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joinable.v
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joinable.v
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(* Copyright (c) 2014, Robert Dockins *)
Require Import basics.
Require Import preord.
Require Import categories.
Require Import sets.
Require Import finsets.
Require Import esets.
Require Import effective.
Require Import directed.
Require Import plotkin.
(** * Joinable relations
Joinable relations represent the compact elements of the
continuous function space, and form the exponential object
for profinite domains.
Here we carry out the techincal proofs required
to show the joinable relations form an effective Plotkin preorder.
*)
(** This definition of joinable relations is a minor modification
of a definition given by Samson Abramsky. It is more obviously effective
than the definition given by Carl Gunter in his dissertation.
*)
Definition is_joinable_relation {A B:preord} hf (R:finset (A × B)) :=
inh hf R /\
forall (G:finset (A × B)) (HGinh:inh hf G), G ⊆ R ->
forall x, minimal_upper_bound x (image π₁ G) ->
exists y, (x,y) ∈ R /\ upper_bound y (image π₂ G).
Definition joinable_relation hf (A B:preord)
:= { R:finset (A × B) | is_joinable_relation hf R }.
Definition joinable_rel_ord hf (A B:preord) (R S:joinable_relation hf A B) :=
forall a b, (a,b) ∈ proj1_sig R ->
exists a' b', (a',b') ∈ proj1_sig S /\ a' ≤ a /\ b ≤ b'.
Program Definition joinable_rel_ord_mixin hf (A B:preord) :
Preord.mixin_of (joinable_relation hf A B) :=
Preord.Mixin (joinable_relation hf A B) (joinable_rel_ord hf A B) _ _.
Next Obligation.
repeat intro. exists a. exists b. split; auto.
Qed.
Next Obligation.
repeat intro.
destruct (H a b) as [a' [b' [?[??]]]]; auto.
destruct (H0 a' b') as [a'' [b'' [?[??]]]]; auto.
exists a''. exists b''. split; auto.
split; etransitivity; eauto.
Qed.
Canonical Structure joinable_rel_order hf (A B:preord) : preord :=
Preord.Pack (joinable_relation hf A B) (joinable_rel_ord_mixin hf A B).
(** The ordering relation between joinable relations is decidable. *)
Lemma joinable_ord_dec
hf
(A B:preord)
(HAeff : effective_order A)
(HBeff : effective_order B)
(R S:joinable_relation hf A B) :
{ R ≤ S }+{ R ≰ S }.
Proof.
intros.
destruct (finset_find_dec' (A×B)
(fun z => exists w, w ∈ proj1_sig S /\ π₁#w ≤ π₁#z /\ π₂#z ≤ π₂#w))
with (proj1_sig R).
- intros.
destruct H0 as [w [??]].
exists w. intuition; rewrite <- H; auto.
- intro.
destruct (finset_find_dec (A×B)
(fun w => π₁#w ≤ π₁#x /\ π₂#x ≤ π₂#w))
with (proj1_sig S).
+ intros.
destruct H0 as [??].
rewrite H in H0.
rewrite H in H1.
split; auto.
+ intros.
assert (Hdec : ord_dec (A × B)).
{ constructor. intros.
destruct x1 as [a b]. destruct y as [c d].
destruct (eff_ord_dec A HAeff a c).
- destruct (eff_ord_dec B HBeff b d).
left. split; auto.
right. intros [??]. apply n; auto.
- right. intros [??]. apply n; auto.
}
destruct (eff_ord_dec _ HAeff (π₁ # x0) (π₁ # x)).
* destruct (eff_ord_dec _ HBeff (π₂ # x) (π₂ # x0)); auto.
right. intros [??]. apply n; auto.
* right. intros [??]. apply n; auto.
+ destruct s as [z [?[??]]].
left. exists z. split; auto.
+ right.
intros [w [?[??]]].
apply (n w); auto.
- destruct s as [z [??]].
right.
red. intros.
destruct (H1 (fst z) (snd z)) as [p [q [?[??]]]].
+ destruct z; auto.
+ apply H0. exists (p,q). split; auto.
- left.
repeat intro.
destruct (e (a,b)) as [[p q] [?[??]]]; auto.
exists p. exists q. split; auto.
Qed.
(** Given a finite relation, we can decide if it is joinable or not.
If not, we can find evidence demonstrating where the relation
is insufficently complete.
