(*======================================================== NUMBERING AND CHARACTERISTICS: =========================================================*)
Open Scope Z_scope.
(* 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.
(* The Euler-Poincare characteristic is even: OK: *)

(* THEOREME OF THE GENUS: OK *)
Theorem genus_theorem : forall m:fmap,
inv_qhmap m -> 2 * (nc m) >= (ec m).