Commit e1589fea authored by Ralf Jung's avatar Ralf Jung

sts_ownS_op

parent 74dc65a4
......@@ -26,6 +26,7 @@ Context {sts : stsT}.
(** ** Step relations *)
Inductive step : relation (state sts * tokens sts) :=
| Step s1 s2 T1 T2 :
(* TODO: This asks for ⊥ on sets: T1 ⊥ T2 := T1 ∩ T2 ⊆ ∅. *)
prim_step s1 s2 tok s1 T1 tok s2 T2
tok s1 T1 tok s2 T2 step (s1,T1) (s2,T2).
Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop :=
......@@ -51,10 +52,13 @@ Hint Extern 10 (_ ∈ _) => solve_elem_of : sts.
Hint Extern 10 (_ _) => solve_elem_of : sts.
(** ** Setoids *)
Instance framestep_proper' : Proper (() ==> (=) ==> (=) ==> impl) frame_step.
Proof. intros ?? HT ?? <- ?? <-; destruct 1; econstructor; eauto with sts. Qed.
Instance framestep_mono : Proper (flip () ==> (=) ==> (=) ==> impl) frame_step.
Proof.
intros ?? HT ?? <- ?? <-; destruct 1; econstructor;
eauto with sts; solve_elem_of.
Qed.
Global Instance framestep_proper : Proper (() ==> (=) ==> (=) ==> iff) frame_step.
Proof. by intros ?? [??] ??????; split; apply framestep_proper'. Qed.
Proof. by intros ?? [??] ??????; split; apply framestep_mono. Qed.
Instance closed_proper' : Proper (() ==> () ==> impl) closed.
Proof.
intros ?? HT ?? HS; destruct 1;
......@@ -328,6 +332,25 @@ Lemma sts_op_auth_frag_up s T :
tok s T sts_auth s sts_frag_up s T sts_auth s T.
Proof. intros; apply sts_op_auth_frag; auto using elem_of_up, closed_up. Qed.
Lemma sts_op_frag S1 S2 T1 T2 :
T1 T2 sts.closed S1 T1 sts.closed S2 T2
sts_frag (S1 S2) (T1 T2) sts_frag S1 T1 sts_frag S2 T2.
Proof.
(* Somehow I feel like a very similar proof muts have happened above, when
proving the DRA axioms. After all, we are just reflecting the operation here. *)
intros HT HS1 HS2; split; [split|constructor; solve_elem_of]; simpl.
- intros; split_ands; try done; []. constructor; last solve_elem_of.
by eapply closed_ne.
- intros (_ & _ & H). inversion_clear H. constructor; first done.
+ move=>s /elem_of_intersection
[/(closed_disjoint _ _ HS1) Hs1 /(closed_disjoint _ _ HS2) Hs2].
solve_elem_of +Hs1 Hs2.
+ move=> s1 s2 /elem_of_intersection [Hs1 Hs2] Hstep.
apply elem_of_intersection.
split; eapply closed_step; eauto; [|]; eapply framestep_mono, Hstep;
done || solve_elem_of.
Qed.
(** Frame preserving updates *)
Lemma sts_update_auth s1 s2 T1 T2 :
step (s1,T1) (s2,T2) sts_auth s1 T1 ~~> sts_auth s2 T2.
......
......@@ -63,6 +63,14 @@ Section sts.
sts_own γ s T pvs E E (sts_ownS γ S T).
Proof. intros. by apply own_update, sts_update_frag_up. Qed.
Lemma sts_ownS_op γ S1 S2 T1 T2 :
T1 T2 sts.closed S1 T1 sts.closed S2 T2
sts_ownS γ (S1 S2) (T1 T2) (sts_ownS γ S1 T1 sts_ownS γ S2 T2)%I.
Proof.
intros HT HS1 HS2. rewrite /sts_ownS -own_op.
by apply own_proper, sts_op_frag.
Qed.
Lemma sts_alloc E N s :
nclose N E
φ s pvs E E ( γ, sts_ctx γ N φ sts_own γ s ( sts.tok s)).
......
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