\newpage
\subsubsection{Equivalence}
\begin{theorem}[]{Equivalence Theorem}
    For every statement $s$ of IMP, we have
    \[
        \vdash \langle s, \sigma \rangle \rightarrow \sigma' \Leftrightarrow \langle s, \sigma \rangle \rightarrow_1^* \sigma'
    \]
    {\scriptsize If the execution of $s$ from some state terminates successfully in one of the semantics, than so will it in the other.
    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}
\end{theorem}


\inlinelemma[Equivalence Lemma 1] For every statement $s$ of IMP and states $\sigma$ and $\sigma'$ we have:
\[
    \vdash \langle s, \sigma \rangle \rightarrow \sigma' \implies \langle s, \sigma \rangle \rightarrow_1^* \sigma'
\]
{\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!)}


\inlinelemma[Equivalence Lemma 2] For every statement $s$ of IMP and states $\sigma$ and $\sigma'$ and $k \in \N$, we have:
\[
    \langle s, \sigma \rangle \rightarrow_1^k \sigma' \implies \vdash \langle s, \sigma \rangle \rightarrow \sigma'
\]
{\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!)}