For n-gram models, Property 5 trivially holds since BO n−1 (w i−1 1 ) and Φ n (w i−1 1 ) are defined as sets of sequences ending with w i−1 i−n+2 and w i−1 i−n+1 with the former clearly being a superset of the latter.
when Φ can be arbitrary, e.g., a decision tree, that is not necessarily so.