Definition planar(m:fmap):= genus m = 0.

(* Specifications: *)
Lemma Euler_Poincare: forall m:fmap, inv_qhmap m -> planar m -> ec m / 2 = nc m.
Lemma expf_planar_0: forall (m:fmap)(x y:dart), inv_qhmap m -> planar m -> prec_Lq m di0 x y -> let mc:= clos m in expf mc (A_1 mc di1 x) y -> planar (L m di0 x y).
Lemma expf_planar_1: forall (m:fmap)(x y:dart), inv_qhmap m -> planar m -> prec_Lq m di1 x y -> let mc:= clos m in expf mc x (A mc di0 y) -> planar (L m di1 x y).
Lemma planar_V: planar V.
Lemma planar_I: forall (m:fmap)(x:dart), inv_qhmap m -> planar m -> prec_I m x -> planar (I m x).
Lemma not_eqc_planar: forall (m:fmap)(k:dim)(x y:dart), inv_qhmap m -> planar m -> prec_Lq m k x y -> let mc:= clos m in ~eqc mc x y -> planar (L m k x y).
Lemma expf_planar_L0: forall (m:fmap)(x y:dart), inv_qhmap m -> planar m -> prec_Lq m di0 x y -> let mc:= clos m in (~eqc mc x y \/ expf mc (A_1 mc di1 x) y) -> planar (L m di0 x y).
Lemma expf_planar_L1: forall (m:fmap)(x y:dart), inv_qhmap m -> planar m -> prec_Lq m di1 x y -> let mc:= clos m in (~eqc mc x y \/ expf mc x (A mc di0 y)) -> planar (L m di1 x y).