intuition.
simpl in *.
eapply trans.
eapply refl.
rewrite (sumList_filter_partition e).
eapply eqRat_refl.
eapply add_proper.
eapply trans.
eapply (rel_sumList_compat); intuition.
eapply mult_proper.
eapply refl.
eapply eqRat_refl.
eapply H2.
eapply filter_In.
eauto.
eapply refl.
rewrite sumList_factor_constant_r.
eapply ratMult_eqRat_compat; intuition.
rewrite (sumList_filter_partition e) in H1.
rewrite <- H1.
rewrite ratAdd_0_r.
eapply ratAdd_eqRat_compat.
eapply sumList_body_eq; intuition.
rewrite <- ratMult_1_r at 1.
eapply ratMult_eqRat_compat.
intuition.
destruct ( EqDec_dec bool_EqDec (e a0) true); intuition.
exfalso.
eapply n.
eapply filter_In.
eauto.
symmetry.
eapply sumList_0; intuition.
destruct (EqDec_dec bool_EqDec (e a0) true).
apply filter_In in H4.
intuition.
rewrite e0 in H6.
simpl in *.
congruence.
eapply ratMult_0_r.
eapply trans.
eapply (rel_sumList_compat); intuition.
eapply mult_proper.
eapply refl.
eapply eqRat_refl.
eapply H3.
eapply filter_In in H4.
intuition.
destruct (e a0); intuition.
eapply refl.
rewrite sumList_factor_constant_r.
eapply ratMult_eqRat_compat; intuition.
rewrite <- H1.
rewrite <- (@evalDist_lossless _ c); trivial.
symmetry.
rewrite (sumList_partition e _ (evalDist c)).
rewrite ratAdd_comm.
rewrite ratSubtract_ratAdd_assoc.
rewrite ratSubtract_0.
rewrite <- ratAdd_0_r.
rewrite (sumList_filter_partition e).
symmetry.
rewrite ratAdd_0_l.
eapply ratAdd_eqRat_compat.
symmetry.
eapply sumList_0; intuition.
apply filter_In in H4.
intuition.
rewrite H6.
eapply ratMult_0_r.
eapply sumList_body_eq; intuition.
apply filter_In in H4.
intuition.
destruct (e a0); simpl in *; try discriminate.
symmetry.
eapply ratMult_1_r.
eapply sumList_le; intuition.
destruct (EqDec_dec bool_EqDec (e a0) true).
rewrite e0.
intuition.
destruct (e a0); intuition.
eapply sumList_le; intuition.
destruct ( EqDec_dec bool_EqDec (e a0) true).
destruct (e a0); try discriminate.
intuition.
destruct (e a0); intuition.
Qed.

Lemma rel_sumList_factor_r :
forall (A : Set) (f : A -> Rat) (ls : list A) c rel,
RatRel rel ->
rel (sumList ls (fun a => (f a) * c)) ((sumList ls f) * c).