I'm reading Sistla and Clarke's paper "The Complexity of Propositional Linear Temporal Logics". In section 4 they start with the following set up:
Let $S=(s, \xi), T=(t, \pi)$ be structures such that, for some $m \geq 0$, the following conditions are satisfied: $$ \forall i, 0 \leq i \leq m, t_{i}=s_{i} ; \quad \forall i, i>m+1, t_{i}=s_{i-1} \text {, } $$ and $\pi$ is an extension of $\xi$, that is $$ \pi(t_r) = \begin{cases} \xi(s_r) & 0 \le r \le m \\ \xi(s_{r-1}) & m < r \end{cases} $$ such that $$ \pi\left(t_{m+1}\right)=\pi\left(t_{m}\right) $$ that is, $T$ is obtained by duplicating the $m$-th state in $S$ successively once. The following lemma is easily proved by induction on the formula $f$.
Lemma:
For any $f \in L(\mathbf{U})$,
$T, t_{m} \vDash f$ iff $T, t_{m+1} \vDash f$ and for any $\delta$ in $s$, $S, \delta \vDash f$ iff $T, \delta \vDash f$.
Note that I added what they mean by extension myself.
After the lemma, which I proved, they say:
Note that the lemma is not true for $\mathbf{L}(\mathbf{U}, \mathbf{X})$.
I've been playing around with formulas in $\mathbf{L}(\mathbf{U}, \mathbf{X})$, however, I could not find the counterexample. Is this counterexample standard?
