(*========================================================== ============================================================ TOPOLOGICAL FMAPS, HMAPS -  WITH TAGS AND EMBEDDINGS  PART 7:  expf / B0, nf_L0L0, AND CONSEQUENCES   COMPLETE: OK! ============================================================ ==========================================================*)
(* cd Desktop/JFD/JFD08/GALAPAGOS/DELAUNAY08/  coqtop -opt  Load Jordan1. Load Jordan2. Load Jordan3. Load Jordan4. Load Jordan5. Load Jordan6. *)
Require Export Jordan6.
(*========================================================= cF, expf / B...OK !! (GREAT RESULTS) VERY USEFUL FOR Jordan CT ===========================================================*)

(* Specifications: *)
Lemma cF_B:forall(m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x ->
cF (B m k x) z =
if eq_dim_dec k zero
then
cA_1 (B m zero x) one
(if eq_dart_dec (A m zero x) z
then top m zero x
else if eq_dart_dec (bottom m zero x) z
then x else cA_1 m zero z)
else
(if eq_dart_dec (A m one x) (cA_1 (B m one x) zero z)
then top m one x
else
if eq_dart_dec (bottom m one x)
(cA_1 (B m one x) zero z)
then x
else cA_1 m one (cA_1 (B m one x) zero z)).
Lemma B_B : forall (m:fmap)(k j:dim)(u v:dart),
B (B m k u) j v =  B (B m j v) k u.