(* OK!!: *)

(* Specifications: *)
Lemma nopred_expe_L_i:
forall(m:fmap)(i:nat)(k:dim)(x y z:dart),
inv_hmap (L m k x y) ->
exd m z -> ~pred m zero z ->
let t:= Iter (cA m zero) i z in
(i < MA0.Iter_upb m z)%nat ->
expe (L m k x y) z t.
Lemma expe_bottom_z: forall(m:fmap)(z:dart),
inv_hmap m -> exd m z ->
MA0.expo m (bottom m zero z) z.
Lemma bottom_bottom_expe: forall(m:fmap)(z t:dart),
inv_hmap m -> exd m z -> exd m t ->
bottom m zero z = bottom m zero t ->
MA0.expo m z t.
Lemma top_top_expe: forall(m:fmap)(z t:dart),
inv_hmap m -> exd m z -> exd m t ->
top m zero z = top m zero t ->
MA0.expo m z t.
Lemma pre_ring1_Br1_aux:
forall(m:fmap)(x x':dart)(l:list),
inv_hmap m -> planar m ->
let y:= cA m zero x in
let y':= cA m zero x' in
double_link_list m (cons x x' l) ->
pre_ring0 m (cons x x' l) ->
pre_ring1 m l ->
~expf m y y' -> pre_ring1 (Br1 m x x') l.