(*========================================================== ============================================================ 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! *)
(* DISCONNECTION CRITERION: VERY USEFUL *)
(*==========================================================  2008 JORDAN NEW FORMALIZATION:  ==========================================================*)
(* ======================================================= EQUIVALENCE AND top/bottom =========================================================*)
(* OK: *)
(* SPECIALIZATIONS: *)
(* OK: *)
(* OK: *)
(*========================================================== PROPERTIES ON B ===========================================================*)
(*=========================================================== expe AND bottom ============================================================*)
(* expo and between in an edge: *)
Definition expe(m:fmap)(z t:dart):=
MA0.expo m z t.
Definition betweene(m:fmap)(z v t:dart):=
MA0.between m z v t.
(* TO HAVE expe_bottom: *)
(* THEN, SIMILARLY (OK): *)
(* FINALLY: *)
(*=========================================================== expe, betweene_dec, between_bottom_B0 AND MA0.between_dec ============================================================*)
(* Decidability of betweene, in Prop: *)
(* IT SEEMS TO HAVE A PROBLEM WITH :  MA0.expo1_dec : forall (m : fmap) (z t : dart), inv_hmap m -> exd m z -> {MA0.expo1 m z t} + {~ MA0.expo1 m z t} *)
(* WE ALSO HAVE (EASILY): *)
(* OK! BY INDUCTION ON j: *)
(* THEN, BY INDUCTION ON j: *)
(* THEN, WE HAVE (OK!): *)
(* THEN (OK): *)
(* USEFUL LEMMAS: *)
(* OK: *)
(* OK!!: *)
(* OK!!: *)
(* FINALLY, THE EXPECTED RESULT (OK): *)
(* COROLLARY: OK *)
(* OK: *)
(* DECIDABILITY OF MA0.between: COULD BE GENERALIZED *)
(* OK !! *)
(*========================================================== CHANGING THE OPENING IN AN ORBIT BY L_B: exd, inv_hmap, planar, cA, cA_1, bottom, cF, cF_1, top, expf, expe, eqc... ==========================================================*)
(* OK! *)
(* IDEM, SYMMETRIC: *)

Lemma A_1_L_B_top_bot:forall(m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x ->
A_1 (L (B m k x) k (top m k x) (bottom m k x)) k z =
if eq_dart_dec (A m k x) z then nil
else if eq_dart_dec (bottom m k x) z then (top m k x)
else A_1 m k z.