induction 1; intuition; simpl in *.
rewrite lowDistApprox_Ret_inv; eauto.
eapply leRat_refl.
edestruct (lowDistApprox_bind_left_total); eauto.
eapply leRat_trans.
eapply lowDistApprox_le_bind;
eauto.
inversion H3; clear H3; subst.
eapply sumList_rel_le.
eauto.
eapply sumList_rel_sumList.
intuition.
edestruct lowDistApprox_left_total.
eapply comp_eq_dec; eauto.
eauto.
edestruct lowDistApprox_left_total.
eapply bind_eq_dec; eauto.
eauto.
rewrite H5; eauto.
rewrite <- H6; eauto.
eapply ratMult_leRat_compat; eauto.
destruct (le_dec n n0).
assert (exists n1, n0 = n + n1)%nat.
exists (minus n0 n).
omega.
destruct H0.
subst.
rewrite lowDistApprox_Rnd; eauto.
eapply leRat_terms; intuition.
rewrite lowDistApprox_Rnd_lt.
eapply rat0_le_all.
eauto.
omega.
destruct n.
destruct H1.
destruct H1.
intuition.
simpl in *.
inversion H2; clear H2; subst.
inversion H6; clear H6; subst.
inversion H8; clear H8; subst.
inversion H2; clear H2; subst.
apply evalDet_steps_bind_done_inv in H8.
destruct H8.
destruct H2.
intuition.
apply filter_In in H0.
intuition.
assert (b = x).
eapply evalDet_steps_nil_eq; eauto.
subst.
rewrite H6 in H5.
inversion H5; clear H5; subst.
simpl in *.
inversion H10; clear H10; subst.
inversion H1; clear H1; subst.
inversion H9; clear H9; subst.
inversion H7; clear H7; subst.
rewrite H4.
unfold indicator.
rewrite H6.
rewrite ratMult_1_l.
erewrite evalDet_steps_done_support_singleton; eauto.
unfold sumList; simpl.
rewrite H6.
simpl.
rewrite <- ratMult_1_l.
eapply ratMult_leRat_compat.
rewrite <- ratInverse_1.
eapply eqRat_impl_leRat.
eapply ratInverse_eqRat_compat.
eapply rat1_ne_rat0.
rewrite <- ratAdd_0_l.
rewrite (evalDet_steps_done_evalDist eqd); eauto.
destruct (eqd a0 a0).
intuition.
congruence.
eapply eqRat_impl_leRat.
rewrite (evalDet_steps_done_evalDist eqd); eauto.
destruct (eqd a0 a0).
eapply num_dem_same_rat1.
unfold natToPosnat, posnatToNat.
trivial.
congruence.
inversion H7; clear H7; subst.
rewrite H4.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply rat_num_0.
eapply rat0_le_all.
inversion H1; clear H1; subst.
discriminate.
inversion H6; clear H6; subst.
rewrite H4.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply rat_num_0.
eapply rat0_le_all.
edestruct lowDistApprox_repeat_left_total; eauto.
eapply leRat_trans.
eapply lowDistApprox_le_repeat; eauto.
omega.
unfold lowDistApprox_repeat in *.
unfold ratMult_rel, expRat_rel, indicator_rel in *.
edestruct lowDistApprox_left_total.
eapply comp_eq_dec; eauto.
eauto.
edestruct (sumList_rel_left_total
(fun a1 : A => lowDistApprox c a1 (S n))
(filter (fun a1 : A => negb (P a1)) (getSupport c))).
intuition.
eapply lowDistApprox_left_total; eauto.
edestruct (sumList_rel_left_total
(fun (i : nat) (r : Rat) =>
forall r1' : Rat,
sumList_rel (fun a : A => lowDistApprox c a (S n))
(filter (fun a : A => negb (P a)) (getSupport c)) r1' ->
r == expRat r1' i)
(getNats 0 (S (S n)))).
intuition.
econstructor.
intuition.
eapply expRat_eqRat_compat; eauto.
eapply sumList_rel_func.
eapply H4.
trivial.
intuition.
eapply lowDistApprox_func; eauto.
case_eq (P a); intuition.
rewrite H2; eauto.
2:{
intuition.
eapply ratMult_eqRat_compat.
symmetry.
eauto.
eapply sumList_rel_func.
eapply H5.
eapply H8.
intuition.
rewrite H9.
rewrite H11.
eapply eqRat_refl.
eauto.
trivial.
congruence.
