Merge branch 'ralf/reservation-map' into 'master'

New Camera dyn_reservation_map and generalize namespace_map → reservation_map

See merge request iris/iris!646

CHANGELOG.md

@@ -12,6 +12,16 @@ lemma.
   (`dfrac`) instead of a fraction (`frac`). Normal fractions are now denoted
   `●{#q} a` and `●V{#q} a`. Lemmas affected by this have been renamed such that
   the "frac" in their name has been changed into "dfrac".
+* Generalize `namespace_map` to `reservation_map` which enhances `gmap positive
+  A` with a notion of 'tokens' that enable allocating a particular name in the
+  map. See [algebra.reservation_map](iris/algebra/reservation_map.v) for further
+  information.
+* Add `dyn_reservation_map` which further extends `reservation_map` with the
+  ability to dynamically allocate an infinite set of tokens. This is useful to
+  perform synchronized allocation of the same name in two maps/APIs without
+  dedicated support from one of the involved maps/APIs. See
+  [algebra.dyn_reservation_map](iris/algebra/dyn_reservation_map.v) for further
+  information.
 
 **Changes in `base_logic`:**
 

_CoqProject

@@ -45,7 +45,8 @@ iris/algebra/gmultiset.v
 iris/algebra/coPset.v
 iris/algebra/proofmode_classes.v
 iris/algebra/ufrac.v
-iris/algebra/namespace_map.v
+iris/algebra/reservation_map.v
+iris/algebra/dyn_reservation_map.v
 iris/algebra/lib/excl_auth.v
 iris/algebra/lib/frac_auth.v
 iris/algebra/lib/ufrac_auth.v

iris/algebra/dyn_reservation_map.v

+From iris.algebra Require Export gmap coPset local_updates.
+From iris.algebra Require Import reservation_map updates proofmode_classes.
+From iris.prelude Require Import options.
+
+(** The camera [dyn_reservation_map A] over a camera [A] extends [gmap positive
+A] with a notion of "reservation tokens" for a (potentially infinite) set [E :
+coPset] which represent the right to allocate a map entry at any position [k ∈
+E].  Unlike [reservation_map], [dyn_reservation_map] supports dynamically
+allocating these tokens, including infinite sets of them.  This is useful when
+syncing the keys of this map with another API that dynamically allocates names:
+we can first reserve a fresh infinite set [E] of tokens here, then allocate a
+new name *in [E]* with the other API (assuming it offers the usual "allocate a
+fresh name in an infinite set" API), and then use our tokens to allocate the
+same name here.  In effect, we have performed synchronized allocation of the
+same name across two maps, without the other API having to have dedicated
+support for this.
+
+The key connectives are [dyn_reservation_map_data k a] (the "points-to"
+assertion of this map), which associates data [a : A] with a key [k : positive],
+and [dyn_reservation_map_token E] (the reservation token), which says
+that no data has been associated with the indices in the mask [E]. The important
+properties of this camera are:
+
+- The lemma [dyn_reservation_map_reserve] provides a frame-preserving update to
+  obtain ownership of [dyn_reservation_map_token E] for some fresh infinite [E].
+- The lemma [dyn_reservation_map_alloc] provides a frame preserving update to
+  associate data to a key: [dyn_reservation_map_token E ~~> dyn_reservation_map_data k a]
+  provided [k ∈ E] and [✓ a].
+
+In the future, it could be interesting to generalize this map to arbitrary key
+types instead of hard-coding [positive]. *)
+
+Record dyn_reservation_map (A : Type) := DynReservationMap {
+  dyn_reservation_map_data_proj : gmap positive A;
+  dyn_reservation_map_token_proj : coPset_disj
+}.
+Add Printing Constructor dyn_reservation_map.
+Global Arguments DynReservationMap {_} _ _.
+Global Arguments dyn_reservation_map_data_proj {_} _.
+Global Arguments dyn_reservation_map_token_proj {_} _.
+Global Instance: Params (@DynReservationMap) 1 := {}.
+Global Instance: Params (@dyn_reservation_map_data_proj) 1 := {}.
