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coercion semantics

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@ -508,40 +508,42 @@ while preserving its semantics.
\subsubsection{Inference of Expression Types} \subsubsection{Inference of Expression Types}
The type-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\) is a finite mapping from variables \(\metavariable{x_i} \in \exprvars\) to ground types \(\metavariable{\tau_i} \in \typenonterm{\emptyset}\). As usual, the typing-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\)
is a finite mapping which assigns variables \(\metavariable{x_i} \in \exprvars\) to types \(\metavariable{\tau_i} \in \nonterm{T}\).
Using the inference rules given in \ref{def:typerules}, further typing-judgements Using the inference rules given in \ref{def:typerules}, further typing-judgements
of the form of the form
\begin{itemize} \begin{itemize}
\item \(\metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and \item \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
\item \(\metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)" \item \(\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
\end{itemize} \end{itemize}
can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \exprnonterm{\emptyset}{\exprvars}\) and \(\metavariable{\tau} \in \typenonterm{\emptyset}\) can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
\begin{definition}[Syntactic Well-Typedness] \begin{definition}[Syntactic Well-Typedness]
An expression \(\metavariable{e} \in \exprnonterm{\emptyset}{\emptyset}\) is \textbf{syntactically well-typed} if there exists a type \(\metavariable{\tau} \in \typenonterm{\emptyset}\), An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{syntactically well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}. such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
\end{definition} \end{definition}
\begin{definition}[Semantic Well-Typedness] \begin{definition}[Semantic Well-Typedness]
An expression \(\metavariable{e} \in \exprnonterm{\emptyset}{\emptyset}\) is \textbf{semantically well-typed} if there exists a type \(\metavariable{\tau} \in \typenonterm{\emptyset}\), An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{semantically well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules}. such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules} and \ref{def:semtyperules}.
\end{definition} \end{definition}
\begin{definition}[Inference Rules for the Typing Relation.]
\label{def:typerules}
As usual, each rule is composed of premises (above the horizontal line) and a conclusion (below the line): \begin{definition}[Syntactic Typing Relation]
\label{def:typerules}
\begin{mathpar} \begin{mathpar}
\inferrule[T-Variable]{ \inferrule[T-Variable]{
\metavariable{x} \in \exprvars\\ % \metavariable{x} \in \exprvars\\
\metavariable{\tau} \in \nonterm{T}\\ % \metavariable{\tau} \in \nonterm{T}\\
\metavariable{x}:\metavariable{\tau} \in \Gamma\\ \metavariable{x}:\metavariable{\tau} \in \Gamma\\
}{ }{
\Gamma \vdash \metavariable{x}:\metavariable{\tau} \Gamma \vdash \metavariable{x}:\metavariable{\tau}
}\and }\and
\inferrule[T-LetBinding]{ \inferrule[T-LetBinding]{
\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\ \Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
\Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} : \metavariable{\tau} \Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} : \metavariable{\tau}
@ -549,17 +551,16 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) : \metavariable{\tau} \Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) : \metavariable{\tau}
} }
\inferrule[T-TypeAbs]{ \inferrule[T-TypeAbs]{
\metavariable{\tau} \in \nonterm{T} \\ % \metavariable{\tau} \in \nonterm{T} \\
\metavariable{e} \in \nonterm{E} \\ % \metavariable{e} \in \nonterm{E} \\
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\ \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
}{ }{
\Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau} \Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
} }
\inferrule[T-TypeApp]{ \inferrule[T-TypeApp]{
\metavariable{\tau} \in \nonterm{T} \\ % \metavariable{\tau} \in \nonterm{T} \\
\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\ \Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
\metavariable{\sigma} \in \nonterm{T} \metavariable{\sigma} \in \nonterm{T}
}{ }{
@ -567,42 +568,34 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
} }
\inferrule[T-ValueAbs]{ \inferrule[T-Abs]{
\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\ % \metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
\metavariable{e} \in \nonterm{E} \\ % \metavariable{e} \in \nonterm{E} \\
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\ \Gamma,\metavariable{x}:\metavariable{\sigma} \vdash \metavariable{e} : \metavariable{\tau} \\
}{ }{
\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau} \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
} }
\inferrule[T-ValueApp]{ \inferrule[T-App]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\ \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
\Gamma \vdash \metavariable{a} : \metavariable{\sigma} \\ \Gamma \vdash \metavariable{a} : \metavariable{\sigma} \\
}{ }{
\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau} \Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
