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Prove lemma

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Daniel Hillerström 5 years ago
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  1. 2
      macros.tex
  2. 152
      thesis.tex

2
macros.tex
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@ -133,6 +133,8 @@
\newcommand{\Res}{\keyw{res}}
\newcommand{\Cong}{\mathrm{cong}}
%% Handler projections.
\newcommand{\mret}{\mathrm{ret}}
\newcommand{\mops}{\mathrm{ops}}

152
thesis.tex
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@ -11483,14 +11483,14 @@ the resumption to wrap the handler around its body.
Nevertheless, the simulation proof makes minimal use of this power,
merely using it to rename a single variable.
%
We write $R_{\mathrm{cong}}$ for the compatible closure of relation
We write $R_{\Cong}$ for the compatible closure of relation
$R$, that is the smallest relation including $R$ and closed under term
constructors for $\SCalc$.
%% , otherwise known as \emph{reduction up to
%% congruence}.
\begin{theorem}[Simulation up to congruence]
If $M \reducesto N$ then $\dstrans{M} \reducesto_{\mathrm{cong}}^+
If $M \reducesto N$ then $\dstrans{M} \reducesto_{\Cong}^+
\dstrans{N}$.
\end{theorem}
@ -11670,7 +11670,7 @@ translation. However, this time the administrative overhead is more
significant. Reduction up to congruence is insufficient and we require
a more semantic notion of administrative reduction.
\begin{definition}[Administrative evaluation contexts]
\begin{definition}[Administrative evaluation contexts]\label{def:admin-eval}
An evaluation context $\EC \in \EvalCat$ is administrative,
$\admin(\EC)$, when the following two criteria hold.
\begin{enumerate}
@ -11681,7 +11681,7 @@ a more semantic notion of administrative reduction.
$V \in \ValCat$:
%
\[
\EC[\EC'[\Do\;\ell\;V]] \reducesto^\ast \Let\; x \revto \Do\;\ell\;V \;\In\; \EC[\EC'[\Return\;x]].
\EC[\EC'[\Do\;\ell\;V]] \reducesto_\Cong^\ast \Let\; x \revto \Do\;\ell\;V \;\In\; \EC[\EC'[\Return\;x]].
\]
\end{enumerate}
\end{definition}
@ -11698,7 +11698,7 @@ forwarded.
%% the second property in a uniform way by ensuring that the operation is
%% handled at least once.
\begin{definition}[Approximation up to administrative reduction]
\begin{definition}[Approximation up to administrative reduction]\label{def:approx-admin}
Define $\approxa$ as the compatible closure of the following inference
rules.
%
@ -11708,7 +11708,7 @@ rules.
{M \approxa M}
\inferrule*
{M \reducesto M' \\ M' \approxa N}
{M \reducesto_\Cong M' \\ M' \approxa N}
{M \approxa N}
\inferrule*
@ -11727,7 +11727,7 @@ The following lemma states that the forwarding component of the
translation is administrative.
%
\begin{lemma}\label{lem:sdtrans-admin}
For all shallow handlers $H$, the following context is administrative:
For all shallow handlers $H$, the following context is administrative
%
\[
\Let\; z \revto
@ -11738,13 +11738,142 @@ For all shallow handlers $H$, the following context is administrative:
%
\end{lemma}
%
\begin{proof}
We need to check both conditions of Definition~\ref{def:admin-eval}.
\begin{enumerate}
\item We need to show that for all $V \in \ValCat$
%
\[
\Let\; z \revto
\Handle\; \Return\;V \;\With\; \sdtrans{H}\;
\In\;
\Let\; \Record{f;\_} = z\; \In\; f\,\Unit \reducesto^\ast \Return\;V.
\]
%
We show this by direct calculation using the assumption $\hret = \{\Return\;x \mapsto N\}$.
