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Daniel Hillerström
2021-05-27 13:04:43 +01:00
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thesis.tex
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@@ -14562,6 +14562,12 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
By case analysis on $\reducesto$ and induction on $\approxa$ using
Lemma~\ref{lem:sdtrans-subst} and Lemma~\ref{lem:sdtrans-admin}.
%
By induction on $M' \approxa \sdtrans{M}$ and side induction on
$M \reducesto N$.
%
The interesting case is reflexivity of $\approxa$ where
$M \reducesto N$ is an application of $\semlab{Op^\dagger}$.
\noindent\textbf{Case} $\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto
N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ where
$\ell \notin \BL(\EC)$ and
@@ -14572,14 +14578,63 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
$M' = \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. We
can compute $N'$ by direct calculation starting from $M'$ yielding
%
\begin{derivation}
% \begin{derivation}
% & \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}\\
% =\reducesto^+& \reason{\semlab{Op} ($\ell \notin \BL(\sdtrans{\EC})$), $2\times$\semlab{Let},\semlab{Split},\semlab{App}, Lemma~\ref{lem:sdtrans-subst}}\\
% &\sdtrans{N_\ell[\lambda x.
% \bl
% \Let\;z \revto (\lambda y.\Handle\;\EC[\Return\;y]\;\With\;H)~x\;\In\\
% \Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,V/p]}
% \el\\
% \end{derivation}
\begin{derivation}
& \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}\\
=\reducesto^+& \reason{\semlab{Op} ($\ell \notin \BL(\sdtrans{\EC})$), $2\times$\semlab{Let},\semlab{Split},\semlab{App}, Lemma~\ref{lem:sdtrans-subst}}\\
=& \reason{definition of $\sdtrans{-}$}\\
&\bl
\Let\;z \revto \Handle\;\sdtrans{\EC}[\Do\;\ell~\sdtrans{V}]\;\With\;\sdtrans{H}\;\In\\
\Let\;\Record{f;g} = z\;\In\;g\,\Unit
\el\\
\reducesto^+& \reason{\semlab{Op} using assumption $\ell \notin \BL(\sdtrans{\EC})$, \semlab{Let}, \semlab{Let}}\\
% &\bl
% \Let\;z \revto
% (\bl
% \Let\;r \revto \lambda x.\Let\;z \revto r~x\;\In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit\;\In\\
% \Return\;\Record{
% \bl
% \lambda\Unit.\Let\;x \revto \Do\;\ell~p\;\In\;r~x;\\
% \lambda\Unit.\sdtrans{N_\ell}})[
% \bl
% \sdtrans{V}/p,\\
% \lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H}/r]
% \el
% \el
% \el\\
% \In\;\Let\;\Record{f;g} = z\;\In\;g\,\Unit
% \el\\
% \reducesto^+& \reason{\semlab{Let}, \semlab{Let}}\\
&\bl
\Let\;\Record{f;g} = \Record{
\bl
\lambda\Unit.\Let\;x \revto \Do\;\ell~\sdtrans{V}\;\In\;r~x;\\
\lambda\Unit.\sdtrans{N_\ell}}[\lambda x.
\bl
\Let\;z \revto (\lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
\Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,\sdtrans{V}/p]\;\In\; g\,\Unit
\el
\el\\
\el\\
\reducesto^+ &\reason{\semlab{Split}, \semlab{App}}\\
&\sdtrans{N_\ell}[\lambda x.
\bl
\Let\;z \revto (\lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
\Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,\sdtrans{V}/p]
\el\\
=& \reason{by Lemma~\ref{lem:sdtrans-subst}}\\
&\sdtrans{N_\ell[\lambda x.
\bl
\Let\;z \revto (\lambda y.\Handle\;\EC[\Return\;y]\;\With\;H)~x\;\In\\
\Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,V/p]}
\el\\
\el
\end{derivation}
%
Take the final term to be $N'$. If the resumption
@@ -14628,6 +14683,12 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
desired.
\item Inductive step: Assume $admin(\EC')$ and
$M' \approxa \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$.
%
By the induction the hypothesis $M' \reducesto N''$. Take
$N' = \EC'[N'']$. The result follows by an application of the
admin rule.
\item Compatibility step: We check every syntax constructor,
however, since the relation is compositional\dots
\end{enumerate}
\end{proof}
% \begin{proof}