induction n; intuition; simpl in *.
inversion H0. trivial.
inversion H0; clear H0; subst.
intuition.
eapply IHn; eauto.
Qed.

Lemma pred_count_func : forall (A : Type)(P : A -> Prop)(ls : list A) n1 n2,
pred_count P ls n1 ->
pred_count P ls n2 ->
n1 = n2.