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

(* Specifications: *)
Lemma Euler_Poincare: forall m:fmap, inv_hmap m -> planar m -> ec m / 2 = nc m.
Lemma planar_dec:forall m:fmap, {planar m} + {~planar m}.
Lemma planar_V: planar V.
Lemma planar_I: forall (m:fmap)(x:dart)(t:tag)(p:point), inv_hmap m -> planar m -> prec_I m x -> planar (I m x t p).
Lemma expf_planar_0: forall (m:fmap)(x y:dart), inv_hmap m -> planar m -> prec_L m zero x y -> expf m (cA_1 m one x) y -> planar (L m zero x y).
Lemma expf_planar_1: forall (m:fmap)(x y:dart), inv_hmap m -> planar m -> prec_L m one x y -> expf m x (cA m zero y) -> planar (L m one x y).
Lemma not_eqc_planar: forall (m:fmap)(k:dim)(x y:dart), inv_hmap m -> planar m -> prec_L m k x y -> ~eqc m x y -> planar (L m k x y).