Definition empty_dict {K V} : dict K V := Dict [].
Definition empty_position_map : position_map := empty_dict.
Definition empty_stash : stash := [].
Fixpoint empty_oram n : oram :=
match n with
| 0 => leaf
| S m => node None (empty_oram m) (empty_oram m)
end.
Definition empty_state `{Config} : state := {|
state_position_map := empty_position_map;
state_stash := empty_stash;
state_oram := empty_oram (S LOP);
|}.

Lemma empty_oram_p_b n :
is_p_b_tr (empty_oram n) n.