find_rank_with_minimum_cost_imp goalvalue agg_cost rank_parent parent_reverse_state ref_t direction = do 
    t <- val ref_t 
    [size_left, size_right] <- size_children ref_t 
    aux t size_left size_right
 where
    aux Leaf _         _          = error "we should have found the goal value already"
    aux t    size_left size_right
        | min_cost_in_this_tree == goalvalue = return . return $ current_rank
        | otherwise                          = do
                                                min_cost_l <- compute_min_cost $ l t
                                                min_cost_r <- compute_min_cost $ r t
                                                case compareMaybe min_cost_l min_cost_r of
                                                    EQ -> error "rank not found"
                                                    LT -> find_rank (l t) LeftChild
                                                    GT -> find_rank (r t) RightChild
     where
        min_cost_in_this_tree = icost t + agg_cost
        agg_cost' = agg_cost + icost t
        x_reversed =  parent_reverse_state &#8853; (reversed (item t))
        current_rank = compute_rank rank_parent direction x_reversed parent_reverse_state size_left_node size_right_node
        find_rank = find_rank_with_minimum_cost_imp goalvalue agg_cost' current_rank x_reversed