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Theorem 8.9

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Daniel Hillerström
2021-05-27 13:49:42 +01:00
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thesis.tex
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@@ -14560,35 +14560,26 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By case analysis on $\reducesto$ and induction on $\approxa$ using % By case analysis on $\reducesto$ and induction on $\approxa$ using
Lemma~\ref{lem:sdtrans-subst} and Lemma~\ref{lem:sdtrans-admin}. % Lemma~\ref{lem:sdtrans-subst} and Lemma~\ref{lem:sdtrans-admin}.
% %
By induction on $M' \approxa \sdtrans{M}$ and side induction on By induction on $M' \approxa \sdtrans{M}$ and side induction on
$M \reducesto N$. $M \reducesto N$ using Lemma~\ref{lem:sdtrans-subst} and Lemma~\ref{lem:sdtrans-admin}.
% %
The interesting case is reflexivity of $\approxa$ where The interesting case is reflexivity of $\approxa$ where
$M \reducesto N$ is an application of $\semlab{Op^\dagger}$. $M \reducesto N$ is an application of $\semlab{Op^\dagger}$, which
we will show.
\noindent\textbf{Case} $\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto In the reflexivity case we have $M' \approxa \sdtrans{M}$, where
N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ where $M = \ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H$ and
$\ell \notin \BL(\EC)$ and $N = N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ such that
$H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$. \smallskip\\ $M \reducesto N$ where $\ell \notin \BL(\EC)$ and
There are three subcases to consider. $H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$.
\begin{enumerate} %
\item Base step: Hence by reflexivity of $\approxa$ we have
$M' = \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. We $M' = \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. Now we
can compute $N'$ by direct calculation starting from $M'$ yielding 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}\\ & \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}\\
=& \reason{definition of $\sdtrans{-}$}\\ =& \reason{definition of $\sdtrans{-}$}\\
&\bl &\bl
@@ -14596,23 +14587,6 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
\Let\;\Record{f;g} = z\;\In\;g\,\Unit \Let\;\Record{f;g} = z\;\In\;g\,\Unit
\el\\ \el\\
\reducesto^+& \reason{\semlab{Op} using assumption $\ell \notin \BL(\sdtrans{\EC})$, \semlab{Let}, \semlab{Let}}\\ \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 &\bl
\Let\;\Record{f;g} = \Record{ \Let\;\Record{f;g} = \Record{
\bl \bl
@@ -14642,55 +14616,161 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
$r \notin \FV(N_\ell)$ then the two terms $N'$ and $r \notin \FV(N_\ell)$ then the two terms $N'$ and
$\sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}$ are the $\sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}$ are the
identical, and thus the result follows immediate by reflexivity of identical, and thus the result follows immediate by reflexivity of
the $\approxa$-relation. Otherwise $N'$ approximates the $\approxa$-relation. Otherwise the proof reduces to showing that
$N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ at least up to a use of the larger resumption term simulates the smaller resumption term,
$r$. We need to show that the approximation remains faithful during i.e (note we lift the $\approxa$-relation to value terms).
any application of $r$. Specifically, we proceed to show that for
any value $W \in \ValCat$
% %
\[ \[
(\bl (\bl
\lambda x.\Let\;z \revto (\lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\ \lambda x.\Let\;z \revto (\lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
\Let\;\Record{f;g} = z \;\In\;f\,\Unit)~W \approxa (\lambda y.\sdtrans{\EC}[\Return\;y])~W. \Let\;\Record{f;g} = z \;\In\;f\,\Unit) \approxa (\lambda y.\sdtrans{\EC}[\Return\;y]).
\el \el
\] \]
% %
The right hand side reduces to $\sdtrans{\EC}[\Return\;W]$. Two We use the congruence rules to apply a single $\semlab{App}$ on the
applications of \semlab{App} on the left hand side yield the term left hand side to obtain
% %
\[ \[
\Let\;z \revto \Handle\;\sdtrans{\EC}[\Return\;W]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit. (\bl
\lambda x.\Let\;z \revto \Handle\;\sdtrans{\EC}[\Return\;x]\;\With\;\sdtrans{H}\;\In\\
\Let\;\Record{f;g} = z \;\In\;f\,\Unit) \approxa (\lambda y.\sdtrans{\EC}[\Return\;y]).
