(*========================================================== ============================================================ TOPOLOGICAL FMAPS, HMAPS - WITH TAGS AND EMBEDDINGS PROOF OF THE JORDAN THEOREM  PART 10: FORMALIZATION AND PROOF OF JCT WITH A 2008 NEW RING FORMALIZATION  expf, genus, planar / B, RINGS, JCT   OK: COMPLETE!! ============================================================ ==========================================================*)
(* cd Desktop/JORDAN_2008 coqtop -opt *)
(* ABOUT ... min: Load Jordan1. Load Jordan2. Load Jordan3. Load Jordan4. Load Jordan5. Load Jordan6. Load Jordan7. Load Jordan8. Load Jordan9. *)
Require Export Jordan9.
Open Scope Z_scope.
(*============================================================ genus, planar / L_B0, B0, planarity criterion / B0 =============================================================*)
(* OK: *)
(* OK: *)
(* OK!: *)
(* PLANARITY CRITERION / B0: OK! *)

Theorem planarity_crit_B0: forall (m:fmap)(x:dart),
inv_hmap m -> succ m zero x ->
let m0 := B m zero x in
let y := A m zero x  in
(planar m <->
(planar m0 /\ (~eqc m0 x y \/ expf m0 (cA_1 m0 one x) y))).