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Daniel Hillerström
2020-07-17 03:37:32 +01:00
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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.
\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
latter in a single step. Figure~\ref{fig:base-language-small-step}
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 $N$.
%
\paragraph{Evaluation contexts}
Recall from Section~\ref{sec:base-language-terms},
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
context $[~]$ and let expressions $\Let\;x \revto \EC \;\In\;N$.
For now the choices of using fine-grain call-by-value and evaluation
contexts may seem odd, if not arbitrary at this stage; the reader may
wonder with good reason why we elect to use fine-grain call-by-value
over ordinary call-by-value. In Chapter~\ref{ch:unary-handlers} we
will reap the benefits from our design choices, as we shall see that
the combination of fine-grain call-by-value and evaluation contexts
The choices of using fine-grain call-by-value and evaluation contexts
may seem odd, if not arbitrary at this point; the reader may wonder
with good reason why we elect to use fine-grain call-by-value over
ordinary call-by-value. In Chapter~\ref{ch:unary-handlers} we will
reap the benefits from our design choices, as we shall see that the
combination of fine-grain call-by-value and evaluation contexts
provide the basis for a convenient, simple semantic framework for
working with continuations.
@@ -1221,9 +1222,8 @@ working with continuations.
\label{sec:base-language-metatheory}
Thus far we have defined the syntax, static semantics, and dynamic
semantics of \BCalc{}. In this section, we finish the definition of
\BCalc{} by stating and proving some standard metatheoretic properties
about the language.
semantics of \BCalc{}. In this section, we state and prove some
customary metatheoretic properties about \BCalc{}.
%
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
value and $M \in \CompCat$ a computation such that
$\typ{\Delta;\Gamma}{V : A}$ and $\typ{\Delta;\Gamma}{M : C}$, then
$\typ{\Delta;\sigma~\Gamma}{\sigma~V : \sigma~A}$ and
$\typ{\Delta;\sigma~\Gamma}{\sigma~M : \sigma~C}$.
$\typ{\Delta;\Gamma\sigma}{V\sigma : A\sigma}$ and
$\typ{\Delta;\Gamma\sigma}{M\sigma : C\sigma}$.
\end{lemma}
%
\begin{proof}