347 lines
14 KiB
TeX
347 lines
14 KiB
TeX
\section{Syntax and Typing}
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The input syntax for our algorithm is shown in figure \ref{fig:syntax}
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and the respective type rules in figure \ref{fig:expressionTyping} and \ref{fig:typing}.
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Our algorithm is an extension of the \emph{Global Type Inference for Featherweight Generic Java}\cite{TIforFGJ} algorithm.
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The input language is designed to showcase type inference involving existential types.
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Method call rule T-Call is the most interesting part, because it emulates the behaviour of a Java method call,
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where existential types are implicitly \textit{opened} and \textit{closed}.
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%The T-Elvis rule mimics the type judgement of a branch expression like \texttt{if-else}.
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%and is solely used for examples.
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The calculus does not include method overriding for simplicity reasons.
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Type inference for that is described in \cite{TIforFGJ} and can be added to this algorithm accordingly.
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Our algorithm is designed for extensibility with the final goal of full support for Java.
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\unify{} is the core of the algorithm and can be used for any calculus sharing the same subtype relations as depicted in \ref{fig:subtyping}.
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Additional language constructs can be added by implementing the respective constraint generation functions in the same fashion as described in chapter \ref{chapter:constraintGeneration}.
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%Additional features like overriding methods and method overloading can be added by copying the respective parts from there.
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%Additional features can be easily added by generating the respective constraints (Plümicke hier zitieren)
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% The type system in \cite{WildcardsNeedWitnessProtection} allows a method to \textit{override} an existing method declaration in one of its super classes,
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% but only by a method with the exact same type.
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% The type system presented here does not allow the \textit{overriding} of methods.
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% Our type inference algorithm consumes the input classes in succession and could only do a type check instead of type inference
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% on overriding methods, because their type is already determined.
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% Allowing overriding therefore has no implication on our type inference algorithm.
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\begin{figure}
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$
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\begin{array}{lrcl}
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\hline
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\text{Parameterized classes} & \mv N & ::= & \exptype{C}{\ol{T}} \\
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\text{Types} & \type{S}, \type{T}, \type{U} & ::= & \type{X} \mid \wcNtype{\Delta}{N} \\
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\text{Lower bounds} & \type{K}, \type{L} & ::= & \type{T} \mid \bot \\
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\text{Type variable contexts} & \Delta & ::= & \overline{\wildcard{X}{T}{L}} \\
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\text{Class declarations} & D & ::= & \texttt{class}\ \exptype{C}{\ol{X \triangleleft T}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M}} \\
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\text{Method declarations} & \texttt{M} & ::= & \generics{\ol{X \triangleleft T}} \type{T} \ \texttt{m}(\overline{\type{T}\ x}) = t \\
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\text{Terms} & t & ::= & x \\
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& & \ \ | & \texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})\\
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& & \ \ | & t.f\\
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& & \ \ | & t.\generics{\ol{T}}\texttt{m}(\overline{t})\\
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& & \ \ | & t \elvis{} t\\
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\text{Variable contexts} & \Gamma & ::= & \overline{x:\type{T}}\\
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\hline
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\end{array}
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$
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\caption{Input Syntax}\label{fig:syntax}
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\end{figure}
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% Each class type has a set of wildcard types $\overline{\Delta}$ attached to it.
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% The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$,
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% which can be used inside the type parameters $\ol{T}$.
