(*=========================================================
                 CLOSURES cF, cF_1 OF F, F_1 :
==========================================================*)

(* SUCCESSOR IN A COMPLETED FACE: *)

Definition cF (m:fmap)(z:dart):=
  cA_1 m one (cA_1 m zero z).Definition cF_1 (m:fmap)(z:dart):=
  cA m zero (cA m one z).

(* Specifications: *)
Lemma exd_cF_not_nil : forall (m:fmap)(z:dart), inv_hmap m -> (exd m z <-> cF m z <> nil).
Lemma exd_cF_1_not_nil : forall (m:fmap)(z:dart), inv_hmap m -> (exd m z <-> cF_1 m z <> nil).
Lemma exd_cF : forall (m:fmap)(z:dart), inv_hmap m -> (exd m z <-> exd m (cF m z)).
Lemma exd_cF_1 : forall (m:fmap)(z:dart), inv_hmap m -> (exd m z <-> exd m (cF_1 m z)).
Lemma inj_cF : forall (m:fmap), inv_hmap m -> inj_dart (exd m) (cF m).
Lemma inj_cF_1 : forall (m:fmap), inv_hmap m -> inj_dart (exd m) (cF_1 m).
Lemma cF_cF_1:forall (m:fmap)(z:dart), inv_hmap m -> exd m z -> cF m (cF_1 m z) = z.
Lemma cF_1_cF:forall (m:fmap)(z:dart), inv_hmap m -> exd m z -> cF_1 m (cF m z) = z.
Lemma surj_cF : forall (m:fmap), inv_hmap m -> surj_dart (exd m) (cF m).
Lemma bij_cF : forall (m:fmap), inv_hmap m -> bij_dart (exd m) (cF m).
Lemma surj_cF_1 : forall (m:fmap), inv_hmap m -> surj_dart (exd m) (cF_1 m).
Lemma bij_cF_1 : forall (m:fmap), inv_hmap m -> bij_dart (exd m) (cF_1 m).