(* Number of darts: *)

Fixpoint nd(m:fmap):Z :=
 match m with
    V => 0
  | I m0 x => nd m0 + 1
  | L m0 _ _ _ => nd m0
 end.(* Number of vertices: *)

Fixpoint nv(m:fmap):Z :=
 match m with
    V => 0
  | I m0 x => nv m0 + 1
  | L m0 di0 x y => nv m0
  | L m0 di1 x y => nv m0 -
      if eq_dart_dec (A (clos m0) di1 x) y
      then 0 else 1
 end.(* Number of edges: *)

Fixpoint ne(m:fmap):Z :=
 match m with
    V => 0
  | I m0 x => ne m0 + 1
  | L m0 di0 x y => ne m0 -
      if eq_dart_dec (A (clos m0) di0 x) y
      then 0 else 1
  | L m0 di1 x y => ne m0
 end.(* Number of faces: *)

Fixpoint nf(m:fmap):Z :=
 match m with
    V => 0
  | I m0 x => nf m0 + 1
  | L m0 di0 x y =>
      let mc := clos m0 in
      let x0 := A mc di0 x in
      let x_1:= A_1 mc di1 x in
      nf m0 +
       if eq_dart_dec y x0 then 0
       else if expf_dec mc x_1 y then 1
	    else -1
  | L m0 di1 x y =>
      let mc := clos m0 in
      let x1 := A mc di1 x in
      let y0 := A mc di0 y in
      nf m0 +
       if eq_dart_dec y x1 then 0
       else if expf_dec mc x y0 then 1
	    else -1
 end.(* Number of connected components: *)

Fixpoint nc(m:fmap):Z :=
 match m with
    V => 0
  | I m0 x => nc m0 + 1
  | L m0 _ x y => nc m0 -
       if eqc_dec (clos m0) x y then 0 else 1
 end.(* Euler-Poincare characteristic: *)

Definition ec(m:fmap): Z:=
  nv m + ne m + nf m - nd m.

Theorem genus_theorem : forall m:fmap, inv_qhmap m -> 2 * (nc m) >= (ec m).