Definition compare : forall x y : t, Compare lt eq x y.Definition eq_dec := N_as_OT.eq_dec.Module E:=NUsualOrderedType.Module ME:=OrderedTypeFacts(E).Module PE:=KeyOrderedType(E).Definition key := N.Definition t := Map.Definition empty : t A := M0 A.Definition is_empty (m : t A) : bool := 
    MapEmptyp _ (MapCanonicalize _ m).Definition find (x:key)(m: t A) : option A := MapGet _ m x.Definition mem (x:key)(m: t A) : bool := 
    match find x m with 
     | Some _ => true
     | None => false
    end.Definition add (x:key)(v:A)(m:t A) : t A := MapPut _ m x v.Definition remove (x:key)(m:t A) : t A := MapRemove _ m x.Definition elements (m : t A) : list (N*A) := alist_of_Map _ m.Definition cardinal (m : t A) : nat := MapCard _ m.Definition MapsTo (x:key)(v:A)(m:t A) := find x m = Some v.Definition In (x:key)(m:t A) := exists e:A, MapsTo x e m.Definition Empty m := forall (a : key)(e:A) , ~ MapsTo a e m.Definition eq_key (p p':key*A) := E.eq (fst p) (fst p').Definition eq_key_elt (p p':key*A) := 
          E.eq (fst p) (fst p') /\ (snd p) = (snd p').Definition lt_key (p p':key*A) := E.lt (fst p) (fst p').

Lemma remove_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).