*)
Lemma is_joinable_rel_dec0 hf
(A B:preord)
(HAeff : effective_order A)
(HBeff : effective_order B)
(HAplt : plotkin_order hf A)
(R:finset (A × B)) (Rinh : inh hf R) :
{ is_joinable_relation hf R } +
{ exists (G:finset (A×B)), inh hf G /\ G ⊆ R /\ exists z : A,
minimal_upper_bound z (image π₁ G) /\
(forall q : B, (z, q) ∈ R -> ~ upper_bound q (image π₂ G)) }.
Proof.
assert (Hdec : ord_dec (A × B)).
{ constructor. intros.
destruct x; destruct y.
destruct (eff_ord_dec A HAeff c c1);
[ destruct (eff_ord_dec B HBeff c0 c2) |].
- left; split; auto.
- right; intros [??]; apply n; auto.
- right; intros [??]; apply n; auto.
}
set (H:=I).
set (P (G:finset (A×B)) :=
inh hf G ->
forall x:A, minimal_upper_bound x (image π₁ G) ->
exists y, (x,y) ∈ R /\ upper_bound y (image π₂ G)).
assert (Hrespects : forall x y:finset (A×B), x ≈ y -> P x -> P y).
{ unfold P; intros x y ? ? Hy. intros.
destruct H1 with x0; auto.
- apply inh_eq with y; auto.
- destruct H2.
split.
+ red; intros. apply H2.
apply image_axiom2 in H4.
destruct H4 as [q[??]].
rewrite H5. apply image_axiom1.
rewrite <- H0; auto.
+ intros. apply H3.
* red; intros.
apply image_axiom2 in H6.
destruct H6 as [q [??]].
apply H4.
rewrite H7. apply image_axiom1. rewrite H0; auto.
* auto.
- destruct H3.
exists x1. split; auto.
red; intros.
apply image_axiom2 in H5.
destruct H5 as [a [??]].
apply H4.
rewrite H6.
apply image_axiom1.
rewrite H0. auto.
}
destruct (finsubset_dec' (A × B) Hdec P) with R; auto.
- unfold P. intro G.
destruct (inh_dec _ hf G).
set (Q x0 := minimal_upper_bound x0 (image π₁ G) -> exists y, (x0,y) ∈ R /\ upper_bound y (image π₂ G)).
destruct (finset_find_dec' A Q) with (mub_closure HAplt (image π₁ G)).
+ unfold Q; intros.
destruct H1 as [q [??]].
revert H2. apply minimal_upper_bound_ok; auto.
exists q; split; auto.
apply member_eq with (x,q); auto.
destruct H0; split; split; auto.
+ intro x.
unfold Q.
destruct (mub_finset_dec hf A HAeff HAplt (image π₁ G) x).
* apply inh_image; auto.
* destruct (finset_find_dec (A×B) (fun q => fst q ≈ x /\ upper_bound (snd q) (image π₂ G))) with R.
** intros. destruct H1; split.
*** rewrite <- H1.
destruct H0 as [[??][??]]; auto.
*** revert H2. apply upper_bound_ok.
destruct H0 as [[??][??]]; split; auto.
** intros [a b]. simpl.
destruct (eff_ord_dec A HAeff a x); [
destruct (eff_ord_dec A HAeff x a); [
destruct (upper_bound_dec B HBeff (image π₂ G) b) |] |].
*** left. split; auto.
*** right. intros [??]. apply n; auto.
*** right. intros [[??]?]. apply n; auto.
*** right. intros [[??]?]. apply n; auto.
** destruct s as [z [??]].
destruct H1.
destruct z as [a b]. simpl in H1, H2.
left. intros.
exists b. split; auto.
apply member_eq with (a,b); auto.
destruct H1; split; split; auto.
** right; intro.
destruct H0; auto.
destruct H0.
apply n in H0.
apply H0.
split; auto.
* left. intro. elim n; auto.
+ destruct s as [z [??]].
right. intro.
apply H1. intro.
apply H2; auto.
+ left.
intros HG. intros.
apply q; auto.
apply (mub_clos_mub HAplt (image π₁ G)) with (image π₁ G); auto.
* apply inh_image; auto.
* apply mub_clos_incl.
+ left. intro. contradiction.
- left. split; auto.
intros.
apply p; auto.
- right.
destruct e as [G [??]].
destruct (inh_dec _ hf G).
+ exists G. split; auto.
set (Q x0 := minimal_upper_bound x0 (image π₁ G) -> exists y, (x0,y) ∈ R /\ upper_bound y (image π₂ G)).
destruct (finset_find_dec' A Q) with (mub_closure HAplt (image π₁ G)).