}
eapply ratMult_leRat_compat.
eapply ratMult_leRat_compat.
unfold indicator.
rewrite H6.
intuition.
rewrite (@sum_power_series (S (S n)) (sumList_rel (fun a : A => lowDistApprox c a (S n))
(filter (fun a : A => negb (P a)) (getSupport c)))); eauto; try omega.
2:{
intuition.
eapply sumList_rel_func; eauto.
intuition.
eapply lowDistApprox_func;
eauto.
}
3:{
unfold ratMult_rel, ratSubtract_rel, expRat_rel, ratInverse_rel.
intuition.
rewrite H7.
rewrite H8.
eapply eqRat_refl.
intuition.
eapply ratSubtract_eqRat_compat.
eauto.
eapply sumList_rel_func.
eapply H4.
trivial.
intuition.
eapply lowDistApprox_func; eauto.
eapply eqRat_refl.
intuition.
eapply expRat_eqRat_compat.
eapply sumList_rel_func.
eapply H4.
trivial.
intuition.
eapply lowDistApprox_func; eauto.
}
eapply leRat_trans.
eapply ratMult_leRat_compat.
eapply ratSubtract_le.
eapply leRat_refl.
eapply ratInverse_leRat.
2:{
eapply ratSubtract_leRat.
eapply leRat_refl.
eapply sumList_rel_le.
eauto.
eapply sumList_rel_sumList.
intuition.
rewrite <- H9.
eapply IHwell_formed_comp.
eauto.
}
intuition.
eapply getSupport_In_evalDist.
eapply filter_In; eauto.
eapply sumList_0.
rewrite <- evalDist_lossless in H7.
rewrite sumList_filter_partition in H7.
rewrite ratAdd_comm in H7.
rewrite ratSubtract_ratAdd_inverse in H7.
eapply H7.
trivial.
trivial.
rewrite ratMult_1_l.
eapply eqRat_impl_leRat.
eapply ratInverse_eqRat_compat.
intuition.
eapply getSupport_In_evalDist.
eapply filter_In; eauto.
eapply sumList_0.
rewrite <- evalDist_lossless in H7.
rewrite sumList_filter_partition in H7.
rewrite ratAdd_comm in H7.
rewrite ratSubtract_ratAdd_inverse in H7.
eapply H7.
trivial.
trivial.
rewrite <- (evalDist_lossless); eauto.
rewrite (sumList_filter_partition).
rewrite ratAdd_comm.
rewrite ratSubtract_ratAdd_inverse.
intuition.
intuition.
eapply ratAdd_not_leRat.
erewrite <- sumList_filter_partition.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply evalDist_lossless.
eauto.
eapply leRat_trans.
eapply H8.
eapply sumList_rel_le.
eauto.
eapply sumList_rel_sumList.
intuition.
rewrite <- H11.
eapply IHwell_formed_comp.
eauto.
intuition.
eapply sumList_0 in H9.
eapply getSupport_In_evalDist; eauto.
eapply filter_In; eauto.
eapply filter_In.
intuition.
eapply filter_In; eauto.
rewrite negb_involutive.
eapply filter_In; eauto.
eapply IHwell_formed_comp; eauto.
rewrite H2; eauto.
2:{
intuition.
congruence.
eapply ratMult_eqRat_compat.
symmetry.
eauto.
eapply sumList_rel_func.
eapply H5.
trivial.
intuition.
rewrite H9; eauto.
rewrite H11; eauto.
intuition.
}
repeat rewrite ratMult_0_l.
eapply rat0_le_all.
Qed.

Lemma lowDistApprox_repeat_scale_limit : forall (A : Set)(c : Comp A)(P : A -> bool),
well_formed_comp c ->
(exists a, In a (filter P (getSupport c))) ->
(forall a, rat_inf_limit (lowDistApprox c a) (evalDist c a)) ->
rat_inf_limit
(fun n : nat =>
sumList_rel
(fun i : nat =>
expRat_rel
(sumList_rel (fun a0 : A => lowDistApprox c a0 n)
(filter (fun a0 : A => negb (P a0)) (getSupport c))) i)
(getNats O n))
(ratInverse (sumList (filter P (getSupport c)) (evalDist c))).