Definition forces_t2 (K k : kripke_tree) (a : form) :=
  forces kripke_tree (fun k0 : kripke_tree => Suc k0 K)
    (fun k0 k1 : kripke_tree => Suc k1 k0)
    (fun (k0 : kripke_tree) (i : Int) => forces0_t (Atms k0) i) k a.

Lemma forces_t2_is_local : forall (a : form) (k : kripke_tree), Is_Monotone_kripke_tree k -> forall k' : kripke_tree, Suc k' k -> forces_t2 k k' a -> forall K : kripke_tree, Is_Monotone_kripke_tree K -> Suc k' K -> forces_t2 K k' a.