* unfold Q; intros.
destruct H3 as [q [??]].
{ revert H4. apply minimal_upper_bound_ok; auto. }
exists q; split; auto.
apply member_eq with (x,q); auto.
destruct H2; split; split; auto.
* intro x.
unfold Q.
destruct (mub_finset_dec hf A HAeff HAplt (image π₁ G) x).
** apply inh_image; auto.
** destruct (finset_find_dec (A×B) (fun q => fst q ≈ x /\ upper_bound (snd q) (image π₂ G))) with R.
*** intros. destruct H3; split.
rewrite <- H3.
destruct H2 as [[??][??]]; auto.
revert H4. apply upper_bound_ok.
destruct H2 as [[??][??]]; split; auto.
*** intros [a b]. simpl.
destruct (eff_ord_dec A HAeff a x); [
destruct (eff_ord_dec A HAeff x a); [
destruct (upper_bound_dec B HBeff (image π₂ G) b)
|] |].
**** left. split; auto.
**** right. intros [??]. apply n; auto.
**** right. intros [[??]?]. apply n; auto.
**** right. intros [[??]?]. apply n; auto.
*** destruct s as [z [??]].
destruct H3.
destruct z as [a b]. simpl in H3, H4.
left. intros.
exists b. split; auto.
apply member_eq with (a,b); auto.
destruct H3; split; split; auto.
*** right; intro.
destruct H2; auto.
destruct H2.
apply n in H2.
apply H2.
split; auto.
** left. intro. elim n; auto.
* destruct s as [z [??]].
destruct (mub_finset_dec hf A HAeff HAplt (image π₁ G) z).
** apply inh_image; auto.
** split; auto.
exists z. split; auto.
repeat intro.
apply H3. red. intros.
exists q. split; auto.
** elim H3; red; intros. elim n; auto.
* elim H1. red; intros.
apply q; auto.
apply (mub_clos_mub HAplt (image π₁ G)) with (image π₁ G); auto.
** apply inh_image; auto.
** apply mub_clos_incl.
+ elim H1.
red; intro. contradiction.
Qed.
Lemma is_joinable_rel_dec hf
(A B:preord)
(HAeff : effective_order A)
(HBeff : effective_order B)
(HAplt : plotkin_order hf A)
(R:finset (A × B)) (Rinh : inh hf R) :
{ is_joinable_relation hf R } + { ~is_joinable_relation hf R }.
Proof.
destruct (is_joinable_rel_dec0 hf A B HAeff HBeff HAplt R); auto.
right; intro.
destruct H.
destruct e as [G [? [? [z [??]]]]].
destruct (H0 G H1 H2 z H3) as [y [??]].
apply (H4 y); auto.
Qed.
Lemma is_joinable_rel_dec' hf
(A B:preord)
(HAeff : effective_order A)
(HBeff : effective_order B)
(HAplt : plotkin_order hf A)
(R:finset (A × B)) :
{ is_joinable_relation hf R } + { ~is_joinable_relation hf R }.
Proof.
destruct (inh_dec _ hf R).
- destruct (is_joinable_rel_dec0 hf A B HAeff HBeff HAplt R); auto.
right; intro.
destruct H.
destruct e as [G [? [? [z [??]]]]].
destruct (H0 G H1 H2 z H3) as [y [??]].
apply (H4 y); auto.
- right; intros [??]. contradiction.
Qed.
(** Joinable relations form an effective preorder. We can enumerate
the joinable relations by first enumerating all the finite
relations between A and B and selecting those that are joinable.
*)
Program Definition joinable_rel_effective hf
(A B:preord)
(HAeff : effective_order A)
(HBeff : effective_order B)
(HAplt : plotkin_order hf A) :
effective_order (joinable_rel_order hf A B)
:= EffectiveOrder _ _ _ _.
Next Obligation.
intros.
apply joinable_ord_dec; auto.
Qed.
Next Obligation.
intros.
set (X := finsubsets (A×B) (eprod (eff_enum A HAeff) (eff_enum B HBeff))).
exact (fun n => match X n with
| None => None
| Some a =>
match is_joinable_rel_dec' hf A B HAeff HBeff HAplt a with
| left H => Some (exist _ a H)
| right _ => None
end
end).
Defined.
Next Obligation.
intros.
unfold joinable_rel_effective_obligation_2.
red. simpl.
destruct x as [x H].
assert (x ∈ finsubsets (A×B) (eprod (eff_enum A HAeff) (eff_enum B HBeff))).
