# 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 \(\mathbf{P}^{\prime}\)  
At the start of Stage 1.1, a corner of \(\mathbf{P}\) may have color ready or start. If a corner \(c_{i}\) of \(\mathbf{P}\) becomes active and has color start, then it assumes color ready. The side robots \(\mathcal{Q}_{s}\) of \(\mathbf{P}\) change their color to side1 from start. **Definition 4**: _Let \(r_{a}\) and \(r_{b}\) be the counterclockwise and clockwise neighbors of a corner \(c_{i}\) of \(\mathbf{P}\) in the boundary of \(\mathbf{P}\). If there are no side robots in \(\overline{c_{i}c_{i-1}}\) (\(\overline{c_{i}c_{i+1}}\)), then \(r_{a}\) is \(c_{i-1}\) (\(r_{b}\) is \(c_{i+1}\))._  
After \(c_{i}\) has color ready, if it sees both \(r_{a},r_{b}\) have color \(\in\{\)ready, side1, corner1\(\}\), then it assumes color ready_moving and moves to a point in \(EA(c_{i})\). When \(c_{i}\) becomes active next time, it is already in \(EA(c_{i})\), and each robot in the sets \(\mathcal{Q}_{s}\), \(\mathcal{Q}_{i}\) sees it (Lemma 1). Corner \(c_{i}\) then changes its color to corner1. After that, \(c_{i}\) does not move in any future epochs (but it may assume new colors). After all corners of \(\mathbf{P}\) move to their \(EA(*)\), they form \(\mathbf{P}^{\prime}\). Any robot with color side1 changes its color to special when it becomes a corner (due to the moves of corners of \(\mathbf{P}\)) and it sees at least one other robot with color side1 or special in the side of \(\mathbf{P}\) to which it belongs. If the robot with color side1 is the only robot in that side, then it retains color side1. The robot can easily identify this situation since it is in the corridor of the side of \(\mathbf{P}^{\prime}\) that is formed from the moves of the corners in the side of \(\mathbf{P}\) to which it belongs. For example, if \(s_{i}\) is the only side robot in a side \(S\) of \(\mathbf{P}\), then it sees no other robot in the corridor of \(S\) besides itself. **Lemma 9** (Reference [9]). : _The interior robots of \(\mathbf{P}\) remain as the interior robots of \(\mathbf{P}^{\prime}\)._  
**Lemma 10**. : \(\mathbf{P}^{\prime}\) _has the same number of corners as \(\mathbf{P}\)._  
Proof. : A corner \(c_{i}\) of \(\mathbf{P}\) moves to \(EA(c_{i})\) only after it sees both \(r_{a},r_{b}\) have color \(\in\{\texttt{ready},\texttt{side1},\texttt{corner1}\}\). Since the interior robots of \(\mathbf{P}\) do nothing, it only remains to show that side robots of \(\mathbf{P}\) remain in their original positions. This is immediate since, if \(r_{a}\) and/or \(r_{b}\) are side robots of \(\mathbf{P}\), then they take color side1 before \(c_{i}\) moves to \(EA(c_{i})\). Moreover, after \(r_{a}\) and/or \(r_{b}\) take color side1, they do not move to their eligible areas even if they become corners of \(\mathbf{P}\). The only possibility is that \(r_{a},r_{b}\) might change their color to special. **Lemma 11**. : _The corners of \(\mathbf{P}\) become corners of \(\mathbf{P}^{\prime}\) and take color corner1 in (at most) three epochs._  
Proof. : All the robots in the sets \(R_{c}\) and \(R_{s}\) (corners and sides of \(\mathbf{P}\), respectively) change their colors to ready and side1, respectively, in, at most, one epoch. All the corners in \(R_{c}\) then move to their \(EA(*)\) by the end of the next epoch with color ready_moving. By the end of the third epoch, the robots that moved to \(EA(*)\) change their color to corner1. We now describe how the corners of \(\mathbf{P}^{\prime}\) change their colors from corner1 to corner2, corner3, or corner. For a corner \(c_{i}\) with color corner1, we will define conditions to be satisfied on both adjacent sides that will depend on \(r_{a}\), its side \(S_{a}\), and the neighboring corner on that side \(c_{i-1}\) and, likewise, \(r_{b}\), \(S_{b}\), and \(c_{i+1}\). Figure 10 shows \(c_{i}\) (as \(c_{0}\)) and its neighboring sides \(c_{i-1}\) and \(c_{i+1}\) (as \(w^{\prime}\) and \(w^{\prime\prime}\)) as it moves from \(\mathbf{P}\) to \(\mathbf{P}^{\prime}\).