(*========================================================== ============================================================ 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 ===========================================================*)
(* expf / B *)
(* ========== 1ST PART OF CS: CASE expf m x0 xb0 =========== *)
(* OK : *)
(* OK:*)
(* SYMMETRICALLY, WE HAVE: *)

Lemma expf_B0_CS_1_b_prel1:forall (m:fmap)(x:dart)(i j:nat),
inv_hmap m -> succ m zero x ->
let x0 := cA m zero x in
let xb0 := bottom m zero x in
let xh0 := top m zero x in
let xh0_1 := cA_1 m one xh0 in
let p := MF.Iter_upb m xh0_1 in
let z := Iter (cF m) i xh0_1 in
x0 = Iter (cF m) j xh0_1 ->
(i <= j < p)%nat
-> expf (B m zero x) xh0_1 z.