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\begin{figure}[tp]
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\begin{center}
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$\begin{array}{l}
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\typerule{S-Refl}\\
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\begin{array}{@{}c}
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\Delta \vdash \type{T} <: \type{T}
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\end{array}
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\end{array}$
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$\begin{array}{l}
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\typerule{S-Trans}\\
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\begin{array}{@{}c}
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\Delta \vdash \type{S} <: \type{T}' \quad \quad
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\Delta \vdash \type{T}' <: \type{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \type{S} <: \type{T}
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\end{array}
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\end{array}$
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$\begin{array}{l}
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\typerule{S-Upper}\\
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\begin{array}{@{}c}
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\wildcard{X}{U}{L} \in \Delta \\
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\vspace*{-0.9em}\\
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\hline
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\vspace*{-0.9em}\\
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\Delta \vdash \type{X} <: \type{U}
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\end{array}
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\end{array}$
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$\begin{array}{l}
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\typerule{S-Lower}\\
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\begin{array}{@{}c}
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\wildcard{X}{U}{L} \in \Delta \\
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\vspace*{-0.9em}\\
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\hline
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\vspace*{-0.9em}\\
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\Delta \vdash \type{L} <: \type{X}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{S-Extends}\\
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\begin{array}{@{}c}
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\texttt{class}\ \exptype{C}{\overline{\type{X} \triangleleft \type{U}}} \triangleleft \exptype{D}{\ol{S}} \ \{ \ldots \} \\
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\ol{X} \cap \text{dom}(\Delta) = \emptyset
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \wcNtype{\Delta'}{\type{N}} <: \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}}
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\end{array}
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\end{array}$
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$\begin{array}{l}
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\typerule{S-Exists}\\
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\begin{array}{@{}c}
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\Delta', \Delta \vdash [\ol{T}/\ol{\type{X}}]\ol{L} <: \ol{T} \quad \quad
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\Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\
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\text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad
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\text{dom}(\Delta') \cap \text{fv}(\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}) = \emptyset
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}} <:
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\wcNtype{\ol{\wildcard{X}{U}{L}}}{N}
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\end{array}
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\end{array}$
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\end{center}
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\caption{Subtyping}\label{fig:subtyping}
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\end{figure}
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\begin{figure}[tp]
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\begin{center}
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$\begin{array}{l}
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\typerule{WF-Bot}\\
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\begin{array}{@{}c}
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\Delta \vdash \bot \ \ok
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\end{array}
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\end{array}$
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$\begin{array}{l}
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\typerule{WF-Top}\\
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\begin{array}{@{}c}
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\Delta \vdash \ol{L}, \ol{U} \ \ok
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \overline{\wildcard{W}{U}{L}}.\texttt{Object}
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\end{array}
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\end{array}$
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$\begin{array}{l}
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\typerule{WF-Var}\\
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\begin{array}{@{}c}
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\wildcard{W}{U}{L} \in \Delta \quad \quad
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\Delta \vdash \ol{L}, \ol{U} \ \ok
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \wildcard{W}{U}{L} \ \ok
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{WF-Class}\\
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\begin{array}{@{}c}
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\Delta' = \ol{\wildcard{W}{U}{L}} \quad \quad
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\Delta, \Delta' \vdash \ol{T}, \ol{L}, \ol{U} \ \ok \quad \quad
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\Delta, \Delta' \vdash \ol{L} <: \ol{U} \\
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\Delta, \Delta' \vdash \ol{T} <: [\ol{T}/\ol{X}] \ol{U'} \quad \quad
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\texttt{class}\ \exptype{C}{\ol{X \triangleleft U'}} \triangleleft \type{N} \ \{ \ldots \}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok
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\end{array}
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\end{array}$
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\end{center}
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\caption{Well-formedness}\label{fig:well-formedness}
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\end{figure}
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\begin{figure}[tp]
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\begin{center}
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$\begin{array}{l}
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\typerule{T-Var}\\
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\begin{array}{@{}c}
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\texttt{x} : \type{T} \in \Gamma
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\\
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\hline
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\vspace*{-0.3cm}\\
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\triangle | \Gamma \vdash \texttt{x} : \type{T}
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\end{array}
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\end{array}$ \hfill
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$\begin{array}{l}
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\typerule{T-Field}\\
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\begin{array}{@{}c}
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\Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad
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\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
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\textit{fields}(\type{N}) = \ol{U\ f} \\
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\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
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\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
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\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-New}\\
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\begin{array}{@{}c}
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\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad
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\text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
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\Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
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\Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
