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 zero x y => nv m0
  | L m0 one x y => nv m0 - 1
 end.(* Number of edges: *)

Fixpoint ne(m:fmap):Z :=
 match m with
    V => 0
  | I m0 x _ _ => ne m0 + 1
  | L m0 zero x y => ne m0 - 1
  | L m0 one 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 zero x y =>
      let x_1:= cA_1 m0 one x in
      nf m0 +
       if expf_dec m0 x_1 y then 1 else -1
  | L m0 one x y =>
      let y0 := cA m0 zero y in
      nf m0 +
       if expf_dec m0 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 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_hmap m -> 2 * (nc m) >= (ec m).