# Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case +
## 6 Stage 1-Interior Depletion
### Stage 1.4-Positioning the Robots on the Eligible Line to the Sides of \(\mathbf{P}^{\prime}\)  
Let \(r\) be a robot on the corner line segment \(L_{i}\) or the triangle line segment \(\overline{x_{i}y_{i}}\). Note that, if there are robots in \(L_{i}\), then there are no robots in \(\overline{x_{i}y_{i}}\) (and vice versa). Therefore, for simplicity, we denote by \(\overline{x_{i}y_{i}}\) both the corner and triangle line segments of \(c_{i}\). The goal is to move \(r\) to a position on side \(\overline{c_{i}c_{i-1}}\) or \(\overline{c_{i}c_{i+1}}\) of \(\mathbf{P}^{\prime}\) between points \(c_{i}\) and \(x_{i}\) (\(\overline{c_{i}x_{i}}\)) or \(c_{i}\) and \(y_{i}\) (\(\overline{c_{i}y_{i}}\)). At the end of Stage 1.4, all the transit colored robots are on the sides of \(\mathbf{P}^{\prime}\) with color side2. To move robots on \(\overline{x_{i}y_{i}}\) to \(\overline{c_{i}x_{i}}\) or \(\overline{c_{i}y_{i}}\), each robot on \(\overline{x_{i}y_{i}}\) should be able to recognize sides \(\overline{c_{i}c_{i-1}}\) and \(\overline{c_{i}c_{i+1}}\) of \(\mathbf{P}^{\prime}\). Let \(y_{i-1}\) and \(x_{i+1}\) be the points in sides \(\overline{c_{i}c_{i-1}}\) and \(\overline{c_{i}c_{i+1}}\) at distance \(\operatorname{length}(\overline{c_{i}c_{i-1}})/8\) and \(\operatorname{length}(\overline{c_{i}c_{i+1}})/8\) from \(c_{i-1}\) and \(c_{i+1}\), respectively. It is easy to show that each robot on \(\overline{x_{i}y_{i}}\) sees all robots that are positioned on \(\overline{c_{i}c_{i-1}}\) and \(\overline{c_{i}c_{i+1}}\) between points \(x_{i}\) and \(y_{i-1}\) and \(y_{i+1}\). Therefore, we first move two robots on (i) \(\overline{x_{i}y_{i}}\) and \(\overline{x_{i+1}y_{i+1}}\) to \(\overline{y_{i}x_{i+1}}\) and (ii)\(\overline{x_{i}y_{i}}\) and \(\overline{x_{i-1}y_{i-1}}\) to \(\overline{x_{i}y_{i-1}}\) and color them side2. After that, all robots in \(\overline{x_{i}y_{i}}\) can move to \(\overline{c_{i}x_{i}}\) or \(\overline{c_{i}y_{i}}\). Figure 13 illustrates the ideas of this stage. Lemma 21: _Let \(\alpha\) be the robot on \(\overline{x_{i}y_{i}}\) and \(\beta\) be the robot on \(\overline{x_{i+1}y_{i+1}}\) that are closest to side \(\overline{c_{i}c_{i+1}}\) among the robots in \(\overline{x_{i}y_{i}}\) and \(\overline{x_{i+1}y_{i+1}}\), respectively. At least one of \(\alpha\), \(\beta\) sees both \(c_{i}\) and \(c_{i+1}\)._  
Proof: Robot \(\alpha\) can see \(c_{i}\) as it is closest to \(\overline{x_{i}y_{i}}\), and no robots are in the triangle \(\{c_{i}x_{i}y_{i}\}\). Similarly, \(\beta\) can see \(c_{i+1}\). If \(\alpha\) can also see \(c_{i+1}\), then the lemma is satisfied. Otherwise, a robot on \(\overline{x_{i+1}y_{i+1}}\) blocks the view. In that case, \(\beta\), at the end of \(\overline{x_{i+1}y_{i+1}}\), can see \(c_{i}\) as its line of sight passes between \(\alpha\) and \(\overline{c_{i}c_{i+1}}\), and the lemma is satisfied. Suppose \(\alpha\) sees both \(c_{i}\) and \(c_{i+1}\) (the case of \(\beta\) seeing \(c_{i}\), \(c_{i+1}\) is analogous). Robot \(\alpha\) assumes color transit_moving and moves to the point \(\rho\) at distance \(\mathsf{length}(\overline{c_{i}c_{i+1}})/4\) from \(c_{i}\) in \(\overline{c_{i}c_{i+1}}\). Then, \(\alpha\) changes its color to side2. Repeat this process so that one more robot \(\alpha^{\prime}\) places itself at point \(\varrho\) at distance \(\mathsf{length}(\overline{c_{i}c_{i+1}})/4\) from \(c_{i+1}\) in \(\overline{c_{i}c_{i+1}}\). This robot \(\alpha^{\prime}\) will be \(\beta\) unless the next robot \(\gamma\) on \(\overline{x_{i}y_{i}}\) after \(\alpha\) determines that \(\beta\) cannot see \(c_{i}\), and then \(\alpha^{\prime}\) will be \(\gamma\). Lemma 22: _Each robot on \(\overline{x_{i}y_{i}}\) sees the robots with color side2 placed at points \(\rho\) and \(\varrho\) of side \(\overline{c_{i}c_{i+1}}\).