unfold rat_inf_limit, inf_limit.
intuition.
assert (forall (epsilon : Rat),
~ epsilon == 0 ->
exists n : nat,
forall n': nat,
n' >= n ->
(forall (b : B) r,
lowDistApprox c1 b n' r ->
(ratDistance r (evalDist c1 b)) <= epsilon)).
assert (forall (epsilon : Rat),
(epsilon == 0 -> False) ->
forall (a : B),
exists n : nat,
forall n' : nat,
n' >= n ->
forall r,
lowDistApprox c1 a n' r ->
(ratDistance r (evalDist c1 a) <= epsilon)).
intuition.
intuition.
specialize (H4 _ H5).
edestruct (rel_map_left_total _ (getSupport c1) H4).
Local Open Scope rat_scope.
econstructor.
intuition.
assert (n' >= (maxList x)).
eapply H7.
edestruct (in_dec (comp_eq_dec c1) b (getSupport c1)).
edestruct (rel_map_in_inv H6); eauto.
intuition.
eapply H12.
eapply le_trans.
eapply maxList_correct; eauto.
eauto.
trivial.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply (ratIdentityIndiscernables r).
eapply eqRat_trans.
eapply getSupport_not_In_lowDistApprox; eauto.
eapply eqRat_symm.
eapply getSupport_not_In_evalDist; eauto.
eapply rat0_le_all.
clear H1.
assert (forall (epsilon : Rat),
~ epsilon == 0 ->
exists n : nat,
forall n' : nat,
n' >= n ->
(forall a b r,
In b (getSupport c1) ->
lowDistApprox (c2 b) a n' r ->
(ratDistance r (evalDist (c2 b) a)) <= epsilon)).
assert (forall (epsilon : Rat),
(epsilon == 0 -> False) ->
forall (x : (B * A)),
In (fst x) (getSupport c1) ->
exists n : nat,
forall n' : nat,
n' >= n ->
forall r,
lowDistApprox (c2 (fst x)) (snd x) n' r ->
(ratDistance r (evalDist (c2 (fst x)) (snd x)) <= epsilon)).
intuition.
intuition.
specialize (H1 epsilon0).
intuition.
edestruct (@rel_map_left_total_strong' _ nat (getSupport_bind_cp c1 c2) (fun y => In (fst y) (getSupport c1)) (fun x n => forall n',
n' >= n ->
forall r : Rat,
lowDistApprox (c2 (fst x)) (snd x) n' r ->
(ratDistance r (evalDist (c2 (fst x)) (snd x))) <= epsilon0)).
eapply H6.
eapply in_getSupport_bind_cp_fst.
exists (maxList x).
intuition.
edestruct (in_dec (EqDec_dec (pair_EqDec (comp_EqDec c1) (bind_EqDec c1 c2))) (b, a0) (getSupport_bind_cp c1 c2)).
edestruct (rel_map_in_inv H1); eauto.
intuition.
simpl in *.
eapply H12.
eapply le_trans.
eapply maxList_correct; eauto.
eauto.
trivial.
eapply leRat_trans.
eapply eqRat_impl_leRat.
eapply (ratIdentityIndiscernables r).
apply not_In_getSupport_bind_cp in n.
edestruct n.
intuition.
eapply eqRat_trans.
eapply getSupport_not_In_lowDistApprox; eauto.
eapply eqRat_symm.
eapply getSupport_not_In_evalDist; eauto.
eauto.
eapply comp_eq_dec; eauto.
eapply bind_eq_dec; eauto.
eapply rat0_le_all.
clear H2.
assert (nz (length (getSupport c1))).
econstructor.
eapply getSupport_length_nz.
trivial.
(*----------------------------- choose epsilon here---------------------------- *)
specialize (H4 ((1 / 3 ) * (epsilon * (RatIntro 1 (natToPosnat H2))))).
specialize (H1 ((1 / 3 ) * (epsilon * (RatIntro 1 (natToPosnat H2))))).
(* ---------------------------- choose epsilon here ------------------------------- *)
edestruct H4.
eapply ratMult_nz;
econstructor.
rattac.
eapply ratMult_nz; econstructor; intuition.
clear H4.
edestruct H1.
eapply ratMult_nz; econstructor.
rattac.
eapply ratMult_nz; econstructor; intuition.
clear H1.
exists (max x x0).
intuition.
inversion H6; clear H6; subst.
eapply (@leRat_trans _ (epsilon * (RatIntro 1 (natToPosnat H2)) * (length (getSupport c1) / 1))).
simpl.
eapply sumList_rel_distance; eauto.
intuition.
2:{
eapply sumList_rel_sumList.
}
simpl in *.
subst.
assert (well_formed_comp (c2 a0)).
eapply H0.
trivial.
destruct (@lowDistApprox_left_total _ (comp_eq_dec c1) c1 a0 H n').
destruct (@lowDistApprox_left_total _ (bind_eq_dec c1 c2) (c2 a0) a H9 n').
erewrite H8; eauto.
eapply leRat_trans.
eapply ratDistance_ratMult_le.
eapply H5.
eapply Max.max_lub_l.
eapply H1.
trivial.
eapply H4.
eapply Max.max_lub_r.
eapply H1.
trivial.
trivial.
eapply lowDistApprox_le_1; eauto.
eapply lowDistApprox_le_1; eauto.
eapply evalDist_le_1.
eapply evalDist_le_1.
rewrite <- ratMult_assoc.
rewrite <- ratMult_num_den.
rewrite num_dem_same_rat1.
rewrite ratMult_1_l.
eapply leRat_refl.
unfold posnatMult, natToPosnat, posnatToNat.
trivial.
rewrite ratMult_assoc.
rewrite ratMult_comm.
rewrite <- ratMult_num_den.
rewrite num_dem_same_rat1.
rewrite ratMult_1_l.
eapply leRat_refl.
unfold posnatMult, natToPosnat, posnatToNat.
rewrite mult_comm.
trivial.
Qed.

Lemma datMap_left_total : forall (A B : Set)(eqdb : eq_dec A)(t : DistApproxTree A)(f : A -> DistApproxTree B -> Prop),
(forall a, In a (getTreeSupport eqdb t) -> exists b, f a b) ->
exists t',
datMap f t t'.