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.