pi$ reflects this
approach. It starts with an empty
method environment and applies class typing to each class in the
sequence provided. Each processed class adds its method typings to the method
environment which is threaded through to constitute the program type
as the final method environment $\mv\Pi$.
Expression typing $\mathtt{\Pi; \Delta; \Gamma \vdash e
: T}$ changes subtly (see Figure~\ref{fig:expression-typing}). As
\TFGJ omits some type annotations, we are forced to adapt some of
FGJ's typing rules. The new rules infer omitted types and disable
polymorphic recursion.
The new method
environment $\mv\Pi$ is only used in the revised
rule for method invocation
\rulename{GT-INVK}, where it is passed as an additional parameter to
\textit{mtype}. The revised definition of \textit{mtype} (Figure~\ref{fig:auxiliary-functions}) locates the class that contains
the method definition by traversing the subtype hierarchy and looks up the
method type in environment $\mv\Pi$, which contains the method types
that were already inferred. Our definition of \textit{mtype} does not support
overloading as $\mv\Pi$ relate at most one type to each method
definition (cf.\ rule \rulename{GR-CLASS}). The instantiation of the
method's type parameters is inferred in \TFGJ.
The rule \rulename{GT-NEW} changes to infer the
instantiation of the class's type parameters: the rule
simply assumes a suitable instantiation by some $\ol U$.
Finally, \rulename{GT-CAST} replaces the three rules
\rulename{GT-UCAST'}, \rulename{GT-DCAST'}, and \rulename{GT-SCAST'}
of FGJ. This is a slight simplification with respect to FGJ. While
the three original rules cover disjoint use cases (upcast, downcast,
and stupid cast that is sure to fail) of the cast
operation, they are not exhaustive! The rule \rulename{GT-DCAST'} only
admits downcasts that work the same in a type-passing semantics as in
a type erasure semantics. We elide this distinction for simplicity,
though it could be handled by introducing constraints analogous to the
\textit{dcast} function from FGJ.
% The input for our type inference algorithm is based on Featherweight Generic Java (FGJ).
% FGJ is defined by syntax and typing rules.
% We already changed the syntax to allow typeless FGJ programs as input for our algorithm.
% Additionally we alter the typing rules slightly, which is presented in this chapter.
%Our type inference algorithm takes typeless FGJ classes as input.
%The generated output is correct FGJ, although we have to alter some rules.
%This chapter defines the typing rules for our version of FGJ,
%which our type inference algorithm is able to process.
The typing rule for a method $\mv m$, \rulename{GT-METHOD}, changes
significantly. By our assumption on the order, in which classes are
processed, the typing of $\mv m$ is already provided by the
method environment $\mv\Pi$. The type environment $\mv\Delta$ is
also provided as an input. Moreover, to rule out polymorphic
recursion, the assumptions about the local methods of class $\mv{C}$
are monomorphic at this stage. The rule type checks the body for the
inferred type of method $\mv m$.
All this information is provided and generated by the rule for class
typing, \rulename{GT-CLASS}. A class typing for \mv{C} receives an incoming
method type environment $\mv\Pi$ and generates an extended one
$\mv{\Pi''}$ which additionally contains the method types inferred for
\mv{C}.
In $\mv{\Pi'}$, we generate some monomorphic types for all
methods of class $\mv C$. We use these types to check the methods. Afterwards, we
return generalized versions of these same types in $\mv{\Pi''}$. All
method types use the same generic type variables $\ol Y$ with the
same constraints $\ol P$. It is safe to make this assumption in the absence of
polymorphic recursion as we will show in
Proposition~\ref{prop:polymorphi-recursion}.
\if0
\begin{itemize}
\item We omit the rules for valid downcasts.
\todo[inline]{Andi: Is this correct? We need another constraint to be correct:
$N \lessdot_{downcast} a$ for a cast of the form: \texttt{(N)a}.
