intros.
apply kripke_model.
unfold Suc in |- *.
intros.
apply succs_refl.
unfold Suc in |- *.
intros.
apply succs_trans with (t1 := k1); assumption.
unfold Suc in |- *.
unfold Atms in |- *.
intros k0 k1 suc0 suc1 suc1' i forces_k1.
cut (Is_Monotone atoms Int forces0_t k).
intro is_mon_k.
inversion_clear is_mon_k.
apply H0 with (t0 := k0); assumption.
apply is_monotone_tree_is_monotone.
assumption.
Qed.

Lemma kripke_tree_succ :
forall K : kripke_tree,
Is_Monotone_kripke_tree K ->
forall k : kripke_tree, Suc k K -> Is_Monotone_kripke_tree k.