induction n; intuition; simpl in *.
intuition.
intuition.
Qed.

(* Specifications: *)
Lemma pred_count_app : forall (A : Type)(P : A -> Prop)(ls1 ls2 : list A) n1 n2,
pred_count P ls1 n1 ->
pred_count P ls2 n2 ->
pred_count P (ls1 ++ ls2) (n1 + n2).