\el
\] \]
% %
Define Now the trick is to define the following context
% %
$\EC' \defas \Let\;z \revto \Handle\; [\,]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit$ \[
\EC' \defas \Let\;z \revto \Handle\; [\,]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit.
\]
% %
such that $\EC'$ is an administrative evaluation context by The context $\EC'$ is an administrative evaluation context by
Lemma~\ref{lem:sdtrans-admin}. Then it follows by Lemma~\ref{lem:sdtrans-admin}. Now it follows by
Defintion~\ref{def:approx-admin} that Defintion~\ref{def:approx-admin} that
$\EC'[\sdtrans{\EC}[\Return\;W]] \approxa $(\lambda x.\EC'[\sdtrans{\EC}[\Return\;x]]) \approxa
\sdtrans{\EC}[\Return\;W]$. (\lambda y.\sdtrans{\EC}[\Return\;y])$.
%
% \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
% $H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$. \smallskip\\
% There are three subcases to consider.
% \begin{enumerate}
% \item Base step:
% $M' = \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. We
% can compute $N'$ by direct calculation starting from $M'$ yielding
% %
% % \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}\\
% =& \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
% \end{derivation}
% %
% Take the final term to be $N'$. If the resumption
% $r \notin \FV(N_\ell)$ then the two terms $N'$ and
% $\sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}$ are the
% identical, and thus the result follows immediate by reflexivity of
% the $\approxa$-relation. Otherwise $N'$ approximates
% $N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ at least up to a use of
% $r$. We need to show that the approximation remains faithful during
% any application of $r$. Specifically, we proceed to show that for
% any value $W \in \ValCat$
% %
% \[
% (\bl
% \lambda x.\Let\;z \revto (\lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
% \Let\;\Record{f;g} = z \;\In\;f\,\Unit)~W \approxa (\lambda y.\sdtrans{\EC}[\Return\;y])~W.
% \el
% \]
% %
% The right hand side reduces to $\sdtrans{\EC}[\Return\;W]$. Two
% applications of \semlab{App} on the left hand side yield the term
% %
% \[
% \Let\;z \revto \Handle\;\sdtrans{\EC}[\Return\;W]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit.
% \]
% %
% Define
% %
% $\EC' \defas \Let\;z \revto \Handle\; [\,]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit$
% %
% such that $\EC'$ is an administrative evaluation context by
% Lemma~\ref{lem:sdtrans-admin}. Then it follows by
% Defintion~\ref{def:approx-admin} that
% $\EC'[\sdtrans{\EC}[\Return\;W]] \approxa
% \sdtrans{\EC}[\Return\;W]$.
\item Inductive step: Assume $M' \reducesto M''$ and % \item Inductive step: Assume $M' \reducesto M''$ and
$M'' \approxa \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. Using a similar argument to above we get that % $M'' \approxa \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. Using a similar argument to above we get that
\[ % \[
\sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H} % \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}
\reducesto^+ \sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}. % \reducesto^+ \sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}.
\] % \]
Take $N' = M''$ then by the first induction hypothesis % Take $N' = M''$ then by the first induction hypothesis
$M' \reducesto N'$ and by the second induction hypothesis % $M' \reducesto N'$ and by the second induction hypothesis
$N' \approxa \sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}$ as % $N' \approxa \sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}$ as
desired. % desired.
\item Inductive step: Assume $admin(\EC')$ and % \item Inductive step: Assume $admin(\EC')$ and
$M' \approxa \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. % $M' \approxa \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$.
% % %
By the induction the hypothesis $M' \reducesto N''$. Take % By the induction the hypothesis $M' \reducesto N''$. Take
$N' = \EC'[N'']$. The result follows by an application of the % $N' = \EC'[N'']$. The result follows by an application of the
admin rule. % admin rule.
\item Compatibility step: We check every syntax constructor, % \item Compatibility step: We check every syntax constructor,
however, since the relation is compositional\dots % however, since the relation is compositional\dots
\end{enumerate} % \end{enumerate}
\end{proof} \end{proof}
% \begin{proof} % \begin{proof}
% By case analysis on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst} % By case analysis on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst}