Section WellOrderConstruction.
Variable T:Type.
Definition restriction_relation (R:relation T) (S:Ensemble T) :
relation ({z:T | In S z}) :=
fun (x y:{z:T | In S z}) => R (proj1_sig x) (proj1_sig y).
Record partial_WO : Type := {
pwo_S: Ensemble T;
pwo_R: relation T;
pwo_R_lives_on_S: forall (x y:T), pwo_R x y -> In pwo_S x /\ In pwo_S y;
pwo_wo: well_order (restriction_relation pwo_R pwo_S)
}.
(* the last condition below says that S WO1 is a downward closed subset of S WO2 *)
Record partial_WO_ord (WO1 WO2:partial_WO) : Prop := {
pwo_S_incl: Included (pwo_S WO1) (pwo_S WO2);
pwo_restriction: forall x y:T, In (pwo_S WO1) x -> In (pwo_S WO1) y ->
(pwo_R WO1 x y <-> pwo_R WO2 x y);
pwo_downward_closed: forall x y:T, In (pwo_S WO1) y -> In (pwo_S WO2) x ->
pwo_R WO2 x y -> In (pwo_S WO1) x
}.
Definition partial_WO_chain_ub: forall C:Ensemble partial_WO,
chain partial_WO_ord C -> partial_WO.
refine (fun C H => let US := [ x:T | exists WO:partial_WO,
In C WO /\ In (pwo_S WO) x ] in
let UR := fun x y:T => exists WO:partial_WO, In C WO /\
pwo_R WO x y in
Build_partial_WO US UR _ _).
- intros.
unfold UR in H0.
destruct H0.
destruct H0.
split.
+ constructor.
exists x0.
split.
* assumption.
* pose proof (pwo_R_lives_on_S x0 x y).
tauto.
+ constructor.
exists x0.
split.
* assumption.
* pose proof (pwo_R_lives_on_S x0 x y).
tauto.
- constructor.
+ assert (forall (WO:partial_WO) (x:{z:T | In (pwo_S WO) z}),
In C WO -> In US (proj1_sig x)).
{ intros.
constructor.
exists WO.
split.
- assumption.
- exact (proj2_sig x).
}
assert (forall (WO:partial_WO) (iC:In C WO) (x:{z:T | In (pwo_S WO) z}),
Acc (restriction_relation (pwo_R WO) (pwo_S WO)) x ->
Acc (restriction_relation UR US)
(exist _ (proj1_sig x) (H0 WO x iC))).
{ intros.
induction H1.
constructor.
intros.
destruct x as [x ix].
destruct y as [y iy].
unfold restriction_relation in H3.
simpl in H3.
assert (In (pwo_S WO) y).
{ destruct H3.
destruct H3.
pose proof (H WO x0 iC H3).
case H5.
- intro.
destruct H6.
apply pwo_downward_closed0 with x.
+ assumption.
+ pose proof (pwo_R_lives_on_S x0 y x).
tauto.
+ assumption.
- intro.
destruct H6.
apply pwo_S_incl0.
pose proof (pwo_R_lives_on_S x0 y x).
tauto.
}
pose proof (H2 (exist (In (pwo_S WO)) y H4)).
simpl in H5.
assert (iy = H0 WO (exist (In (pwo_S WO)) y H4) iC).
{ apply proof_irrelevance. }
rewrite <- H6 in H5.
apply H5.
unfold restriction_relation.
simpl.
destruct H3.
destruct H3.
pose proof (H WO x0 iC H3).
case H8.
- intros.
destruct H9.
apply <- pwo_restriction0.
+ assumption.
+ assumption.
+ assumption.
- intro.
destruct H9.
apply -> pwo_restriction0.
+ assumption.
+ pose proof (pwo_R_lives_on_S x0 y x).
tauto.
+ pose proof (pwo_R_lives_on_S x0 y x).
tauto.
}
red. intro.
destruct a.
inversion i.
destruct H2.
destruct H2.
pose proof (H1 x0 H2 (exist _ x H3)).
simpl in H4.
assert (i = H0 x0 (exist _ x H3) H2).
{ apply proof_irrelevance. }
rewrite <- H5 in H4.
apply H4.
apply (wo_well_founded (pwo_wo x0)).
+ unfold total_strict_order.
intros.
destruct x.
destruct y.
unfold restriction_relation.
simpl.
destruct i.
destruct e.
destruct a.
destruct i0.
destruct e.
destruct a.
case (H x1 x2 i i0).
* intro.
assert (In (pwo_S x2) x).
{ apply H0. assumption. }
case (wo_total_strict_order (pwo_wo x2) (exist _ x H1) (exist _ x0 i2)).
-- unfold restriction_relation.
simpl.
left.
exists x2.
tauto.
-- intro.
case H2.
++ right; left.
apply subset_eq_compat.
injection H3.
trivial.
++ unfold restriction_relation.
simpl.
right; right.
exists x2. tauto.
* intro.
assert (In (pwo_S x1) x0).
{ apply H0. assumption. }
case (wo_total_strict_order (pwo_wo x1) (exist _ x i1) (exist _ x0 H1)).
-- unfold restriction_relation.
simpl.
left.
exists x1.
tauto.
-- intro.
case H2.
++ right; left.
apply subset_eq_compat.
injection H3.
trivial.
++ unfold restriction_relation; simpl.
right; right.
exists x1.
tauto.
Defined.

Lemma partial_WO_chain_ub_correct: forall (C:Ensemble partial_WO)
(c:chain partial_WO_ord C), forall WO:partial_WO, In C WO ->
partial_WO_ord WO (partial_WO_chain_ub C c).