This is a different constraint then the normal subtype relationship (see FJ paper)
}
\todo[inline]{PJT 20220202: we should have a single
rule for all casts without any distinction between the three cases. It has to be
split in three cases in building the completion, but this
distinction is not necessary for type inference.}
%The downcast problem:
%This is wrong:
%m(List<SortedList> p, Object o){
% p.add((SortedList) o);
%}
%This is correct:
%m(List<SortedList> p, List<Object> o){
% p.add((SortedList) o);
%}
%Currently we do not generate constraints for casts
\end{itemize}
\fi
\begin{figure}[tp]
\fbox{
\begin{minipage}{0.9\textwidth }
\begin{small}
\textbf{Subtyping:}\\[1em]
\begin{tabularx}{\linewidth}{X c X r}
& $\mathtt{ \Delta \vdash T \subeq T } $
& & \rulename{S-REFL} \\
& \\
&
$\mathtt{\ddfrac{ \Delta \vdash S \subeq T \quad \quad
\Delta \vdash T \subeq U }{ \Delta \vdash S \subeq U
}}$ & & \rulename{S-TRANS} \\
& \\
& $\mathtt{ \Delta \vdash X \subeq \Delta(X)
}$ & & \rulename{S-VAR} \\
& \\
&
$\mathtt{\ddfrac{ \texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft \mv N \set{ \ldots
} }{ \Delta \vdash \exptype{C}{\ol{T}} \subeq
[\ol{T}/\ol{X}]\mv N
}}$ & & \rulename{S-CLASS} \\
& & & \\
\hline
\multicolumn{1}{l}{\textbf{Well-formed types:}}\\
& & & \\
& $\mathtt{ \Delta \vdash \texttt{Object}\ \texttt{ok}
}$ & & \rulename{WF-OBJECT}\\
& \\
&
$\mathtt{\ddfrac{ X \in \textit{dom}(\Delta) }{ \Delta
\vdash X \ \texttt{ok} } }
$ & & \rulename{WF-VAR} \\
& \\
& $\mathtt{\ddfrac{\begin{array}{c}
\texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft \mv N \{ \ldots \} \\
\mathtt{\Delta} \vdash \ol{T} \ \texttt{ok}
\quad \quad \mathtt{\Delta} \vdash \ol{T} \subeq
[\ol{T}/\ol{X}]\ol{N}
\end{array}
}{ \Delta \vdash \exptype{C}{\ol{T}} \
\texttt{ok} } } $ & & \rulename{WF-CLASS}
\end{tabularx}
\end{small}
\end{minipage}
}
\caption{Well-formedness and subtyping}
\label{fig:well-formedness-and-subtyping}
\end{figure}
\begin{figure}[tp]
\fbox{
\begin{minipage}{0.9\textwidth}
\begin{small}
\textbf{Expression typing:}\\
\begin{tabularx}{\textwidth}{c X r}
%\begin{tabular}{l@{\quad}l}
$\mathtt{
\environmentvdash x : \Gamma(x)
}$ & & \rulename{GT-VAR} \\
& \\
$\mathtt{\ddfrac{
\Pi; \Delta; \Gamma \vdash e_0:T_0 \qquad
\mathit{fields}(\mathit{bound}_\Delta(T_0)) = \overline{T} \ \overline{f}}
{\Pi; \Delta; \Gamma \vdash e_0.\mathtt{f}_i : T_i}
}
$ & & \rulename{GT-FIELD} \\
& \\
$\mathtt{\ddfrac{\begin{array}{c}
\mathtt{\environmentvdash e_0 : T_0 } \quad \quad
\mathtt{\mathit{mtype}(m, \mathit{bound}_\Delta (T_0), \Pi) = \exptype{}{\ol{Y} \triangleleft \ol{P}} \ol{U} \to U } \\
\mathtt{\Delta \vdash \ol{V} \ \texttt{ok} } \quad \quad
\mathtt{\Delta \vdash \ol{V} \subeq [\ol{V}/\ol{Y}]\ol{P} } \qquad
\mathtt{\environmentvdash \ol{e} : \ol{S} } \quad \quad
\mathtt{\Delta \vdash \ol{S} \subeq [\ol{V}/\ol{Y}]\ol{U}}
