# 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 \(L_{i}\)
It then moves toward that corner to be positioned on the corner line segment if it sees at least two robots that are positioned on that corner line segment with color eligible. The \(\mathcal{O}(\log N)\)-epoch argument for this approach is in the lines of the argument we provided in Stage 1.2. We now describe the \(\mathcal{O}(1)\)-epoch approach developed in SSS'17 [14]. At the end of Stage 1.2, all interior robots of \(\mathbf{P}^{\prime}\) that are on the eligible lines have color eligible and the rest have color internal. Let \(c_{i}\) be a corner of \(\mathbf{P}^{\prime}\) colored corner4, and let \(r\) be a robot with color internal that is inside the corner polygon \(CP(c_{i})\). Note that \(r\) is closer to \(c_{i}\) than other corners of \(\mathbf{P}^{\prime}\) and it always sees \(c_{i}\) (Lemma 16). Robot \(r\) moves as follows. **Condition 1.3.1:** Robot \(r\) has color internal and it can see at least two eligible colored robots towards \(c_{i}\). **Action 1.3.1:** Let \(L_{r}\) be the line formed by those eligible robots. Robot \(r\) assumes color internal_moving and moves to the intersection point \(w\) of lines \(L_{r}\) and \(\overline{c_{i}r}\). **Condition 1.3.2:** Robot \(r\) has color internal_moving. **Action 1.3.2:** Robot \(r\) assumes color eligible. As soon as \(c_{i}\) does not see any robot with color internal or internal_moving (i.e., all robots in the interior of \(\mathbf{P}^{\prime}\) are placed in the eligible lines), it assumes color corner5. We can prove the following two lemmas. **Lemma 19**: _During Stage 1.3, the eligible colored robots positioned on \(EL_{i}\) of a corner \(c_{i}\) of \(\mathbf{P}^{\prime}\) are seen by all the internal colored robots inside \(CP(c_{i})\) (and vice versa), if there is no transient robot towards \(EL_{i}\)._  
We have from Lemmas 16 and 17 that the robots inside \(CP(c_{i})\) see all the robots on \(EL_{i}\) at the end of Stage 1.2. The direct extension of those lemmas shows that, if a robot inside \(CP(c_{i})\) moves to \(EL_{i}\), it will also be visible to the remaining robots inside \(CP(c_{i})\) waiting to move to \(EL_{i}\). Therefore, if some interior robot \(r_{i}\) inside \(CP(c_{i})\) does not see any robot \(r_{e}\) on \(EL_{i}\), then there must be some robot \(r_{f}\) that is blocking the view of \(r_{e}\) from \(r_{i}\). Since the interior robots do not block the view while they are waiting to move to \(EL_{i}\), there must be a robot currently in transit to \(EL_{i}\). **Lemma 20**: _During Stage 1.3, all the robots in the interior of \(\mathbf{P}^{\prime}\) (with color internal) are correctly positioned on the eligible lines of the corners of \(\mathbf{P}^{\prime}\) and colored eligible. Moreover, the corners of \(\mathbf{P}^{\prime}\) are colored corner5. Furthermore, Stage 1.3 runs for \(\mathcal{O}(1)\) epochs avoiding collisions and Stage 1.3 starts only after Stage 1.2.2 finishes._  
The idea is to show that the configuration at the end of Stage 1.2 satisfies all the conditions of Definition 2 to apply the Beacon-Directed Curve Positioning framework of Section 3 to run this stage in \(\mathcal{O}(1)\) epochs. We have that the final positions of the robots inside \(CP(c_{i})\) on the eligible line \(EL_{i}\) (a 2-point curve) are in the positions enclosed within the cone area \(QC(c_{i})\) and two eligible robots (that serve as left beacons) are in the left of \(QC(c_{i})\) and two eligible robots (that serve as right beacons) are in the right of \(QC(c_{i})\). We argue that all four conditions of Definition 2 are satisfied in Stage 1.3. Condition (a) is satisfied since the paths \(p_{i},p_{j}\) of two different robots \(r_{i},r_{j}\) in the interior of \(CP(c_{i})\) do not intersect while moving to \(EL_{i}\). Condition (b) is also satisfied since the robots move in straight lines joining them with \(c_{i}\). Condition (c) is satisfied since the paths \(p_{i},p_{j}\) intersect \(EL_{i}\) at one point.