induction 1; intuition.
inversion H0; clear H0; subst.
econstructor; eauto.
econstructor; eauto.
inversion H; clear H; subst.
inversion H2; clear H2; subst.
econstructor; eauto.
econstructor; eauto.
eapply pc_no; eauto.
econstructor; eauto.
inversion H2; clear H2; subst.
econstructor; eauto.
econstructor; eauto.
econstructor; eauto.
econstructor; eauto.
Qed.

(* Specifications: *)
Lemma pred_count_eq_none_inv : forall (A : Set)(P : A -> Prop)(ls : list A) c,
pred_count P ls c ->
(forall a, In a ls -> ~P a) ->
c = O.