Notation no_invG :=
  (fun s : S_Ck SG =>
   match s with
   | (sg, z) =>
       sg = Lowering /\ ~ lt_Ck z kg1 \/ sg = Raising /\ ~ lt_Ck z kg3
   end) (only parsing).

Lemma No_invG : forall s : S_Ck SG, ~ InvG s -> (fun s : S_Ck SG => match s with | (sg, z) => sg = Lowering /\ ~ lt_Ck z kg1 \/ sg = Raising /\ ~ lt_Ck z kg3 end) s.