induction 1; intuition; simpl in *.
unfold sumList; simpl in *.
destruct (pf a a); intuition.
rewrite <- ratAdd_0_l.
intuition.
eapply eqRat_trans.
apply evalDist_sum_bind_eq.
eapply comp_eq_dec; eauto.
eapply bind_eq_dec; eauto.
eapply eqRat_trans.
eapply sumList_body_eq.
intuition.
apply ratMult_eqRat_compat.
apply eqRat_refl.
apply H1.
trivial.
eapply eqRat_trans.
eapply sumList_body_eq; intuition.
apply ratMult_1_r.
trivial.
rewrite sumList_body_const.
rewrite getAllBvectors_length.
rewrite <- ratMult_num_den.
rewrite mult_1_l.
eapply num_dem_same_rat1.
rewrite posnatMult_1_r.
unfold posnatToNat, natToPosnat.
trivial.
eapply eqRat_trans.
eapply sumList_body_eq.
intuition.
eapply ratMult_eqRat_compat.
assert (P a = true).
eapply filter_In; eauto.
unfold indicator.
rewrite H2.
eapply ratMult_1_l.
eapply eqRat_refl.
eapply ratInverse_scale_sum_1.
intuition.
eapply getSupport_In_evalDist.
eapply filter_In.
eapply H1.
trivial.
destruct (filter P (getSupport c)); simpl in *; intuition.
Qed.

(* Specifications: *)
Lemma sumList_filter_evalDist_le_1 : forall (A : Set)(c : Comp A)(P : A -> bool) a,
well_formed_comp c ->
In a (filter P (getSupport c)) ->
1 <= sumList (filter (fun a' => negb (P a')) (getSupport c)) (evalDist c) ->
False.