Commit 7c1de72a authored by Robbert Krebbers's avatar Robbert Krebbers

Non-disjoint CMRA structure on gset and coPset disj.

I had to perform some renaming to avoid name clashes.
parent f038b880
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From iris.algebra Require Export cmra.
From iris.algebra Require Import updates local_updates.
From iris.prelude Require Export collections coPset.
(** This is pretty much the same as algebra/gset, but I was not able to
generalize the construction without breaking canonical structures. *)
(* The union CMRA *)
Section coPset.
Implicit Types X Y : coPset.
Canonical Structure coPsetC := leibnizC coPset.
Instance coPset_valid : Valid coPset := λ _, True.
Instance coPset_op : Op coPset := union.
Instance coPset_pcore : PCore coPset := λ _, Some .
Lemma coPset_op_union X Y : X Y = X Y.
Proof. done. Qed.
Lemma coPset_core_empty X : core X = .
Proof. done. Qed.
Lemma coPset_included X Y : X Y X Y.
Proof.
split.
- intros [Z ->]. rewrite coPset_op_union. set_solver.
- intros (Z&->&?)%subseteq_disjoint_union_L. by exists Z.
Qed.
Lemma coPset_ra_mixin : RAMixin coPset.
Proof.
apply ra_total_mixin; eauto.
- solve_proper.
- solve_proper.
- solve_proper.
- intros X1 X2 X3. by rewrite !coPset_op_union assoc_L.
- intros X1 X2. by rewrite !coPset_op_union comm_L.
- intros X. by rewrite coPset_op_union coPset_core_empty left_id_L.
- intros X1 X2 _. by rewrite coPset_included !coPset_core_empty.
Qed.
Canonical Structure coPsetR := discreteR coPset coPset_ra_mixin.
Lemma coPset_ucmra_mixin : UCMRAMixin coPset.
Proof. split. done. intros X. by rewrite coPset_op_union left_id_L. done. Qed.
Canonical Structure coPsetUR :=
discreteUR coPset coPset_ra_mixin coPset_ucmra_mixin.
Lemma coPset_opM X mY : X ? mY = X from_option id mY.
Proof. destruct mY; by rewrite /= ?right_id_L. Qed.
Lemma coPset_update X Y : X ~~> Y.
Proof. done. Qed.
Lemma coPset_local_update X Y mXf : X Y X ~l~> Y @ mXf.
Proof.
intros (Z&->&?)%subseteq_disjoint_union_L.
intros; apply local_update_total; split; [done|]; simpl.
intros mZ _. rewrite !coPset_opM=> HX. by rewrite (comm_L _ X) -!assoc_L HX.
Qed.
End coPset.
(* The disjoiny union CMRA *)
Inductive coPset_disj :=
| CoPset : coPset coPset_disj
| CoPsetBot : coPset_disj.
Section coPset.
Section coPset_disj.
Arguments op _ _ !_ !_ /.
Canonical Structure coPset_disjC := leibnizC coPset_disj.
......@@ -27,7 +80,7 @@ Section coPset.
repeat (simpl || case_decide);
first [apply (f_equal CoPset)|done|exfalso]; set_solver by eauto.
Lemma coPset_included X Y : CoPset X CoPset Y X Y.
Lemma coPset_disj_included X Y : CoPset X CoPset Y X Y.
Proof.
split.
- move=> [[Z|]]; simpl; try case_decide; set_solver.
......@@ -60,4 +113,4 @@ Section coPset.
Proof. split; try apply _ || done. intros [X|]; coPset_disj_solve. Qed.
Canonical Structure coPset_disjUR :=
discreteUR coPset_disj coPset_disj_ra_mixin coPset_disj_ucmra_mixin.
End coPset.
End coPset_disj.
......@@ -2,13 +2,68 @@ From iris.algebra Require Export cmra.
From iris.algebra Require Import updates local_updates.
From iris.prelude Require Export collections gmap.
(* The union CMRA *)
Section gset.
Context `{Countable K}.
Implicit Types X Y : gset K.
Canonical Structure gsetC := leibnizC (gset K).
Instance gset_valid : Valid (gset K) := λ _, True.
Instance gset_op : Op (gset K) := union.
Instance gset_pcore : PCore (gset K) := λ _, Some .
Lemma gset_op_union X Y : X Y = X Y.
Proof. done. Qed.
Lemma gset_core_empty X : core X = .
Proof. done. Qed.
Lemma gset_included X Y : X Y X Y.
Proof.
split.
- intros [Z ->]. rewrite gset_op_union. set_solver.
- intros (Z&->&?)%subseteq_disjoint_union_L. by exists Z.
