# 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 \(\mathbf{P}^{\prime}\)
Moreover, they see two robots placed at points \(\rho^{\prime}\) and \(\varrho^{\prime}\) on \(\overline{c_{i}c_{i-1}}\) defined analogously to \(\varrho\) and \(\varrho\)._  
Proof: As noted above, 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 \(y_{i}\) and \(x_{i+1}\). Robots at \(\rho\), \(\varrho\), \(\rho^{\prime}\), and \(\varrho^{\prime}\) are in these ranges. Robot \(r\) in \(\overline{x_{i}y_{i}}\), after seeing two side2 colored robots each in \(\overline{c_{i}c_{i-1}}\) and \(\overline{c_{i}c_{i+1}}\), moves as follows. Let \(L\) and \(L^{\prime}\) be the lines perpendicular to \(\overline{\varrho^{\prime}}\) and \(\overline{\varrho^{\prime}}\overline{\varrho^{\prime}}\) passing through \(r\), respectively. Robot \(r\) assumes color transit_moving and moves to the intersection point \(w\) of \(\overline{\varrho^{\prime}}\) and \(L\) or \(w^{\prime}\) of \(\overline{\varrho^{\prime}}\overline{\varrho^{\prime}}\) and \(L\), whichever is closest to it. When \(r\) becomes active next, it changes it color to side2 from transit_moving. Lemma 23: _The robots in the corner and/or triangle line segments of \(\mathbf{P}^{\prime}\) move to the sides of \(\mathbf{P}^{\prime}\) and assume color side2 in \(\mathcal{O}(1)\) epochs. Furthermore, Stage 1.4 starts only after Stage 1.3 finishes._  
Proof: The movement to a side of \(\mathbf{P}^{\prime}\) is an instance of Beacon-Directed Curve Positioning. Robots move to points \(\rho\) and \(\varrho\) on each side \(\overline{c_{i}c_{i+1}}\) of \(\mathbf{P}^{\prime}\) and take color side2 in, at most,four epochs. The remaining robots on corner line segments or triangle line segments move to a side of \(\mathbf{P}^{\prime}\) and take color side2 in two more epochs. Finally, Stage 1.4 starts only after this stage finishes since the eligible colored robots on the eligible lines see at least a robot with color internal or internal_moving until Stage 1.3 finishes. The corners can change their color to corner5 since they do not see any robot with color internal or internal_moving after all robots in the interior of \(\mathbf{P}^{\prime}\) moved to the eligible lines of the corners of \(\mathbf{P}^{\prime}\). Stage 1.5-Making Side Robots of \(\mathbf{P}^{\prime}\) the Side Robots of \(\mathbf{P}^{\prime\prime}\)  
Let \(S\) be a side of \(\mathbf{P}\) with \(c_{i},c_{i+1}\) its endpoints. There is a side \(S^{\prime}\) in \(\mathbf{P}^{\prime}\) with \(c_{i},c_{i+1}\) its endpoints and all the side robots in \(S\) are in the corridor of \(S^{\prime}\) with color either side1 or special. The robots that become side robots in Stage 1.4 are on \(S^{\prime}\) with color side2. Stage 1.5 is for the side robots on \(S^{\prime}\). Stage 1.5 starts for the side robots on \(S^{\prime}\) when they do not see any other robot with color except corner, side2, side1, special. We assume in the description below that there are at least two side robots on \(S\) and at least a side robot on \(S^{\prime}\). The cases of number of robots on \(S\) and \(S^{\prime}\) not satisfying this assumption can be treated as special cases and can be executed in a constant number of epochs. Let \(\{s_{1},s_{2},\ldots,s_{w}\}\) be the side robots on \(S\) with \(s_{1}\) being closer to \(c_{i}\). Robots \(s_{1},s_{w}\) have color special and the others have side1. Since \(c_{i},c_{i+1}\) are endpoints of \(S^{\prime}\), divide \(S\) into three sides \(S_{l},S_{m},S_{r}\), where \(S_{l}=\overline{c_{i}s_{1}}\), \(S_{m}=\overline{s_{1}s_{w}}\), and \(S_{r}=\overline{s_{w}c_{i+1}}\). Let \(L_{l},L_{r}\) be the lines perpendicular to \(S^{\prime}\) passing through \(s_{1},s_{w}\), respectively. Let \(s^{\prime}_{1},s^{\prime}_{w}\) be the intersection points of \(L_{l},S^{\prime}\) and \(L_{r},S^{\prime}\), respectively.