Set Implicit Arguments.
Module Type VWS.
Declare Module E : DecidableType.
Definition key := E.t.
Hint Transparent key.
Parameter t : Type -> Type.
Section Types.
Variable elt:Type.
Parameter empty : t elt.
Parameter add : key -> elt -> t elt -> t elt.
Parameter find : key -> t elt -> option elt.
Parameter remove : key -> t elt -> t elt.
Parameter empty_o : forall x, find x empty = None.
Parameter add_eq_o : forall m x y e, E.eq x y -> find y (add x e m) = Some e.
Parameter add_neq_o : forall m x y e, ~ E.eq x y -> find y (add x e m) = find y m.
Parameter remove_eq_o : forall m x y, E.eq x y -> find y (remove x m) = None.
Parameter remove_neq_o : forall m x y, ~ E.eq x y -> find y (remove x m) = find y m.
End Types.
End VWS.
Module WS_to_VWS (Map : WS) <: VWS.
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.
End WS_to_VWS.
Module StringMapList := FMapList.Make string_lex_as_OT_compat.
Module LinearTimeStringMap <: VWS := WS_to_VWS StringMapList.
Module Type IndexedType.
Parameter t: Type.
Parameter index: t -> positive.
Parameter index_inj: forall (x y: t), index x = index y -> x = y.
Parameter eq: forall (x y: t), {x = y} + {x <> y}.
End IndexedType.
Module IT_to_DT (I : IndexedType) <: DecidableType.
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.
End IT_to_DT.
Module IndexedPositiveMap (X : IndexedType) <: VWS.
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 empty_o : forall A x, find x (empty A) = None.