unfold Is_Monotone_kripke_tree in |- *.
intros K mon k suc.
apply is_monotone_tree_successor with K; assumption.
Qed.
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.