(* Connectivity : Definition in Prop "a la" Warshall *)

Fixpoint eqc(m:fmap)(x y:dart){struct m}:Prop:=
 match m with
     V => False
  |  I m0 x0 => x=x0 /\ y=x0 \/ eqc m0 x y
  |  L m0 _ x0 y0 =>
      eqc m0 x y
     \/ eqc m0 x x0 /\ eqc m0 y0 y
     \/ eqc m0 x y0 /\ eqc m0 x0 y
 end.

Lemma eqc_eqc_A_1_l : forall m:fmap, inv_qhmap m -> forall (k:dim)(x y:dart), eqc m x y -> pred m k x -> eqc m (A_1 m k x) y.