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

Lemma in_up_oram_tr : forall o id lvl dlt p p',
NoDup (List.map block_blockid (get_all_blks_tree o)) ->
In id (List.map block_blockid (get_all_blks_tree (up_oram_tr o lvl dlt p'))) ->
In id (List.map block_blockid (concat (lookup_path_oram o p))) ->
In id (List.map block_blockid (concat (lookup_path_oram (up_oram_tr o lvl dlt p') p))).