If I understand the chronology correctly, Tywin orders the invasion of the Riverlands either before Robert’s death or so soon afterwards that he is unlikely to know of the demise of the crown. As Tywin likely assumes that Robert is still King as his armies flood over the Golden Tooth, what is he thinking, what is his plan re Robert? He is a knowing traitor – Greyjoy raids, Dornish invasion, and civil wars excluded, this is the only post-Aegon invasion by one lord paramount of another’s domain.