{ apply finsubsets_complete.
red; intros.
destruct a as [a b].
apply eprod_elem.
split; apply eff_complete.
}
destruct H0 as [n ?].
exists n.
destruct (finsubsets (A×B) (eprod (eff_enum A HAeff) (eff_enum B HBeff)) n); auto.
destruct (is_joinable_rel_dec' hf A B HAeff HBeff HAplt c).
- destruct H0.
split; red; simpl; auto.
+ red. simpl; intros.
apply H0 in H2.
exists a. exists b. split; auto.
+ red; simpl; intros.
apply H1 in H2.
exists a. exists b. auto.
- apply n0; auto.
split.
+ destruct H.
apply inh_eq with x; auto.
+ intros.
destruct H.
destruct (H3 G) with x0 as [y [??]]; auto.
* rewrite H0; auto.
* exists y. split; auto.
rewrite <- H0; auto.
Qed.
(** In this section, we show that, given an arbitrary finite relation M
and an approximable relation R above M, we can "swell up" M into
a joinable relation M' where M ⊆ M' ⊆ R.
This is done by a recursive procedure that, at each step, asks
if M is joinable. If so, we are done. If not, there is some
subset of M lacking the necessary upper bound. We add a new
pair to M to fix the deficiency and recurse. The termination
of this procedure relies critically on the fact that the
MUB closure of a finite set is finite in Plotkin orders. Thus, at
some point we will run out of elements that could possibly be added
to M, and the procedure must terminate giving the desired joinable
relation.
Then we can then show that the set of joinable relations below some
approximable relation is directed; this will be important later
for the proof that currying is an approximable relation.
*)
Section directed_joinables.
Variable hf:bool.
Variables A B:preord.
Variable HAeff : effective_order A.
Variable HAplt : plotkin_order hf A.
Variable HBeff : effective_order B.
Variable HBplt : plotkin_order hf B.
Let OD := OrdDec A (eff_ord_dec A HAeff).
Variable R:erel A B.
Hypothesis HR : forall x x' y y', x ≤ x' -> y' ≤ y -> (x,y) ∈ R -> (x',y') ∈ R.
Hypothesis HRdir : forall a, directed hf (erel_image A B OD R a).
Section swell.
Variable RS : finset (A×B).
Hypothesis RSinh : inh hf RS.
Let RS' := finprod
(mub_closure HAplt (image π₁ RS))
(mub_closure HBplt (image π₂ RS)).
Section swell_relation.
Variable G:finset (A×B).
Hypothesis HG : G ⊆ R.
Hypothesis HG' : G ⊆ RS'.
Variable G0:finset (A×B).
Hypothesis HG0 : G0 ⊆ G.
Hypothesis G0inh : inh hf G0.
Variable z:A.
Hypothesis Hz : minimal_upper_bound z (image π₁ G0).
Lemma swell_row :
exists q,
(z,q) ∈ RS' /\ (z,q) ∈ R /\
upper_bound q (image π₂ G0).
Proof.
destruct (HRdir z (image π₂ G0)) as [q [??]].
- apply inh_image; auto.
- red; intros.
apply image_axiom2 in H.
destruct H as [q[??]].
apply erel_image_elem.
destruct q as [x y]. simpl in H0.
apply HR with x y.
+ apply Hz. change x with (π₁#((x,y):A×B)).
apply image_axiom1. auto.
+ destruct H0; auto.
+ apply HG; auto.
- apply erel_image_elem in H0.
destruct (mub_complete HBplt (image π₂ G0) q) as [q0 [??]]; auto.
{ apply inh_image; auto. }
exists q0. split.
unfold RS'. apply finprod_elem. split.
+ apply (mub_clos_mub HAplt (image π₁ RS)) with (image π₁ G0); auto.
* apply inh_image. auto.
* red; intros.
apply image_axiom2 in H3. destruct H3 as [y [??]].
apply HG0 in H3.
apply HG' in H3.
unfold RS' in H3.
destruct y as [x y].
apply finprod_elem in H3.
rewrite H4. simpl.
destruct H3; auto.
+ apply (mub_clos_mub HBplt (image π₂ RS)) with (image π₂ G0); auto.
* apply inh_image. auto.
* red; intros.
apply image_axiom2 in H3. destruct H3 as [y [??]].
apply HG0 in H3.
apply HG' in H3.
unfold RS' in H3.
destruct y as [x y].
apply finprod_elem in H3.
rewrite H4. simpl.
destruct H3; auto.
