Non-disjoint CMRA structure on gset and coPset disj.

I had to perform some renaming to avoid name clashes.

algebra/coPset.v

 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.

algebra/gset.v

@@ -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 _ {_ _}.

heap_lang/lib/ticket_lock.v

@@ -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]").

program_logic/ownership.v

@@ -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 & ?).

program_logic/thread_local.v

@@ -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.