Module E := Map.E.Module F := Facts Map.Definition key := E.t.Definition t := Map.t.Definition empty := Map.empty.Definition add := Map.add.Definition find := Map.find.Definition remove := Map.remove.Definition empty_o := F.empty_o.Definition add_eq_o := F.add_eq_o.Definition add_neq_o := F.add_neq_o.Definition remove_eq_o := F.remove_eq_o.Definition remove_neq_o := F.remove_neq_o.Module StringMapList := FMapList.Make string_lex_as_OT_compat.Module LinearTimeStringMap <: VWS := WS_to_VWS StringMapList.Definition t := I.t.Definition eq := @eq t.Definition eq_refl := @eq_refl t.Definition eq_sym := @eq_sym t.Definition eq_trans := @eq_trans t.Definition eq_dec := I.eq.Module E := IT_to_DT X.Module M := PositiveMap.Module F := Facts M.Definition key := X.t.Definition t := M.t.Definition empty := M.empty.Definition add (A : Type) (i : key) (v : A) (m : t A) : t A :=
    M.add (X.index i) v m.Definition find (A : Type) (i : key) (m : t A) : option A :=
    M.find (X.index i) m.Definition remove (A : Type) (i : key) (m : t A) : t A :=
    M.remove (X.index i) m.

Lemma add_eq_o : forall A m x y e, E.eq x y -> @find A y (add x e m) = Some e.