Definition on_path b o p :=
    In b (concat (lookup_path_oram o p)).

Lemma at_lvl_in_path_blocks_selection : forall s p p' lvl lvl' b, (lvl < lvl')%nat -> at_lvl_in_path (state_oram s) lvl' p b -> at_lvl_in_path (state_oram (blocks_selection p' lvl s)) lvl' p b.