(*==========================================================  TOPOLOGICAL FMAPS, HMAPS - WITH TAGS AND EMBEDDINGS  PART 6:.  PROPERTIES OF MERGING AND SPLITTING FACES expf / L1  COMPLETE: OK! ============================================================*)
(* Load Jordan1. Load Jordan2. Load Jordan3. Load Jordan4. Load Jordan5. *)
Require Export Jordan5.
(*========================================================== 1. MERGING FACES BY L1 (CASE ~expf m x y0) ===========================================================*)
(* CASE k = one IN cF_L (from Jordan5) : OK *)
(* 1. CASE ~expf m x y0: FACE MERGING *)
(* FIRST, SOME LEMMAS: *)
(* OK: *)
Open Scope nat_scope.
(* i-SUCCS OF x BY cF_L1, FOR i < DEG OF x, ARE INVARIANT: *)
(* OK: *)
(* i-SUCCS OF x BY cF_L1, FOR i < DEG x, DIFFER FROM x *)
(* OK: *)
(* THE dx_SUCC OF x BY cF_L1 IS y_1: *)
(* IT IS DIFFRENT FROM x: *)
(* j-SUCCS OF y BY cF_L1, FOR j < DEGREE OF y, ARE INVARIANT: *)
(* j-SUCCS OF y_1 BY cF_L1, FOR j < DEG OF y0 (OR y_1), ARE INVARIANT:*)

Lemma cF_L1_y_1_y0:forall(m:fmap)(x y:dart)(j:nat),
inv_hmap m -> prec_L m one x y ->
let m1:= L m one x y in
let y0 := cA m zero y in
let y_1 := cA_1 m one y in
let dx := MF.degree m x in
let dy0 := MF.degree m y0 in
~expf m x y0 ->
j <= dy0 -1 ->
Iter (cF m1) (dx + j) x = Iter (cF m) j y_1.