Definition MapsTo (i:key)(v:A)(m:t A) := find i m = Some v.Definition In (i:key)(m:t A) := exists e:A, MapsTo i 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').Global Instance eqk_equiv : Equivalence eq_key := _.Global Instance eqke_equiv : Equivalence eq_key_elt := _.Global Instance ltk_strorder : StrictOrder lt_key := _.

Lemma remove_1 : E.eq x y -> ~ In y (remove x m).