(* FINALLY: *)

(* Specifications: *)
Lemma expe_top: forall(m:fmap)(z t:dart),
inv_hmap m -> MA0.expo m z t ->
top m zero z = top m zero t.
Theorem incr_genus:forall(m:fmap)(k:dim)(x y:dart),
inv_hmap m -> prec_L m k x y ->
genus m <= genus (L m k x y).
Theorem inversion_planarity:forall(m:fmap)(k:dim)(x y:dart),
inv_hmap m -> prec_L m k x y ->
planar (L m k x y) -> planar m.
Theorem planarity_crit_0: forall (m:fmap)(x y:dart),
inv_hmap m -> prec_L m zero x y ->
(planar (L m zero x y) <->
(planar m /\ (~ eqc m x y \/ expf m (cA_1 m one x) y))).
Theorem planarity_crit_1: forall (m:fmap)(x y:dart),
inv_hmap m -> prec_L m one x y ->
(planar (L m one x y) <->
(planar m /\ (~ eqc m x y \/ expf m x (cA m zero y)))).