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I'm currently reading a paper on First-order definable languages by Volker Diekert and Paul Gastin. Im having trouble understanding a part of the proof for lemma 3.2 (splitting lemma). The part I'm having a problem with, is about the complement:

(Note: I've substituted $B^\infty = B^* \cup B^\omega$ with $B^*$ which shouldn't cause problems)

From the paper:

It remains to deal with the complement $\Sigma^* \setminus L$ of a star-free set. By induction, we have $L \cap B^* A B^* = \bigcup_{1 \leq i \leq n} K_iAL_i$. If some $K_i$ and $K_j$ are not disjoint then we can rewrite $$K_iAL_i \cup K_jAL_j = (K_i \setminus K_j)AL_i \cup (K_j \setminus K_i)AL_j \cup (K_i \cap K_j)A(L_i\cup L_j)$$ We can also add $(B^* \setminus \bigcup_{i} K_i)A \emptyset$ in case $\bigcup_{i}K_i$ is strictly contained in $B^*$. Therefore, we may assume that $\{K_i \mid 1 \leq i \leq n\}$ forms a partition of $B^*$. This yields: $$(\Sigma^* \setminus L) \cap B^*AB^* = \bigcup_{1\leq 1 \leq n} K_iA(B^* \setminus L_i)$$

Questions:

  1. In the last part of the proof $\bigcup_{1\leq 1 \leq n} K_iA(B^* \setminus L_i)$, why is it enough to only build the complement for $L_i$ with $B^* \setminus L_i$? Why don't we need $B^* \setminus K_i$?
  2. Why does $\{K_i \mid 1 \leq i \leq n\}$ form a partition of $B^*$?
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  1. For the set $\{K_i \mid 1 \leq i \leq n\}$ to form a partition of $B^*$, it is necessary, that all $K_i$ are pairwise disjoint and every element in $B^*$ is cointained in some $K_i$. You already showed that they are disjoint, as is is possible to split two overlapping $K_i$ into three disjoint ones by using the parts that don't overlap and the intersection instead. The second condition, that everything is contained, is achieved by adding $(B^* \setminus \bigcup_i K_i)$ to the set $\{K_i \mid 1 \leq i \leq n\}$ so $n$ is incresed by one and still is finite. And this $(B^* \setminus \bigcup_i K_i)$ contains everything that was not previously contained.

  2. As $\{K_i \mid 1 \leq i \leq n\}$ forms a partition, everything in $B^*$ is contained in some $K_i$ and is, therefore, also contained in the complement of another one, which means, it doesn't change anything to use the complement of the $K_i$s.

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