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 lookup_empty {K V} `{Ord K} (v_def : V) (k : K) :
lookup_dict v_def k (empty_dict) = v_def.