(* 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).
(* Decidability of succf: *)

(* Specifications: *)
Lemma predf_dec :
forall (m:fmap)(x:dart),
{predf m x}+{~predf m x}.
Lemma predf_exd : forall (m:fmap)(x:dart),
inv_qhmap m -> predf m x -> exd m x.