\end{array}}
{\environmentvdash \mathtt{e_0.\mv{m}(\overline{e}) : [\ol{V}/\ol{Y}]U }}
}$ & & \rulename{GT-INVK} \\
& \\
$\mathtt{ \ddfrac{\Delta \vdash {\mv N} \ \texttt{ok} \quad
\mv N = \exptype{C}{\ol{U}} \quad
\textit{fields}(\mv N) = \ol{T}\ \ol{f} \quad
\environmentvdash \ol{e} : \ol{S} \quad \Delta \vdash \ol{S} \subeq \ol{T}
}{
\environmentvdash \texttt{new C}(\ol{e}): \mv N
}
}$ & & \rulename{GT-NEW} \\
& \\
$\ddfrac{\mathtt{\environmentvdash e_0 : T_0}}
{\mathtt{\environmentvdash (N) e_0 : N}}
$ & & \rulename{GT-CAST} \\
%
% & \\
%
% $\ddfrac{\mathtt{\environmentvdash e_0 : T_0 \quad \quad \Delta \vdash \textit{bound}_\Delta(T_0) \subeq N}}
% {\mathtt{\environmentvdash (N) e_0 : N}}
% $ & & \rulename{GT-UCAST} \\
%
% & \\
%
% $ \ddfrac{\begin{array}{c}
% \mathtt{\environmentvdash e_0 : T_0 \quad \quad \Delta \vdash N\ \texttt{ok} \quad \quad \Delta \vdash N \subeq \textit{bound}_\Delta(T_0) } \\
% \mathtt{N = \exptype{C}{\ol{T}} \quad \quad \textit{bound}_\Delta(T_0) = \exptype{D}{\ol{U}} \quad \quad \textit{dcast}(C,D)}
% \end{array}
% }{\mathtt{\environmentvdash (N) e_0 : N}}$ & & \rulename{GT-DCAST} \\
%
% & \\
%
% $\ddfrac{\begin{array}{c}
% \mathtt{\environmentvdash e_0 : T_0 \quad \quad \Delta \vdash N\ \texttt{ok} \quad \quad N = \exptype{C}{\ol{T}} } \\
% \mathtt{\textit{bound}_\Delta(T_0) = \exptype{D}{\ol{U}} \quad \quad C \ntrianglelefteq D \quad \quad D \ntrianglelefteq C \quad \quad \textit{stupid warning}}
% \end{array}}
% {\mathtt{\environmentvdash (N) e_0 : N}}
% $ & & \rulename{GT-SCAST}
\end{tabularx}\\[1em]
% \end{small}
% \end{minipage}
% % }
% % \fbox{
% \begin{minipage}{\textwidth}
% \begin{small}
\textbf{Method typing:}\\[1em]
\begin{tabularx}{\textwidth}{c X r}
$\ddfrac{
\begin{array}{c}
\mathtt{\forall \ol{T}, T: \exptype{}{} \ol{T} \to T \in
\Pi(\exptype{C}{\ol{X} \triangleleft \ol{N}}.m) } \quad \quad
\mathtt{\Delta \vdash S \subeq T }
\\
\mathtt{\Pi; \Delta ; \ol{x}:\ol{T},\ this : \exptype{C}{\ol{X}} \vdash e_0 : S}
\\
\mathtt{
\textit{override}(m, N, \exptype{}{\ol{Y} \triangleleft
\ol{P}}\ol{T} \to T, \Pi) }
\end{array}} {
\mathtt{\Pi, \Delta \vdash \texttt{m}(\ol{\mathtt{x}}) \{\texttt{return}\ \mathtt{e}_0;\}
\texttt{ OK in }\exptype{C}{\ol{X} \triangleleft \ol{N}} \extends N \texttt{
with } \exptype{}{\ol{Y} \triangleleft \ol{P}} }
}$ & & \rulename{GT-METHOD}\\
\end{tabularx}\\[1em]
\textbf{Class typing:}\\[1em]
\begin{tabularx}{\textwidth}{X c X r}
%GT-CLASS: - This rule is modified by us
& $\ddfrac{
\begin{array}{c}
\mathtt{\Pi' = \Pi \cup \set{(\exptype{C}{ \ol{X} \triangleleft \ol{N}}.m, \ol{T} ) \mapsto \exptype{}{} \ol{T_m} \to T_m \mid m \in \ol{M}} } \\
\mathtt{\Pi'' = \Pi \cup \set{(\exptype{C}{ \ol{X} \triangleleft \ol{N}}.m, \ol{T} ) \mapsto [\ol{T}/\ol{X}]
\exptype{}{\ol{Y} \triangleleft \ol{P}} \ol{T_m} \to T_m \mid m \in \ol{M}} } \\
