induction 1; intuition; subst.
econstructor.
econstructor.
eapply IHrel_map; eauto.
intuition.
eapply H2.
simpl.
intuition.
simpl.
intuition.
trivial.
Qed.

(* Specifications: *)
Theorem pred_count_eq_0 : forall (A B : Set)(ls : list B)(ls' : list A)(f : B -> A -> Prop)(P : A -> Prop) v,
(forall a b, In b ls -> f b a -> ~ P a) ->
rel_map f ls ls' ->
pred_count P ls' v ->
v = O.