intros a; elim a; clear a.
unfold forces_t2 in |- *.
simpl in |- *; trivial.
unfold forces_t2 in |- *; simpl in |- *.
trivial.
intros a ih_a b ih_b k mon_k k' suc_k'_k forces_k_k'_ab K mon_K suc_k'_K.
elim forces_k_k'_ab.
fold (forces_t2 k k' a) in |- *.
fold (forces_t2 k k' b) in |- *.
clear forces_k_k'_ab.
intros u0 u1.
split.
change (forces_t2 K k' a) in |- *.
apply ih_a with (k := k); assumption.
change (forces_t2 K k' b) in |- *.
apply ih_b with (k := k); assumption.
intros a ih_a b ih_b k mon_k k' suc_k'_k forces_k_k'_ab K mon_K suc_k'_K.
elim forces_k_k'_ab; clear forces_k_k'_ab.
fold (forces_t2 k k' a) in |- *.
intro u0.
left.
change (forces_t2 K k' a) in |- *.
apply ih_a with (k := k); assumption.
fold (forces_t2 k k' b) in |- *.
intros u0.
right.
change (forces_t2 K k' b) in |- *.
apply ih_b with (k := k); assumption.
intros a ih_a b ih_b.
intros k mon_k k' suc_k'_k forces_k_k'_ab K mon_K suc_k_K.
unfold forces_t2 in |- *; simpl in |- *.
intros k''.
fold (forces_t2 K k'' a) in |- *.
fold (forces_t2 K k'' b) in |- *.
intros.
apply ih_b with (k := k); clear ih_b.
assumption.
unfold Suc in |- *; apply succs_trans with k'; assumption.
apply (forces_k_k'_ab k''); clear forces_k_k'_ab.
unfold Suc in |- *; apply succs_trans with k'; assumption.
assumption.
change (forces_t2 k k'' a) in |- *.
apply ih_a with (k := K); clear ih_a.
assumption.
assumption.
assumption.
assumption.
unfold Suc in |- *; apply succs_trans with k'; assumption.
assumption.
assumption.
Qed.
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).