induction 1; intuition; simpl in *.
f_equal.
eauto.
specialize (H1 a).
intuition.
Qed.

Lemma pred_count_first_skip : forall (A : Type)(P : A -> Prop)(ls : list A)(c : nat),
pred_count P ls c ->
forall n,
exists c1 c2,
pred_count P (firstn n ls) c1 /\
pred_count P (skipn n ls) c2 /\
(c1 + c2 = c)%nat.