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