\begin{derivation}
&\Let\; z \revto
\Handle\; \Return\;V \;\With\; \sdtrans{H}\;
\In\;
\Let\; \Record{f;\_} = z\; \In\; f\,\Unit\\
\reducesto^+& \reason{\semlab{Lift}, \semlab{Ret}, \semlab{Let}, \semlab{Split}}\\
% &\Let\;\Record{f;\_} = \Record{\lambda\Unit.\Return\;x;\lambda\Unit.\sdtrans{N}}\;\In\;f\,\Unit
& (\lambda\Unit.\Return\;V)\,\Unit \reducesto \Return\;V
\end{derivation}
\item We need to show that for all evaluation contexts
$\EC' \in \EvalCat$, operations
$\ell \in \BL(\EC) \backslash \BL(\EC')$, and values
$V \in \ValCat$
%
\[
\ba{@{~}l@{~}l}
&\Let\; z \revto
\Handle\; \EC'[\Do\;\ell\;V]\;\With\;\sdtrans{H} \;\In\;\Record{f;\_} = z\;\In\;f\,\Unit\\
\reducesto_\Cong^\ast& \Let\; x \revto \Do\;\ell\;V \;\In\; \Let\; z \revto \Handle\; \EC'[\Return\;x]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;\_} = z\;\In\;f\,\Unit
\ea
\]
%
We show this by direct calculation using the assumption that
$\ell \notin \BL(\EC')$.
\begin{derivation}
&\Let\; z \revto
\Handle\; \EC'[\Do\;\ell\;V]\;\With\;\sdtrans{H}\;\In\;
\Let\;\Record{f;\_} = z\;\In\;f\,\Unit\\
\reducesto^+& \reason{\semlab{Lift}, \semlab{Op} using assumption $\ell \notin \EC'$}\\
&\bl \Let\; z \revto
\bl
\Let\;r\revto
\lambda x.\bl\Let\;z \revto (\lambda x.\Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H})~x\\
\In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit
\el\\
\In\;\Return\;\Record{\lambda\Unit.\Let\;x\revto \Do\;\ell~V\;\In\;r~x;\lambda\Unit.\sdtrans{N}}
\el\\
\In\;\Record{f;\_} = z\;\In\;f\,\Unit
\el\\
\reducesto^+& \reason{\semlab{Lift}, \semlab{Let}, \semlab{Split}, \semlab{App}}\\
&(\Let\;x\revto \Do\;\ell~V\;\In\;r~x)[(\lambda x.\bl
\Let\;z \revto (\lambda x.\Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H})\,x\\
\In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit)/r]\el\\
\reducesto_\Cong &\reason{\semlab{App} tail position reduction}\\
&\bl
\Let\;x \revto \Do\;\ell~V\;\In\;\Let\;z \revto (\lambda x.\Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H})\,x\; \In\\\Let\;\Record{f;g} = z\;\In\;f\,\Unit
\el\\
\reducesto_\Cong& \reason{\semlab{App} reduction under binder}\\
&\bl \Let\;x \revto \Do\;\ell~V\;\In\;\Let\;z \revto \Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H}\; \In\\\Let\;\Record{f;g} = z\;\In\;f\,\Unit\el
\end{derivation}
% The proof proceeds by induction on the structure of $\EC'$.
% %
% \begin{description}
% \item[Base step] $\EC' = [\,]$. The proof proceeds by direct calculation.
% \begin{derivation}
% &\Let\; z \revto
% \Handle\; \Do\;\ell\;V\;\With\;\sdtrans{H} \;\In\;\Record{f;\_} = z\;\In\;f\,\Unit\\
% \reducesto^+& \reason{\semlab{Lift}, \semlab{Op}}\\
% &\bl \Let\; z \revto
% \bl
% \Let\;r\revto
% \lambda x.\bl\Let\;z \revto (\lambda x.\Handle\;\Return\;x\;\With\;\sdtrans{H})~x\\
% \In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit
% \el\\
% \In\;\Return\;\Record{\lambda\Unit.\Let\;x\revto \Do\;\ell~V\;\In\;r~x;\lambda\Unit.\sdtrans{N}}
% \el\\
% \In\;\Record{f;\_} = z\;\In\;f\,\Unit
% \el\\
% \reducesto^+ & \reason{\semlab{Lift}, \semlab{Let}, \semlab{Split}, \semlab{App}}\\
% &(\Let\;x\revto \Do\;\ell~V\;\In\;r~x)[(\lambda x.\bl
% \Let\;z \revto (\lambda x.\Handle\;\Return\;x\;\With\;\sdtrans{H})\,x\\
% \In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit)/r]\el\\
% \reducesto_\Cong& \reason{\semlab{App} tail position reduction}\\
% &\bl
% \Let\;x \revto \Do\;\ell~V\;\In\;\Let\;z \revto (\lambda x.\Handle\;\Return\;x\;\With\;\sdtrans{H})\,x\; \In\\\Let\;\Record{f;g} = z\;\In\;f\,\Unit
% \el\\
% \reducesto_\Cong& \reason{\semlab{App} reduction under binder}\\
% &\bl \Let\;x \revto \Do\;\ell~V\;\In\;\Let\;z \revto \Handle\;\Return\;x\;\With\;\sdtrans{H}\; \In\\\Let\;\Record{f;g} = z\;\In\;f\,\Unit\el
% \end{derivation}
% \item[Inductive step] There are two subcases to consider.
