The correct Timing of a Rook Exchange

In the following endgame White forced a better rook endgame ( with an imprisoned rook) and waited then for the right moment to exchange the rooks. The correct timing was here important, because the direct exchange would have forced a loss. So he first had to improve the conditions for the transposition into a pawn endgame.

Part 1: White forced a rook endgame, cut off black king and brought black rook in an immobile position. He now could exchange the rooks  at any time

Part 2: The  direct rook exchange with Rc2 would have failed cause of black tempo moves with the king side pawns

Part 3: White first improved the situation on king side and exchanged then the rooks