(*========================================================== ============================================================ 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: *)
(* OK: USEFULL?  Lemma cA_cA_1_L_B_top_bot:forall(m:fmap)(k:dim)(x:dart), inv_hmap m -> succ m k x -> (forall z:dart, exd m z -> cA (L (B m k x) k (top m k x) (bottom m k x)) k z = cA m k z) -> (forall z:dart, exd m z -> cA_1 (L (B m k x) k (top m k x) (bottom m k x)) k z = cA_1 m k z). *)
(* OK: *)
(* OK! *)
(* OK !! *)
(* OK!! *)
(* OK! *)
(* OK: *)
(* OK!! *)
(*========================================================== (PRE-)DOUBLE-LINKS AND BREAKING ===========================================================*)
(* BREAKING A PRE-DOUBLE-LINK (x x'): GENERAL *)
Definition Br1(m:fmap)(x x':dart):fmap:=
if succ_dec m zero x
then if succ_dec m zero x'
then B (L (B m zero x)
zero (top m zero x) (bottom m zero x)) zero x'
else B m zero x
else B m zero x'.
(* OK !! *)
(* TO BE A DOUBLE-LINK: *)
Definition double_link(m:fmap)(x x':dart):Prop:=
x <> x' /\ expe m x x'.
(* cA0 / Br1 : *)
(* SYMMETRICALLY: *)
(* IDEM, WITH cF, cF_1 / Br1: *)
(* GREAT!! *)
(* THE SYMMETRIC: *)
(* OK!! VERY IMPORTANT: *)

Theorem nc_Br1: forall (m:fmap)(x x':dart),
inv_hmap m -> planar m -> double_link m x x' ->
let y  := cA m zero x in
let y' := cA m zero x' in
nc (Br1 m x x') = nc m +
if expf_dec m y y' then 1 else 0.