+Global Instance: Params (@dyn_reservation_map_token_proj) 1 := {}.
+
+Definition dyn_reservation_map_data {A : cmra} (k : positive) (a : A) : dyn_reservation_map A :=
+  DynReservationMap {[ k := a ]} ε.
+Definition dyn_reservation_map_token {A : cmra} (E : coPset) : dyn_reservation_map A :=
+  DynReservationMap ∅ (CoPset E).
+Global Instance: Params (@dyn_reservation_map_data) 2 := {}.
+
+(** We consruct the OFE and CMRA structure via an isomorphism with
+[reservation_map]. *)
+
+Section ofe.
+  Context {A : ofe}.
+  Implicit Types x y : dyn_reservation_map A.
+
+  Local Definition to_reservation_map x : reservation_map A :=
+    ReservationMap (dyn_reservation_map_data_proj x) (dyn_reservation_map_token_proj x).
+  Local Definition from_reservation_map (x : reservation_map A) : dyn_reservation_map A :=
+    DynReservationMap (reservation_map_data_proj x) (reservation_map_token_proj x).
+
+  Local Instance dyn_reservation_map_equiv : Equiv (dyn_reservation_map A) := λ x y,
+    dyn_reservation_map_data_proj x ≡ dyn_reservation_map_data_proj y ∧
+    dyn_reservation_map_token_proj x = dyn_reservation_map_token_proj y.
+  Local Instance dyn_reservation_map_dist : Dist (dyn_reservation_map A) := λ n x y,
+    dyn_reservation_map_data_proj x ≡{n}≡ dyn_reservation_map_data_proj y ∧
+    dyn_reservation_map_token_proj x = dyn_reservation_map_token_proj y.
+
+  Global Instance DynReservationMap_ne :
+    NonExpansive2 (@DynReservationMap A).
+  Proof. by split. Qed.
+  Global Instance DynReservationMap_proper :
+    Proper ((≡) ==> (=) ==> (≡)) (@DynReservationMap A).
+  Proof. by split. Qed.
+  Global Instance dyn_reservation_map_data_proj_ne :
+    NonExpansive (@dyn_reservation_map_data_proj A).
+  Proof. by destruct 1. Qed.
+  Global Instance dyn_reservation_map_data_proj_proper :
+    Proper ((≡) ==> (≡)) (@dyn_reservation_map_data_proj A).
+  Proof. by destruct 1. Qed.
+
+  Definition dyn_reservation_map_ofe_mixin : OfeMixin (dyn_reservation_map A).
+  Proof.
+    by apply (iso_ofe_mixin to_reservation_map).
+  Qed.
+  Canonical Structure dyn_reservation_mapO :=
+    Ofe (dyn_reservation_map A) dyn_reservation_map_ofe_mixin.
+
+  Global Instance DynReservationMap_discrete a b :
+    Discrete a → Discrete b → Discrete (DynReservationMap a b).
+  Proof. intros ?? [??] [??]; split; unfold_leibniz; by eapply discrete. Qed.
+  Global Instance dyn_reservation_map_ofe_discrete :
+    OfeDiscrete A → OfeDiscrete dyn_reservation_mapO.
+  Proof. intros ? [??]; apply _. Qed.
+End ofe.
+
+Global Arguments dyn_reservation_mapO : clear implicits.
+
+Section cmra.
+  Context {A : cmra}.
+  Implicit Types a b : A.
+  Implicit Types x y : dyn_reservation_map A.
+  Implicit Types k : positive.
+
+  Global Instance dyn_reservation_map_data_ne i : NonExpansive (@dyn_reservation_map_data A i).
+  Proof. intros ? ???. rewrite /dyn_reservation_map_data. solve_proper. Qed.
+  Global Instance dyn_reservation_map_data_proper N :
+    Proper ((≡) ==> (≡)) (@dyn_reservation_map_data A N).
+  Proof. solve_proper. Qed.
+  Global Instance dyn_reservation_map_data_discrete N a :
+    Discrete a → Discrete (dyn_reservation_map_data N a).
+  Proof. intros. apply DynReservationMap_discrete; apply _. Qed.
