(*========================================================== ============================================================ 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: *)

Lemma cF_L:forall(m:fmap)(k:dim)(x y z:dart),
inv_hmap m -> prec_L m k x y ->
cF (L m k x y) z =
if eq_dim_dec k zero
then
cA_1 m one
(if eq_dart_dec y z
then x
else
if eq_dart_dec (cA m zero x) z
then cA_1 m zero y
else cA_1 m zero z)
else
(if eq_dart_dec y (cA_1 m zero z)
then x
else
if eq_dart_dec (cA m one x) (cA_1 m zero z)
then cA_1 m one y
else cA_1 m one (cA_1 m zero z)).