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\Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad
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\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
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\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
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\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok
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\\
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\hline
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\vspace*{-0.3cm}\\
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\triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Call}\\
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\begin{array}{@{}c}
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\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
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\generics{\ol{X \triangleleft U'}} \type{N} \to \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
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\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
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\\
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\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
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\Delta | \Gamma \vdash \texttt{e} : \type{T}_r \quad \quad
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\Delta | \Gamma \vdash \ol{e} : \ol{T} \quad \quad
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\Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
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\Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}}
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\\
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\Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
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\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
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\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
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\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Call}\\
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\begin{array}{@{}c}
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\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
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\generics{\ol{X \triangleleft U'}} \type{N} \to \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
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\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'} \quad \quad
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\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok
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\\
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\Delta | \Gamma \vdash \ol{e} : \ol{T} \quad \quad
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\Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}} \quad \quad
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\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
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\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
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\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash \texttt{m}(\ol{e}) : \type{T}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Elvis}\\
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\begin{array}{@{}c}
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\triangle | \Gamma \vdash \texttt{t} : \type{T}_1 \quad \quad
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\triangle | \Gamma \vdash \texttt{t}_2 : \type{T}_2 \quad \quad
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\triangle \vdash \type{T}_1 <: \type{T} \quad \quad
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\triangle \vdash \type{T}_2 <: \type{T} \quad \quad
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\\
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\hline
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\vspace*{-0.3cm}\\
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\triangle | \Gamma \vdash \texttt{t}_1 \ \texttt{?:} \ \texttt{t}_2 : \type{T}
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\end{array}
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\end{array}$
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\end{center}
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\caption{Expression Typing}\label{fig:expressionTyping}
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\end{figure}
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\begin{figure}
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\begin{center}
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$\begin{array}{l}
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\typerule{T-Method}\\
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\begin{array}{@{}c}
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\exptype{C}{\ol{X}} \to \ol{T} \to \type{T} \in \mathtt{\Pi}(\texttt{m})\quad \quad
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\triangle' = \overline{\type{Y} : \bot .. \type{P}} \quad \quad
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\triangle, \triangle' \vdash \ol{P}, \type{T}, \ol{T} \ \ok \\
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\text{dom}(\triangle) = \ol{X} \quad \quad
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%\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \ \{ \ldots \} \\
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\mathtt{\Pi} | \triangle, \triangle' | \ol{x : T}, \texttt{this} : \exptype{C}{\ol{X}} \vdash \texttt{e} : \type{S} \quad \quad
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\triangle \vdash \type{S} <: \type{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\mathtt{\Pi} | \triangle \vdash \texttt{m}(\ol{x}) = \texttt{e} \ \ok \text{ in C with } \generics{\ol{Y \triangleleft P}}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Class}\\
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\begin{array}{@{}c}
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\mathtt{\Pi}' = \mathtt{\Pi} \cup \set{ \exptype{C}{\ol{X}}.\texttt{m} : \ol{T}_\texttt{m} \to \type{T}_\texttt{m} \mid \texttt{m} \in \ol{M}} \\
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\mathtt{\Pi}'' = \mathtt{\Pi} \cup \set{ \exptype{C}{\ol{X}}.\texttt{m} :
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\generics{\ol{X \triangleleft \type{N}}, \ol{Y \triangleleft P}}\ol{T}_\texttt{m} \to \type{T}_\texttt{m} \mid \texttt{m} \in \ol{M} } \\
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\triangle = \overline{\type{X} : \bot .. \type{U}} \quad \quad
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\triangle \vdash \ol{U}, \ol{T}, \type{N} \ \ok \quad \quad
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\mathtt{\Pi}' | \triangle \vdash \ol{M} \ \ok \text{ in C with} \ \generics{\ol{Y \triangleleft P}}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \{ \ol{T\ f}; \ol{M} \} : \mathtt{\Pi}''
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Program}\\
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\begin{array}{@{}c}
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\emptyset \vdash \texttt{L}_1 : \mathtt{\Pi}_1 \quad \quad
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\mathtt{\Pi}_1 \vdash \texttt{L}_2 : \mathtt{\Pi}_1 \quad \quad
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\ldots \quad \quad
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\mathtt{\Pi}_{n-1} \vdash \texttt{L}_n : \mathtt{\Pi}_n \quad \quad
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\\
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\hline
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\vspace*{-0.3cm}\\
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\vdash \ol{L} : \mathtt{\Pi}_n
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\end{array}
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\end{array}$
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\end{center}
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\caption{Class and Method Typing rules}\label{fig:typing}
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\end{figure}
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\begin{figure}
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$\text{fields}(\exptype{Object}{}) = \emptyset$
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\quad \quad
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$\begin{array}{l}
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\typerule{F-Class}\\
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\begin{array}{@{}c}
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\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ol{S\ f}; \ol{M}} \quad \quad
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\text{fields}([\ol{T}/\ol{X}]\type{N}) = \ol{U\ g}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f}
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\end{array}
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\end{array}$
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\end{figure} |