Require Export VST.veric.base.
Local Open Scope Z_scope.
Local Open Scope pred.
Program Definition kind_at (k: kind) (l: address) : pred rmap :=
fun m => exists rsh, exists sh, exists pp, m @ l = YES rsh sh k pp.
Next Obligation.
try intro; intros.
destruct H0 as [rsh [sh [pp ?]]].
generalize (eq_sym (resource_at_approx a l)); intro.
generalize (age1_resource_at a a'  H l (a@l) H1); intro.
rewrite H0 in H2. simpl in H2. eauto.
Qed.
Definition spec : Type :=  forall (sh: Share.t) (l: AV.address), pred rmap.
Program Definition yesat_raw (pp: preds) (k: kind)
(sh: share) (rsh: readable_share sh) (l: address) : pred rmap :=
fun phi => phi @ l = YES sh rsh k (preds_fmap (approx (level phi)) (approx (level phi)) pp).
Next Obligation.
try intro; intros.
apply (age1_resource_at a a' H l (YES sh rsh k pp) H0).
Qed.
Obligation Tactic := idtac.
Program Definition yesat (pp: preds) (k: kind) : spec :=
fun (sh: share) (l: AV.address) (m: rmap) =>
exists rsh, yesat_raw pp k sh rsh l m.
Next Obligation.
intros; intro; intros.
destruct H0 as [p ?]; exists p.
apply pred_hereditary with a; auto.
Qed.
Program Definition pureat (pp: preds) (k: kind) (l: AV.address): pred rmap :=
fun phi => phi @ l = PURE k (preds_fmap (approx (level phi)) (approx (level phi)) pp).
Next Obligation.
intros; intro; intros.
apply (age1_resource_at a a' H l (PURE k pp) H0).
Qed.
Ltac do_map_arg :=
match goal with |- ?a = ?b =>
match a with context [map ?x _] =>
match b with context [map ?y _] => replace y with x; auto end end end.
(* NOT TRUE, because the shares might not match Lemma extensionally_yesat: forall pp k sh l, extensionally (yesat pp k sh l) = yesat pp k sh l. *)
Program Definition noat (l: AV.address) : pred rmap :=
fun m => identity (m @ l).
Next Obligation.
intros; intro; intros.
apply (age1_resource_at_identity _ _ l H); auto.
Qed.
Definition resource_share (r: resource) : option share :=
match r with
| YES sh _ _ _ => Some sh
| NO sh _ => Some sh
| PURE _ _ => None
end.
Definition nonlock (r: resource) : Prop :=
match r with
| YES _ _ k _ => isVAL k \/ isFUN k
| NO _ _ => True
| PURE _ _ => False
end.
Program Definition nonlockat (l: AV.address): pred rmap :=
fun m => nonlock (m @ l).
Next Obligation.
intros; intro; intros.
unfold resource_share in *.
destruct (a @ l) eqn:?H.
+ rewrite (necR_NO a a' l _ n) in H1 by (constructor; auto).
rewrite H1; assumption.
+ eapply necR_YES in H1; [ | constructor; eassumption].
rewrite H1; assumption.
+ eapply necR_PURE in H1; [ | constructor; eassumption].
rewrite H1; assumption.
Qed.
Program Definition shareat (l: AV.address) (sh: share): pred rmap :=
fun m => resource_share (m @ l) = Some sh.
Next Obligation.
intros; intro; intros.
unfold resource_share in *.
destruct (a @ l) eqn:?H.
+ rewrite (necR_NO a a' l _ n) in H1 by (constructor; auto).
rewrite H1; assumption.
+ eapply necR_YES in H1; [ | constructor; eassumption].
rewrite H1; assumption.
+ inv H0.
Qed.
Program Definition jam {A} {JA: Join A}{PA: Perm_alg A}{agA: ageable A}{AgeA: Age_alg A} {B: Type} {S': B -> Prop} (S: forall l, {S' l}+{~ S' l}) (P Q: B -> pred A) : B -> pred A :=
fun (l: B) m => if S l then P l m else Q l m.
Next Obligation.
intros; intro; intros.
if_tac; try (eapply pred_hereditary; eauto).
Qed.
Definition is_resource_pred (p: address -> pred rmap) (q: resource -> address -> nat -> Prop) :=
forall l w, (p l) w = q (w @ l) l (level w).
Definition resource_stable (p: address -> pred rmap) :=
forall l w w', w @ l = w' @ l -> level w = level w' -> (p l) w = (p l) w'.
Program Definition noghost : pred rmap := fun m => identity (ghost_of m).
Next Obligation.

Lemma allp_jam_overlap: forall (P Q: address -> Prop) (p q: address -> pred rmap)
(P_DEC: forall l, {P l} + {~ P l})
(Q_DEC: forall l, {Q l} + {~ Q l}),
(exists resp, is_resource_pred p resp) ->
(exists resp, is_resource_pred q resp) ->
(forall l w1 w2, p l w1 -> q l w2 -> joins w1 w2 -> False) ->
(exists l, P l /\ Q l) ->
allp (jam P_DEC p noat) * allp (jam Q_DEC q noat) |-- FF.