+ split.
* apply HR with z q; auto.
* destruct H1; auto.
Qed.
Variable XS : finset (A×B).
Hypothesis HXS : forall x, x ∈ XS <-> x ∈ RS' /\ x ∉ G.
Let ODAB := OrdDec (A×B) (eff_ord_dec _ (effective_prod HAeff HBeff)).
Hypothesis noub : forall q, (z,q) ∈ G -> ~upper_bound q (image π₂ G0).
Lemma swell_relation :
exists G':finset (A×B), exists XS':finset (A×B),
length XS' < length XS /\
G ⊆ G' /\ G' ⊆ R /\ G' ⊆ RS' /\
(forall x, x ∈ XS' <-> x ∈ RS' /\ x ∉ G').
Proof.
destruct swell_row as [r [?[??]]].
exists ((z,r)::G)%list.
exists (finset_remove ODAB XS (z,r)).
repeat split.
- apply finset_remove_length2.
apply HXS. split; auto.
red; intro. apply (noub r); auto.
- red; intros. apply cons_elem; auto.
- red; intros.
apply cons_elem in H2. destruct H2.
rewrite H2; auto. apply HG; auto.
- red; intros.
apply cons_elem in H2. destruct H2.
rewrite H2; auto. apply HG'; auto.
- apply finset_remove_elem in H2.
destruct H2.
apply HXS in H2. destruct H2.
auto.
- intro.
apply finset_remove_elem in H2.
destruct H2.
apply HXS in H2. destruct H2.
apply cons_elem in H3. destruct H3.
apply H4; auto.
apply H5; auto.
- intros.
apply finset_remove_elem.
destruct H2.
split.
+ apply HXS.
split; auto.
intro. apply H3. apply cons_elem. auto.
+ intro. apply H3. apply cons_elem. auto.
Qed.
End swell_relation.
Lemma swell_inductive_step
(G:finset (A×B))
(Ginh : inh hf G)
(HG : G ⊆ R) (HG': G ⊆ RS')
(XS:finset (A×B))
(HXS: forall x, x ∈ XS <-> x ∈ RS' /\ x ∉ G) :
is_joinable_relation hf G \/
exists G':finset (A×B), exists XS':finset (A×B),
length XS' < length XS /\
G ⊆ G' /\ G' ⊆ R /\ G' ⊆ RS' /\
(forall x, x ∈ XS' <-> x ∈ RS' /\ x ∉ G').
Proof.
destruct (is_joinable_rel_dec0 hf A B HAeff HBeff HAplt G); auto.
destruct e as [G0 [? [? [z [??]]]]].
destruct swell_relation with G G0 z XS
as [G' [XS' [?[?[?[??]]]]]]; auto.
right. exists G'. exists XS'. intuition.
Qed.
Lemma swell_aux (n:nat) : forall
(G:finset (A×B))
(Ginh : inh hf G)
(HG : G ⊆ R)
(HG': G ⊆ RS')
(XS : finset (A×B))
(HXS: forall x, x ∈ XS <-> x ∈ RS' /\ x ∉ G)
(Hn : n = length XS),
exists G':finset (A×B),
G ⊆ G' /\ G' ⊆ R /\ is_joinable_relation hf G'.
Proof.
induction n using (well_founded_induction Wf_nat.lt_wf); intros.
destruct (swell_inductive_step G Ginh HG HG' XS HXS).
- exists G. split; auto. red; auto.
- destruct H0 as [G' [XS' [?[?[?[??]]]]]].
destruct (H (length XS')) with G' XS' as [G'' [?[??]]]; auto.
+ rewrite Hn. auto.
+ apply inh_sub with G; auto.
+ exists G''. split; auto.
red; intros. apply H5. apply H1. auto.
Qed.
Hypothesis HRS : RS ⊆ R.
Lemma swell : exists G:finset (A×B),
RS ⊆ G /\ G ⊆ R /\ is_joinable_relation hf G.
Proof.
assert (HRdec : forall x, { x ∉ RS }+{ ~x ∉ RS }).
{ intro. destruct (eff_in_dec (effective_prod HAeff HBeff) RS x).
- right; auto.
- left; auto.
}
set (XS := finsubset _ (fun x => x ∉ RS) HRdec RS').
destruct (swell_aux (length XS)) with RS XS as [G [?[??]]]; auto.
- red; intros.
unfold RS'.
destruct a. apply finprod_elem.
split.
