Commit 2f75bf0d authored by Giuseppe Castagna's avatar Giuseppe Castagna
Browse files

eliminated contexts

parent e8424c98
......@@ -144,34 +144,35 @@ well as the proof of its type safety.
\subsection{Parallel semantics}\label{app:parallel}
One technical difficulty in the proof of the subject reduction
theorem is that, when reducing an expression $e$ into $v$ in a type
property is that, when reducing an expression $e$ into $v$ in a type
case, the expression $e$ disappears ($e$ is not a
subexpression of the test anymore) and, thus, we can no longer refine the
expression $e$ in the ``then'' and ``else'' branches (which might
contain occurences of $e$). To circumvent this issue, we introduce a notion of parallel
reduction which essentially reduces all occurrences of a
sub-expression appearing in a type cases also in the ``then'' and
``else'' branch at the same time. The semantics based on parallel
reduction is given below.
\[
\begin{array}{lrcl}
\textbf{Expressions} & e & ::= & c\alt x \alt \lambda^{\bigwedge \arrow t t}x.e \alt e e \alt \pi_i e \alt (e,e) \alt \ite e t e e\\
\textbf{Values} & v & ::= & c \alt \lambda^{\bigwedge \arrow t t}x.e \alt (v,v)
\end{array}
\]
\[
\begin{array}{lrcl}
\textbf{Context} & C[] & ::= & e [] \alt [] v \alt (e,[]) \alt ([],v)
\end{array}
\]
``else'' branch at the same time.
%% \[
%% \begin{array}{lrcl}
%% \textbf{Expressions} & e & ::= & c\alt x \alt \lambda^{\bigwedge \arrow t t}x.e \alt e e \alt \pi_i e \alt (e,e) \alt \ite e t e e\\
%% \textbf{Values} & v & ::= & c \alt \lambda^{\bigwedge \arrow t t}x.e \alt (v,v)
%% \end{array}
%% \]
%% \[
%% \begin{array}{lrcl}
%% \textbf{Context} & C[] & ::= & e [] \alt [] v \alt (e,[]) \alt ([],v)
%% \end{array}
%% \]
The idea is to label each step of reduction done by a context rule
with the notion of reduction (one of the notions of reduction of
Section~\ref{sec:opsem}) that caused the context to reduce. In
case of a reduction of the expression tested in the type case,
that same reduction is applied in parallel to both branches.
The semantics based on parallel
reduction is given below where expressions, value and contexts are defined as in Sections~\ref{sec:syntax} and~\ref{sec:opsem}.
For convenience, we denote $e\xleadsto{e\mapsto e'}e'$ by
$e\idleadsto e'$ and by $e\uleadsto e'$ a step of reduction of the
......
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