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paper: rename inference rules to match coq definitions

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Michael Sippel 2024-08-22 09:05:20 +02:00
parent 4323e7d09f
commit 9c17e9e642
Signed by: senvas
GPG key ID: F96CF119C34B64A6
1 changed files with 33 additions and 22 deletions
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paper/main.tex
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@ -387,30 +387,30 @@ Transitive closure over \(\rightarrow_\text{distribute}\), \(\rightarrow_\text{c
\begin{mathpar} \begin{mathpar}
\inferrule[E-Refl]{ \inferrule[T-Eq-Refl]{
\metavariable{\tau} \in \nonterm{T} \metavariable{\tau} \in \nonterm{T}
}{ }{
\metavariable{\tau} \equiv \metavariable{\tau} \metavariable{\tau} \equiv \metavariable{\tau}
}\and }\and
\inferrule[E-Trans]{ \inferrule[T-Eq-Trans]{
\metavariable{\tau_1} \equiv \metavariable{\tau_2}\\ \metavariable{\tau_1} \equiv \metavariable{\tau_2}\\
\metavariable{\tau_2} \equiv \metavariable{\tau_3} \metavariable{\tau_2} \equiv \metavariable{\tau_3}
}{ }{
\metavariable{\tau_1} \equiv \metavariable{\tau_3} \metavariable{\tau_1} \equiv \metavariable{\tau_3}
} }
\inferrule[E-Rename]{ \inferrule[T-Eq-Alpha]{
\metavariable{\tau_1} \rightarrow_\alpha \metavariable{\tau_2} \metavariable{\tau_1} \rightarrow_\alpha \metavariable{\tau_2}
}{ }{
\metavariable{\tau_1} \equiv \metavariable{\tau_2} \metavariable{\tau_1} \equiv \metavariable{\tau_2}
} }
\inferrule[E-Distribute]{ \inferrule[T-Eq-Distribute]{
\metavariable{\tau_1} \rightarrow_\text{distribute} \metavariable{\tau_2} \metavariable{\tau_1} \rightarrow_\text{distribute} \metavariable{\tau_2}
}{ }{
\metavariable{\tau_1} \equiv \metavariable{\tau_2} \metavariable{\tau_1} \equiv \metavariable{\tau_2}
}\and }\and
\inferrule[E-Condense]{ \inferrule[T-Eq-Condense]{
\metavariable{\tau_1} \rightarrow_\text{condense} \metavariable{\tau_2} \metavariable{\tau_1} \rightarrow_\text{condense} \metavariable{\tau_2}
}{ }{
\metavariable{\tau_1} \equiv \metavariable{\tau_2} \metavariable{\tau_1} \equiv \metavariable{\tau_2}
@ -425,7 +425,7 @@ See \hyperref[coq:type-equiv]{equiv.v:\ref{coq:type-equiv}}.
\begin{lemma}[Symmetry of \(\equiv\)] \begin{lemma}[Symmetry of \(\equiv\)]
\begin{mathpar} \begin{mathpar}
\inferrule[E-Symm]{ \inferrule[T-Eq-Symm]{
\metavariable{\tau_1} \equiv \metavariable{\tau_2} \metavariable{\tau_1} \equiv \metavariable{\tau_2}
}{ }{
\metavariable{\tau_2} \equiv \metavariable{\tau_1} \metavariable{\tau_2} \equiv \metavariable{\tau_1}
@ -450,17 +450,24 @@ PNF reached by exhaustive application of \(\rightarrow_\text{condense}\).
