(* Predecessor in a face - direct version: *)

Definition F_1 (m:fmap)(x:dart):=
  A m di0 (A m di1 x).(* To have a predecessor in a (direct) face for x: *)

Definition predf(m:fmap)(y:dart):Prop:=
  succ m di1 y /\ succ m di0 (A m di1 y).

Lemma not_exd_F_1_nil : forall (m:fmap)(x:dart), inv_qhmap m -> ~exd m x -> F_1 m x = nil.