intros A World le forces0 kripke k a w; elim a; clear a.
simpl in |- *.
trivial.
intros.
elim kripke.
intros refl trans mon.
simpl in |- *; apply (mon k k'); assumption.
intros a ih_a b ih_b.
simpl in |- *.
intros u0; elim u0; clear u0; intros u0 u1.
intros; split.
apply ih_a; assumption.
apply ih_b; assumption.
intros a ih_a b ih_b.
simpl in |- *.
intros u0; elim u0; clear u0; intros u0.
intros; left.
apply ih_a; assumption.
intros; right.
apply ih_b; assumption.
intros a ih_a b ih_b.
simpl in |- *.
intros.
apply H.
assumption.
elim kripke.
intros refl trans mon.
apply (trans k k' k'0); assumption.
assumption.
Qed.

Lemma soundness :
forall (t : proof_term) (context : flist) (a : form),
derives context t a ->
forall (A : Set) (World : A -> Type) (le : A -> A -> Type)
(forces0 : A -> Int -> Prop),
Kripke_Model A World le forces0 ->
forall k : A,
World k ->
(forall c : form, In c context -> forces A World le forces0 k c) ->
forces A World le forces0 k a.