Build an automaton from a given automaton to prove regularity of more complex strings

Question

let $L$ be a regular language, and let $A=\{\Sigma, Q, q_0, F, \delta\}$ be a DFA such that $L = L(A)$.

I need to prove that $$L_p=\{xy\in\Sigma^*\mid\delta(q_0, y)=p\text{ and } \delta(p, x)\in F\}$$ is regular for every $p\in Q$ by defining an automaton $A_p$.

I tried to define it like so $$A_p=\{\Sigma, Q\times Q, (p, q_0), F\times \{p\}, \delta_p\}$$ where $\delta_p$ is defined like so $$ \delta_p((q_1, q_2), \sigma) = (\delta(q_1, \sigma), q_2)\text{ if }q_1 \not\in F, \\ \delta_p((q_1, q_2), \sigma) = (q_1, \delta(q_2, \sigma))\text{ if }q_1 \in F. $$ But then I have a problem where $x=uv$, $\delta(p, u)\in F$ and $\delta(p, x)\in F$.

I thought maybe I define $A_p$ as a non-deterministic automaton and then $$\delta_p((q_1, q_2), \sigma) = \{(\delta(q_1, \sigma), q_2), (q_1, \delta(q_2, \sigma))\}.$$ However, word $z=uwv$ where $$\delta_p((p, q_0), u) = (p_1, q_0) \Longrightarrow \delta_p((p_1, q_0), w) = (p_1, p) \\ \Longrightarrow \delta_p((p_1, p), v) = (p_2, p), p_2\in F\times \{p\}$$ is accepted while it shouldn't be part of $L_p$

Answer 1

Since how a word $w$ is recognized as a member of $L_p$ depends on a successful choice of partition of $w$ into a prefix ($x$) and a suffix ($y$) among many possible partitions, a DFA with $\epsilon$-moves that accepts $L_p$ should be easier to build than a DFA.

To distinguish whether a symbol is used as in $x$ or in $y$, we can employ two copies of $A$, one copy for reading $x$ and one copy for reading $y$. When the state after reading some $x$ is one of the final states of the first copy, switch to the initial state $q_0$ of the second copy optionally, continuing reading the remaining $y$. At the end of reading all input, accept if the second copy is at state $p$.

Two copies of $A$ can be built on the cartesian product $Q\times \{1,2\}$, where the copy index $1$ and $2$ indicates which copy of $A$.

Here is a DFA with $\epsilon$-moves that accepts $L_p$.
$D_p=\{\Sigma, Q\times\{1,2\}, (p, 1), \{(p,2)\}, \mu\}$, where $\mu$ is defined below. $\quad\mu((q,1), \sigma) = (\delta(q, \sigma), 1)\quad\forall q \in Q,\ \sigma\in\Sigma,$
$\quad\mu((q,2), \sigma) = (\delta(q, \sigma), 2)\quad\forall q \in Q,\ \sigma\in\Sigma,$
$\quad\mu((f,1), \epsilon) = (q_0, 2)\quad\forall f \in F.$
Note that when the copy index is fixed, $D_p$ behaves exactly as $A$.