+ apply (mub_clos_incl HAplt (image π₁ RS)).
change c with (π₁#((c,c0):A×B)).
apply image_axiom1. auto.
+ apply (mub_clos_incl HBplt (image π₂ RS)).
change c0 with (π₂#((c,c0):A×B)).
apply image_axiom1. auto.
- split; intros.
+ unfold XS in H.
apply finsubset_elem in H; auto.
intros. rewrite <- H1. auto.
+ unfold XS.
apply finsubset_elem; auto.
intros. rewrite <- H0. auto.
- exists G. auto.
Qed.
End swell.
Variable M:finset (joinable_rel_order hf A B).
Hypothesis Minh : inh hf M.
Hypothesis HM : forall S, S ∈ M -> proj1_sig S ⊆ R.
Let RS := concat _ (List.map (fun R => proj1_sig R) M) : finset (A×B).
Lemma RS_elem :
forall a, a ∈ RS -> (exists S, S ∈ M /\ a ∈ proj1_sig S).
Proof.
intro a.
unfold RS.
clear -HM.
induction M; simpl in *; intros.
- apply nil_elem in H. elim H.
- apply app_elem in H. destruct H.
+ exists a0. split; auto.
exists a0; split; simpl; auto.
+ destruct IHc.
* intros. apply HM.
apply cons_elem; auto.
* auto.
* exists x.
destruct H0.
split; auto.
apply cons_elem; auto.
Qed.
Lemma RS_elem' : forall S a, List.In S M -> a ∈ proj1_sig S -> a ∈ RS.
Proof.
intros. unfold RS.
clear -M HM H H0.
induction M; simpl.
- elim H.
- destruct H.
+ subst a0.
apply app_elem. left. auto.
+ apply app_elem. right.
apply IHc; auto.
intros. apply HM; auto.
apply cons_elem; auto.
Qed.
Theorem directed_joinables :
exists Q, upper_bound Q M /\ proj1_sig Q ⊆ R.
Proof.
destruct (swell RS) as [G [?[??]]]; auto.
- generalize RS_elem'.
revert M HM Minh RS.
case_eq hf; intros; auto.
destruct Minh as [S ?].
assert (inh true (proj1_sig S)).
{ destruct S. destruct i. auto. }
destruct H1 as [q [??]].
destruct H2.
destruct x.
red.
destruct H3.
destruct (H3 c c0) as [m [n [?[??]]]]; auto.
exists (m,n).
apply H0 with q; auto.
- red; intros.
apply RS_elem in H.
destruct H as [S [??]].
apply (HM S); auto.
- exists (exist _ G H1). split.
+ red; intros.
destruct H2 as [q [??]].
rewrite H3.
red; simpl; intros.
red; simpl; intros.
exists a. exists b. split; auto.
apply H.
apply RS_elem' with q; auto.
+ simpl. auto.
Qed.
End directed_joinables.
Section joinable_plt.
Variable hf:bool.
Variables A B:preord.
Variable HAeff : effective_order A.
Variable HAplt : plotkin_order hf A.
Variable HBeff : effective_order B.
Variable HBplt : plotkin_order hf B.
(** The intersection of an approximable relation with the cartesian
product of MUB closed sets is a joinable relation.
*)
Lemma intersect_approx
(R:A×B -> Prop)
(Hdec : forall x, {R x}+{~R x})
(HR : forall a a' b b', a ≤ a' -> b' ≤ b -> R (a,b) -> R (a',b'))
(HRdir : forall a (M:finset B) (HMinh:inh hf M),
(forall b, b ∈ M -> R (a,b)) ->
exists z, R (a,z) /\ upper_bound z M)
(P:finset A) (Q:finset B)
(Hinh : if hf then exists x y, x ∈ P /\ y ∈ Q /\ R (x,y) else True)
(HP:mub_closed hf A P) (HQ:mub_closed hf B Q) :
is_joinable_relation hf (finsubset (A×B) R Hdec (finprod P Q)).
Proof.
assert (HR2 : forall x y : Preord_Eq (A × B), x ≈ y -> R x -> R y).
{ intros.
destruct x; destruct y.
apply HR with c c0; auto.
destruct H as [[??][??]]; auto.
destruct H as [[??][??]]; auto.
}
split.
- destruct hf; auto.
destruct Hinh as [x [y [?[??]]]].
red.
exists (x,y).
apply finsubset_elem; auto.
split; auto.
apply finprod_elem. split; auto.
