(* Changing k-links: effect: breaking/merging of orbits *)
(* Precondition for hmaps: *)
Definition prec_lh (m:fmap)(k:dim)(x y:dart)
:= exd m x /\ exd m y /\ y <> A m k x.
(* Note that the construction of fixpoints is allowed: *)
Definition l (m:fmap)(k:dim)(x y:dart) :=
let xk  := (A m k x) in
let y_k := (A_1 m k y) in
let m1 := (B_1 (B m k x) k y) in
(L (L m1 k x y) k y_k xk).

Lemma y_xk_vs_yk_x : forall(m:fmap)(k:dim)(x y:dart),
inv_hmap m -> exd m x -> exd m y ->
let xk:= A m k x in
let yk:= A_1 m k y in
y <> xk <-> yk <> x.