# 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 \(L_{i}\)
Let \(L^{\prime}\) be a line connecting \(c_{i}\) with an internal colored robot \(r\) in the left or right of \(\overleftrightarrow{c_{i}t_{1}}\) such that, in the _cone area_\(QC(r)\) formed by \(L^{\prime}\) and \(\overleftrightarrow{c_{i}t_{1}}\), there is no other internal colored robot. Let \(w\) be the intersection point of \(L^{\prime}_{i}\) and \(L^{\prime}\). Robot \(t_{1}\) moves to the midpoint \(m\) of the line segment that connects it with \(w\) (note that all three points \(w,t_{1}\), and \(m\) are in \(L^{\prime}_{i}\)) assuming color transit2_moving (Figure 11g). It then changes its color to eligible when it becomes active next time. After \(t_{2}\) and \(t_{3}\) see both \(t_{1}\) and \(t_{4}\) with color eligible, \(t_{2}\) moves to point \(op_{1}\) (the position of \(t_{1}\) in \(L^{\prime}_{i}\) before it moved to point \(m\)) and \(t_{3}\) moves to \(op_{4}\) (the position of \(t_{4}\) in \(L^{\prime}_{i}\) before it moved) (Figure 11h). Robots \(t_{2},t_{3}\) then assume color eligible. After \(c_{i}\) sees all \(t_{1},t_{2},t_{3},t_{4}\) are on \(L^{\prime}_{i}\) with color eligible, it assumes color corner3. During Stage 1.2.1, four interior robots of \(\mathbf{P}^{\prime}\) inside the corner polygon \(CP(c_{i})\) are correctly placed on the eligible line \(EL_{i}\) of \(c_{i}\) and colored eligible and the corners of \(\mathbf{P}^{\prime}\) are colored \(\in\{\texttt{corner3},\texttt{corner5},\texttt{corner}\}\). Stage 1.2.1 runs for \(\mathcal{O}(1)\) epochs avoiding collisions and Stage 1.2.1 starts only after Stage 1.1 finishes. Proof. : It is easy to see that, if the robot \(r\) in the interior of \(\mathbf{P}^{\prime}\) is not inside \(CP(c_{i})\), it does not move to \(EL_{i}\) since \(r\) does not find \(c_{i}\) closest to it among the corners of \(\mathbf{P}^{\prime}\). We first show that four robots on \(CP(c_{i})\) correctly move to \(L_{i}\) (which is \(TL_{i}\) or \(CL_{i}\)) and then show that they are then correctly positioned on the eligible line \(EL_{i}\). To prove the first case, it is sufficient to show that an internal robot \(r_{i}\) always sees its closest corner \(c_{i}\) and two neighboring corners \(c_{i-1}\) and \(c_{i+1}\) of \(c_{i}\) in \(\mathbf{P}^{\prime}\) during Stage 1.2.1. This will allow \(r_{i}\) to correctly compute \(L_{i}\) and move to it if it is closer to \(L_{i}\) than any other interior robot of \(\mathbf{P}^{\prime}\). Notice that \(r_{i}\) sees all the corners of \(\mathbf{P}^{\prime}\) in the beginning of Stage 1.2.1 as no robot has moved to lines \(CL_{i}\) or \(TL_{i}\) at that time. Let \(r_{i-1},r_{i+1}\) be the closest interior robots to corners \(c_{i-1},c_{i+1}\), respectively. Let \(r_{i-1}\) (\(r_{i+1}\)) be currently moving or have moved to \(L_{i-1}\) (\(L_{i+1}\)). We will show that \(r_{i-1},r_{i+1}\) do not block \(r_{i}\) to see \(c_{i-1},c_{i+1}\), respectively. Observe that, since \(r_{i-1}\) (\(r_{i+1}\)) is currently moving to or has moved to \(L_{i-1}\) (\(L_{i+1}\)), that means \(r_{i-1}\) (\(r_{i+1}\)) is closer to \(c_{i-1}\) (\(c_{i+1}\)) than any other internal robot of \(\mathbf{P}^{\prime}\). Moreover, according to the algorithm for Stage 1.2.1.1, except \(r_{i-1}\) (\(r_{i+1}\)), no other robot is currently moving to \(c_{i-1}\) (\(c_{i+1}\)). Let \(L_{r_{i}}\) be the line parallel to \(\overleftrightarrow{c_{i-1}c_{i+1}}\) passing through \(r_{i}\). Define lines \(L_{r_{i-1}},L_{r_{i+1}}\) similarly. Note that, if \(r_{i-1}\) (\(r_{i+1}\)) is moving to or has already moved to \(L_{i-1}\) (\(L_{i+1}\)), then no other robot with color start is inside the triangle divided by line \(L_{r_{i-1}}\) (\(L_{r_{i+1}}\)) towards \(c_{i-1}\) (\(c_{i+1}\)). Moreover, since the corners \(c_{i},c_{i-1},c_{i+1}\) have already moved to their eligible areas, a unique line connects every internal robot with these corners. Therefore, as \(r_{i},r_{i-1}\), and \(r_{i+1}\) are moving toward different corners of \(\mathbf{P}^{\prime}\), they do not block the view of \(r_{i}\) to see corners \(c_{i-1}\) and \(c_{i+1}\).