intros k mon a b forces_a_b foreach_succs.
unfold forces_t in |- *.
unfold forces_t2 in |- *.
simpl in |- *.
intro k'.
fold (forces_t2 k k' a) in |- *.
fold (forces_t2 k k' b) in |- *.
intros suc_k'_k; clear suc_k'_k.
intros suc_k'_k.
inversion_clear suc_k'_k.
assumption.
intro forces_a.
apply forces_t2_is_local with (k := t1).
unfold Is_Monotone_kripke_tree in |- *.
apply is_monotone_tree_successor with k.
assumption.
apply successor_trans with t1.
assumption.
apply succs_refl.
assumption.
apply (foreach_succs t1 H k').
assumption.
assumption.
change (forces_t2 t1 k' a) in |- *.
apply forces_t2_is_local with (k := k).
assumption.
unfold Suc in |- *; apply successor_trans with t1.
assumption.
assumption.
assumption.
unfold Is_Monotone_kripke_tree in |- *.
apply is_monotone_tree_successor with k.
assumption.
apply successor_trans with t1.
assumption.
apply succs_refl.
assumption.
assumption.
unfold Suc in |- *; apply successor_trans with t1.
assumption.
assumption.
Qed.

Lemma forces_t_mon :
forall k : kripke_tree,
Is_Monotone_kripke_tree k ->
forall a : form,
forces_t k a -> forall k' : kripke_tree, Suc k' k -> forces_t k' a.