(*========================================================== ============================================================ TOPOLOGICAL FMAPS, HMAPS -  WITH TAGS AND EMBEDDINGS  PART 5:  characteristics, eqc, expf / L0, L_B  COMPLETE FOR DIM 0 (DIM 1 IN THE NEXT FILE): OK! ============================================================ ==========================================================*)
Require Export Jordan4.
(* OK: *)
(* OK: *)
(* LEMMA, USEFUL: *)
(*=========================================================== cF, expf on L and B ============================================================*)
(* OK, TRIVIAL: USEFUL? *)
(*========================================================= cF / L, expf / L, CNS ===========================================================*)
(* expf IS AN EQUIVALENCE (NEW PLACE): *)
(* OK!: *)
(* OK: *)
(* OK: *)
(* OK: *)
(* RENAMING of MF.between: *)
Definition betweenf:= MF.between.
(* IDEM: USED IN between_expf_L0_2:*)
(* OK!: *)
(* SYMMETRICALLY, USEFUL: *)
(* THEN, OK: *)
(* SYMMETRIC, OK: *)
(* OK: BUT REPLACE  betweenf m x_1 z x0 BY expf m x_1 z, WHICH IS EQUIVALENT: *)
(* WITH THE REPLACEMENT, OK: *)
(* IDEM (TO PROVE), SYMMETRIC: BUT REPLACE betweenf m y_0_1 z y BY expf m y z which is EQUIVALENT *)
(* NEW FORMULATION!! *)

Lemma between_expf_L0_4_prel:forall(m:fmap)(x y:dart)(i:nat),
inv_hmap (L m zero x y) ->
let x_1 := cA_1 m one x in
let y_0 := cA_1 m zero y in
let y_0_1 := cA_1 m one y_0 in
let z := Iter (cF m) i y_0_1 in
~ expf m x_1 y ->
betweenf m y_0_1 z y ->
expf (L m zero x y) y_0_1 z.