- repeat intro.
assert (x ∈ P).
{ apply HP with (image π₁ G); auto.
apply inh_image; auto.
red; intros.
apply image_axiom2 in H1. destruct H1 as [y [??]].
apply H in H1.
apply finsubset_elem in H1.
destruct H1.
destruct y.
apply finprod_elem in H1.
destruct H1. simpl in H2.
rewrite <- H2 in H1. auto.
intros.
destruct x0; destruct y0.
eapply HR. 3: apply H5.
destruct H4 as [[??][??]]; auto.
destruct H4 as [[??][??]]; auto.
}
destruct (HRdir x (image π₂ G)).
+ apply inh_image; auto.
+ intros.
apply image_axiom2 in H2.
destruct H2 as [q [??]].
generalize H2; intro H2'.
apply H in H2.
apply finsubset_elem in H2.
destruct H2.
destruct q.
simpl in H3.
apply HR with c c0; auto.
apply H0.
change c with (π₁#((c,c0):A×B)).
apply image_axiom1. auto.
{ intros.
destruct x0; destruct y.
apply HR with c c0; auto.
destruct H5 as [[??][??]]; auto.
destruct H5 as [[??][??]]; auto.
}
+ destruct H2.
destruct (mub_complete HBplt (image π₂ G) x0) as [y [??]]; auto.
* apply inh_image; auto.
* exists y. split; auto.
apply finsubset_elem.
intros.
destruct x1. destruct y0.
apply HR with c c0; auto.
destruct H6 as [[??][??]]; auto.
destruct H6 as [[??][??]]; auto.
split.
** apply finprod_elem; split; auto.
apply (HQ (image π₂ G)); auto.
apply inh_image; auto.
red; intros.
apply image_axiom2 in H6.
destruct H6 as [q [??]].
apply H in H6.
apply finsubset_elem in H6.
destruct H6.
destruct q.
apply finprod_elem in H6.
destruct H6.
rewrite H7; simpl; auto.
intros.
destruct x1; destruct y0.
apply HR with c c0; auto.
destruct H9 as [[??][??]]; auto.
destruct H9 as [[??][??]]; auto.
** apply HR with x x0; auto.
** destruct H4; auto.
Qed.
Definition mkrel (R:joinable_relation hf A B) (x:A×B) :=
exists a b, (a,b) ∈ proj1_sig R /\ a ≤ fst x /\ snd x ≤ b.
Lemma mkrel_dec R : forall x, {mkrel R x}+{~mkrel R x}.
Proof.
intros.
unfold mkrel.
destruct (finset_find_dec (A×B) (fun y => fst y ≤ fst x /\ snd x ≤ snd y))
with (proj1_sig R).
- intros.
destruct H. destruct H0.
destruct H; destruct H1.
split; eauto.
- intros.
destruct (eff_ord_dec A HAeff (fst x0) (fst x)); [
destruct (eff_ord_dec B HBeff (snd x) (snd x0)) |].
+ left. split; auto.
+ right; intros [??]; apply n; auto.
+ right; intros [??]; apply n; auto.
- destruct s.
left. exists (fst x0). exists (snd x0). auto.
- right; intros [a [b [?[??]]]].
apply (n (a,b)); auto.
Qed.
Lemma mkrel_dir R :
forall a (M:finset B) (HMinh:inh hf M), (forall b, b ∈ M -> mkrel R (a,b)) ->
exists z, mkrel R (a,z) /\ upper_bound z M.
Proof.
intros.
set (P q := mkrel R q /\ snd q ∈ M /\ fst q ≤ a).
assert (POK : forall x y, x ≈ y -> P x -> P y).
{ clear. unfold P; intros.
destruct H as [[??][??]].
intuition.
- destruct H4 as [m [n [?[??]]]].
exists m. exists n. intuition eauto.
- apply member_eq with (snd x); auto.
- eauto.
}
assert (Pdec : forall q, { P q }+{ ~P q }).
{ intro q.
destruct (mkrel_dec R q); [
destruct (eff_in_dec HBeff M (snd q)); [
destruct (eff_ord_dec A HAeff (fst q) a) |] |].
- left. split; auto.
- right; intros [?[??]]; apply n; auto.
- right; intros [?[??]]; apply n; auto.
- right; intros [?[??]]; apply n; auto.
}
set (R' := finsubset (A×B) (fun q => fst q ≤ a)
(fun q => eff_ord_dec A HAeff (fst q) a) (proj1_sig R)).
assert (R'inh : inh hf R').
