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

Lemma kv_in_delta_in_tree : forall (o : oram) (id : block_id) (v : nat) (del : list block) (lvl: nat )(p :path), In (Block id v) del -> coord_in_bound o p lvl -> at_lvl_in_path (up_oram_tr o lvl del p) lvl p (Block id v).