induction c; intuition; simpl in *.
unfold sumList; simpl in *.
destruct (e a a); intuition.
rewrite <- ratAdd_0_l.
intuition.
eapply leRat_trans.
eapply eqRat_impl_leRat.
apply evalDist_sum_bind_eq.
eapply comp_eq_dec; eauto.
eapply bind_eq_dec; eauto.
eapply leRat_trans.
eapply sumList_le.
intuition.
apply ratMult_leRat_compat.
apply leRat_refl.
apply H.
eapply leRat_trans.
eapply sumList_le; intuition.
eapply eqRat_impl_leRat.
eapply ratMult_1_r.
trivial.
rewrite sumList_body_const.
rewrite getAllBvectors_length.
rewrite <- ratMult_num_den.
rewrite mult_1_l.
eapply eqRat_impl_leRat.
eapply num_dem_same_rat1.
rewrite posnatMult_1_r.
unfold posnatToNat, natToPosnat.
trivial.
destruct (gt_dec (length (filter b (getSupport c))) 0).
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply sumList_body_eq.
intuition.
eapply ratMult_eqRat_compat.
assert (b a = true).
eapply filter_In; eauto.
unfold indicator.
rewrite H0.
eapply ratMult_1_l.
eapply eqRat_refl.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply ratInverse_scale_sum_1.
intuition.
eapply getSupport_In_evalDist.
eapply filter_In.
eapply H.
trivial.
trivial.
intuition.
destruct (filter b (getSupport c)); simpl in *.
unfold sumList.
simpl.
eapply rat0_le_all.
omega.
Qed.

Lemma evalDist_le_1 : forall (A : Set)(c : Comp A) a,
evalDist c a <= 1.