\mathtt{\Delta = \ol{X} \subeq \ol{N}, \ol{Y} \subeq \ol{P}}
\qquad \mathtt{\Delta \vdash\ol{P} \
\texttt{ok}}
\qquad \forall \mv{m}: \mathtt{\Delta \vdash \ol{T_m}, T_m \ \texttt{ok}}\\
\mathtt{\ol{X} \subeq \ol{N} \vdash \ol{N}, N, \ol{T}\ \texttt{ok}
\quad\quad \mathit{fields}(\mathtt{N}) = \ol{\mathtt{U}} \ \ol{\mathtt{g}}} \\\
\mathtt{\Pi', \Delta \vdash \ol{M} \ \texttt{OK IN}\
\exptype{C}{\ol{X} \triangleleft \ol{N}} \extends N \texttt{
with } \exptype{}{\ol{Y} \triangleleft \ol{P}}}\\
\mathtt{K = C(\overline{U} \ \overline{g}, \overline{T} \ \overline{f}) \{ \texttt{super}(\overline{g}); \ \texttt{this}.\overline{f}=\overline{f}; \} }\\
%\quad \quad \overline{M} \ \texttt{OK IN C} \\
\end{array}
}
{\mathtt{ \Pi \vdash \texttt{class}\ \exptype{C}{\ol{X}
\triangleleft \ol{N}} \triangleleft N\ \{ \overline{T} \
\overline{f}; \ K \ \overline{M} \} \ \texttt{OK} : \Pi''}}
$ & & \rulename{GT-CLASS} \\
\end{tabularx}\\[1em]
\textbf{Program typing:}\\[1em]
\begin{tabularx}{\textwidth}{X c X r}
& $\ddfrac{
\mathtt{\emptyset \vdash L_1 : \Pi_1} \quad
\mathtt{\Pi_1 \vdash L_2 : \Pi_2} \quad \dots \quad
\mathtt{\Pi_{n-1} \vdash L_n : \Pi_n}
}{
\mathtt{\vdash \ol L : \Pi_n}
}$
&& \rulename{GT-PROGRAM}
\end{tabularx}
\end{small}
\end{minipage}
}
\caption{Typing rules}
\label{fig:expression-typing}
\end{figure}
\begin{figure}[tp]
\fbox{
\begin{minipage}{0.9\textwidth}
\begin{small}
\textbf{Field lookup:} \\[1em]
\begin{tabularx}{\textwidth}{cXr}
$\mathit{fields}(\mv{Object}) = \bullet$
&& \rulename{F-OBJECT} \\
&& \\
$\ddfrac{
\mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}\triangleleft
\ol{N}\ \{ \overline{S} \ \overline{f}; \ K \ \overline{M} \}
} \qquad
\mathtt{\mathit{fields} ([\ol T/\ol X]N) = \ol U\ \ol g}
}{
\mathit{fields}(\exptype{C}{\ol T}) = \ol U\ \ol g, [\ol T/\ol
X]\ol S\ \ol f
}$
&& \rulename{F-CLASS}
\end{tabularx}\\[1em]
\textbf{Method type lookup:} \\[1em]
\begin{tabularx}{\textwidth}{cXr}
$\ddfrac{
\begin{array}{c}
\mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft
\ol{N}}\ \triangleleft \mv{N}\ \{ \overline{C} \ \overline{f}; \ K \ \overline{M} \}
\qquad m \in \ol{M}} \\
% \mathtt{
% \texttt{m}(\ol{\mathtt{x}}) \{\texttt{return}\ \mathtt{e}_0;\} \in \ol{M}
% } \\
\mathtt{
\exptype{}{\ol{Y} \triangleleft \ol{P}} \ol{U} \to U \in \Pi (\exptype{C}{\ol{X} \triangleleft \ol{N}}.m, \ol{T}) }
\end{array}
} {\mathtt{
\textit{mtype}(m, \exptype{C}{\ol{T}}, \Pi) = \exptype{}{\ol{Y} \triangleleft \ol{P}}\ol{U} \to U
}}$ & & \rulename{MT-CLASS}\\
& & \\
$\ddfrac{\mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}\triangleleft
\mv{N}\ \{ \overline{C} \ \overline{f}; \ K \ \overline{M} \}
\qquad m \notin \ol{M}} }{
\mathtt{\textit{mtype}(m, \exptype{C}{\ol{T}}, \Pi) =
\textit{mtype}(m, [\ol{T}/\ol{X}]N, \Pi)}
}$ & & \rulename{MT-SUPER} \\
\end{tabularx}\\[1em]