{ subst R'.
clear -H HMinh.
unfold mkrel in H.
destruct hf; auto.
destruct R.
destruct i. simpl.
destruct HMinh.
destruct (H x0) as [p [q [??]]]; auto.
simpl in *.
exists (p,q).
apply finsubset_elem; auto.
- intros.
transitivity (fst x1); auto.
destruct H3 as [[??][??]]; auto.
- split; auto.
simpl. destruct H2; auto.
}
destruct (mub_complete HAplt (image π₁ R') a).
- apply inh_image. auto.
- red; intros.
apply image_axiom2 in H0.
destruct H0. destruct H0.
unfold R' in H0.
apply finsubset_elem in H0.
destruct H0.
rewrite H1; auto.
intros.
transitivity (fst x1); auto.
destruct H3 as [[??][??]]; auto.
- destruct H0.
assert (is_joinable_relation hf (proj1_sig R)) by apply proj2_sig.
destruct H2 as [_ ?].
destruct (H2 R') with x; auto.
+ red; intros.
unfold R' in H3.
apply finsubset_elem in H3.
destruct H3; auto.
intros.
transitivity (fst x0); auto.
destruct H5 as [[??][??]]; auto.
+ destruct H3.
exists x0. split.
exists x. exists x0.
split; auto.
red; intros.
generalize (H x1 H5). intros.
destruct H6 as [p [q [?[??]]]].
simpl in H7, H8.
transitivity q; auto.
apply H4.
change q with (π₂#((p,q):A×B)).
apply image_axiom1.
unfold R'.
apply finsubset_elem; auto.
intros.
transitivity (fst x2); auto.
destruct H9 as [[??][??]]; auto.
Qed.
Lemma mkrel_ord R : forall a a' b b', a ≤ a' -> b' ≤ b -> mkrel R (a,b) -> mkrel R (a',b').
Proof.
unfold mkrel; intros.
destruct H1 as [p [q [?[??]]]].
exists p. exists q. split; auto. split.
- transitivity a; auto.
- transitivity b; auto.
Qed.
Lemma mkrel_mono : forall R S, R ≤ S ->
forall x, mkrel R x -> mkrel S x.
Proof.
unfold mkrel; intros.
destruct H0 as [a [b [?[??]]]].
destruct (H a b) as [a' [b' [?[??]]]]; auto.
exists a'. exists b'. repeat split; auto.
- transitivity a; auto.
- transitivity b; auto.
Qed.
Lemma mkrel_mono' : forall (R S:joinable_relation hf A B),
(forall x, x ∈ proj1_sig R -> mkrel S x) -> R ≤ S.
Proof.
repeat intro.
destruct (H (a,b)) as [a' [b' [?[??]]]]. auto.
exists a'. exists b'. split; auto.
Qed.
(** Joinable relations have normal sets. In particular, the set of all
joinable relations that are subsets of the cartesian product of two
MUB-closed sets is a normal set.
*)
Section join_rel_normal.
Variable (P:finset A).
Variable (Q:finset B).
Variable (HP:mub_closed hf A P).
Variable (HQ:mub_closed hf B Q).
Fixpoint select_jrels (l:finset (finset (A×B))) : finset (joinable_rel_order hf A B) :=
match l with
| nil => nil
| cons R l' =>
match is_joinable_rel_dec' hf A B HAeff HBeff HAplt R with
| left H => cons (exist _ R H) (select_jrels l')
| right _ => select_jrels l'
end
end.
Lemma select_jrels_elem1 : forall R (H:is_joinable_relation hf R) XS,
R ∈ XS -> (exist _ R H : joinable_rel_order hf A B) ∈ select_jrels XS.
Proof.
induction XS; simpl; intros.
- apply nil_elem in H0. elim H0.
- destruct (is_joinable_rel_dec' hf A B HAeff HBeff HAplt a).
+ apply cons_elem in H0. destruct H0.
* apply cons_elem. left.
split.
** red; simpl; intros.
red; simpl; intros.
exists a0. exists b. split; auto.
destruct H0. apply H0; auto.
** red; simpl; intros.
red; simpl; intros.
exists a0. exists b. split; auto.
destruct H0. apply H2; auto.
* apply cons_elem. right.
apply IHXS. auto.
+ apply cons_elem in H0.
destruct H0.
* elim n.
destruct H.
split.
** apply inh_eq with R; auto.
** intros.
destruct (e G) with x; auto.