intros t context a der_t A World le forces0 kripke_model0.
elim der_t; clear der_t t a context.
(* t=(Var n) *)
intros context n a nth k world forces_context.
apply forces_context.
apply nth_in with n; assumption.
(* t=(Efq r b) *)
intros context r a der_t ih k world forces_context.
elimtype False.
apply (ih k world forces_context).
(* t=(Abs a0 r) *)
intros context a r b der_t ih k world forces_context.
simpl in |- *.
intros k' world_k' le_k' forces_a0.
apply (ih k' world_k').
intros c in_c.
inversion_clear in_c.
rewrite <- H.
assumption.
apply forces_mon with k; try assumption.
apply forces_context; assumption.
(* t=(App r s) *)
intros context r s a b der_r ih_r der_s ih_s k world forces_context.
apply (ih_r k world forces_context k).
assumption.
elim kripke_model0; auto.
apply (ih_s k world forces_context).
(* t=(Pair r s) *)
intros context r s a b der_r ih_r der_s ih_s k world forces_context.
split.
apply (ih_r k world forces_context).
apply (ih_s k world forces_context).
(* t=(PrL r) *)
intros context r a b der_r ih k world forces_context.
generalize (ih k world forces_context).
intros u0; elim u0; intros; assumption.
(* t=(PrR r) *)
intros context r a b der_r ih k world forces_context.
generalize (ih k world forces_context).
intros u0; elim u0; intros; assumption.
(* t=(OrFL r b) *)
intros context r a b der_r ih k world forces_context.
left.
apply (ih k world forces_context).
(* t=(OrFR r b) *)
intros context r a b der_r ih k world forces_context.
right.
apply ih; assumption.
(* t=(Cas r s t) *)
intros context r s t a b c der_r ih_r der_s ih_s der_t ih_t k world
forces_context.
generalize (ih_r k world forces_context); clear ih_r.
intro u0; elim u0; clear u0.
intro forces_a0.
apply ih_s; clear ih_s ih_t.
assumption.
intros c0 in_c0.
inversion_clear in_c0.
rewrite <- H; assumption.
apply forces_context; assumption.
intros forces_b.
apply ih_t; clear ih_s ih_t.
assumption.
intros c0 in_c0; inversion_clear in_c0.
rewrite <- H; assumption.
apply forces_context; assumption.
(* t=(Shift r) *)
intros context r a b der_r ih k world forces_context.
apply ih.
assumption.
intros c in_c.
apply forces_context.
right; assumption.
Qed.

Lemma forces_b__forces_a_imp_b :
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 b : form,
forces A World le forces0 k b ->
forall a : form, forces A World le forces0 k (Imp a b).