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

Lemma write_back_in_stash_kv_rel_aux : forall n s p id v start, well_formed s -> length p = LOP -> Nat.le (start + n) LOP -> blk_in_stash id v s -> blk_in_stash id v (write_back_r start p n s) \/ exists k, (start <= k /\ at_lvl_in_path (state_oram (write_back_r start p n s)) k p (Block id v) /\ at_lvl_in_path (state_oram (write_back_r start p n s)) k (calc_path id s) (Block id v))%nat.