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EvaluationContexts: proofreading

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Adam Chlipala 2021-03-28 16:59:59 -04:00
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frap_book.tex
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@ -3304,10 +3304,10 @@ Last chapter's type-system definition may be reused unchanged.
We just need to make a small modification to the sequence of results leading to type soundness.
\begin{lemma}\label{preservation0}
If $\smallstepo{e}{e'}$ and $\hasty{}{e}{\tau}$, then $\hasty{}{e'}{\tau}$.
If $\smallstepo{e_1}{e_2}$ and $\hasty{}{e_1}{\tau}$, then $\hasty{}{e_2}{\tau}$.
\end{lemma}
\begin{proof}
By inversion on the derivation of $\smallstepo{e}{e'}$.
By inversion on the derivation of $\smallstepo{e_1}{e_2}$.
\end{proof}
\begin{lemma}\label{preservation_prime}
@ -3374,8 +3374,8 @@ $$\begin{array}{rrcl}
The $\mathsf{match}$ form, following pattern-matching in Coq and other languages, accounts for most of the syntactic complexity.
Two new small-step rules explain its behavior.
$$\infer{\smallstepo{\match{\inl{v}}{x_1}{e_1}{x_2}{e_2}}{\subst{v}{x_1}{e_1}}}{}$$
$$\infer{\smallstepo{\match{\inr{v}}{x_1}{e_1}{x_2}{e_2}}{\subst{v}{x_2}{e_2}}}{}$$
$$\infer{\smallstepo{\match{\inl{v}}{x_1}{e_1}{x_2}{e_2}}{\subst{e_1}{x_1}{v}}}{}$$
$$\infer{\smallstepo{\match{\inr{v}}{x_1}{e_1}{x_2}{e_2}}{\subst{e_2}{x_2}{v}}}{}$$
And the typing rules:
$$\infer{\hasty{\Gamma}{\inl{e}}{\tau_1 + \tau_2}}{
@ -3409,7 +3409,7 @@ We also introduce metavariable $C^-$ to stand for an evaluation context that doe
It is handy to express the idea of exceptions \emph{bubbling up} to the nearest enclosing $\mathsf{catch}$ constructs.
Specifically, here are three rules to define exception behavior.
$$\infer{\smallstepo{\catch{v}{x}{e}}{v}}{}
\quad \infer{\smallstepo{\catch{\throw{v}}{x}{e}}{\subst{v}{x}{e}}}{}
\quad \infer{\smallstepo{\catch{\throw{v}}{x}{e}}{\subst{e}{x}{v}}}{}
\quad \infer{\smallstepo{C^-[\throw{v}]}{\throw{v}}}{}$$
And the typing rules, where the biggest twist is that a $\mathsf{throw}$ expression can have any type, since it never terminates normally:
@ -3479,17 +3479,17 @@ Then we can adapt the type-preservation proof as follows.
(The progress proof works essentially the same as before.)
\begin{lemma}\label{mutable_preservation0}
If $\smallstepo{e}{e'}$ and $\hasty{}{e}{\tau}$, then $\hasty{}{e'}{\tau}$.
If $\smallstepo{e_1}{e_2}$ and $\hasty{}{e_1}{\tau}$, then $\hasty{}{e_2}{\tau}$.
\end{lemma}
\begin{proof}
By inversion on the derivation of $\smallstepo{e}{e'}$.
By inversion on the derivation of $\smallstepo{e_1}{e_2}$.
\end{proof}
\begin{lemma}\label{preservation1}
If $\smallstepw{(\sigma, e)}{(\sigma', e')}$, $\rhasty{\Delta}{\mempty}{e}{\tau}$, and $\Delta \simeq \sigma$, then $\rhasty{\Delta}{\mempty}{e'}{\tau}$ and $\Delta \simeq \sigma'$.
If $\smallstepw{(\sigma_1, e_1)}{(\sigma_2, e_2)}$, $\rhasty{\Delta}{\mempty}{e_1}{\tau}$, and $\Delta \simeq \sigma_1$, then $\rhasty{\Delta}{\mempty}{e_2}{\tau}$ and $\Delta \simeq \sigma_2$.
\end{lemma}
\begin{proof}
By inversion on the derivation of $\smallstepw{(\sigma, e)}{(\sigma', e')}$, with appeal to Lemma \ref{mutable_preservation0}.
By inversion on the derivation of $\smallstepw{(\sigma_1, e_1)}{(\sigma_2, e_2)}$, with appeal to Lemma \ref{mutable_preservation0}.
\end{proof}
\begin{lemma}\label{mutable_preservation_prime}
@ -3500,10 +3500,10 @@ Then we can adapt the type-preservation proof as follows.
\end{proof}
\begin{lemma}[Preservation]\label{mutable_preservation}
If $\smallstepw{(\sigma_1, e_1)}{(\sigma_2, e_2)}$, $\rhasty{\Delta}{}{e_1}{\tau}$, and $\Delta \simeq \sigma_1$, then $\rhasty{\Delta}{\mempty}{e_2}{\tau}$ and $\Delta \simeq \sigma_2$.
If $\smallstep{(\sigma_1, e_1)}{(\sigma_2, e_2)}$, $\rhasty{\Delta}{}{e_1}{\tau}$, and $\Delta \simeq \sigma_1$, then $\rhasty{\Delta}{\mempty}{e_2}{\tau}$ and $\Delta \simeq \sigma_2$.
\end{lemma}
\begin{proof}
By inversion on the derivation of $\smallstepw{(\sigma_1, e_1)}{(\sigma_2, e_2)}$, with appeal to Lemma \ref{mutable_preservation_prime}.
By inversion on the derivation of $\smallstep{(\sigma_1, e_1)}{(\sigma_2, e_2)}$, with appeal to Lemma \ref{mutable_preservation_prime}.
\end{proof}
Everything can be brought together in type-safety proof with a strengthened invariant that also asserts the $\simeq$ relation, just as in the various statements of preservation.
@ -3511,7 +3511,7 @@ Everything can be brought together in type-safety proof with a strengthened inva
\section{Concurrency Again}
Mutable variables provide a means for communication between threads, so we can bring concurrency to our language with remarkably little effort.
We choose a \emph{concurrent pair}\index{concurrent pair operator} operator, which builds a pair through simultaneous evaluation of the expressions for the two elements.
We choose a \emph{concurrent pairing}\index{concurrent pairing} operator, which builds a pair through simultaneous evaluation of the expressions for the two elements.
$$\begin{array}{rrcl}
\textrm{Expressions} & e &::=& \ldots \mid e || e \\
\textrm{Contexts} & C &::=& \ldots \mid C || e \mid e || C