+  Global Instance dyn_reservation_map_token_discrete E : Discrete (@dyn_reservation_map_token A E).
+  Proof. intros. apply DynReservationMap_discrete; apply _. Qed.
+
+  Local Instance dyn_reservation_map_valid_instance : Valid (dyn_reservation_map A) := λ x,
+    match dyn_reservation_map_token_proj x with
+    | CoPset E =>
+      ✓ (dyn_reservation_map_data_proj x) ∧ set_infinite (⊤ ∖ E) ∧
+      (* dom (dyn_reservation_map_data_proj x) ⊥ E *)
+      (∀ i, dyn_reservation_map_data_proj x !! i = None ∨ i ∉ E)
+    | CoPsetBot => False
+    end.
+  Global Arguments dyn_reservation_map_valid_instance !_ /.
+  Local Instance dyn_reservation_map_validN_instance : ValidN (dyn_reservation_map A) := λ n x,
+    match dyn_reservation_map_token_proj x with
+    | CoPset E =>
+      ✓{n} (dyn_reservation_map_data_proj x) ∧ set_infinite (⊤ ∖ E) ∧
+      (* dom (dyn_reservation_map_data_proj x) ⊥ E *)
+      (∀ i, dyn_reservation_map_data_proj x !! i = None ∨ i ∉ E)
+    | CoPsetBot => False
+    end.
+  Global Arguments dyn_reservation_map_validN_instance !_ /.
+  Local Instance dyn_reservation_map_pcore_instance : PCore (dyn_reservation_map A) := λ x,
+    Some (DynReservationMap (core (dyn_reservation_map_data_proj x)) ε).
+  Local Instance dyn_reservation_map_op_instance : Op (dyn_reservation_map A) := λ x y,
+    DynReservationMap (dyn_reservation_map_data_proj x ⋅ dyn_reservation_map_data_proj y)
+                   (dyn_reservation_map_token_proj x ⋅ dyn_reservation_map_token_proj y).
+
+  Definition dyn_reservation_map_valid_eq :
+    valid = λ x, match dyn_reservation_map_token_proj x with
+                 | CoPset E =>
+                   ✓ (dyn_reservation_map_data_proj x) ∧ set_infinite (⊤ ∖ E) ∧
+                   (* dom (dyn_reservation_map_data_proj x) ⊥ E *)
+                   ∀ i, dyn_reservation_map_data_proj x !! i = None ∨ i ∉ E
+                 | CoPsetBot => False
+                 end := eq_refl _.
+  Definition dyn_reservation_map_validN_eq :
+    validN = λ n x, match dyn_reservation_map_token_proj x with
+                    | CoPset E =>
+                      ✓{n} (dyn_reservation_map_data_proj x) ∧ set_infinite (⊤ ∖ E) ∧
+                      (* dom (dyn_reservation_map_data_proj x) ⊥ E *)
+                      ∀ i, dyn_reservation_map_data_proj x !! i = None ∨ i ∉ E
+                    | CoPsetBot => False
+                    end := eq_refl _.
+
+  Lemma dyn_reservation_map_included x y :
+    x ≼ y ↔
+      dyn_reservation_map_data_proj x ≼ dyn_reservation_map_data_proj y ∧
+      dyn_reservation_map_token_proj x ≼ dyn_reservation_map_token_proj y.
+  Proof.
+    split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
+    intros [[z1 Hz1] [z2 Hz2]]; exists (DynReservationMap z1 z2); split; auto.
+  Qed.
+
+  Lemma dyn_reservation_map_data_proj_validN n x : ✓{n} x → ✓{n} dyn_reservation_map_data_proj x.
+  Proof. by destruct x as [? [?|]]=> // -[??]. Qed.
+  Lemma dyn_reservation_map_token_proj_validN n x : ✓{n} x → ✓{n} dyn_reservation_map_token_proj x.
+  Proof. by destruct x as [? [?|]]=> // -[??]. Qed.
+
+  Lemma dyn_reservation_map_cmra_mixin : CmraMixin (dyn_reservation_map A).
+  Proof.
+    apply (iso_cmra_mixin_restrict from_reservation_map to_reservation_map); try done.
