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

Lemma isEqvPath_lookup_path_oram : forall o n id lvl dlt p p', is_p_b_tr o (S n) -> length p = n -> length p' = n -> (lvl < n)%nat -> In id (List.map block_blockid dlt) -> isEqvPath p p' lvl = true -> In id (List.map block_blockid (concat (lookup_path_oram (up_oram_tr o lvl dlt p) p'))).