(* Definition of "Planar formation" of fmap: *)

Fixpoint plf(m:fmap):Prop:=
  match m with 
     V => True
   | I m0 x => plf m0
   | L m0 di0 x y => plf m0 /\ 
      (let mc := (clos m0) in
        ~eqc mc x y \/ expf mc (A_1 mc di1 x) y)
   | L m0 di1 x y => plf m0 /\ 
      (let mc := (clos m0) in
        ~eqc mc x y \/ expf mc x (A mc di0 y))
  end.

Lemma planar_I_rcp: forall(m:fmap)(x:dart), inv_qhmap (I m x) -> planar (I m x) -> planar m.