+    - intros n [m [E|]];
+        rewrite dyn_reservation_map_validN_eq reservation_map_validN_eq /=;
+        naive_solver.
+    - intros n [m1 [E1|]] [m2 [E2|]] [Hm ?]=> // -[?[??]]; split; simplify_eq/=.
+      + by rewrite -Hm.
+      + split; first done. intros i. by rewrite -(dist_None n) -Hm dist_None.
+    - intros [m [E|]]; rewrite dyn_reservation_map_valid_eq dyn_reservation_map_validN_eq /=
+        ?cmra_valid_validN; naive_solver eauto using O.
+    - intros n [m [E|]]; rewrite dyn_reservation_map_validN_eq /=;
+        naive_solver eauto using cmra_validN_S.
+    - intros n [m1 [E1|]] [m2 [E2|]]=> //=; rewrite dyn_reservation_map_validN_eq /=.
+      rewrite {1}/op /cmra_op /=. case_decide; last done.
+      intros [Hm [Hinf Hdisj]]; split; first by eauto using cmra_validN_op_l.
+      split.
+      + rewrite ->difference_union_distr_r_L in Hinf.
+        eapply set_infinite_subseteq, Hinf. set_solver.
+      + intros i. move: (Hdisj i). rewrite lookup_op.
+        case: (m1 !! i); case: (m2 !! i); set_solver.
+  Qed.
+
+  Canonical Structure dyn_reservation_mapR :=
+    Cmra (dyn_reservation_map A) dyn_reservation_map_cmra_mixin.
+
+  Global Instance dyn_reservation_map_cmra_discrete :
+    CmraDiscrete A → CmraDiscrete dyn_reservation_mapR.
+  Proof.
+    split; first apply _.
+    intros [m [E|]]; rewrite dyn_reservation_map_validN_eq dyn_reservation_map_valid_eq //=.
+      by intros [?%cmra_discrete_valid ?].
+  Qed.
+
+  Local Instance dyn_reservation_map_empty_instance : Unit (dyn_reservation_map A) :=
+    DynReservationMap ε ε.
+  Lemma dyn_reservation_map_ucmra_mixin : UcmraMixin (dyn_reservation_map A).
+  Proof.
+    split; simpl.
+    - rewrite dyn_reservation_map_valid_eq /=. split; [apply ucmra_unit_valid|]. split.
+      + rewrite difference_empty_L. apply top_infinite.
+      + set_solver.
+    - split; simpl; [by rewrite left_id|by rewrite left_id_L].
+    - do 2 constructor; [apply (core_id_core _)|done].
+  Qed.
+  Canonical Structure dyn_reservation_mapUR :=
+    Ucmra (dyn_reservation_map A) dyn_reservation_map_ucmra_mixin.
+
+  Global Instance dyn_reservation_map_data_core_id N a :
+    CoreId a → CoreId (dyn_reservation_map_data N a).
+  Proof. do 2 constructor; simpl; auto. apply core_id_core, _. Qed.
+
+  Lemma dyn_reservation_map_data_valid N a :
+    ✓ (dyn_reservation_map_data N a) ↔ ✓ a.
+  Proof.
+    rewrite dyn_reservation_map_valid_eq /= singleton_valid.
+    split; first naive_solver. intros Ha.
+    split; first done. split; last set_solver.
+    rewrite difference_empty_L. apply top_infinite.
+  Qed.
+  Lemma dyn_reservation_map_token_valid E :
+    ✓ (dyn_reservation_map_token E) ↔ set_infinite (⊤ ∖ E).
+  Proof.
+    rewrite dyn_reservation_map_valid_eq /=. split; first naive_solver.
+    intros Hinf. do 2 (split; first done). by left.
+  Qed.
+  Lemma dyn_reservation_map_data_op N a b :
+    dyn_reservation_map_data N (a ⋅ b) = dyn_reservation_map_data N a ⋅ dyn_reservation_map_data N b.
+  Proof.
+      by rewrite {2}/op /dyn_reservation_map_op_instance /dyn_reservation_map_data /= singleton_op left_id_L.
+  Qed.
