(* 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 predf_F_1: forall (m:fmap)(y:dart), inv_qhmap m -> (predf m y <-> F_1 m y <> nil).