(* ==========================================================  PLANARITY CRITERIA (NECESSARY CONDITIONS)  OK, modulo expf SYMMETRY!! =============================================================*)
(* OK: *)

(* Specifications: *)
Lemma eq_genus_I : forall(m:fmap)(x:dart),
inv_qhmap (I m x) -> genus (I m x) = genus m.
Lemma incr_genus_L0:forall(m:fmap)(x y:dart),
inv_qhmap (L m di0 x y) ->
genus m <= genus (L m di0 x y).
Lemma incr_genus_L1:forall(m:fmap)(x y:dart),
inv_qhmap (L m di1 x y) ->
genus m <= genus (L m di1 x y).
Lemma planar_I_rcp: forall(m:fmap)(x:dart),
inv_qhmap (I m x) -> planar (I m x) -> planar m.
Lemma planar_L0_rcp: forall(m:fmap)(x y:dart),
inv_qhmap (L m di0 x y) ->
planar (L m di0 x y) -> planar m.
Lemma planar_L1_rcp: forall(m:fmap)(x y:dart),
inv_qhmap (L m di1 x y) ->
planar (L m di1 x y) -> planar m.
Lemma succf_expf_F: forall(m:fmap)(x:dart),
inv_qhmap m -> succf m x -> expf m x (F m x).