induction ls1; intuition; simpl in *.
inversion H; subst; clear H.
simpl.
trivial.
inversion H; clear H; subst.
simpl.
econstructor; eauto.
econstructor; eauto.
Qed.

(* Specifications: *)
Lemma pred_count_permutation : forall (A : Set)(P : A -> Prop)(ls1 ls2 : list A),
Permutation ls1 ls2 ->
forall c,
pred_count P ls1 c ->
pred_count P ls2 c.