fact: Fixpoint stepsouth (b : Board) : Board := match b with | R r b' => R r (stepsouth b') | K l1 (R l2 (R l3 b')) => match Tristep' l1 l2 l3 with | TrS l1' l2' l3' => R l1' (K l2' (R l3' b')) | TrF l1' l2' l3' => K l1' (R l2' (R l3' b')) end | _ => b end.