+  Lemma dyn_reservation_map_data_mono N a b :
+    a ≼ b → dyn_reservation_map_data N a ≼ dyn_reservation_map_data N b.
+  Proof. intros [c ->]. rewrite dyn_reservation_map_data_op. apply cmra_included_l. Qed.
+  Global Instance dyn_reservation_map_data_is_op N a b1 b2 :
+    IsOp a b1 b2 →
+    IsOp' (dyn_reservation_map_data N a) (dyn_reservation_map_data N b1) (dyn_reservation_map_data N b2).
+  Proof. rewrite /IsOp' /IsOp=> ->. by rewrite dyn_reservation_map_data_op. Qed.
+
+  Lemma dyn_reservation_map_token_union E1 E2 :
+    E1 ## E2 →
+    dyn_reservation_map_token (E1 ∪ E2) = dyn_reservation_map_token E1 ⋅ dyn_reservation_map_token E2.
+  Proof.
+    intros. by rewrite /op /dyn_reservation_map_op_instance
+      /dyn_reservation_map_token /= coPset_disj_union // left_id_L.
+  Qed.
+  Lemma dyn_reservation_map_token_difference E1 E2 :
+    E1 ⊆ E2 →
+    dyn_reservation_map_token E2 = dyn_reservation_map_token E1 ⋅ dyn_reservation_map_token (E2 ∖ E1).
+  Proof.
+    intros. rewrite -dyn_reservation_map_token_union; last set_solver.
+    by rewrite -union_difference_L.
+  Qed.
+  Lemma dyn_reservation_map_token_valid_op E1 E2 :
+    ✓ (dyn_reservation_map_token E1 ⋅ dyn_reservation_map_token E2) ↔
+      E1 ## E2 ∧ set_infinite (⊤ ∖ (E1 ∪ E2)).
+  Proof.
+    split.
+    - rewrite dyn_reservation_map_valid_eq /= {1}/op /cmra_op /=. case_decide; last done.
+      naive_solver.
+    - intros [Hdisj Hinf]. rewrite -dyn_reservation_map_token_union //.
+      apply dyn_reservation_map_token_valid. done.
+  Qed.
+
+  Lemma dyn_reservation_map_reserve (Q : dyn_reservation_map A → Prop) :
+    (∀ E, set_infinite E → Q (dyn_reservation_map_token E)) →
+    ε ~~>: Q.
+  Proof.
+    intros HQ. apply cmra_total_updateP=> n [mf [Ef|]];
+      rewrite left_id {1}dyn_reservation_map_validN_eq /=; last done.
+    intros [Hmap [Hinf Hdisj]].
+    (* Pick a fresh set disjoint from the existing tokens [Ef] and map [mf],
+       such that both that set [E1] and the remainder [E2] are infinite. *)
+    edestruct (coPset_split_infinite (⊤ ∖ (Ef ∪ dom coPset mf))) as
+        (E1 & E2 & HEunion & HEdisj & HE1inf & HE2inf).
+    { rewrite -difference_difference_L.
+        by apply difference_infinite, dom_finite. }
+    exists (dyn_reservation_map_token E1).
+    split; first by apply HQ. clear HQ.
+    rewrite dyn_reservation_map_validN_eq /=.
+    rewrite coPset_disj_union; last set_solver.
+    split; first by rewrite left_id_L. split.
+    - eapply set_infinite_subseteq, HE2inf. set_solver.
+    - intros i. rewrite left_id_L. destruct (Hdisj i) as [?|Hi]; first by left.
+      destruct (mf !! i) as [p|] eqn:Hp; last by left.
+      apply (elem_of_dom_2 (D:=coPset)) in Hp. right. set_solver.
+  Qed.
+  Lemma dyn_reservation_map_reserve' :
+    ε ~~>: (λ x, ∃ E, set_infinite E ∧ x = dyn_reservation_map_token E).
+  Proof. eauto using dyn_reservation_map_reserve. Qed.
+
+  Lemma dyn_reservation_map_alloc E k a :
+    k ∈ E → ✓ a → dyn_reservation_map_token E ~~> dyn_reservation_map_data k a.