}\and }\and
\inferrule[T-Compatible]{
\Gamma \vdash \metavariable{e} : \metavariable{\tau}
}{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}
}\and
\inferrule[T-MorphAbs]{ \inferrule[T-MorphAbs]{
\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\ % \metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
\metavariable{e} \in \nonterm{E} \\ % \metavariable{e} \in \nonterm{E} \\
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\ \Gamma,\metavariable{x}:\metavariable{\tau} \vdash \metavariable{e} : \metavariable{\tau'} \\
}{
\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
}\and
\inferrule[T-MorphApp]{
\Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
\exists \metavariable{h} . \Gamma \vdash \metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_{morph}} \metavariable{\tau'}\\
\metavariable{\tau} \precsim \metavariable{\tau'} \metavariable{\tau} \precsim \metavariable{\tau'}
}{ }{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'} \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\tau}\typeterminal{\rightarrow_{morph}}\metavariable{\tau'}
}\and
\inferrule[T-MorphFun]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
}{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
}\and }\and
\inferrule[T-Ascension]{ \inferrule[T-Ascension]{
@ -617,16 +610,98 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
\metavariable{\tau} \leq \metavariable{\tau'} \metavariable{\tau} \leq \metavariable{\tau'}
}{ }{
\Gamma \vdash \metavariable{e} : \metavariable{\tau'} \Gamma \vdash \metavariable{e} : \metavariable{\tau'}
}\and }
\end{mathpar} \end{mathpar}
\end{definition} \end{definition}
\begin{definition}[Semantic Typing Relation]
\label{def:semtyperules}
\begin{mathpar}
\inferrule[T-NativeRepr]{
\Gamma\vdash \metavariable{e} : \metavariable{\tau}
}{
\Gamma\vdash \metavariable{e} :\approx \metavariable{\tau}
}
\inferrule[T-CoercedRepr]{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}\\
% \metavariable{\tau} \precsim \metavariable{\tau'}\\
%\exists \metavariable{h} \text{ s.t. }
\Gamma \vdash \metavariable{h}: \typeterminal{\metavariable{\tau}\rightarrow_\text{morph}\metavariable{\tau'}}
}{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
}
\inferrule[T-CompatibleApp]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \rightarrow \metavariable{\tau}\\
\Gamma \vdash \metavariable{a} :\approx \metavariable{\sigma}
}{
\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
}
\end{mathpar}
\end{definition}
\subsection{Coercion Semantics}
We define the translation function \(\llbracket . \rrbracket\) which completes a \emph{semantically well-typed} expression
by inserting all required coercions based on the typing derivation of the expression.
The result shall be a \emph{syntactically well-typed} expression.
We write \(C :: \sigma \precsim \tau\) to mean "C is a subtyping derivation tree whose conclusion is \(\sigma \precsim \tau\)".
Similarly, we write \(D :: \Gamma \vdash e : \tau\) to mean "D is a typing derivation whose conclusion is \(\Gamma \vdash e : \tau\)"
\begin{definition}[Translation]
\begin{mathpar}
\Big{\llbracket} \inferrule[T-SemanticSubtype]{
D_1 :: \Gamma \vdash \metavariable{h}:\metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}\\
D_2 :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\\
% C :: \metavariable{\tau} \precsim \metavariable{\tau'}
}{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
}\Big{\rrbracket} = \exprterminal{(}
\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
%\metavariable{h} \llbracket D_2 \rrbracket
\exprterminal{)}
\Big{\llbracket} \inferrule[T-CoercedApp]{
D_1 :: \Gamma \vdash \metavariable{f}:\metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
D_2 :: \Gamma \vdash \metavariable{a}:\approx\metavariable{\sigma}
}{
\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
}\Big{\rrbracket} = \exprterminal{(}
%\exprterminal{(}\metavariable{f} \llbracket D_2 \rrbracket \exprterminal{)}
\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
\exprterminal{)}
\Big{\llbracket}
\inferrule[\emph{Otherwise}]{}{
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}
}
\Big{\rrbracket} = \metavariable{e}
\end{mathpar}
\end{definition}
\begin{lemma}[Elimination of \(:\approx\)]
\label{lemma:translation}
For all \emph{semantically well-typed} expressions \metavariable{e} with the typing derivation \(D :: \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\),
the translation \(\llbracket D \rrbracket = \metavariable{e'}\), yields a \emph{syntactically well-typed} expression \metavariable{e'} with
\(\emptyset \vdash \metavariable{e'} : \metavariable{\tau} \)
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\subsection{Evaluation} \subsection{Evaluation}
Evaluation of an expression \(\metavariable{e} \in \exprnonterm{\emptyset}{\emptyset}\) is defined by exhaustive application of the rewrite rules \(\rightarrow_\beta\) and \(\rightarrow_\delta\), Evaluation of an expression \(\metavariable{e} \in \nonterm{E}\) is defined by exhaustive application of the rewrite rule \(\rightarrow_\beta\) as in \ref{def:evalrules}.
which are given in \ref{def:evalrules}.
\begin{definition}[Inference Rules for Evaluation] \begin{definition}[Inference Rules for Evaluation]
\label{def:evalrules} \label{def:evalrules}
@ -644,13 +719,13 @@ which are given in \ref{def:evalrules}.
