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.