Next Obligation.
repeat split; intros; intro H; decompose [and] H; congruence.
Defined.
Next Obligation.
repeat split; intros; intro H; decompose [and] H; congruence.
Defined.
Next Obligation.
repeat split; intros; intro H; decompose [and] H; congruence.
Defined.
Next Obligation.
repeat split; intros; intro H; decompose [and] H; congruence.
Defined.
Next Obligation.
exists (fun _ => None); auto.
intros [[]|]; constructor.
Defined.
Next Obligation.
constructor.
- intros [[]|] [[]|] [[]|] [[]|]; unfold join; simpl; auto; try contradiction; try congruence.
intros (? & ? & ? & ?) (? & ? & ? & ?); f_equal; f_equal; eapply join_eq; eauto.
- intros [[]|] [[]|] [[]|] [[]|] [[]|]; try contradiction; unfold join; simpl;
intros; decompose [and] H; decompose [and] H0;
repeat match goal with H : (_, _) = (_, _) |- _ => inv H end;
try solve [eexists (Some _); split; auto; simpl; auto]; try solve [exists None; split; auto].
+ destruct (join_assoc H2 H6) as (sh' & ? & ?), (join_assoc H5 H9) as (a' & ? & ?).
exists (Some (sh', a')); repeat (split; auto).
intro; subst.
apply join_Bot in H8 as []; auto.
+ exists (Some (s2, g2)); auto.
- intros [[]|] [[]|] [[]|]; try contradiction; unfold join; auto.
intros (? & ? & ? & ?); split; auto; split; auto; split; apply join_comm; auto.
- intros [[]|] [[]|] [[]|] [[]|]; try contradiction; auto.
intros (? & ? & ? & ?) (? & ? & ? & ?); f_equal; f_equal; eapply join_positivity; eauto.
Defined.
Next Obligation.
destruct a as [[]|]; destruct b as [[]|]; destruct c as [[]|]; try trivial;
unfold join in *; try contradiction.
- decompose [and] H; assumption.
- congruence.
Defined.
Definition completable {P : Ghost} (a: @G (pos_PCM P)) r := exists x, join a x (Some (Tsh, r)).
Local Obligation Tactic := idtac.
Global Program Instance ref_PCM (P : Ghost) : Ghost :=
{ valid a := valid (fst a) /\ match snd a with Some r => completable (fst a) r | None => True end;
Join_G a b c := @Join_G (pos_PCM P) (fst a) (fst b) (fst c) /\
@psepalg.Join_lower _ (psepalg.Join_discrete _) (snd a) (snd b) (snd c) }.
Next Obligation.
intros P; apply sepalg_generators.Sep_prod.
+ apply @Sep_G.
+ apply psepalg.Sep_option.
Defined.
Next Obligation.
intros P; apply sepalg_generators.Perm_prod.
+ apply @Perm_G.
+ apply psepalg.Perm_option.
Defined.
Next Obligation.
intros P ??? [? J] []; split; [eapply join_valid; eauto|].
destruct a, b, c; simpl in *; inv J; auto.
+ destruct o1; auto.
destruct H1.
destruct (join_assoc H H1) as (? & ? & ?); eexists; eauto.
+ inv H2.
Defined.
End Reference.
Program Instance exclusive_PCM A : Ghost := { valid a := True; Join_G := Join_lower (Join_discrete A) }.
Definition ext_PCM Z : Ghost := ref_PCM (exclusive_PCM Z).

Lemma valid_ext_both : forall {Z} (ora : Z), @valid (ext_PCM _) (Some (Tsh, Some ora), Some (Some ora)).
Definition ext_both {Z} (ora : Z) : {g : ghost.Ghost & {a : ghost.G | ghost.valid a}} :=
existT _ (ext_PCM _) (exist _ _ (valid_ext_both ora)).