(* OK!: *)

(* Specifications: *)
Lemma planar_B0:forall(m:fmap)(x:dart),
inv_hmap m -> succ m zero x ->
planar m -> planar (B m zero x).
Lemma pre_ring0_Br1: forall(m:fmap)(l:list),
inv_hmap m -> planar m ->
pre_ring0 m l ->
let (x,x') := first l in
pre_ring0 (Br1 m x x') (tail l).
Lemma expf_I:forall(m:fmap)(x z t:dart)(u:tag)(p:point),
inv_hmap m -> prec_I m x ->
exd m z -> x <> z ->
(expf (I m x u p) z t <-> expf m z t).
Lemma between_expf_L0_1: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 z := Iter (cF m) i x_1 in
betweenf m x_1 z y ->
expf (L m zero x y) x_1 z.
Lemma expf_L0_CS:forall (m:fmap)(x y z t:dart),
inv_hmap (L m zero x y) -> exd m z ->
let x0 := cA m zero x in
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
(if expf_dec m x_1 y
then
betweenf m x_1 z y /\ betweenf m x_1 t y
\/ betweenf m y_0_1 z x0 /\ betweenf m y_0_1 t x0
\/ ~ expf m x_1 z /\ expf m z t
else
expf m x_1 z /\ expf m y t
\/ expf m x_1 t /\ expf m y z
\/ expf m z t)
-> expf (L m zero x y) z t.