(* Partial injection on dart, Df giving the domain of f: *)

Definition inj_dart(Df:dart->Prop)(f:dart->dart):Prop:=
    forall x x':dart,
     (Df x)->(Df x')->(f x)=(f x')->x=x'.

Lemma succ_pred_clos : forall (m:fmap)(k:dim)(z:dart), inv_hmap m -> exd m z -> (cA m k z <> nil /\ cA_1 m k z <> nil).