intros k mon a forces_a k' suc.
unfold forces_t in |- *.
apply forces_t2_is_local with (k := k).
assumption.
assumption.
unfold forces_t2 in |- *.
apply forces_mon with (k := k).
apply (kripke_tree_kripke_model k mon).
unfold Suc in |- *; apply succs_refl.
assumption.
assumption.
assumption.
unfold Is_Monotone_kripke_tree in |- *.
apply is_monotone_tree_successor with k.
assumption.
assumption.
unfold Suc in |- *; apply succs_refl.
Qed.

Lemma soundness_t :
forall (t : proof_term) (context : flist) (a : form),
derives context t a ->
forall k : kripke_tree,
Is_Monotone_kripke_tree k ->
(forall c : form, In c context -> forces_t k c) -> forces_t k a.