Definition forces_t (k : kripke_tree) := forces_t2 k k.

Lemma forces_t_imp : forall k : kripke_tree, Is_Monotone_kripke_tree k -> forall a b : form, (forces_t k a -> forces_t k b) -> (forall k' : kripke_tree, In_Forest atoms k' (Succs k) -> forces_t k' (Imp a b)) -> forces_t k (Imp a b).