# Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case +
## 6 Stage 1-Interior Depletion
### Stage 1.1-Making Corners of \(\mathbf{P}\) the Corners of \(\mathbf{P}^{\prime}\)
#### 6.2.1 Stage 1.2.1: Moving Four Interior Robots in \(\mathbf{P^{\prime}}\) to \(L_{i}\)
\(QC^{\prime\prime}(r_{i})\) for \(r_{i}\) formed by lines \(\overleftarrow{c_{i}r_{i}}\) and \(\overrightarrow{c_{i}c_{i+1}}\) can be defined similarly; the left of Figure 12 also shows \(QC^{\prime\prime}(r_{1})\) and \(QC^{\prime\prime}(r_{2})\) of two internal robots \(r_{1}\) and \(r_{2}\). If there is no other robot with color internal in \(QC^{\prime}(r_{i})\) (and/or \(QC^{\prime\prime}(r_{i})\)) that is closer to \(c_{i}\) than \(r_{i}\), then \(r_{i}\) moves to \(EL_{i}\) at the intersection point of \(EL_{i}\) and \(\overleftarrow{c_{i}r_{i}}\) assuming color internal_moving. As depicted in the left of Figure 12, \(r_{1}\) does not see any other robot closer to \(c_{i}\) in \(QC^{\prime\prime}(r_{1})\) and moves to \(EL_{i}\). It then assumes color eligible when it becomes active next time. Robot \(r_{i}\) determines whether there is some other robot \(r_{i}\) in the cone area \(QC^{\prime}(r_{i})\) (and/or \(QC^{\prime\prime}(r_{i})\)) that is closer to \(c_{i}\) than itself as follows. Let \(L^{\prime},L^{\prime\prime}\) be two lines perpendicular to \(\overleftarrow{c_{i}c_{i-1}}\) and \(\overleftarrow{c_{i}c_{i+1}}\), respectively, passing through their motions, forming the convex polygon \(CP(c_{i})\). If there is a robot \(r_{i}\) with color internal in \(QC^{\prime}(r_{i})\) (\(QC^{\prime\prime}(r_{i})\)) divided by line \(L^{\prime}\) (\(L^{\prime\prime}\)) towards \(c_{i}\), then \(r_{i}\) assumes that \(r_{j}\) is closer to \(c_{i}\) in \(QC^{\prime}(r_{i})\) (\(QC^{\prime\prime}(r_{i})\)). Note that \(r_{i}\) always sees \(c_{i}\) (Lemma 16). However, \(r_{i}\) may not see \(c_{i-1}\) or \(c_{i+1}\) when there is another robot \(r^{\prime}\) in \(CP(c_{i})\) that it currently transient to \(EL_{i}\). Nevertheless, observe that the robots moving to \(EL_{i-1}\) and \(EL_{i+1}\) do not block the view of \(r_{i}\) to see \(c_{i-1}\) and \(c_{i+1}\). Therefore, \(r_{i}\) decides to move to \(EL_{i}\) if and only if the first robot it sees in both the left and right of \(c_{i}\) has color \(\in\) {corner3, corner4, corner5, corner}. Finally, when \(r_{i}\) sees two robots with color eligible in its cone area \(QC^{\prime}(r_{i})\) (and/or \(QC^{\prime\prime}(r_{i})\)), it does not move to \(EL_{i}\) (even if there is no other robot \(r^{\prime}\) closer to \(c_{i}\) in \(QC^{\prime}(r_{i})\) and/or \(QC^{\prime\prime}(r_{i})\) than itself). As soon as the corner \(c_{i}\) sees (at least) four eligible robots in \(EL_{i}\), it assumes color corner4 if the following holds. Let \(t^{\prime}\) (\(t^{\prime\prime}\)) be the second robot in \(EL_{i}\) from the either end \(x_{i},y_{i}\) of \(EL_{i}\) (Figure 12). Let \(QC(c_{i})\) be the cone area formed by lines \(\overleftarrow{c_{i}t^{\prime}}\) and \(\overleftarrow{c_{i}t^{\prime\prime}}\) (the area between two thick dotted lines in the right of Figure 12). If all the robots with color internal inside the corner polygon \(CP(c_{i})\) lie within \(QC(c_{i})\), \(c_{i}\) assumes color corner4. The right of Figure 12 illustrates these ideas, where all the internal robots (shown as black) are within \(QC(c_{i})\). **Lemma 17**. : _Let there be at least four eligible colored robots on \(EL_{i}\). All the robots with color internal inside the corner polygon \(CP(c_{i})\) see \(c_{i}\) and the robots on \(EL_{i}\). Furthermore, \(c_{i}\) sees all the robots on \(EL_{i}\)._  
Proof. : The proof is to extend the argument of Lemma 16 for the robots that move to \(EL_{i}\) during Stage 1.2.2. We have from Lemma 16 that the points on \(EL_{i}\) are visible to all the interior robots of \(\mathbf{P}\). Moreover, since any interior robot \(r_{j}\) moves to \(EL_{i}\) following the line \(\overleftarrow{c_{i}r_{j}}\) joining \(c_{i}\) with \(r_{j}\), it does not block the visibility of \(c_{i}\) to see the robots inside \(CP(c_{i})\)  
Figure 12: An illustration of how the interior robots of \(\mathbf{P}^{\prime}\) move to the eligible lines during Stage 1.2.2.