\textbf{Valid method overriding:}\\[1em]
\begin{tabularx}{\textwidth}{c}
$\ddfrac{\mathtt{\textit{mtype}(m, N, \Pi) = \exptype{}{\ol{Z} \triangleleft \ol{Q}} \ol{U} \to \mv U \
\text{implies}\ \ol{P}, \ol{T} = [\ol{Y}/\ol{Z}](\ol{Q},\ol{U}) \
\text{and}\ \ol{Y} \subeq \ol{P} \vdash T_0 \subeq [\ol{Y}/\ol{Z}]U_0}
} {
\mathtt{\textit{override}(m, N, \exptype{}{\ol{Y} \triangleleft \ol{P}} \ol{T} \to T_0, \Pi)}
}$
\end{tabularx}
\end{small}
\end{minipage}
}
\caption{Auxiliary functions}
\label{fig:auxiliary-functions}
\end{figure}
% \fbox{
% \begin{minipage}{\textwidth}
% \begin{small}
% \textbf{Method Typing (broken):}\\[1em]
% \begin{tabularx}{\textwidth}{X c X r}
% & $\mathtt{\ddfrac{\begin{array}{c}
% \mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}} \triangleleft N \{ \ldots\ \ol{M}\ \ldots\}
% \quad \quad \Delta = \ol{X} \subeq \ol{N}, \ol{Y} \subeq \ol{P} }\\
% \mathtt{\Delta \vdash \ol{T}, T, \ol{P} \ \texttt{ok} \quad \quad
% \textit{override}(m, N, \exptype{}{\ol{Y} \triangleleft \ol{P}}\ol{T} \to T) } \\
% \mathtt{\Pi(\exptype{C}{\ol{X}}.m) = \exptype{}{\ol{Y} \triangleleft \ol{P}}\ \ol{T} \to T }\\
% \mathtt{\Pi; \Delta ; \ol{x}:\ol{T},\ this : \exptype{C}{\ol{X}} \vdash e_0 : S \quad \quad
% \Delta \vdash S \subeq T } \\
% \end{array}} {
% %{\exptype{}{\ol{Y} \triangleleft \ol{P}}\ T \ m(\ol{T}\ \ol{x}) \{
% %\texttt{return} \ e_0; \} \ \texttt{OK IN}\ \exptype{C}{\ol{X} \triangleleft
% %\ol{N}}}
% \mathtt{\Pi \vdash \exptype{}{\ol{Y} \triangleleft \ol{P}}\ T\ \texttt{meth}(\ol{T}\ \ol{\mathtt{x}}) \{\texttt{return}\ \mathtt{e}_0;\}
% \texttt{ OK in }\exptype{C}{\ol{X} \triangleleft \ol{N}} }
% }}$ & & GT-METHOD\\
% \end{tabularx}\\[1em]
% \textbf{Class Typing (broken):}\\[1em]
% \begin{tabularx}{\textwidth}{X c X r}
% %GT-CLASS: - This rule is modified by us
% & $\ddfrac{
% \begin{array}{c}
% \mathtt{\Pi = \set{\exptype{C}{\ol{X}}.m : \ol{T_m} \to T_m \, |\, m \in \ol{M}} } \\
% \mathtt{\ol{X} \subeq \ol{N} \vdash \ol{N}, N, \ol{T}\ \texttt{ok}
% \quad\quad fields(\mathtt{N}) = \ol{\mathtt{U}} \ \ol{\mathtt{g}}} \quad \quad
% \mathtt{\Pi \vdash \ol{M} \ \texttt{OK IN}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}}\\
% \mathtt{K = C(\overline{U} \ \overline{g}, \overline{T} \ \overline{f}) \{ \texttt{super}(\overline{g}); \ \texttt{this}.\overline{f}=\overline{f}; \} }\\
% %\quad \quad \overline{M} \ \texttt{OK IN C} \\
% \end{array}
% }
% {\mathtt{ \texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}} \triangleleft N\ \{ \overline{T} \ \overline{f}; \ K \ \overline{M} \} \ \texttt{OK}}}
% $ & & GT-CLASS
% \end{tabularx}
% \end{small}
% \end{minipage}
% }
% \fbox{
% \begin{minipage}{\textwidth}
% \begin{small}
% \textbf{Method type lookup (broken):} \\[1em]
% \begin{tabularx}{\textwidth}{cXr}
% $\ddfrac{
% \begin{array}{c}
