(*======================================================== "NEW" RINGS: ==========================================================*)
(* THE ELEMENTS OF l ARE DOUBLE_LINKS: double_link_list m l *)
(* THE DARTS OF l DO NOT BELONG TO THE EDGE OF x *)
Fixpoint distinct_edge_list
(m:fmap)(x:dart)(l:list){struct l}:Prop:=
match l with
lam => True
| cons xs _ l0 =>
distinct_edge_list m x l0 /\ ~expe m x xs
end.
(* CONDITION (0): THE EDGES OF l ARE DISTINCTS: *)
Fixpoint pre_ring0(m:fmap)(l:list){struct l}:Prop:=
match l with
lam => True
| cons x _ l0 => pre_ring0 m l0 /\
distinct_edge_list m x l0
end.
(*�TWO double_links (x,x'), (xs,xs') ARE ADJACENT BY A FACE:*)
Definition face_adjacent(m:fmap)(x x' xs xs':dart):=
let y':= cA m zero x' in
let ys:= cA m zero xs in
expf m y' ys.
(*�CONDITION (1) OF CONTINUITY: SUCCESSIVE double_link IN l ARE face_adjacent:*)
Fixpoint pre_ring1(m:fmap)(l:list){struct l}:Prop:=
match l with
lam => True
| cons x x' l0 =>
match l0 with
lam => True
| cons xs xs' l' =>
pre_ring1 m l0 /\ face_adjacent m x x' xs xs'
end
end.
(* CONDITION (2) OF CLOSURE: THE FIST AND LAST FACES ARE ADJACENT *)
Definition pre_ring2(m:fmap)(l:list):Prop:=
match l with
lam => True
| cons x x' l0 =>
match (last l) with (xs,xs') =>
face_adjacent m xs xs' x x'
end
end.
(* CONDITION (3) OF SIMPLICITY: THE FACES OF y' AND ys ARE DISTINCT : *)
Definition distinct_faces(m:fmap)(x x' xs xs':dart):Prop:=
let y := cA m zero x in
let ys:= cA m zero xs in
~expf m y ys.
(* THE FACE OF y' IS DISTINCT FROM ALL FACES OF l: *)
Fixpoint distinct_face_list
(m:fmap)(x x':dart)(l:list){struct l}:Prop:=
match l with
lam => True
| cons xs xs' l0 => distinct_face_list m x x' l0
/\ distinct_faces m x x' xs xs'
end.
(* RING SIMPLICITY: *)
Fixpoint pre_ring3(m:fmap)(l:list){struct l}:Prop:=
match l with
lam => True
| cons x x' l0 =>
pre_ring3 m l0 /\ distinct_face_list m x x' l0
end.
(*TO BE A RING: *)
Definition ring(m:fmap)(l:list):Prop:=
~emptyl l /\ double_link_list m l /\
pre_ring0 m l /\
pre_ring1 m l /\
pre_ring2 m l /\
pre_ring3 m l.
(*============================================================ FIRST PROPERTIES WITH Jordan FOR A RING OF LENGTH 1 =============================================================*)
(* OK !! *)

Lemma Jordan1:forall(m:fmap)(x x':dart),
inv_hmap m -> planar m ->
let l:= cons x x' lam in
ring m l -> nc (Br m l) = nc m + 1.