diff --git a/_CoqProject b/_CoqProject index 2c9ab829cf80db90aed905e07f45fc74cb3bd818..fb5889c20b53fae29629e4b1eb6a8d52653c4fee 100644 --- a/_CoqProject +++ b/_CoqProject @@ -46,6 +46,7 @@ iris/algebra/coPset.v iris/algebra/proofmode_classes.v iris/algebra/ufrac.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 diff --git a/iris/algebra/dyn_reservation_map.v b/iris/algebra/dyn_reservation_map.v new file mode 100644 index 0000000000000000000000000000000000000000..53affc428b6befbe502e57c1e353a2111989b372 --- /dev/null +++ b/iris/algebra/dyn_reservation_map.v @@ -0,0 +1,349 @@ +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 of tokens here, then allocate a new +*in that set* with the other 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 in Hinf. + eapply set_infinite_subseteq; last done. + set_solver. + + intros i. move: (Hdisj i). rewrite lookup_op. + case: (m1 !! i)=> [a|]; last auto. + move=> []. + { by 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. 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. 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. + apply difference_infinite; first done. + apply gset_to_coPset_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. split. + - eapply set_infinite_subseteq; last by apply 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 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; last done. 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.