(* 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.
(* Constructive Sufficient Condition of planarity: *)

(* Specifications: *)
Theorem plf_planar:forall (m:fmap),
inv_qhmap m -> plf m -> planar m.