% \mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}\ \texttt{extends N}\{ \overline{C} \ \overline{f}; \ K \ \overline{M} \} \ \texttt{OK}} \quad \quad \mathtt{m \in \ol{M}} \\
% \mathtt{
% \exptype{}{\ol{Y} \triangleleft \ol{P}}\ T\ \texttt{m}(\ol{T}\ \ol{\mathtt{x}}) \{\texttt{return}\ \mathtt{e}_0;\}
% }
% \end{array}
% } {\mathtt{
% \textit{mtype}(m, \exptype{C}{\ol{T}}) = [\ol{T}/\ol{X}]\exptype{}{\ol{Y} \triangleleft \ol{P}}\ol{U} \to U
% }}$ & & MT-CLASS\\
% & & \\
% $\ddfrac{\mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}\triangleleft
% N\{ \overline{C} \ \overline{f}; \ K \ \overline{M} \} \ \mathtt{OK}
% \quad \quad m \notin \ol{M}} }{
% %{\exptype{}{\ol{Y} \triangleleft \ol{P}}\ T \ m(\ol{T}\ \ol{x}) \{ \texttt{return} \ e_0; \} \ \texttt{OK IN}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}}
% \textit{mtype}(m, \exptype{C}{\ol{T}}) = \textit{mtype}(m, [\ol{T}/\ol{X}]N)
% }$ & & MT-SUPER \\
% \end{tabularx}\\[1em]
% \textbf{Valid method overriding (broken):}\\[1em]
% \begin{tabularx}{\textwidth}{c}
% $\ddfrac{
% \textit{mtype}(m, N) = \exptype{}{\ol{Y} \triangleleft \ol{Q}} \ol{U} \to U \
% \text{implies}\ \ol{P}, \ol{T} = [\ol{Y}/\ol{Z}](\ol{Q},\ol{U}) \
% \text{and}\ \ol{Y} \subeq \ol{P} \vdash T_0 \subeq [\ol{Y}/\ol{Z}]U_0
% } {
% %{\exptype{}{\ol{Y} \triangleleft \ol{P}}\ T \ m(\ol{T}\ \ol{x}) \{ \texttt{return} \ e_0; \} \ \texttt{OK IN}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}}
% \textit{override}(m, N, \exptype{}{\ol{Y} \triangleleft \ol{P}} \ol{T} \to T_0)
% }$
% \end{tabularx}
% % PT: this is not used anywhere!
% % \\[1em]
% % \textbf{Method body lookup:} \\[1em]
% % \begin{tabularx}{\textwidth}{cXr}
% % $\ddfrac{\begin{array}{c}
% % \texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}\triangleleft
% % N\{ \overline{S} \ \overline{f}; \ K \ \overline{M} \} \\
% % \mathtt{\exptype{}{\ol{Y} \triangleleft \ol{P}}\ T\ \texttt{m}(\ol{T}\ \ol{\mathtt{x}}) \{\texttt{return}\ \mathtt{e}_0;\}}
% % \in \ol{M}
% % \end{array}} {
% % %{\exptype{}{\ol{Y} \triangleleft \ol{P}}\ T \ m(\ol{T}\ \ol{x}) \{ \texttt{return} \ e_0; \} \ \texttt{OK IN}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}}
% % \textit{mbody}(m, \exptype{C}{\ol{Z}}) = [\ol{Z} / \ol{X}](\exptype{}{\ol{Y}} \ol{T} \to T)
% % }$ & & MB-CLASS \\
% % & & \\
% % $\ddfrac{
% % \mathtt{\texttt{class}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}\triangleleft
% % \ol{N}\ \{ \overline{S} \ \overline{f}; \ K \ \overline{M} \} \quad \quad m \notin \ol{M}}
% % } {
% % %{\exptype{}{\ol{Y} \triangleleft \ol{P}}\ T \ m(\ol{T}\ \ol{x}) \{ \texttt{return} \ e_0; \} \ \texttt{OK IN}\ \exptype{C}{\ol{X} \triangleleft \ol{N}}}
% % \mathtt{\textit{mbody}(m, \exptype{C}{\ol{T}}) = \textit{mbody}(m, [\ol{T}/\ol{X}]N)}
% % }$ & & MB-SUPER \\
% % \end{tabularx}
% \end{small}
% \end{minipage}
% }
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