\inferrule[E-App2]{ \inferrule[E-App2]{
\metavariable{e_2} \rightarrow_\beta \metavariable{e_2'} \metavariable{e_2} \rightarrow_\beta \metavariable{e_2'}
}{ }{
\metavariable{e_1} \metavariable{e_2} \metavariable{v_1} \metavariable{e_2}
\rightarrow_\beta \rightarrow_\beta
\metavariable{e_1} \metavariable{e_2'} \metavariable{v_1} \metavariable{e_2'}
} }\and
\inferrule[E-TypApp]{ \inferrule[E-TypApp]{
\metavariable{\tau} \in \typenonterm{\emptyset}\\ % \metavariable{\tau} \in \typenonterm{\emptyset}\\
\metavariable{e} \rightarrow_\beta \metavariable{e'} \metavariable{e} \rightarrow_\beta \metavariable{e'}
}{ }{
\metavariable{e} \metavariable{e}
@ -671,7 +746,6 @@ which are given in \ref{def:evalrules}.
\rightarrow_\beta \rightarrow_\beta
\{ \metavariable{\alpha} \mapsto \metavariable{\tau} \} \metavariable{e} \{ \metavariable{\alpha} \mapsto \metavariable{\tau} \} \metavariable{e}
}\and }\and
\inferrule[E-AppLam]{ \inferrule[E-AppLam]{
}{ }{
\exprterminal{(\lambda} \metavariable{x} \exprterminal{(\lambda} \metavariable{x}
@ -690,64 +764,27 @@ which are given in \ref{def:evalrules}.
\exprterminal{\text{ in }}\metavariable{e} \exprterminal{\text{ in }}\metavariable{e}
\rightarrow_\beta \rightarrow_\beta
\{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e} \{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e}
} }\and
\inferrule[E-Ascribe]{
\inferrule[E-ImplicitCast]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
\Gamma \vdash \metavariable{h} : \metavariable{\sigma'} \typeterminal{\rightarrow_{morph}} \metavariable{\sigma} \\
\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}
}{ }{
\exprterminal{(} \metavariable{f} \quad \metavariable{a} \exprterminal{)} \metavariable{e}
\rightarrow_\delta \exprterminal{\text{ as }}
\exprterminal{(} \metavariable{f} \quad \exprterminal{(} \metavariable{h} \quad \metavariable{a} \exprterminal{))} \metavariable{\tau}
\rightarrow_\beta
\metavariable{e}
} }
\end{mathpar} \end{mathpar}
\end{definition} \end{definition}
\begin{lemma}[\(\beta\)-reduction preserves \(\delta\)-normalform]
\label{lemma:preserve-delta-normalform}
Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow_\beta \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[\(\delta\)-normalform eliminates compatibility]
\label{lemma:eliminate-compat}
Assume \(\emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\) and \(\metavariable{e} \rightarrow_{\delta}^* \metavariable{e'}\) such that \(\metavariable{e'}\) is in \(\delta\)-normalform.
Then \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\)
\begin{proof}
\end{proof}
\end{lemma}
\subsection{Soundness} \subsection{Soundness}
\begin{lemma}[\(\beta\)-Preservation]
\label{lemma:beta-preservation}
Assume the expression \(\metavariable{e}\) is \textbf{syntactically well-typed}, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\) for some type \(\metavariable{\tau}\). Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\) it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[\(\delta\)-Preservation]
\label{lemma:delta-preservation}
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\begin{lemma}[Preservation] \begin{lemma}[Preservation]
\label{lemma:preservation} \label{lemma:preservation}
Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\) for some type \(\metavariable{\tau}\). Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\) it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well. Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\)
for some type \(\metavariable{\tau}\).
Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
\begin{proof} \begin{proof}
\todo{} \todo{}
@ -756,23 +793,44 @@ Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdas
\begin{lemma}[Progress] \begin{lemma}[Progress]
\label{lemma:progress} \label{lemma:progress}
If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\), then either \(\metavariable{e}\) is a value or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\) If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\),
then either \(\metavariable{e}\) is a value
or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
\begin{proof} \begin{proof}
\todo{} \todo{}
\end{proof} \end{proof}
\end{lemma} \end{lemma}
\begin{theorem}[Soundness] \begin{theorem}[Syntactic Type Soundness]
If \(\emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value. \label{theorem:syntactic-soundness}
No syntactically well-typed expression is stuck.
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\metavariable{\tau}\).
Then it never occurs that \(\metavariable{e} \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
\begin{proof} \begin{proof}
By \ref{lemma:}
Follows from \ref{lemma:progress} and \ref{lemma:preservation}. Follows from \ref{lemma:progress} and \ref{lemma:preservation}.
\end{proof} \end{proof}
\end{theorem} \end{theorem}
\begin{theorem}[Semantic Type Soundness]
\label{theorem:semantic-soundness}
No semantically well-typed expression is stuck.
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
Then it never occurs that \(\llbracket D \rrbracket \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
\begin{proof}
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
By \ref{lemma:translation}, \(\emptyset \vdash \llbracket D \rrbracket : \metavariable{\tau}\)
and thus it follows by \ref{theorem:syntactic-soundness} that \metavariable{e} is not stuck.
\end{proof}
\end{theorem}
\newpage
\section{Boehm-Berarducci Encoding} \section{Boehm-Berarducci Encoding}