intuition.
simpl in *.
rewrite (sumList_filter_partition (@snd A bool)).
rewrite (sumList_filter_partition (@snd A bool) (getSupport c2)).
rewrite ratDistance_add_same_r_gen.
eapply leRat_trans.
eapply ratDistance_le_max_triv.
eapply maxRat_leRat_same.
eapply leRat_trans.
2:{
eapply sumList_filter_le.
}
eapply sumList_le; intuition.
apply filter_In in H1.
intuition.
eapply ratMult_leRat_compat; intuition.
destruct a0.
simpl in *.
subst.
destruct (EqDec_dec bool_EqDec true true); try congruence.
destruct (EqDec_dec eqda a0 a); intuition.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
symmetry.
eapply H.
}
eapply leRat_trans.
2:{
eapply sumList_filter_le.
}
eapply sumList_le; intuition.
apply filter_In in H1.
intuition.
eapply ratMult_leRat_compat; intuition.
destruct a0.
simpl in *.
subst.
destruct (EqDec_dec bool_EqDec true true); try congruence.
destruct (EqDec_dec eqda a0 a); intuition.
assert (eq_dec (A * bool)).
unfold eq_dec.
eapply (EqDec_dec (pair_EqDec _ _)).
assert (NoDup (filter (fun a0 : A * bool => negb (snd a0)) (intersect H1 (getSupport c1) (getSupport c2)))).
eapply filter_NoDup.
eapply filter_NoDup; eapply getSupport_NoDup.
eapply eqRat_trans.
symmetry.
eapply sumList_subset'.
trivial.
eapply H2.
eapply filter_NoDup.
eapply getSupport_NoDup.
intuition.
apply filter_In in H3.
intuition.
eapply filter_In.
split.
eapply in_intersect.
eapply Permutation_in.
eapply intersect_comm.
eapply getSupport_NoDup.
2:{
eauto.
}
eapply getSupport_NoDup.
trivial.
intuition.
apply filter_In in H3.
intuition.
destruct a0.
simpl in *.
destruct (EqDec_dec eqda a0 a); subst.
rewrite ratMult_1_r.
destruct b; simpl in *; try discriminate.
rewrite H0.
eapply getSupport_not_In_evalDist.
intuition.
eapply H4.
eapply filter_In.
intuition.
eapply in_intersect.
intuition.
apply ratMult_0_r.
symmetry.
eapply eqRat_trans.
symmetry.
eapply sumList_subset'.
trivial.
eapply H2.
eapply filter_NoDup.
eapply getSupport_NoDup.
intuition.
apply filter_In in H3.
intuition.
eapply filter_In.
split.
eapply in_intersect.
eauto.
trivial.
intuition.
apply filter_In in H3.
intuition.
destruct a0.
simpl in *.
destruct (EqDec_dec eqda a0 a); subst.
rewrite ratMult_1_r.
destruct b; simpl in *; try discriminate.
rewrite <- H0.
eapply getSupport_not_In_evalDist.
intuition.
eapply H4.
eapply filter_In.
intuition.
eapply in_intersect.
intuition.
apply ratMult_0_r.
eapply sumList_body_eq.
intuition.
apply filter_In in H3.
intuition.
destruct a0; simpl in *.
destruct b; simpl in *; try discriminate.
rewrite H0.
intuition.
Qed.

Theorem evalDist_right_ident : forall (A : Set)(eqd : EqDec A)(c : Comp A) a,
evalDist (x <-$ c; ret x) a == evalDist c a.