### 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 ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment