induction 1; intuition; simpl in *.
exfalso.
eapply H1.
eauto.
trivial.
eapply IHpred_count.
intuition.
eapply H1; eauto.
Qed.

(* Specifications: *)
Lemma pred_count_eq_1_inv : forall (A : Set)(eqd : eq_dec A)(P : A -> Prop)(ls : list A) c a,
pred_count P ls c ->
NoDup ls ->
P a ->
In a ls ->
(forall a', In a' ls -> a <> a' -> ~P a') ->
c = (S O).