Let $X$ be a smooth manifold, let $K \subset X$ be a relatively compact open subset with smooth boundary, and let
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\begin{align*}
\pi:T^*X \setminus 0 \to X
\end{align*}
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denote the cotangent projection. For each distribution $v \in \mathcal{D}'(X)$, write $WF(v) \subset T^*X \setminus 0$ for its wave front set. Let $P \in \Psi_{\mathrm{cl}}^m(X)$ be a properly supported classical pseudodifferential operator of order $m$ with real homogeneous principal symbol
be the maximal integral curve of $H_p$ satisfying $0 \in I_q$ and $\gamma_q(0)=q$.
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Assume the following backward exit alternative holds for $K$: whenever $q \in \operatorname{Char}(P)\cap T^*K$, $q\notin WF(Pu)$, and there is no $t_s\in I_q$ with $t_s\le 0$, $\gamma_q(t_s)\in WF(Pu)\cap T^*K$, and $\gamma_q([t_s,0])\subset T^*K$, then either $\gamma_q(t)\in T^*K$ for every $t\in I_q\cap(-\infty,0]$, or there exists $t_b\in I_q$ with $t_b<0$, $\pi(\gamma_q(t_b))\in \partial K$, and $\pi(\gamma_q((t_b,0]))\subset K$.