Require Export VST.veric.base.
Local Open Scope pred.
Notation ghost_approx m := (ghost_fmap (approx (level m)) (approx (level m))).
(* Ownership construction based on "Iris from the ground up", Jung et al. *)
Program Definition ghost_is g: pred rmap :=
fun m => ghost_of m = ghost_approx m g.
Next Obligation.
intros ???? Hg.
rewrite (age1_ghost_of _ _ H), Hg.
pose proof (age_level _ _ H).
rewrite ghost_fmap_fmap, approx_oo_approx', approx'_oo_approx by omega; eauto.
Qed.
Definition Own g: pred rmap := allp noat && ghost_is g.
Fixpoint make_join (a c : ghost) : ghost :=
match a, c with
| nil, _ => c
| _, nil => nil
| None :: a', x :: c' => x :: make_join a' c'
| _ :: a', None :: c' => None :: make_join a' c'
| Some (ga, pa) :: a', Some (gc, _) :: c' => Some (gc, pa) :: make_join a' c'
end.
Program Definition bupd (P: pred rmap): pred rmap :=
fun m => forall c, joins (ghost_of m) (ghost_approx m c) ->
exists b, joins b (ghost_approx m c) /\
exists m', level m' = level m /\ resource_at m' = resource_at m /\ ghost_of m' = b /\ P m'.
Next Obligation.
Definition ghost_fp_update_ND a B :=
forall n c, joins (ghost_fmap (approx n) (approx n) a) c ->
exists b, B b /\ joins (ghost_fmap (approx n) (approx n) b) c.
Definition ghost_fp_update (a b : ghost) :=
forall n c, joins (ghost_fmap (approx n) (approx n) a) c ->
joins (ghost_fmap (approx n) (approx n) b) c.
Instance ghost_fp_update_preorder: RelationClasses.PreOrder ghost_fp_update.
Definition singleton {A} k (x : A) : list (option A) := repeat None k ++ Some x :: nil.
Definition gname := nat.
Definition own {RA: Ghost} (n: gname) (a: G) (pp: preds) :=
EX v : _, Own (singleton n (existT _ RA (exist _ a v), pp)).
Definition list_set {A} (m : list (option A)) k v : list (option A) :=
firstn k m ++ repeat None (k - length m) ++ Some v :: skipn (S k) m.

Lemma ghost_valid_2: forall {RA: Ghost} g a1 a2 pp,
own g a1 pp * own g a2 pp |-- !!ghost.valid_2 a1 a2.