+  Proof.
+    intros ??. apply cmra_total_update=> n [mf [Ef|]] //.
+    rewrite dyn_reservation_map_validN_eq /= {1}/op /cmra_op /=. case_decide; last done.
+    rewrite left_id_L {1}left_id. intros [Hmf [Hinf Hdisj]]; split; last split.
+    - destruct (Hdisj k) as [Hmfi|]; last set_solver.
+      move: Hmfi. rewrite lookup_op lookup_empty left_id_L=> Hmfi.
+      intros j. rewrite lookup_op.
+      destruct (decide (k = j)) as [<-|].
+      + rewrite Hmfi lookup_singleton right_id_L. by apply cmra_valid_validN.
+      + by rewrite lookup_singleton_ne // left_id_L.
+    - eapply set_infinite_subseteq, Hinf. set_solver.
+    - intros j. destruct (decide (k = j)); first set_solver.
+      rewrite lookup_op lookup_singleton_ne //.
+      destruct (Hdisj j) as [Hmfi|?]; last set_solver.
+      move: Hmfi. rewrite lookup_op lookup_empty; auto.
+  Qed.
+  Lemma dyn_reservation_map_updateP P (Q : dyn_reservation_map A → Prop) k a :
+    a ~~>: P →
+    (∀ a', P a' → Q (dyn_reservation_map_data k a')) →
+    dyn_reservation_map_data k a ~~>: Q.
+  Proof.
+    intros Hup HP. apply cmra_total_updateP=> n [mf [Ef|]] //.
+    rewrite dyn_reservation_map_validN_eq /= left_id_L. intros [Hmf [Hinf Hdisj]].
+    destruct (Hup n (mf !! k)) as (a'&?&?).
+    { move: (Hmf (k)).
+      by rewrite lookup_op lookup_singleton Some_op_opM. }
+    exists (dyn_reservation_map_data k a'); split; first by eauto.
+    rewrite /= left_id_L. split; last split.
+    - intros j. destruct (decide (k = j)) as [<-|].
+      + by rewrite lookup_op lookup_singleton Some_op_opM.
+      + rewrite lookup_op lookup_singleton_ne // left_id_L.
+        move: (Hmf j). rewrite lookup_op. eauto using cmra_validN_op_r.
+    - done.
+    - intros j. move: (Hdisj j).
+      rewrite !lookup_op !op_None !lookup_singleton_None. naive_solver.
+  Qed.
+  Lemma dyn_reservation_map_update k a b :
+    a ~~> b →
+    dyn_reservation_map_data k a ~~> dyn_reservation_map_data k b.
+  Proof.
+    rewrite !cmra_update_updateP. eauto using dyn_reservation_map_updateP with subst.
+  Qed.
+End cmra.
+
+Global Arguments dyn_reservation_mapR : clear implicits.
+Global Arguments dyn_reservation_mapUR : clear implicits.

iris/algebra/namespace_map.v

-From stdpp Require Import namespaces.
-From iris.algebra Require Export gmap coPset local_updates.
-From iris.algebra Require Import updates proofmode_classes.
-From iris.prelude Require Import options.
-
-(** The camera [namespace_map A] over a camera [A] provides the connectives
-[namespace_map_data N a], which associates data [a : A] with a namespace [N],
-and [namespace_map_token E], which says that no data has been associated with
-the namespaces in the mask [E]. The important properties of this camera are:
-
-- The lemma [namespace_map_token_union] enables one to split [namespace_map_token]
-  w.r.t. disjoint union. That is, if we have [E1 ## E2], then we get
-  [namespace_map_token (E1 ∪ E2) = namespace_map_token E1 ⋅ namespace_map_token E2]
-- The lemma [namespace_map_alloc_update] provides a frame preserving update to
-  associate data to a namespace [namespace_map_token E ~~> namespace_map_data N a]
-  provided [↑N ⊆ E] and [✓ a]. *)
-
-Record namespace_map (A : Type) := NamespaceMap {
-  namespace_map_data_proj : gmap positive A;
-  namespace_map_token_proj : coPset_disj
-}.