\subsubsection{Subtype Relations} \subsubsection{Subtype Relations}
We define two relations: first the syntatic subtype relation \(\leq\) and second the semantic subtype relation \(\precsim\). We define two relations: first the representation subtype relation \(\leq\) and second the transformation subtype relation \(\precsim\).
\begin{definition}[Syntactic Subtype (\(\tau_1\leq\tau_2\))] \begin{definition}[Representation Subtype (\(\tau_1\leq\tau_2\))]
\begin{mathpar} \begin{mathpar}
\inferrule[S-Refl]{ \inferrule[TSubRepr-Refl]{
\metavariable{\tau} \equiv \metavariable{\tau'} \metavariable{\tau} \equiv \metavariable{\tau'}
}{ }{
\metavariable{\tau} \leq \metavariable{\tau'} \metavariable{\tau} \leq \metavariable{\tau'}
} }
\inferrule[S-Syntactic]{ \inferrule[TSubRepr-Trans]{
\metavariable{\sigma} \leq \metavariable{\tau}\\
\metavariable{\tau} \leq \metavariable{\nu}
}{
\metavariable{\sigma} \leq \metavariable{\nu}
}
\inferrule[TSubRepr-Ladder]{
\metavariable{\sigma} \leq \metavariable{\tau} \metavariable{\sigma} \leq \metavariable{\tau}
}{ }{
\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \leq \metavariable{\tau} \metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \leq \metavariable{\tau}
@ -468,40 +475,44 @@ We define two relations: first the syntatic subtype relation \(\leq\) and second
\end{mathpar} \end{mathpar}
\end{definition} \end{definition}
\begin{definition}[Semantic Subtype (\(\tau_1\precsim\tau_2\))] \begin{definition}[Transformation Subtype (\(\tau_1\precsim\tau_2\))]
\begin{mathpar} \begin{mathpar}
\inferrule[TSub-Refl]{
\inferrule[S-Refl]{
\metavariable{\tau} \equiv \metavariable{\tau'} \metavariable{\tau} \equiv \metavariable{\tau'}
}{ }{
\metavariable{\tau} \precsim \metavariable{\tau'} \metavariable{\tau} \precsim \metavariable{\tau'}
}\and
\inferrule[TSub-Trans]{
\metavariable{\sigma} \precsim \metavariable{\tau}\\
\metavariable{\tau} \precsim \metavariable{\nu}
}{
\metavariable{\sigma} \precsim \metavariable{\nu}
} }
\inferrule[S-Syntactic]{ \\
\inferrule[TSub-Ladder]{
\metavariable{\sigma} \precsim \metavariable{\tau} \metavariable{\sigma} \precsim \metavariable{\tau}
}{ }{
\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \precsim \metavariable{\tau} \metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \precsim \metavariable{\tau}
} }\and
\inferrule[TSub-Morph]{
\inferrule[S-Semantic]{
\metavariable{\sigma} \equiv \metavariable{\tau} \metavariable{\sigma} \equiv \metavariable{\tau}
}{ }{
\metavariable{\sigma} \typeterminal{\sim} \metavariable{\sigma'} \precsim \metavariable{\tau} \typeterminal{\sim} \metavariable{\tau'} \metavariable{\sigma} \typeterminal{\sim} \metavariable{\sigma'} \precsim \metavariable{\tau} \typeterminal{\sim} \metavariable{\tau'}
} }
\end{mathpar} \end{mathpar}
\end{definition} \end{definition}
\begin{example}[Syntactic \& Semantic Subtypes]$\\$ \begin{example}[Representation \& Transformation Subtypes]$\\$
\begin{enumerate} \begin{enumerate}
\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \leq \quad \) Char }\\ \item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \leq \quad \) Char }\\
.. is a \emph{syntactic subtype}, because the representation of \typeterminal{<Digit 10>} is \emph{embedded} into \typeterminal{Char}.\\ .. is a \emph{representation subtype}, because the representation of \typeterminal{<Digit 10>} is \emph{embedded} into \typeterminal{Char}.\\
\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \precsim \quad \) <Digit 10> \(\sim\) machine.UInt64}\\ \item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \precsim \quad \) <Digit 10> \(\sim\) machine.UInt64}\\
.. is a \emph{semantic subtype}, because the \typeterminal{Char} based representation can be transformed into a representation based on \typeterminal{machine.UInt64}, .. is a \emph{transformation subtype}, because the \typeterminal{Char} based representation can be transformed into a representation based on \typeterminal{machine.UInt64},
while preserving its semantics. while preserving its semantics.
\end{enumerate} \end{enumerate}
\end{example} \end{example}