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Daniel Hillerström
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thesis.tex
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@@ -1165,7 +1165,7 @@ require a change of name of the bound type variable
% to occur in the same substitution map. % to occur in the same substitution map.
\paragraph{Reduction semantics} \paragraph{Reduction semantics}
The reduction relation $\reducesto \in \CompCat \times \CompCat$ The reduction relation $\reducesto \subseteq \CompCat \times \CompCat$
relates a computation term to another if the former can reduce to the relates a computation term to another if the former can reduce to the
latter in a single step. Figure~\ref{fig:base-language-small-step} latter in a single step. Figure~\ref{fig:base-language-small-step}
depicts the reduction rules. The application rules \semlab{App} and depicts the reduction rules. The application rules \semlab{App} and
@@ -1189,6 +1189,7 @@ The \semlab{Let} rule eliminates a trivial computation term
$\Return\;V$ by substituting $V$ for $x$ in the continuation $N$. $\Return\;V$ by substituting $V$ for $x$ in the continuation $N$.
% %
\paragraph{Evaluation contexts} \paragraph{Evaluation contexts}
Recall from Section~\ref{sec:base-language-terms}, Recall from Section~\ref{sec:base-language-terms},
Figure~\ref{fig:base-language-term-syntax} that the syntax of let Figure~\ref{fig:base-language-term-syntax} that the syntax of let
@@ -1208,12 +1209,12 @@ the context to that particular $N$. In our formalism, we call this
rule \semlab{Lift}. Evaluation contexts are generated from the empty rule \semlab{Lift}. Evaluation contexts are generated from the empty
context $[~]$ and let expressions $\Let\;x \revto \EC \;\In\;N$. context $[~]$ and let expressions $\Let\;x \revto \EC \;\In\;N$.
For now the choices of using fine-grain call-by-value and evaluation The choices of using fine-grain call-by-value and evaluation contexts
contexts may seem odd, if not arbitrary at this stage; the reader may may seem odd, if not arbitrary at this point; the reader may wonder
wonder with good reason why we elect to use fine-grain call-by-value with good reason why we elect to use fine-grain call-by-value over
over ordinary call-by-value. In Chapter~\ref{ch:unary-handlers} we ordinary call-by-value. In Chapter~\ref{ch:unary-handlers} we will
will reap the benefits from our design choices, as we shall see that reap the benefits from our design choices, as we shall see that the
the combination of fine-grain call-by-value and evaluation contexts combination of fine-grain call-by-value and evaluation contexts
provide the basis for a convenient, simple semantic framework for provide the basis for a convenient, simple semantic framework for
working with continuations. working with continuations.
@@ -1221,9 +1222,8 @@ working with continuations.
\label{sec:base-language-metatheory} \label{sec:base-language-metatheory}
Thus far we have defined the syntax, static semantics, and dynamic Thus far we have defined the syntax, static semantics, and dynamic
semantics of \BCalc{}. In this section, we finish the definition of semantics of \BCalc{}. In this section, we state and prove some
\BCalc{} by stating and proving some standard metatheoretic properties customary metatheoretic properties about \BCalc{}.
about the language.
% %
We begin by showing that substitutions preserve typability. We begin by showing that substitutions preserve typability.
@@ -1232,8 +1232,8 @@ We begin by showing that substitutions preserve typability.
Let $\sigma$ be any type substitution and $V \in \ValCat$ be any Let $\sigma$ be any type substitution and $V \in \ValCat$ be any
value and $M \in \CompCat$ a computation such that value and $M \in \CompCat$ a computation such that
$\typ{\Delta;\Gamma}{V : A}$ and $\typ{\Delta;\Gamma}{M : C}$, then $\typ{\Delta;\Gamma}{V : A}$ and $\typ{\Delta;\Gamma}{M : C}$, then
$\typ{\Delta;\sigma~\Gamma}{\sigma~V : \sigma~A}$ and $\typ{\Delta;\Gamma\sigma}{V\sigma : A\sigma}$ and
$\typ{\Delta;\sigma~\Gamma}{\sigma~M : \sigma~C}$. $\typ{\Delta;\Gamma\sigma}{M\sigma : C\sigma}$.
\end{lemma} \end{lemma}
% %
\begin{proof} \begin{proof}