-Add Printing Constructor namespace_map.
-Global Arguments NamespaceMap {_} _ _.
-Global Arguments namespace_map_data_proj {_} _.
-Global Arguments namespace_map_token_proj {_} _.
-Global Instance: Params (@NamespaceMap) 1 := {}.
-Global Instance: Params (@namespace_map_data_proj) 1 := {}.
-Global Instance: Params (@namespace_map_token_proj) 1 := {}.
-
-(** TODO: [positives_flatten] violates the namespace abstraction. *)
-Definition namespace_map_data {A : cmra} (N : namespace) (a : A) : namespace_map A :=
-  NamespaceMap {[ positives_flatten N := a ]} ε.
-Definition namespace_map_token {A : cmra} (E : coPset) : namespace_map A :=
-  NamespaceMap ∅ (CoPset E).
-Global Instance: Params (@namespace_map_data) 2 := {}.
-
-(* Ofe *)
-Section ofe.
-Context {A : ofe}.
-Implicit Types x y : namespace_map A.
-
-Local Instance namespace_map_equiv : Equiv (namespace_map A) := λ x y,
-  namespace_map_data_proj x ≡ namespace_map_data_proj y ∧
-  namespace_map_token_proj x = namespace_map_token_proj y.
-Local Instance namespace_map_dist : Dist (namespace_map A) := λ n x y,
-  namespace_map_data_proj x ≡{n}≡ namespace_map_data_proj y ∧
-  namespace_map_token_proj x = namespace_map_token_proj y.
-
-Global Instance NamespaceMap_ne : NonExpansive2 (@NamespaceMap A).
-Proof. by split. Qed.
-Global Instance NamespaceMap_proper : Proper ((≡) ==> (=) ==> (≡)) (@NamespaceMap A).
-Proof. by split. Qed.
-Global Instance namespace_map_data_proj_ne: NonExpansive (@namespace_map_data_proj A).
-Proof. by destruct 1. Qed.
-Global Instance namespace_map_data_proj_proper :
-  Proper ((≡) ==> (≡)) (@namespace_map_data_proj A).
-Proof. by destruct 1. Qed.
-
-Definition namespace_map_ofe_mixin : OfeMixin (namespace_map A).
-Proof.
-  by apply (iso_ofe_mixin
-    (λ x, (namespace_map_data_proj x, namespace_map_token_proj x))).
-Qed.
-Canonical Structure namespace_mapO :=
-  Ofe (namespace_map A) namespace_map_ofe_mixin.
-
-Global Instance NamespaceMap_discrete a b :
-  Discrete a → Discrete b → Discrete (NamespaceMap a b).
-Proof. intros ?? [??] [??]; split; unfold_leibniz; by eapply discrete. Qed.
-Global Instance namespace_map_ofe_discrete :
-  OfeDiscrete A → OfeDiscrete namespace_mapO.
-Proof. intros ? [??]; apply _. Qed.
-End ofe.
-
-Global Arguments namespace_mapO : clear implicits.
-
-(* Camera *)
-Section cmra.
-Context {A : cmra}.
-Implicit Types a b : A.
-Implicit Types x y : namespace_map A.
-
-Global Instance namespace_map_data_ne i : NonExpansive (@namespace_map_data A i).
-Proof. solve_proper. Qed.
-Global Instance namespace_map_data_proper N :
-  Proper ((≡) ==> (≡)) (@namespace_map_data A N).
-Proof. solve_proper. Qed.
-Global Instance namespace_map_data_discrete N a :
-  Discrete a → Discrete (namespace_map_data N a).
-Proof. intros. apply NamespaceMap_discrete; apply _. Qed.
-Global Instance namespace_map_token_discrete E : Discrete (@namespace_map_token A E).
-Proof. intros. apply NamespaceMap_discrete; apply _. Qed.
-
-Local Instance namespace_map_valid_instance : Valid (namespace_map A) := λ x,
-  match namespace_map_token_proj x with
-  | CoPset E =>
-     ✓ (namespace_map_data_proj x) ∧
-     (* dom (namespace_map_data_proj x) ⊥ E *)
-     ∀ i, namespace_map_data_proj x !! i = None ∨ i ∉ E
-  | CoPsetBot => False
-  end.
