paper: add type equivalence

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Michael Sippel 2024-08-08 13:18:36 +02:00
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@ -333,29 +333,120 @@ in an expression \(e \in \nonterm{E} \) with the \(\psi_e(\metavariable{x_i})\)
\end{definition}
\subsection{Typing}
\subsubsection{Equivalence of Type Terms}
\begin{definition}[Distributivity]
\todo{}
%We want distributivity of ladders over type-specialization as well as over function/morphism types.
\begin{definition}[Distributivity in Types]
\begin{mathpar}
\typeterminal{< \metavariable{\sigma}\sim\metavariable{\sigma'} \quad \metavariable{\tau} >}
\rightarrow_\text{distribute}
\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma'} \quad \metavariable{\tau} > }
\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau}\sim\metavariable{\tau'} >}
\rightarrow_\text{distribute}
\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma} \quad \metavariable{\tau'} > }
\typeterminal{ \metavariable{\sigma}\sim\metavariable{\sigma'} \rightarrow \metavariable{\tau} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow \metavariable{\tau} ) \sim ( \metavariable{\sigma'} \rightarrow \metavariable{\tau} ) }
\typeterminal{ \metavariable{\sigma} \rightarrow \metavariable{\tau}\sim\metavariable{\tau'} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow \metavariable{\tau} ) \sim ( \metavariable{\sigma} \rightarrow \metavariable{\tau'} ) }
\typeterminal{ \metavariable{\sigma}\sim\metavariable{\sigma'} \rightarrow_\text{morph} \metavariable{\tau} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau} ) \sim ( \metavariable{\sigma'} \rightarrow_\text{morph} \metavariable{\tau} ) }
\typeterminal{ \metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau}\sim\metavariable{\tau'} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau} ) \sim ( \metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau'} ) }
\end{mathpar}
Let \(\rightarrow_\text{condense}\) be the inverse to \(\rightarrow_\text{distribute}\).
See \hyperref[coq:type-dist]{equiv.v:\ref{coq:type-dist}}.
\end{definition}
\begin{definition}[Equivalence Relation]
\todo{}
\begin{definition}[Alpha Conversion in Types]
\begin{mathpar}
\typeterminal{\forall \metavariable{\alpha} . \metavariable{\tau}}
\rightarrow_{\alpha}
\typeterminal{\forall \metavariable{\alpha'} . } \{ \metavariable{\alpha} \mapsto \metavariable{\alpha'} \} \metavariable{\tau}
\end{mathpar}
\end{definition}
\begin{definition}[Equivalence Relation]
Transitive closure over \(\rightarrow_\text{distribute}\), \(\rightarrow_\text{condense}\) and \(\rightarrow_\alpha\).
\begin{mathpar}
\inferrule[E-Refl]{
\metavariable{\tau} \in \nonterm{T}
}{
\metavariable{\tau} \equiv \metavariable{\tau}
}\and
\inferrule[E-Trans]{
\metavariable{\tau_1} \equiv \metavariable{\tau_2}\\
\metavariable{\tau_2} \equiv \metavariable{\tau_3}
}{
\metavariable{\tau_1} \equiv \metavariable{\tau_3}
}
\inferrule[E-Rename]{
\metavariable{\tau_1} \rightarrow_\alpha \metavariable{\tau_2}
}{
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}
\inferrule[E-Distribute]{
\metavariable{\tau_1} \rightarrow_\text{distribute} \metavariable{\tau_2}
}{
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}\and
\inferrule[E-Condense]{
\metavariable{\tau_1} \rightarrow_\text{condense} \metavariable{\tau_2}
}{
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}
\end{mathpar}
See \hyperref[coq:type-equiv]{equiv.v:\ref{coq:type-equiv}}.
\end{definition}
\begin{lemma}[Symmetry of \(\equiv\)]
\begin{mathpar}
\inferrule[E-Symm]{
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}{
\metavariable{\tau_2} \equiv \metavariable{\tau_1}
}
\end{mathpar}
\begin{proof}
\(\rightarrow_{distribute}\) is the inverse of \(\rightarrow_{condense}\) and \(\rightarrow_{\alpha}\) is symmetric by itself.
\end{proof}
\end{lemma}
\subsubsection{Normal Forms}
\begin{definition}[Ladder Normal Form]
\todo{}
LNF is reached by exhaustive application of \(\rightarrow_\text{distribute}\).
\end{definition}
\subsubsection{Subtyping}
\begin{definition}[Parameter Normal Form]
PNF reached by exhaustive application of \(\rightarrow_\text{condense}\).
\end{definition}
\begin{definition}[Syntactic Subtyping]
\todo{}