Definition InvInside (s : StGlobal) :=
  match s with
  | ((st, x), _) => st = Inside -> gt_Ck x kt1
  end.

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