End Snapshot.
Definition pos_PCM := ghost_PCM.pos_PCM.
Definition ref_PCM := ghost_PCM.ref_PCM.
Notation completable := ghost_PCM.completable.
Section Reference.
Context {P : Ghost}.
Definition ghost_reference g a := own(RA := ref_PCM P) g (None, Some a) NoneP.
Definition ghost_part g sh a := own(RA := ref_PCM P) g (Some (sh, a), None) NoneP.
Definition ghost_part_ref g sh a r :=
own(RA := ref_PCM P) g (Some (sh, a), Some r) NoneP.

Lemma ref_sub : forall g sh a b pp,
own(RA := ref_PCM P) g (Some (sh, a), None) pp * own(RA := ref_PCM P) g (None, Some b) pp |--
!!(if eq_dec sh Tsh then a = b else exists x, join a x b).