(* 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 inj_A_1_qhmap : forall (m:fmap)(k:dim), inv_qhmap m -> inj_dart (pred m k) (A_1 m k).