Qed.
Lemma gset_ra_mixin : RAMixin (gset K).
Proof.
apply ra_total_mixin; eauto.
- solve_proper.
- solve_proper.
- solve_proper.
- intros X1 X2 X3. by rewrite !gset_op_union assoc_L.
- intros X1 X2. by rewrite !gset_op_union comm_L.
- intros X. by rewrite gset_op_union gset_core_empty left_id_L.
- intros X1 X2 _. by rewrite gset_included !gset_core_empty.
Qed.
Canonical Structure gsetR := discreteR (gset K) gset_ra_mixin.
Lemma gset_ucmra_mixin : UCMRAMixin (gset K).
Proof. split. done. intros X. by rewrite gset_op_union left_id_L. done. Qed.
Canonical Structure gsetUR :=
discreteUR (gset K) gset_ra_mixin gset_ucmra_mixin.
Lemma gset_opM X mY : X ? mY = X from_option id mY.
Proof. destruct mY; by rewrite /= ?right_id_L. Qed.
Lemma gset_update X Y : X ~~> Y.
Proof. done. Qed.
Lemma gset_local_update X Y mXf : X Y X ~l~> Y @ mXf.
Proof.
intros (Z&->&?)%subseteq_disjoint_union_L.
intros; apply local_update_total; split; [done|]; simpl.
intros mZ _. rewrite !gset_opM=> HX. by rewrite (comm_L _ X) -!assoc_L HX.
Qed.
End gset.
(* The disjoint union CMRA *)
Inductive gset_disj K `{Countable K} :=
| GSet : gset K gset_disj K
| GSetBot : gset_disj K.
Arguments GSet {_ _ _} _.
Arguments GSetBot {_ _ _}.
Section gset.
Section gset_disj.
Context `{Countable K}.
Arguments op _ _ !_ !_ /.
......@@ -28,7 +83,7 @@ Section gset.
repeat (simpl || case_decide);
first [apply (f_equal GSet)|done|exfalso]; set_solver by eauto.
Lemma coPset_included X Y : GSet X GSet Y X Y.
Lemma gset_disj_included X Y : GSet X GSet Y X Y.
Proof.
split.
- move=> [[Z|]]; simpl; try case_decide; set_solver.
......@@ -63,7 +118,7 @@ Section gset.
Arguments op _ _ _ _ : simpl never.
Lemma gset_alloc_updateP_strong P (Q : gset_disj K Prop) X :
Lemma gset_disj_alloc_updateP_strong P (Q : gset_disj K Prop) X :
( Y, X Y j, j Y P j)
( i, i X P i Q (GSet ({[i]} X))) GSet X ~~>: Q.
Proof.
......@@ -74,43 +129,46 @@ Section gset.
- apply HQ; set_solver by eauto.
- apply gset_disj_valid_op. set_solver by eauto.
Qed.
Lemma gset_alloc_updateP_strong' P X :
Lemma gset_disj_alloc_updateP_strong' P X :
( Y, X Y j, j Y P j)
GSet X ~~>: λ Y, i, Y = GSet ({[ i ]} X) i X P i.
Proof. eauto using gset_alloc_updateP_strong. Qed.
Proof. eauto using gset_disj_alloc_updateP_strong. Qed.
Lemma gset_alloc_empty_updateP_strong P (Q : gset_disj K Prop) :
Lemma gset_disj_alloc_empty_updateP_strong P (Q : gset_disj K Prop) :
( Y : gset K, j, j Y P j)
( i, P i Q (GSet {[i]})) GSet ~~>: Q.
Proof.
intros. apply (gset_alloc_updateP_strong P); eauto.
intros. apply (gset_disj_alloc_updateP_strong P); eauto.
intros i; rewrite right_id_L; auto.
Qed.
Lemma gset_alloc_empty_updateP_strong' P :
Lemma gset_disj_alloc_empty_updateP_strong' P :
( Y : gset K, j, j Y P j)
GSet ~~>: λ Y, i, Y = GSet {[ i ]} P i.
Proof. eauto using gset_alloc_empty_updateP_strong. Qed.
Proof. eauto using gset_disj_alloc_empty_updateP_strong. Qed.
Section fresh_updates.
Context `{Fresh K (gset K), !FreshSpec K (gset K)}.
Lemma gset_alloc_updateP (Q : gset_disj K Prop) X :
Lemma gset_disj_alloc_updateP (Q : gset_disj K Prop) X :
( i, i X Q (GSet ({[i]} X))) GSet X ~~>: Q.
