Definition f := fun(m:fmap)(z:dart) => cA m Md.k z.Definition f_1 := fun(m:fmap)(z:dart) => cA_1 m Md.k z.Definition exd_f := 
   fun(m:fmap)(z:dart) => exd_cA m Md.k z.Definition exd_f_1 := 
   fun(m:fmap)(z:dart) => exd_cA_1 m Md.k z.Definition bij_f := 
   fun(m:fmap)(h:inv_hmap m) => bij_cA m Md.k h.Definition bij_f_1 := 
   fun(m:fmap)(h:inv_hmap m) => bij_cA_1 m Md.k h.Definition f_1_f := fun(m:fmap)(z:dart) => cA_1_cA m Md.k z.Definition f_f_1 := fun(m:fmap)(z:dart) => cA_cA_1 m Md.k z.Definition k:=zero.Definition k:=one.(* OK, BUT THE DIRECT COMPOSITION IS IMPOSSIBLE... : *)

Module McA0:=McA Md0.Module MA0:= Mf McA0.Module McA1:=McA Md1.Module MA1:= Mf McA1.(*========================================================
       FACE OPERATIONS IN HMAPS: F, F_1, cF, cF_1
========================================================*)

(* Following operations are designed for hmaps,
but are defined on fmap for convenience *)

(* Successor F in a face - direct version, 
designed for hmaps: *)

Definition F(m:fmap)(z:dart):=
  A_1 m one (A_1 m zero z).(* To have a successor in a (direct) face for x: *)

Definition succf(m:fmap)(z:dart):Prop:=
  pred m zero z /\ pred m one (A_1 m zero z).

Lemma succf_exd : forall (m:fmap)(z:dart), inv_hmap m -> succf m z -> exd m z.