% \begin{enumerate}[1)]
% \item $\EC' = \Let\;y \revto \EC''\;\In\;N$.
% \item $\EC' = \Handle\;\EC''\;\With\;\sdtrans{H'}$
% \begin{derivation}
% &\bl
% \Let\; z \revto
% \Handle\; (\Handle\;\EC''[\Do\;\ell\;V]\;\With\;\sdtrans{H'})\;\With\;\sdtrans{H}\;\In\\
% \Let\;\Record{f;\_} = z\;\In\;f\,\Unit
% \el\\
% \reducesto^+& \reason{\semlab{Lift}, \semlab{Op} using assumptions $\ell \notin \EC''$ and $H'^\ell = \emptyset$}\\
% &\bl \Let\; z \revto
% \bl
% \Let\;r\revto
% \lambda x.\bl\Let\;z \revto (\lambda x.\Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H})~x\\
% \In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit
% \el\\
% \In\;\Return\;\Record{\lambda\Unit.\Let\;x\revto \Do\;\ell~V\;\In\;r~x;\lambda\Unit.\sdtrans{N}}
% \el\\
% \In\;\Record{f;\_} = z\;\In\;f\,\Unit
% \el\\
% \reducesto^+& \reason{\semlab{Lift}, \semlab{Let}, \semlab{Split}, \semlab{App}}\\
% &(\Let\;x\revto \Do\;\ell~V\;\In\;r~x)[(\lambda x.\bl
% \Let\;z \revto (\lambda x.\Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H})\,x\\
% \In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit)/r]\el\\
% \reducesto_\Cong &\reason{\semlab{App} tail position reduction}\\
% &\bl
% \Let\;x \revto \Do\;\ell~V\;\In\;\Let\;z \revto (\lambda x.\Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H})\,x\; \In\\\Let\;\Record{f;g} = z\;\In\;f\,\Unit
% \el\\
% \reducesto_\Cong& \reason{\semlab{App} reduction under binder}\\
% &\bl \Let\;x \revto \Do\;\ell~V\;\In\;\Let\;z \revto \Handle\;\EC'[\Return\;x]\;\With\;\sdtrans{H}\; \In\\\Let\;\Record{f;g} = z\;\In\;f\,\Unit\el
% \end{derivation}
% \end{enumerate}
% \end{description}
\end{enumerate}
\end{proof}
%
\begin{theorem}[Simulation up to administrative reduction]
If $M' \approxa \sdtrans{M}$ and $M \reducesto N$ then there exists
$N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
\end{theorem}
%
\begin{proof}
By induction on $\reducesto$ using a substitution lemma and
By induction on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst} and
Lemma~\ref{lem:sdtrans-admin}. The interesting case is
$\semlab{Op^\dagger}$, which uses Lemma~\ref{lem:sdtrans-admin} to
approximate the body of the resumption up to administrative reduction.
@ -11858,10 +11987,9 @@ handlers in Section~\ref{sec:deep-as-shallow}, this simulation is only
up to congruence due to the need for an application of a pure function
to a variable to be reduced.
%
\begin{theorem}[Simulation of parameterised handlers by deep
handlers]
\begin{theorem}[Simulation up to congruence]
\label{thm:param-simulation}
If $M \reducesto N$ then $\PD{M} \reducesto^+_{\mathrm{cong}} \PD{N}$.
If $M \reducesto N$ then $\PD{M} \reducesto^+_{\Cong} \PD{N}$.
\end{theorem}
%
\begin{proof}
@ -11907,7 +12035,7 @@ to a variable to be reduced.
\PD{V}/p,\PD{W}/q, \\
\lambda \Record{x,q'}. (\lambda x. \Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~x~q'/r]\\
\el \\
\reducesto_{\mathrm{cong}}& \reason{$\semlab{App}$}\\
\reducesto_{\Cong}& \reason{$\semlab{App}$}\\
&\PD{N}[\PD{V}/p,\PD{W}/q,\lambda \Record{x,q'}. (\Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~q'/r]\\
=& \reason{definition of $\PD{-}$}\\
&\PD{N}[\PD{V}/p,\PD{W}/q,\lambda \Record{x,q'}. \PD{\ParamHandle\;\EC[\Return\;x]\;\With\;(q.~H)(q')}/r]\\

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