(*==========================================================
             (PRE-)DOUBLE-LINKS AND BREAKING
===========================================================*)

(* BREAKING A PRE-DOUBLE-LINK (x x'): GENERAL *) 

Definition Br1(m:fmap)(x x':dart):fmap:=
  if succ_dec m zero x 
  then if succ_dec m zero x' 
       then B (L (B m zero x) 
            zero (top m zero x) (bottom m zero x)) zero x'
       else B m zero x
  else B m zero x'.

(* Specifications: *)
Lemma exd_Br1:forall(m:fmap)(x x' z:dart), exd m z <-> exd (Br1 m x x') z.
Lemma inv_hmap_Br1:forall(m:fmap)(x x':dart), inv_hmap m -> inv_hmap (Br1 m x x').
Lemma planar_Br1:forall(m:fmap)(x x':dart), inv_hmap m -> planar m -> x <> x' -> planar (Br1 m x x').