-Global Arguments namespace_map_valid_instance !_ /.
-Local Instance namespace_map_validN_instance : ValidN (namespace_map A) := λ n x,
-  match namespace_map_token_proj x with
-  | CoPset E =>
-     ✓{n} (namespace_map_data_proj x) ∧
-     (* dom (namespace_map_data_proj x) ⊥ E *)
-     ∀ i, namespace_map_data_proj x !! i = None ∨ i ∉ E
-  | CoPsetBot => False
-  end.
-Global Arguments namespace_map_validN_instance !_ /.
-Local Instance namespace_map_pcore_instance : PCore (namespace_map A) := λ x,
-  Some (NamespaceMap (core (namespace_map_data_proj x)) ε).
-Local Instance namespace_map_op_instance : Op (namespace_map A) := λ x y,
-  NamespaceMap (namespace_map_data_proj x ⋅ namespace_map_data_proj y)
-               (namespace_map_token_proj x ⋅ namespace_map_token_proj y).
-
-Definition namespace_map_valid_eq :
-  valid = λ x, match namespace_map_token_proj x with
-               | CoPset E =>
-                  ✓ (namespace_map_data_proj x) ∧
-                  (* dom (namespace_map_data_proj x) ⊥ E *)
-                  ∀ i, namespace_map_data_proj x !! i = None ∨ i ∉ E
-               | CoPsetBot => False
-               end := eq_refl _.
-Definition namespace_map_validN_eq :
-  validN = λ n x, match namespace_map_token_proj x with
-                  | CoPset E =>
-                     ✓{n} (namespace_map_data_proj x) ∧
-                     (* dom (namespace_map_data_proj x) ⊥ E *)
-                     ∀ i, namespace_map_data_proj x !! i = None ∨ i ∉ E
-                  | CoPsetBot => False
-                  end := eq_refl _.
-
-Lemma namespace_map_included x y :
-  x ≼ y ↔
-    namespace_map_data_proj x ≼ namespace_map_data_proj y ∧
-    namespace_map_token_proj x ≼ namespace_map_token_proj y.
-Proof.
-  split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
-  intros [[z1 Hz1] [z2 Hz2]]; exists (NamespaceMap z1 z2); split; auto.
-Qed.
-
-Lemma namespace_map_data_proj_validN n x : ✓{n} x → ✓{n} namespace_map_data_proj x.
-Proof. by destruct x as [? [?|]]=> // -[??]. Qed.
-Lemma namespace_map_token_proj_validN n x : ✓{n} x → ✓{n} namespace_map_token_proj x.
-Proof. by destruct x as [? [?|]]=> // -[??]. Qed.
-
-Lemma namespace_map_cmra_mixin : CmraMixin (namespace_map A).
-Proof.
-  apply cmra_total_mixin.
-  - eauto.
-  - by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
-  - solve_proper.
-  - intros n [m1 [E1|]] [m2 [E2|]] [Hm ?]=> // -[??]; split; simplify_eq/=.
-    + by rewrite -Hm.
-    + intros i. by rewrite -(dist_None n) -Hm dist_None.
-  - intros [m [E|]]; rewrite namespace_map_valid_eq namespace_map_validN_eq /=
-      ?cmra_valid_validN; naive_solver eauto using O.
-  - intros n [m [E|]]; rewrite namespace_map_validN_eq /=;
-      naive_solver eauto using cmra_validN_S.
-  - split; simpl; [by rewrite assoc|by rewrite assoc_L].
-  - split; simpl; [by rewrite comm|by rewrite comm_L].
-  - split; simpl; [by rewrite cmra_core_l|by rewrite left_id_L].
-  - split; simpl; [by rewrite cmra_core_idemp|done].
-  - intros ??; rewrite! namespace_map_included; intros [??].
-    by split; simpl; apply: cmra_core_mono. (* FIXME: FIXME(Coq #6294): needs new unification *)
-  - intros n [m1 [E1|]] [m2 [E2|]]=> //=; rewrite namespace_map_validN_eq /=.