Proof.
intro; eapply gset_alloc_updateP_strong with (λ _, True); eauto.
intro; eapply gset_disj_alloc_updateP_strong with (λ _, True); eauto.
intros Y ?; exists (fresh Y); eauto using is_fresh.
Qed.
Lemma gset_alloc_updateP' X : GSet X ~~>: λ Y, i, Y = GSet ({[ i ]} X) i X.
Proof. eauto using gset_alloc_updateP. Qed.
Lemma gset_disj_alloc_updateP' X :
GSet X ~~>: λ Y, i, Y = GSet ({[ i ]} X) i X.
Proof. eauto using gset_disj_alloc_updateP. Qed.
Lemma gset_alloc_empty_updateP (Q : gset_disj K Prop) :
Lemma gset_disj_alloc_empty_updateP (Q : gset_disj K Prop) :
( i, Q (GSet {[i]})) GSet ~~>: Q.
Proof. intro. apply gset_alloc_updateP. intros i; rewrite right_id_L; auto. Qed.
Lemma gset_alloc_empty_updateP' : GSet ~~>: λ Y, i, Y = GSet {[ i ]}.
Proof. eauto using gset_alloc_empty_updateP. Qed.
Proof.
intro. apply gset_disj_alloc_updateP. intros i; rewrite right_id_L; auto.
Qed.
Lemma gset_disj_alloc_empty_updateP' : GSet ~~>: λ Y, i, Y = GSet {[ i ]}.
Proof. eauto using gset_disj_alloc_empty_updateP. Qed.
End fresh_updates.
Lemma gset_alloc_local_update X i Xf :
Lemma gset_disj_alloc_local_update X i Xf :
i X i Xf GSet X ~l~> GSet ({[i]} X) @ Some (GSet Xf).
Proof.
intros ??; apply local_update_total; split; simpl.
......@@ -118,13 +176,13 @@ Section gset.
- intros mZ ?%gset_disj_valid_op HXf.
rewrite -gset_disj_union -?assoc ?HXf ?cmra_opM_assoc; set_solver.
Qed.
Lemma gset_alloc_empty_local_update i Xf :
Lemma gset_disj_alloc_empty_local_update i Xf :
i Xf GSet ~l~> GSet {[i]} @ Some (GSet Xf).
Proof.
intros. rewrite -(right_id_L _ _ {[i]}).
apply gset_alloc_local_update; set_solver.
apply gset_disj_alloc_local_update; set_solver.
Qed.
End gset.
End gset_disj.
Arguments gset_disjR _ {_ _}.
Arguments gset_disjUR _ {_ _}.
......@@ -141,7 +141,7 @@ Section proof.
- wp_cas_suc.
iDestruct "Hainv" as (s) "[Ho %]"; subst.
iVs (own_update with "Ho") as "Ho".
{ eapply auth_update_no_frag, (gset_alloc_empty_local_update n).
{ eapply auth_update_no_frag, (gset_disj_alloc_empty_local_update n).
rewrite elem_of_seq_set; omega. }
iDestruct "Ho" as "[Hofull Hofrag]".
iVs ("Hclose" with "[Hlo' Hln' Haown Hofull]").
......
......@@ -156,7 +156,7 @@ Proof.
iIntros (Hfresh) "[Hw HP]". iDestruct "Hw" as (I) "[? HI]".
iVs (own_empty (A:=gset_disjUR positive) disabled_name) as "HE".
iVs (own_updateP with "HE") as "HE".
{ apply (gset_alloc_empty_updateP_strong' (λ i, I !! i = None φ i)).
{ apply (gset_disj_alloc_empty_updateP_strong' (λ i, I !! i = None φ i)).
intros E. destruct (Hfresh (E dom _ I))
as (i & [? HIi%not_elem_of_dom]%not_elem_of_union & ?); eauto. }
iDestruct "HE" as (X) "[Hi HE]"; iDestruct "Hi" as %(i & -> & HIi & ?).
......
......@@ -56,7 +56,7 @@ Section proofs.
iVs (own_empty (A:=prodUR coPset_disjUR (gset_disjUR positive)) tid) as "Hempty".
iVs (own_updateP with "Hempty") as ([m1 m2]) "[Hm Hown]".
{ apply prod_updateP'. apply cmra_updateP_id, (reflexivity (R:=eq)).
apply (gset_alloc_empty_updateP_strong' (λ i, i nclose N)).
apply (gset_disj_alloc_empty_updateP_strong' (λ i, i nclose N)).
intros Ef. exists (coPpick (nclose N coPset.of_gset Ef)).
rewrite -coPset.elem_of_of_gset comm -elem_of_difference.
apply coPpick_elem_of=> Hfin.
......
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