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25 lines
1.3 KiB
TeX
25 lines
1.3 KiB
TeX
\newpage
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\subsubsection{Equivalence}
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\begin{theorem}[]{Equivalence Theorem}
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For every statement $s$ of IMP, we have
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\[
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\vdash \langle s, \sigma \rangle \rightarrow \sigma' \Leftrightarrow \langle s, \sigma \rangle \rightarrow_1^* \sigma'
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\]
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{\scriptsize If the execution of $s$ from some state terminates successfully in one of the semantics, than so will it in the other.
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The execution fails to terminate in the big-step semantics if and only if it either gets stuck, or runs forever in the small-step semantics}
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\end{theorem}
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\inlinelemma[Equivalence Lemma 1] For every statement $s$ of IMP and states $\sigma$ and $\sigma'$ we have:
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\[
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\vdash \langle s, \sigma \rangle \rightarrow \sigma' \implies \langle s, \sigma \rangle \rightarrow_1^* \sigma'
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\]
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{\scriptsize If the execution of $s$ from $\sigma$ terminates successfully in the big-step semantics, so will it in the small-step semantics (in the same state, too!)}
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\inlinelemma[Equivalence Lemma 2] For every statement $s$ of IMP and states $\sigma$ and $\sigma'$ and $k \in \N$, we have:
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\[
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\langle s, \sigma \rangle \rightarrow_1^k \sigma' \implies \vdash \langle s, \sigma \rangle \rightarrow \sigma'
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\]
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{\scriptsize If the execution of $s$ from $\sigma$ terminates successfully in the small-step semantics, so will it in the big-step semantics (in the same state, too!)}
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